Locally Trivial Deformations of Toric Varieties
Nathan Ilten Sharon Robins

arabic
§ ABSTRACT 
	We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan Σ, we construct a deformation functor _Σ by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, _Σ is isomorphic to '_X_Σ, the functor of locally trivial deformations for the toric variety X_Σ associated to Σ. In particular, for any complete toric variety X that is smooth in codimension 2 and -factorial in codimension 3, there exists a fan Σ such that _Σ is isomorphic to _X, the functor of deformations of X. We apply these results to give a new criterion for a smooth complete toric variety to have unobstructed deformations, and to compute formulas for higher order obstructions, generalizing a formula of Ilten and Turo for the cup product. We use the functor _Σ to explicitly compute the deformation spaces for a number of toric varieties, and provide examples exhibiting previously unobserved phenomena. In particular, we classify exactly which toric threefolds arising as iterated ^1-bundles have unobstructed deformation space.

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§ INTRODUCTION



 §.§ Background and motivation

Let  be an algebraically closed field of characteristic zero and X a variety over . The functor _X of isomorphism classes of (infinitesimal) deformations of X provides useful information on how X might fit into a moduli space. 
In the setting where X is smooth, _X coincides with the functor _X' of isomorphism classes of locally trivial deformations of X.
In general, locally trivial first order deformations are described by H^1(X,_X) and obstructions to lifting locally trivial deformations live in H^2(X,_X). In fact, all locally trivial deformations of X are controlled by the Čech complex of the tangent sheaf _X, see <ref> below.


Despite this seemingly concrete description of _X', it is still quite challenging to explicitly understand _X' for specific examples.
In this paper, we will give a purely combinatorial description of the deformation functor _X' when X is a -factorial complete toric variety. 
First order deformations of smooth complete toric varieties were described combinatorially by N. Ilten in <cit.>. In <cit.> Ilten and R. Vollmert showed that homogeneous first order deformations can be extended to one-parameter families over ^1, providing a sort of skeleton of the versal deformation (see also work by A. Mavlyutov <cit.> and A. Petracci <cit.>). However, it turns out that in general there are obstructions to combining these one-parameter families. This was observed by Ilten and C. Turo in <cit.>, which contains a combinatorial description of the cup product.

These results place the deformation theory of smooth complete toric varieties in a strange liminal space: although they can be obstructed, unlike say Calabi-Yau varieties <cit.>, they do not satisfy “Murphy's law” <cit.>, that predicts deformation spaces will have arbitrarily bad singularities. It thus remains an interesting challenge to determine exactly what spaces can occur as the deformation space of a smooth complete toric variety.



 §.§ Main results

We now summarize the main results of the paper. Let X=X_Σ be a -factorial toric variety with corresponding fan Σ. See <ref> for notation and details on toric varieties.
Our starting point is the combinatorial description of H^1(X,_X) and H^2(X,_X) from <cit.>.
Indeed, when X is complete, for every k≥ 1 there are isomorphisms
  
    H^k(X,_X)≅⊕_ρ, H^k-1(V_ρ,,),

  see prop:cohom2.
  Here, ρ ranges over rays of Σ,  ranges over characters of the torus,  and V_ρ, is a particular simplicial complex contained in Σ, see <ref>.

By the above, locally trivial first order deformations of X and obstructions to lifting locally trivial deformations can be described via cohomology of the simplicial complexes V_ρ,.
Hence, it is natural to try to completely understand the functor _X' in terms of Čech complexes for the V_ρ,.
The fan Σ induces a closed cover _ρ, of each V_ρ, and we will consider the Čech complex Č^∙(_ρ,,) with respect to this cover.


For any local Artinian -algebra A with residue field  and maximal ideal _A, we will define a natural map

    :	⊕_ρ,Č^0(_ρ,,_A)→⊕_ρ,Č^1(_ρ,,_A).

We then set
 	
    _Σ(A)   = {α∈⊕_ρ, Č^0(_ρ,,_A): (α)=0 } /∼

	where ∼ is a certain equivalence relation. See Definition <ref> for details. This can be made functorial in the obvious way; we call the resulting functor the combinatorial deformation functor.

Our primary result is the following:

thm:main
Let X=X_Σ be a -factorial toric variety without any torus factors and assume that H^1(X,)=H^2(X,)=0, for example, X is a smooth complete toric variety. Then the functor '_X of locally trivial deformations of X is isomorphic to the combinatorial deformation functor _Σ.


By utilizing a comparison theorem, we obtain a similar combinatorial description of _X for complete toric varieties that are smooth in codimension two and -factorial in codimension three, see mildsing for the precise statement.

Our main motivation in introducing the combinatorial deformation functor _Σ was to be able to effectively compute the hull of _X  when X=X_Σ is a toric variety with sufficiently mild singularities. Using the combinatorial deformation functor, we introduce the combinatorial deformation equation and show that one can compute a hull of _X by solving this equation to higher and higher order through a combinatorial process, see <ref>. We also show that in some cases, this process can be simplified further by removing certain maximal cones from the fan Σ (thm:samehull) and limiting those pairs of maximal cones on which we must consider obstruction terms (prop:closure and prop:codimone).


Given two first order deformations of X=X_Σ over [t_1]/t_1^2 and [t_2]/t_2^2, the cup product computes the obstruction to combining them to a deformation over [t_1,t_2]/⟨ t_1^2,t_2^2⟩. This cup product has been described in combinatorial terms by Ilten and Turo in <cit.>. Using the functor _Σ, we are able provide explicit combinatorial formulas for the obstructions to lifting deformations to arbitrary order, see thm:obsformula. In particular, we easily recover the cup product formula of <cit.>. In contrast to the situation of the cup product, our higher order obstruction formulas include not only first order combinatorial deformation data, but also higher order data. 

The combinatorial deformation functor _Σ exhibits a large amount of structure. Utilizing this structure, we provide non-trivial conditions guaranteeing that X_Σ has unobstructed deformations:
(See thm:unobstructed)
	Let X_Σ be a complete toric variety that is smooth in codimension 2 and -factorial in codimension 3. Let  consist of all pairs (ρ,) of rays and characters for which H^0(V_ρ,,) does not vanish.
	Set 
	
    := { (ρ,+)∈Σ(1)× M |
    			(ρ,)∈𝒜; ∈∑_(ρ',')∈𝒜_≥0·' 
    		}.

	If H^1(V_ρ,,)=0 for all pairs (ρ,)∈, then X_Σ is unobstructed.
	
In Example <ref>, we give an example of a smooth toric threefold whose unobstructedness follows from our conditions but cannot be deduced by degree reasons alone.

Finally, we put our machinery to use to calculate the hull of _X for numerous examples.
It makes sense to start with examples of low Picard rank.
By analyzing H^1(X,_X) and H^2(X,_X) for X a smooth complete toric variety of Picard rank two we show:

Let X be a smooth complete toric variety of Picard rank one or two. Then X has unobstructed deformations. Furthermore, X is rigid unless it is the projectivization of a direct sum of line bundles on ^1 such that the largest and smallest degrees differ by at least two.


The first interesting case of examples to consider is thus toric threefolds X_Σ of Picard rank three (toric surfaces are always unobstructed by <cit.>). We focus on the case where Σ is a splitting fan, that is, X_Σ is an iterated ^1-bundle. 
These ^1-bundles have the form 

    (__e⊕__e(aF+bH))

for e,a,b∈, e,b≥ 0
where _e=(_^1⊕_^1(e)) is the eth Hirzebruch surface, and F and H respectively represent the classes in (_e) of the fiber and _𝔽_e(1) in the ^1-bundle fibration of _e over ^1. 
We discover that such threefolds may have obstruction equations whose lowest terms are quadratic or cubic, or they may be unobstructed:
thm:obstructed
	Let 
    X= (__e⊕__e(aF+bH))

	with e,b≥ 0.
	Then X is obstructed in exactly the following cases: 
	
		
		
  * The case e=1, a≤ -2, and b≥ 3-a. In this case, the minimal degree of obstructions is three. 
		
  * The case e≥ 2, a≤ -e, and b ≥  1+2-ae. If a≡ 1  e, then the minimal degree of obstructions is three. In all other cases, the minimal degree of obstructions is two.
		
		 
	
	

By computing the hull of _X for a number of such examples, we find examples whose hulls exhibit the following behaviour:

	
  * The hull has a generically non-reduced component (Example <ref>).
	
  * The hull is irreducible but has a singularity  at the origin (Example <ref>).
	
  * The hull has a pair of irreducible components whose difference in dimension is arbitrarily large (Example <ref>).

None of these phenomena had previously been observed for deformation spaces of smooth toric varieties.

In <cit.>, it was asked if the deformation space of a smooth toric variety X is cut out by quadrics. By thm:obstructed, we see that the answer to this question is negative. In particular, any differential graded lie algebra controlling _X cannot be formal.



 §.§ Our approach

As mentioned above, for any variety X the functor _X' is controlled by the Čech complex
 Č^∙ (,_X) for the tangent sheaf _X with respect to an affine open cover  of X. Indeed, isomorphism classes of deformations of X over a local Artinian -algebra A with residue field  are given by

    _X'(A)≅{x∈Č^1(,_X)⊗_A: x_ij x_jk -x_ik =0}/∼

where _A is the maximal ideal of A,  is the Baker-Campbell-Hausdorff (BCH) product, and ∼ is an equivalence relation induced by an action of Čech zero-cochains, see <ref>, <ref>, and <ref> for details.


More generally, any sheaf of Lie algebras  on a topological space X naturally gives rise to a deformation functor _, see liefun1 and <ref>.
The first step in proving thm:main is to replace the tangent sheaf _X by a simpler sheaf  of Lie algebras. For -factorial toric X, the generalized Euler sequence gives a surjection 

    :=⊕_ρ(D_ρ)→_X

where the D_ρ are the toric boundary divisors, see  <ref>. Moreover, the sheaf  has a natural bracket and the above map is a map of sheaves of Lie algebras (thm:liebracket). This  induces a map of functors _→__X≅_X' that is an isomorphism under appropriate cohomological vanishing conditions.







The Lie algebra structure on  may be seen as coming from the Cox torsor of X, see Remark <ref>. In fact, this is a manifestation of the fact that, under the hypotheses of thm:main, invariant deformations of the Cox torsor are equivalent to locally trivial deformations of X, see rem:cox2 for an even more general statement. This is very much inspired by the approach of J. Christophersen and J. Kleppe <cit.>.

The second step in proving thm:main involves a homotopy fiber analogue in our setting. The sheaf  is a subsheaf of a constant sheaf
 which also comes with a natural Lie bracket. Given such an inclusion ↪ of sheaves of Lie algebras on any space X, we construct a new deformation functor _↪ (functornew2) that can be understood as a certain quotient of the deformation functor controlled by the homotopy fiber of the inclusion ↪ (even though the homotopy fiber doesn't exist in the category of sheaves of Lie algebras).
Under appropriate hypotheses, _ and _↪ are isomorphic (compa2 and prop:isosecfunctor).
We believe that this construction will be of independent interest for studying other deformation problems only involving locally trivial deformations.

In our specific setting of locally trivial deformations of a toric variety X_Σ, it turns out that _↪ is exactly the combinatorial deformation functor _Σ from above, and our primary result follows.



 §.§ Other related literature and motivation

K. Altmann began a systematic study of the deformation theory of affine toric singularities in the 1990's, giving combinatorial descriptions of first order deformations <cit.>, obstructions <cit.>, and homogeneous deformations <cit.>. For the special case of a Gorenstein toric threefold X with an isolated singularity, he gave a combinatorial description of the hull of _X and its irreducible components <cit.>. This has recently been revisited by Altmann, A. Constantinescu, and M. Filip in <cit.> with similar results in a slightly more general setting using different methods. Filip has also given a combinatorial description of the cup product in the affine case, see <cit.>.

There are numerous motivations for studying deformations of toric varieties, both in the affine and in the global situations; we now mention several. Deformations of toric varieties are useful in mirror symmetry for studying deformations of embedded Calabi-Yau hypersurfaces <cit.> and for classifying deformation families of Fano varieties <cit.>. Deformation theory of toric varieties has  been used to study singularities on the boundary of the K-moduli spaces of Fano varieties <cit.>, extremal metrics <cit.>, the boundary of Gieseker moduli spaces <cit.>, and to show that deformations of log Calabi-Yau pairs can be obstructed <cit.>.
We hope that our results here will find similar applications.




 §.§ Organization

We now describe the organization of the remainder of this paper. Section <ref> concerns itself with deformation functors governed by a sheaf of Lie algebras. In <ref> and <ref> we recall preliminaries on deformation functors and Baker-Campbell-Hausdorff products, respectively. In <ref> we define the deformation functor controlled by a sheaf of Lie algebras and study various properties of it. This is similar to the contents of <cit.>, but we work with alternating instead of singular or ordered Čech cochains. Subsection <ref> contains the first significant innovation of the paper: our definition and study of the homotopy fiber analogue deformation functor _↪. 


In Section <ref> we describe a procedure to algorithmically construct the hull of our homotopy fiber analogue _↪ by iteratively solving a deformation equation. The idea of constructing a hull via iterated lifting goes back at least to <cit.> and is made more algorithmic in certain settings in <cit.>, but our setting is distinct enough that we provide a thorough treatment. We set up the situation in <ref>, introduce the deformation equation in <ref>, and show in <ref> that iteratively solving it produces a hull of our homotopy fiber analogue.

In Section <ref> we turn our attention to toric varieties. We recall preliminaries and set notation in <ref>. We then discuss cohomology of the structure sheaf in <ref>, introduce the simplicial complexes V_ρ, and discuss their relation to boundary divisors in <ref>, and introduce the Euler sequence in <ref>.


In Section <ref> we study  deformations of toric varieties. We define the combinatorial deformation functor _Σ in <ref> and prove our main result thm:main. We also discuss connections to Cox torsors. In <ref> we specialize the discussion of <ref> to the toric setting and discuss how to compute the hull of _Σ by solving the combinatorial deformation equation. We discuss formulas for higher order obstructions in <ref>.  In <ref> we discuss how to further simplify computations by removing certain cones from the fan Σ.
We prove our criterion for unobstructedness in <ref>.

In <ref> we turn our attention to examples. In <ref> we introduce primitive collections and prove a sufficient criterion for rigidity (lemma:rigid). We show that smooth complete toric varieties of Picard rank less than three are unobstructed in <ref>. In <ref>  we study toric threefolds X that are iterated ^1-bundles, obtaining very explicit descriptions of H^1(X,_X) and H^2(X,_X). We continue this study in <ref>, providing several examples whose deformation spaces exhibit interesting behaviour and proving thm:obstructed.

We conclude with three appendices. In Appendix <ref>, we provide a proof of compa which gives a criterion for deformation functors controlled by two different sheaves of Lie algebras to be isomorphic. In Appendix <ref> we state a folklore theorem (CMiso) comparing deformations of a scheme X with deformations of an open subscheme U; for lack of a suitable reference we provide a proof. The theorem implies that in particular, for X a Cohen-Macaulay variety that is smooth in codimension two, deformations of X may be identified with deformations of the non-singular locus.
Finally, in Appendix <ref> we show that the deformation equation of <ref> can in fact be iteratively solved.

For the reader who is interested in understanding the precise definition of the combinatorial deformation functor _Σ and the statement of Theorem <ref> as quickly as possible, we recommend reading <ref>, <ref>, <ref>, <ref>, <ref>, and <ref>.




 §.§ Acknowledgements

We thank A. Petracci and F. Meazzini for productive discussions. Both authors were partially supported by NSERC.







§ DEFORMATION FUNCTORS






 §.§ Preliminaries

In this section, we will provide a brief review of the definitions and notation concerning functors of Artin rings. For detailed information regarding the functor of Artin rings, we refer to  <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.

Let  be an algebraically closed field of characteristic zero. Let  be the category of complete local noetherian -algebras with the residue field . For every R∈ we denote by _R the maximal ideal of R. We also consider the subcategory , which consists of local Artinian -algebras with the residue field . We denote by  the category of sets.

A functor of Artin rings is a covariant functor : → such that  F() is singleton. For any morphisms A'→ A and A”→ A in ,  we consider the  natural map induced from the fiber product:

    (A' ×_A A”) →(A')×_(A)(A”).
    

	A functor of Artin rings  is called a deformation functor  if:
	
		
  * (<ref>) is surjective whenever A'→ A is surjective; and 
		
  * (<ref>) is bijective whenever A=.
	
	We say  is homogeneous if (<ref>) is bijective whenever A'→ A is surjective.




The set 

    ^1 = ([t]/t^2),
 known as the tangent space of , exhibits a natural -vector space structure if  is a deformation functor.


	A small extension in  is an exact sequence  
	
    0→ I → A' A → 0,
 
	where π is a morphism in  and I is an ideal of A' such that _A'· I=0. The A'-module structure on I induces a -vector space structure on I.

Definition <ref> is standard, but there is some inconsistency in the literature about the terminology. We opt to maintain consistency with the terminology used in <cit.>,<cit.>.

An important tool for studying deformation functors is obstruction theory, defined as follows:


	A complete obstruction theory (W,ϕ) for a functor of Artin rings  consists of a -vector space W, called the obstruction space, and a function ϕ  that assigns to every small extension 
	
    0→ I → A'  A → 0
 
	and any ζ∈(A) an element ϕ(ζ,A')∈ W⊗ I such that:
	
		
  * (completeness) ϕ(ζ,A')=0 if and only if ζ
		 lifts to (A'); and
		
  * (functoriality) for every morphism of small extensions
		
    0 [r]     I_1 [r] [d, "π_I"]     A'_1 [r] [d,"π_A'"]     A_1 [r] [d, "π_A"]   0
    
    			0 [r]     I_2 [r]      A'_2 [r]    A_2 [r]     0

		and ζ∈(A_1), we have
		
    ϕ(π_A(ζ),A_2')= ⊗π_I (ϕ(ζ,A_1')).

		Here,  is the identity map on W.
		
	

rem:vanishing
	If ζ lifts to (A'), then the vanishing of ϕ(ζ,A') is an immediate consequence of the functoriality property, see <cit.>.


When comparing two deformation functors, it is beneficial to ensure that their associated obstruction theories are comparable, as precisely defined by the following:

obstrutionmapb/w
	Let (,ob_, ϕ_) and (,ob_,ϕ_) be functors of Artin rings together with associated complete obstruction theories, and let f: → be a morphism of functors. We say that a linear map ob_f: ob_→ ob_ is an obstruction map for f if it is compatible with the obstruction theories of  and . In other words, for every small extension 
	
    0→ I → A'  A → 0
 
	and any ζ∈(A), we have
	
    ϕ_(f(ζ),A')= ob_f (ϕ_(ζ,A')).
 


Recall that a map of functors of Artin rings → is smooth if for every surjection A'→ A in , the induced map 

    (A')→(A)×_(A)(A')

is surjective.
We will frequently make use of the standard smoothness criterion to show that a morphism of functors is surjective:

 standardsmooth
	Let (,ob_, ϕ_) and (,ob_,ϕ_) be deformation functors together with associated complete obstruction theories, and let f: → be a morphism of functors that admits an obstruction map ob_f: ob_→ ob_. Assume that:
	
		
  * ^1→^1 is surjective; and
		
  * ob_f is injective.
	
	 Then f is smooth. In particular, it is surjective.
  










 §.§ The Baker-Campbell-Hausdorff product

In this section, our main aim is to lay down the necessary groundwork for defining the deformation functors in <ref> and <ref>. We will begin by giving an overview of the definition of the Baker-Campbell-Hausdorff product for a Lie algebra. Additionally, we will present proofs of several basic properties that will be useful in the subsequent sections. For more details about Lie algebras, we refer to <cit.>, <cit.>.


Let  be a Lie algebra. By possibly embedding  into a universal enveloping algebras (see <cit.>), we can always assume that the Lie bracket on  is given by the commutator in some associative algebra. Through this embedding into an associative algebra, we define exp(x) for any A∈ and x∈⊗_A using the formal power series of the exponential map.  These maps are clearly convergent in the _A-adic topology. 



The Baker-Campbell-Hausdorff (BCH) product x y of x,y ∈⊗_A is defined by the relation

    exp(x)·exp(y)=exp(x y).

The non-linear terms of x y can be expressed as nested commutators of x and y with rational coefficients (see e.g. <cit.>). The expression in nested commutators is not unique, and we will call any expression of this form a BCH formula. The first few terms are easily calculated and well known:

    x y= x+y+12[x,y]+112[x,[x,y]]-112[y,[x,y]]+ (Higher order terms).

An explicit and complete description of a BCH formula is provided by <cit.>


    x y=∑ _n=1^∞(-1)^n-1/n∑ _[ r_1+s_1>0;         ⋮; r_n+s_n>0 ][x^r_1y^s_1x^r_2y^s_2… x^r_ny^s_n]/(∑ _j=1^n(r_j+s_j))·∏ _i=1^nr_i!s_i!.

In this expression, the summation extends to all nonnegative integer values of s_i and r_i, and the following notation is utilized:

    [x^r_1y^s_1… x^r_ny^s_n]=[x,[x,… [x^r_1,[y,[y,… [y ^s_1, … [x,[x,… [x ^r_n,[y,[y,… y ^s_n]]… ]]

with the definition [x] := x.



For x,y,z ∈⊗_A, it is immediate to see that

    (x y) z= x (y z);    x0=0 x=x;   x-x=0,

and (⊗_A,) is a group (see also <cit.>, <cit.>).

We will repeatedly use the following result to establish other calculations.

comm
	Let  be a Lie algebra. Given a small extension 
	
    0→ I → A' → A → 0

	in  and elements x ∈⊗_A', w ∈⊗ I we have the following:
	
		
  * x w = w  x = x+w; 
		
  * (-x) w  x= w. 
	
	Furthermore, if  x,y,z ∈⊗_A' such that  x y  z ∈⊗ I then
	
		
  * x y  z= y z  x= z x  y.
		
		
	


	Since x· w=0, the BCH product reduces to exp(x)·exp(w)=exp(x+w), and the proof of claim <ref> follows. Through an immediate application of claim <ref>, we have
	
    (-x) w  x    = (-x)  x w=w.

	This establishes claim <ref>.  By applying claim <ref> of the lemma, we obtain
	
    x y z   =(-x) (x y z) x  
       = y z x .

	By symmetry, we also obtain y z x= z x  y, and claim <ref> follows.


Now, we will show that  commutes with fiber products in . Let x'∈⊗_A' and A' A be a morphism in . 
For convenience, we also refer to the map  ⊗π': ⊗_A'→⊗_A as π' when the context is clear. Since  is A'-linear, it is immediate that

    π'(x' y')= π'(x') π'(y').


BCHfiberprod
	Let A' A, A” A be morphisms in . Consider x',y' ∈⊗_A' and x”,y”∈⊗_A” with 
	
    π'(x') = π”(x”),   π'(y') = π”(y”).

	Then (x',x”), (y',y”)∈⊗_A'×_A A” and 
	
    (x',x”)  (y',y”)= (x' y', x” y”)



	Consider the following commutative diagram:
	
    ⊗_A'×_A A”[r,"pr_2"] [d,"pr_1"]    ⊗_A”[d, "π”"]
    ⊗_A'[r,"π'"]    ⊗_A
	
The proof involves a diagram chase of the diagram above, utilizing the fact that  commutes with pr_1, pr_2, π', π”, which follows from (<ref>). 


Using  we will define maps that generalize the differential of Čech complexes. These new maps will be used to define the deformation functors in <ref>. 

Let X be a topological space, and ={U_i} be an open or closed cover of X. For any sheaf of abelian groups  on X, we denote by Č^∙_(,) the Čech complex of singular cochains with respect to the cover . It is often advantageous to consider the subcomplex Č^∙(,) ⊆Č^∙_(,) of alternating Cech cochains. We will present our results in alternating cochains whenever possible. Let 
    d:Č^k(,)→Č^k+1(,)
 denote the Čech differential, let Ž^k(,) denote the group of k-cocycles, and let Ȟ^k(,) denote k-th Čech cohomology group. In this section we will only consider Čech complexes for an open cover , but in <ref> we will also consider Čech complexes for a constant sheaf on a simplicial complex with respect to a closed cover.



For the remainder of this section, we take  to be a sheaf of Lie algebras on a topological space X, and ={U_i} to be an open cover of X. For any open set U⊆ X, H^0(U,) is a Lie algebra. Hence, we can apply all the discussions presented at the beginning of this section to H^0(U,).
 
action
	Let A∈. We define a left action of the group (Č^0(,)⊗_A,) on the set Č^1_(,)⊗_A given by
	
    Č^0(,)⊗_A ×Č^1_(,)⊗_A    →Č^1_(,)⊗_A
    
    		(a,x)   ↦ a⊙ x

	via
	
    (a⊙ x)_ij= a_i  x_ij -a_j.



If x∈Č^1(,)⊗_A,
it can be readily verified that

    (a⊙ x)_ij (a⊙ x)_ji=0

for all a∈Č^0(,)⊗_A. Therefore, according to (<ref>), we have (a⊙ x)_ji=-(a⊙ x)_ij. As a result, action can be restricted to alternating cochains.
	


defn:omaps
	Let A∈. We use the product  to define the maps     
	
    :Č^0(,)⊗_A   →Č^1(,)⊗_A,
    :Č^1(,)⊗_A   →Č^2_(,)⊗_A
   
	via
	
    (a)_ij   =a_i -a_j, 
    (x)_ijk   =x_ij x_jk -x_ik.
  


obs:alt
	For every a ∈Č^0(,)⊗_A, we have
	
    (a)_ij(a)_ji=0.

	Since (H^0(U_i∩ U_j,)⊗_A, ) is a group, we can conclude that (a)_ji=-(a)_ij and
	(a) ∈Č^1(,)⊗_A. However, it is important to note that (x) may not be alternating. 


rem:obs
	The maps  and  exhibit several notable properties.
	
		
  *   Both  and , as well as the action described in action, are compatible with the morphisms in , as can be seen from  (<ref>).
		
  *  Compatibility with the fiber products in  holds for both  and , as implied by BCHfiberprod.
		
  *  
					Consider a small extension
		
    0→ I→ A' → A→ 0.

		Let x ∈Č^1(,)⊗_A'. 
		For any y∈Č^1(,)⊗ I, we have
		
    (x+y)=(x)+d(y).

		This result follows by a repeated application of claims <ref> and <ref> of comm.
		A similar result holds for .
		
  *  
		For every x ∈Č^0(,)⊗_A, we observe that ((a)) is zero. This follows from a direct computation, utilizing the fact that (a) is alternating.
		
  *  If (x)=0, then (a⊙ x)=0 for every a∈Č^0(,)⊗_A.
	







 §.§ Deformation functors governed by the sheaves of Lie algebras

In this section, we employ sheaves of Lie algebras as a source for constructing deformation functors. In <cit.>, the authors consider the deformation functors controlled by semicosimplicial Lie algebras (see also <cit.>). As a special case, this includes semicosimplicial Lie algebras arising from the singular Čech complex of a sheaf of Lie algebras (see <cit.>). Since we are working primarily with the alternating Čech complex, we adapt some of the arguments of <cit.> to our setting. 

liefun1
	Let  be a sheaf of Lie algebras on a topological space X, and let  be an open cover of X.
	We define the functor _,: → on objects as follows:
	
    _,(A)= {x ∈Č^1(,)⊗_A: (x)=0 }/∼

	where ∼ is the equivalence relation induced by the action of Č^0(,)⊗_A on Č^1(,)⊗_A as given by action. The functor sends a morphism
	π': A' → A to the map 
    _,(π'): _,(A') →_,(A)

	induced by the map Č^1(,)⊗_A'→Č^1(,)⊗_A. For convenience, we also refer to _,(π') as π'.


By claims <ref> and <ref> of rem:obs, it is evident that the functor is well-defined on both objects and morphisms. Given that we are working over alternating cochains instead of singular cochains (cf. <cit.>), the following lemma is needed for discussing an obstruction theory of _,.


obsL
	Let A∈ and x∈Č^1(,)⊗_A such that (x)=0. Let 
	
    0→ I → A' → A → 0
 be a small extension in , and let x'∈Č^0(,)⊗_A' be any lift of x. Then:
	
		
  * (x') is an alternating 2-cocycle; 
		
  * for any other lifting y'∈Č^1(,)⊗_A' of x, we have
		
    (y')=(x')+d(y'-x');
 
		
  * (a'⊙ x')=(x') for every a'∈Č^1(,)⊗_A'. 
	


First, we will prove that (x') is a 2-cocycle. To do that, we need to show the cocycle condition 

    -(x')_ijk+ (x')_ijℓ-(x')_ikℓ+(x')_jkℓ=0
	
for all U_i,U_j,U_k,U_ℓ in . It is immediate that (x')∈Č^2(,)⊗ I, since its image in Č^2(,)⊗_A is zero. Hence, by comm<ref>, it is enough to show that

    -(x')_ijk(x')_ijℓ -(x')_ikℓ(x')_jkℓ=0.

	
Note that,

    -(x')_ijk(x')_ijℓ -(x')_ikℓ(x')_jkℓ   = x'_ik-x'_jk-x'_ij
          x'_ij x'_jℓ -x'_iℓ
          x'_iℓ-x'_kℓ-x'_ik
          x'_jk x'_kℓ -x'_jℓ.

Since (x') ∈Č^2(,)⊗ I, we can apply comm<ref>. By permuting the terms, it is evident that the right-hand side of the above equation is zero. Thus, (<ref>) and (<ref>) follows.  

We will now show that (x') is alternating as well. It is enough to show that

    (x')_iik=0;  (x')_ijk=- (x')_jik;   (x')_ijk= -(x')_ikj.

Since x' is alternating, it is straightforward to verify that

    (x')_iik= x'_ii x'_ik -x'_ik=0.

Similarly, using x' is alternating, we obtain that

    (x')_ijk(x')_ikj=x'_ij x'_jk-x'_ik x'_ik x'_kj-x'_ij=0.
  
Thus, we obtain (x')_ijk=-(x')_ikj.
By applying comm<ref> (since (x')∈Č^1(,)⊗ I), and using the fact that x' is alternating, we obtain

    (x')_ijk(x')_jik= x'_ij x'_jk -x'_ik x'_ji x'_ik -x'_ik=0.

Hence,  (x')_ijk=-(x')_jik and
this completes the proof of claim <ref>.

To show the claim <ref>, note that, y'-x' ∈Č^1(,)⊗ I, since its image in Č^1(,)⊗_A is zero. Hence,

    (y')   =(x'+ y'-x')
       =(x')+d(y'-x').

Here, the last equation follows from rem:obs<ref>.


Finally, claim <ref> follows by an application of comm<ref>, since (x')∈Č^2(,)⊗ I.




thm:deffunctor1
	Let  be a sheaf of Lie algebras on a topological space X, and let  be an open cover of X. Then:
	
		
  * _, is a deformation functor; 
		
  * the tangent space of the functor _, is isomorphic to Ȟ^1(,); 
		
  * _, has a natural complete obstruction theory (Ȟ^2(,), ϕ), where for a given small extension 
		
    0→ I → A'→ A → 0,
 
		and ζ∈_,(A), the obstruction ϕ(ζ,A') is defined to be the class of (x') in Ȟ^2(,). Here, x'∈Č^1(,)⊗_A' is any lift of x∈Č^1(,)⊗_A which represents ζ.   
	


	While one could explicitly establish claim <ref> using liefun1, we opt to provide a proof similar to that of <cit.>. However, we cannot directly apply the result of <cit.> since we are dealing with alternating cochains. We define two functors _,: → and Č^0_,:→ on objects as follows:
	
    _,(A)   = {x ∈Č^1(,)⊗_A: (x)=0 };
    Č^0_,(A)   =Č^0(,)⊗_A.

	Here,  denotes the category of groups, and the group structure on Č^0(,)⊗_A is defined by .
	It is immediate that Č^0_, is a smooth homogeneous functor (see also <cit.>). From rem:obs<ref>, we infer that _, is also homogeneous and indeed a deformation functor. Since 
    _,=_,Č^0_,,

	claim <ref> follows by <cit.>. The proof of claim <ref> is a direct calculation, and it follows from the discussion in <cit.> by replacing singular cochains with alternating cochains. Finally, for the proof of claim <ref>, Lemma <ref> allows us to work with alternating cochains and uses the discussion in  <cit.>.



Next, we will study the morphisms of deformation functors governed by sheaves of Lie algebras. Let f:→ be a  morphism of  sheaves of Lie algebras on X. Let  be an open cover of X and let  be a refinement of . Then f induces a morphism of functors f:_,→_,. 


compa
	Let f:→ be a  morphism of  sheaves of Lie algebras on X. Let  be an open cover of X, and let  be a refinement of .  Assume that the induced morphism of Čech cohomology f:
	Ȟ^i(,) →Ȟ^i(,) is surjective for i=0, bijective for i=1, and injective for i=2.
	Then the induced morphism of functors f:_,→_, is an isomorphism.

We defer a proof of compa until Appendix <ref>. 
	
cor:comparison
	Let X be a separated scheme of finite type over , let , be affine open covers of X, and let  be a quasi-coherent sheaf of Lie algebras on X.  Then 
    _,≅_,.
 


	Čech cohomology coincides with sheaf cohomology for any affine open cover <cit.>, and the proof directly follows from compa.

In the next sections, we work over affine open coverings and in light of cor:comparison, we suppress  and write _ for the functor _,.




rem:applications
	The functors _ are useful to study several geometric deformation problems. We list some of them here.
	
		
  * Let X be variety over , and let _X and  '_X denote the functor of deformations of X and locally trivial deformations of X respectively. When X is smooth they coincide. The locally trivial deformations of X are governed by the tangent sheaf _X and
		
    __X'_X
 is an isomorphism of functors, see <cit.>.  We will study this situation in detail in <ref> when X is a -factorial toric variety. 
		
		
  * Deformations of a vector bundle ℰ on a scheme X over  is governed by the endomorphism bundle ℰnd(ℰ), see <cit.>. 
		
	
		
  * Let Z ⊂ X be a closed subscheme of a smooth variety X. Let _(X,Z) and '_(X,Z) respectively denote the functors of deformations and locally trivial deformations of the pair (X,Z). When Z is smooth the above functors coincide. Locally trivial deformations of the pair (X,Z) are governed by the logarithmic tangent sheaf _X(-log Z), see <cit.> and <cit.>. 
	









 §.§ A homotopy fiber analogue


In this section, we continue to study the deformation functor _ governed by a quasi-coherent sheaf of Lie algebras  on a separated scheme of finite type over . We introduce new deformation functors that arise from an injective morphism of sheaves of Lie algebras →, see functornew. 
A significant result of our paper is the proof that these new functors are isomorphic to _ under the assumption that  H^1(,) vanishes, see compa2. 


Let X be a separated scheme of finite type over , and fix an affine open covering  of X.
Let ι:→ be an injective morphism of quasi-coherent sheaves of Lie algebras on X. Then we have an exact sequence of quasi-coherent sheaves on X given by

    0 [r]    [r,"ι"]    [r,"λ"]    / [r]    0.

It is worth noting that / is not a sheaf of Lie algebras in general.
The exact sequence (<ref>) induces a map of Čech complexes given by

    0 [r]    Č^∙(,) [r,"ι"]    Č^∙(,) [r,"λ"]    Č^∙(,/) [r]      0.

We will use d to refer to the differential of any one of these complexes. Note that  ι and λ are compatible with d. 
Since ι is injective, we will use
the notation ι^-1 to denote the map

    ι(Č^∙(,)) →Č^∙(,)

inverse to ι. 



	Let  be a quasi-coherent torsion-free sheaf of Lie algebras on a variety X over . In this case, we can always construct an exact sequence of the form (<ref>) as follows.  We fix an affine open covering ={U_i} of X and choose a non-empty open set V⊆⋂ U_i. Using the injection ι: V ↪ X, we define
	
    =ι_*(_|V).
 
	Since  is a torsion-free sheaf, the induced map → is injective. Moreover, the Lie bracket on  induces one on .





functornew
	In the situation (<ref>), we define the functor _↪:→ on objects as follows:
	
    _↪(A)   = {α∈Č^0(, )⊗_A : λ((α))=0 } /∼

	where α=β in _↪(A)  if and only if there exists a ∈Č^0(,)⊗_A such that 
	
    ι(a)⊙(α) = (β).

	The functor sends a morphism
	π: A' → A to the map 
    _↪(π): _↪(A') →_↪(A)

	induced by the map Č^0(,)⊗_A'→Č^0(,)⊗_A.


In a sense that we will not make precise here, the functor _↪ can be viewed as a quotient of the deformation functor controlled by the homotopy fiber of ↪. For a similar construction in the DGLA setting, see <cit.>). 


Before studying properties of the functor _↪, we define a related functor _↪. For this, we need to fix the following data: for each affine open set U⊆ X obtained by intersecting elements of , we fix a -linear section 

    s: (/)(U) →(U)

to λ, that is, λ∘ s: (/)(U)→ (/)(U) is the identity map. 

For every A∈, we thus have the following commutative diagram:
[H]
	
    Č^0(,)⊗_A [r,hook,"ι"] [d,"" left]    Č^0(,)⊗_A [r,twoheadrightarrow,"λ"below] [d,"" left]    Č^0(,/)⊗_A  [l, bend right=20,"s" above]   
    Č^1(,)⊗_A [r,hook,"ι"] [d,"" left]    Č^1(,)⊗_A[r,twoheadrightarrow,"λ"] [d,"" left]    Č^1(,/)⊗_A [l, bend right=20,"s" above] 
    		
    Č^2(,)⊗_A[r,hook,"ι"]    Č^2(,)⊗_A [r,twoheadrightarrow,"λ"]     Č^2(,/)⊗_A .

	equation1
	Map between Čech complexes involving the maps  and 
	
	


functornew2
	In the situation (<ref>) with a fixed collection of sections s as above, we define the functor _↪:→ on objects as follows:
	
    _↪(A)   = {α∈Č^0(, /)⊗_A : λ((s(α)))=0 } /∼

	where α∼β in _↪(A)  if and only if there exists a∈Č^0(,)⊗_A such that 
	
    ι(a)⊙(s(α)) = (s(β)).

	The functor sends a morphism
	π: A' → A to the map 
    _↪(π): _↪(A') →_↪(A)

	induced by the map Č^0(,/)⊗_A'→Č^0(,/)⊗_A.


In the remainder of this section, we show that _↪ and _↪ are isomorphic deformation functors, and they are furthermore isomorphic to _ if Ȟ^1(,)=0.
We first observe that the equivalence relation in functornew can also be defined by a group action.

alternatedefn
	We claim that the equivalence relation in functornew can be expressed as follows: α = β in _↪(A) if and only if there exist a ∈Č^0(, ) ⊗_A and γ∈Ȟ^0(, ) ⊗_A such that
	
    ι(a) α -γ = β.

	
	Indeed, note that γ∈Ȟ^0(, ) ⊗_A is equivalent to γ∈Č^0(, ) ⊗_A and γ_i -γ_j=0 for all U_i,U_j ∈. Thus, (<ref>) and γ∈Ȟ^0(, ) ⊗_A   imply that
	
    β_i -β_j   =ι(a)_iα_i  -γ_iγ_j -α_j -ι(a)_j
       =ι(a)_iα_i  -α_j -ι(a)_j.

 	Hence, we obtain
	
    ι(a) ⊙(α) = (β).

	Moreover, we note that if λ((α))=0, there exists x∈Č^1(,)⊗_A such that ι(x)=(α). Hence, we obtain 
	
    λ((β))= λ(ι(a⊙ x))=0.

	
	Conversely, if 
	
    ι(a) ⊙(α) = (β),

	by setting γ = -βι(a) α, we can see that
	
    ι(a) α -γ = β,   γ∈Ȟ^0(, ) ⊗_A .







 To define an obstruction theory for the functor _↪, we require the following proposition.


prop:cocycle:fun2
	 Let α∈Č^0(,)⊗_A such that λ((α))=0, where A∈. Let
	
    0→ I → A' → A → 0

	be a small extension, and let α'∈Č^0(,)⊗_A' be any lift of α. Then:
	
		
  *   λ((α')) is an alternating 1-cocycle;
		
  *   for any other lift ξ'∈Č^0(,)⊗_A' of α, we have
		
    λ((ξ'))=λ((α'))+ d(λ(ξ'-α'));

		
  *   for any a'∈Č^0(,)⊗_A', we have 
    λ(ι(a')⊙(α'))= λ((α')).

	


	By obs:alt, λ((α')) is an alternating cochain. We will show that it is a cocycle as well. Since  λ((α))=0, there exists x ∈Č^1(,)⊗_A such that ι(x)=(α). Furthermore, λ((α'))∈Č^1(,/)⊗ I, since the image of λ((α')) in Č^1(,/)⊗_A is λ((α))=0. We set 
    η'=s(λ((α'))) ∈Č^1(,)⊗ I.

	To prove claim <ref>, it is enough to show that d(λ(η'))=0. Since  λ((α')- η')=0, there exists x'∈Č^1(,)⊗_A' such that ι(x')= (α')- η'.  By the commutativity of Diagram <ref>, we have
	
    ι((x'))   = (ι(x'))
       = ((α')- η')
       = ((α'))- d(η')
       =-d(η').

	Here, the penultimate equation follows from rem:obs<ref>, and the last equation follows from rem:obs<ref>. Then it follows that
	
    d(λ(-η'))=λ(d(-η'))= λ(ι((x'))=0,
  
	completing the proof of claim <ref>.
	
	Now we will prove claim <ref>. Since α',ξ' are liftings of α, we obtain ξ'-α'∈Č^0(,)⊗ I. Then we have
	
    (ξ')   = (α'+ξ'-α')
       = (α')+d(ξ'-α')

	Here, the last equation follows from rem:obs<ref>. By applying λ, and using the fact that λ commutes with d we obtain
	
    λ((ξ'))=λ((α'))+ d(λ(ξ'-α')).

	
	Now we will prove claim <ref>. As in the proof of claim <ref>, we set η'=s(λ((α'))) and find x' such that
	
    ι(x')   = (α')- η'
       = (α')  -η'.

	Here, the last equation follows by an application of comm<ref>.
	Then for any a'∈Č^0(,)⊗_A', we have
	
    ι(a'⊙ x')   = 	ι(a')⊙ι(x')
       = ι(a')⊙ ((α')-η')
       = (ι(a')⊙(α'))-η'.

	Here, the last equation follows by an application of comm<ref>.
	By applying λ to this equation we obtain
	
    0   = λ(ι(a')⊙(α'))-λ(η')
       =λ(ι(a')⊙(α'))- λ( (α')).



showdeffun2
	Let X be a separated scheme of finite type over , and let  be an affine open cover of X. Consider sheaves of Lie algebras  and  on X which fit into an exact sequence of the form (<ref>). Let s be a collection of sections as in (<ref>). Then:
	
		
  * _↪ is a deformation functor; 
		
  * the map
			
    λ: ^1_↪=_↪([t]/t^2)  →Ȟ^0(,/)/ λ(Ȟ^0(,))⊗ t

		is an isomorphism with inverse s; 
		 
  * the functor _↪  has a natural complete obstruction theory 
    (Ȟ^1(,/), ϕ),
 where for a given small extension 
		
    0→ I → A'→ A → 0,
 
		and ζ∈_↪(A), the obstruction ϕ(ζ,A') is defined to be the class of λ((α')) in Ȟ^1(,/)⊗ I. Here, α'∈Č^0(,)⊗_A' is any lift of α∈Č^0(,)⊗_A which represents ζ.  
	


	The proof of claim <ref> is similar to the proof of Theorem <ref>. 
	We define two functors _↪:→ and C_↪: → on objects as follows:
	
    _↪(A)   ={α∈Č^0(,)⊗_A: λ((α))=0};
    
    		C_↪(A)   ={(a,γ)∈ (Č^0(,)⊗_A) × (Ȟ^0(,)⊗_A)}.
 
	Here,  again denotes the category of groups, and the group structure on C_↪(A) is  defined by . Since λ is _A-linear for any A∈, it commutes with fiber products in . Additionally, as remarked in rem:obs<ref>,  also commutes with fiber products in . Hence, the functor _↪ is homogeneous. Also, it is immediate that C_↪ is a smooth homogeneous functor (see also <cit.>). 
	
	We claim that there exists a left action of the functor C_↪ on _↪ as follows:
	
    C_↪(A) ×_↪(A) →_↪(A),    
    		((a,γ),α)↦ι(a)α -γ.

	By alternatedefn, we have ι(a)α -γ∈_↪(A). It is immediate to verify that (<ref>) is a group action, and _↪ is the quotient of _↪ by C_↪. 
	Thus, by <cit.>, claim <ref> follows.
	 
	
	Now we will compute the tangent space of _↪. We have
	
    ^1 _↪   = {α∈Č^0(, ) ⊗ t : λ((α)) = 0  t^2 } / ∼
       = {α∈Č^0(, ) ⊗ t : λ(d(α)) = 0 } / ∼
       = {α∈Č^0(,  ) ⊗ t : d(λ(α)) = 0 } / ∼.

	In order to show the claim <ref>, notice that there are -linear maps given by
	
    λ:  {α∈Č^0(,  ) ⊗ t : d(λ(α)) = 0 }→ H^0(,  / )⊗ t,

	and 
	
    s: H^0(,  / )⊗ t →{α∈Č^0(,  ) ⊗ t : d(λ(α)) = 0 }.

	Since λ∘ s is the identity map, the above map λ in (<ref>) is surjective.  Here, α∼ 0 in ^1_↪ if and only if there exist a ∈Č^0(, ) ⊗ t and γ∈Ȟ^0(, ) ⊗ t such that
	
    ι(a) + α -γ= 0.

	By applying d, we obtain ι(a) + α∈ H^0(, ) ⊗ t and so
	
    λ(α)=λ(ι(a) + α) ∈λ(H^0(, ) ⊗ t).

It follows that α∼ 0 if and only if λ(α)∈λ(H^0(, ) ⊗ t).
	 Thus, the surjective map (<ref>) induces an isomorphism 
	
    λ: ^1 _↪→ H^0(,  / )/λ(H^0(, )) ⊗ t
 
	with inverse s.

	
	
	Now we will show that _↪ carries a complete obstruction theory with values in Ȟ^1(,/). Let ζ∈_↪ be represented by α∈Č^0(, )⊗_A and let
	
    0→ I → A' → A → 0

	be a small extension. By prop:cocycle:fun2<ref>, λ((α')) defines an element in Ȟ^1(,)⊗ I, where α'∈Č^1(,)⊗_A' is any lift of α.
	By claims <ref> and <ref> of Proposition <ref>, it is evident that φ(ζ,A') is independent of the choice of lifting and the choice of representative of ζ. The functoriality property of the obstruction theory follows from rem:obs<ref>. 
	
	It remains to check the completeness of this obstruction theory. Assume that φ(ζ,A')=0. Then for any representative α of ζ and any lifting α'∈Č^0(, )⊗_A', there exists ξ'∈Č^0(,/)⊗ I  such that d(ξ')= λ((α')). Then α'-s(ξ') is also a lift of α and
	
    λ((α'-s(ξ')))   =λ((α'))-λ(d(s(ξ)))  
       =λ((α'))-d(ξ)  
       =0.

	Here, the first equation follows from rem:obs<ref> and second equation follows from the fact that λ commute with d and λ∘ s is identity. By combining with Remark <ref>, the completeness of the obstruction theory follows. 



Now we will examine the relationship between the deformation functors _ and _→. Let α∈Č^0(, )⊗_A satisfy λ((α))=0. Then, there exists a unique x∈Č^0(, )⊗_A such that ι(x)=(α). Since, ((α))=0 as noted in rem:obs<ref>, by considering the commutativity of Diagram <ref>, we can deduce that (x)=0. This mapping α↦ x=ι^-1((α)) induces a map
	
    ι^-1∘: _→→_,

	 since it is well-behaved under the equivalence relations defined on _ and _→. In fact, for any a∈Č^0(,)⊗_A, we have
	
    ι(a)⊙(α)   = ι(a)⊙ι(x)
       = ι(a⊙ x).

	
	 
	
	

	compa2
		If Ȟ^1(,)=0, then the morphism of functors
		
    ι^-1∘ :_↪→_
 is an isomorphism.
	
	
	First we will show the injectivity of ι^-1∘ :_↪→_. Let A∈, and let α,β∈Č^0(,)⊗_A be representatives of two elements in _↪(A) and
		
    ι^-1∘ (α)= x;   ι^-1∘ (β)=y.

	From liefun1 and functornew, it is immediately seen that if x∼ y, then α∼β.
	
		
	We will show that ι^-1∘ :_↪→_ is smooth; surjectivity then follows. In order to show that ι^-1∘ is smooth, we will verify that it satisfies the assumptions of the standard smoothness criterion (standardsmooth).
		 We have that _↪  and _ are deformation functors (see showdeffun2<ref> and thm:deffunctor1<ref> respectively). Next, we will show the surjectivity of ^1_↪→^1_ and the existence of an injective obstruction map.
		 
		Since Ȟ^1(,)=0, the long exact sequence of Čech cohomology yields that the boundary map δ:Ȟ^0(,/) →Ȟ^1(,) is surjective, and the boundary map δ: Ȟ^1(,/)→Ȟ^2(,) is injective. Here, the boundary map δ can be expressed as ι^-1∘ d ∘ s.
		
		By thm:deffunctor1<ref> and showdeffun2<ref>, we have
		
    H^0(,  / ) ⊗ t _↪([t]/t^2) _([t]/t^2)=Ȟ^1(,) ⊗ t.

		Hence, we obtain ^1 _↪→^1 _ is surjective.	
		
	 
		
		We will now show that δ: Ȟ^1(,/)→Ȟ^2(,) is an obstruction map for the morphism of functors ι^-1∘.
		Let α∈Č^0(,)⊗_A represent an element in _↪(A) and map to an element in _(A), represented by x=ι^-1((α)). Consider the small extension
		
    0 → I → A' → A → 0,
  
		and let α'∈Č^0(,)⊗_A' be a lift of α. By prop:cocycle:fun2<ref>, we have 
    ξ':=λ((α')) ∈Ž^1(,/)⊗ I.

		Moreover, we have
		λ((α')- s(ξ'))=0. Hence, there exists x'∈Č^1(,)⊗_A' such that ι(x')=  (α')- s(ξ'). By the commutativity of Diagram <ref> and by  rem:obs<ref>, and  rem:obs<ref> we have ι((x'))= -d(s(ξ')). Hence,  (x') and  δ(ξ')=ι^-1(d( s(ξ'))) define the same class in Ȟ^1(,)⊗ I.
		 
		Therefore, by the above discussion, the morphism of functors ι^-1∘ satisfies the assumptions in the standard smoothness criterion (standardsmooth) and ι^-1∘  is smooth.
	



	
	rem:H1vanishing
		Let  be a sheaf of Lie algebras on a variety X over , and let  ={U_i} be an affine open cover of X. If  constructed as described in Remark <ref>, we claim that Ȟ^1(,)=0. Since  has a finite subcover, and Čech cohomology coincides with the sheaf cohomology (<cit.>), we may assume that  is a finite cover. As  is constant on the cover , the claim Ȟ^1(,)=0 follows from the contractibility of a simplex. 
	

In situations where we have a good understanding of the quotient sheaf /, we will use the functor _↪ (functornew2) instead of _↪ (functornew). Given a representative α∈Č^0(,/)⊗_A of an element in _↪(A), it is clear that s(α)∈Č^0(,)⊗_A represents an element in _↪(A). In fact, this induces a morphism of functors

    s: _↪→_↪,

since the mapping on the level of cochains is well-behaved under the equivalence relations on  _↪ and _↪. In the following theorem, we will show that this map is an isomorphism.

prop:isosecfunctor
	s: _↪→_↪ is an isomorphism.
 


	Injectivity is an immediate consequence of the definitions of the functors _↪ and _↪.
	
	To show surjectivity, instead of working with the functor _↪,   we consider the functor _↪:→ defined on objects as:
	
    _↪(A)   = {α∈Č^0(, /)⊗_A : λ((s(α)))=0 }.

	Then, we have the following commutative diagram: 
		
    _↪[rr,"s∘ q"] [dr,"q"]      _↪
       _↪[ur, hook, "s"]

	Here, q is the quotient map. We will show that s∘ q:_↪→_↪ is smooth by proving that it satisfies the assumptions of the standard smoothness criterion (standardsmooth). By showing the smoothness of s∘ q:_↪→_↪, we can infer the surjectivity of both
 	s∘ q:_↪→_↪ and s:_↪→_↪.
	
	Since λ and the section s on the cochains are _A-linear for any A∈, they clearly commutes with fiber products in . Additionally, as remarked in  rem:obs<ref>,  also commutes with fiber products in . Hence, the functor _↪ is homogeneous, indeed a deformation functor. Moreover, by  showdeffun2<ref>, _↪ is also a deformation functor. 
	
	 We will compute the tangent space of _↪. We have
	 
    ^1 _↪   ={α∈Č^0(,/)⊗ t: λ((s(α)))=0  t^2 }
       ={α∈Č^0(,/)⊗ t: λ(d(s(α)))=0}
       ={α∈Č^0(,/)⊗ t: d(α)=0}
       =Ȟ^0(,/)⊗ t.

	 Combining with showdeffun2<ref>, we have 
	 
    Ȟ^0(,/)⊗ t = ^1 _↪^1 _↪H^0(,  / )λ(H^0(, ))⊗ t.

	 Since λ∘ s is the identity on Ȟ^0(,/)⊗ t, we obtain that
	 s∘ q:^1 _↪→^1 _↪ is surjective. 
	 
	 It is clear from Proposition <ref> and Theorem <ref> that the identity map on H^1(,/) is an obstruction map for the morphism of functors s: _↪→_↪. Thus, all assumptions in the standard smoothness criterion (standardsmooth) are satisfied, and we obtain that s:_↪→_↪ is smooth.





In light of the above result, we can use either the functor _↪ or _↪. Each functor offers different advantages. The functor _↪ does not require a section s, while computations with _↪ are simpler when there is a better understanding of the quotient sheaf /. In the application we are interested in, as will be described later in <ref> and <ref>, there is a unique section s, so we will prefer to work with the functor _↪.

In comparison to the functor _, the functors _↪ and _↪ offer the advantage of utilizing zero-cochains instead of one-cochains, simplifying computations. Moreover, in some situations (such as the one we will focus on in <ref>), the quotient sheaf / has a very concrete description that makes it nicer to deal with than ; this makes the use of _↪ especially attractive.

	





§ THE DEFORMATION EQUATION



 §.§ Setup

Let X be a separated scheme of finite type over , and let  be an affine open cover of X. Consider sheaves of Lie algebras  and  on X which fit into an exact sequence of the form (<ref>). The goal of this section is to explicitly construct the hull of the deformation functors _ and _↪ (see Definitions <ref> and functornew) under suitable hypotheses.

Although we primarily discuss this section in the context of the functor _→, all considerations apply similarly to the functor _ (see rem:defeq). According to Schlessinger's theorem (see, for example, <cit.>),   _→ has a hull R∈ if 

    ^1 _↪≅ H^0(X,/)/λ(H^0(X,))
 
is finite-dimensional. In other words, there exists a smooth morphism of functors f:(R,-)→_→ such that the induced map on tangent spaces is bijective. By adapting the ideas from <cit.>, we define the deformation equation and use it to explicitly construct the hull of _→, see hull. 
  
  

To construct the deformation equation for _↪, we need a finite-dimensional obstruction space. Hence, we assume that H^1(X,/) is finite-dimensional, but not necessarily that ^1 _↪ is finite-dimensional. We first fix the following data:

	
  * Elements θ_ℓ∈Ž^0(,/), ℓ=1,…,p;
	
  * Elements ω_ℓ∈Ž^1(,/), ℓ=1,…,q whose images in H^1(X,/) form a basis.

We are working here with the Čech complex of alternating cochains Č^k(,/) with respect to the open cover .

Let S=[[t_1,…,t_p]] with maximal ideal =⟨ t_1,…,t_p⟩. We let _k denote the kth graded piece of , and _≤ k (respectively (^2)_≤ k) denote the direct sum of the graded pieces of degree at most k of  (respectively ^2).
It will be useful to fix a graded local monomial order on S (see  e.g. <cit.>). 
For any ideal I⊆ S, the standard monomials of I are those the monomials of S that are not leading monomials for any element of I. Assuming that I contains a power of , the normal form with respect to I of any f∈ S is the unique element f such that f≡f I and f only contains standard monomials.




 §.§ The deformation equation

For each r≥ 1, we inductively construct α^(r)∈Č^0(,/)⊗_≤ r and obstruction polynomials g_1^(r),…, g_q^(r)∈ (^2)_≤ r  such that   

    λ((s(α^(r))))≡ 0   J_r,

where J_r=⟨ g_1^(r),…,g_q^(r)⟩+^r+1.
We begin by setting

    α^(1)= ∑_ℓ=1^pt_ℓ·θ_ℓ  and   g_1^(1)=…= g_q^(1)=0.
	
Since α^(1)∈Ž^0(,/)⊗_≤ 1, it is immediate to see that

    0=d(α^(1)) ≡λ((s(α^(1))))      J_1.

In practice, it is enough to solve the deformation equation 

    λ((s(α^(r))))-∑_ℓ=1^q g_ℓ^(r)·ω_ℓ≡ d(β^(r+1))+∑_ℓ=1^q γ_ℓ^(r+1)·ω_ℓ    ·  J_r

for
	
    β^(r+1)∈Č^0(,/)⊗_r+1;    γ_ℓ^(r+1)∈_r+1.

	We then set 

    α^(r+1)=α^(r)-β^(r+1);     g_ℓ^(r+1)=g_ℓ^(r)+γ_ℓ^(r+1).


In Proposition <ref> we show that

    λ((s(α^(r+1))))≡∑_ℓ=1^q g_ℓ^(r+1)·ω_ℓ    · J_r.

In particular, the desired equation  (<ref>) modulo J_r+1 follows from (<ref>) together with the observation that by construction, · J_r⊆ J_r+1.

As a convention, we set J_0=.
prop:defeqsolving
	Let X be a separated scheme of finite type over . Consider sheaves of Lie algebras  and  on X which fit into an exact sequence of the form (<ref>). Suppose that H^1(X,/) is finite dimensional. Then:
	
		
  * Given a solution α^(r),g_ℓ^(r) of (<ref>)  modulo · J_r-1 with J_r ≡ J_r-1^r, there is a solution of (<ref>) modulo · J_r. 

			
		
  * Given any solution to (<ref>) modulo · J_r the resulting α^(r+1) and g_ℓ^(r+1) satisfy (<ref>) modulo · J_r and J_r+1≡ J_r^r+1. 
	

We defer the proof of prop:defeqsolving to Appendix <ref>.

rem:sm
	By passing to normal forms with respect to · J_r in (<ref>), we may assume that β^(r+1) and γ_ℓ^(r+1) only involve standard monomials of · J_r. By the inductive construction of J_r and the fact that our monomial order is graded, this means that we may assume that the g_ℓ^(r+1) only involve standard monomials of · J_r. 

rem:relevant
	In situations where the cohomology groups H^k(X,/) are multigraded, we may often obtain greater control over which monomials can appear in the obstruction polynomials g_ℓ^(r). This allows us to simplify computations. Given standard monomials t^w for · J_r-1 and t^w' for · J_r, we say that t^w is relevant for t^w' if there exists a monomial t^w” with t^w as a factor such that the monomial t^w' has non-zero coefficient in the normal form of t^w” with respect to · J_r.

	This condition may be used as follows, assuming we are only using standard monomials as in rem:sm: if t^w is not relevant for t^w', then the coefficient of t^w in α^(r) has no effect on the coefficients of t^w' in the possible solutions β^(r+1) and γ_ℓ^(r+1) of (<ref>).

	Indeed, to solve (<ref>), we must consider the normal form of 
	
    λ((s(α^(r))))-∑_ℓ=1^q g_ℓ^(r)ω_ℓ
 with respect to · J_r. By rem:sm the term
∑_ℓ=1^q g_ℓ^(r)ω_ℓ is already in normal form. Furthermore, by the BCH formula (see <ref>) it follows that the coefficient of t^w in α^(r) will only affect the coefficients in λ((s(α^(r)))) of those monomials t^w” with t^w as a factor; passing to the normal form only affects coefficients of monomials for which t^w is relevant.





rem:defeq
	It is also possible to give a similar treatment to the functor _ with the assumption that H^2(X,) is finite-dimensional. In this case, the zero and one-cochains in / we considered above are replaced by one and two-cochains in . Likewise, the map λ∘∘ s is replaced by . We leave it to the reader to fill in the details.














 







 §.§ Versality

Using our solutions to the deformation equation (<ref>), we will construct the hull of _↪. Let g_ℓ
be the projective limit of g_ℓ^(r) in S, and let α be the projective limit of α^(r) in Č^0(,/)⊗ S. We define

    J= ⟨ g_1,…,g_q⟩,    R:=S/J   and   R_r:=S/J_r.
 
Then the pair (α, R)
defines a map of functors of Artin rings 
    f: (R,-)→_↪.
 Indeed, every ζ∈(R,A) factors through a morphism ζ_r : R_r → A for r≫ 0 and then f(ζ_r)= ζ_r(α^(r)) 
defines f.

 

hull
	Assume that the images of θ_1,…,θ_p in 
    Ȟ^0(,/)/λ(Ȟ^0(,))
 are a basis. Then, f:(R,-)→_↪ is a hull, that is, f is smooth and induces an isomorphism on tangent spaces.


Since J⊆^2, we have ^1(R,-)≅ (_R/_R^2)^*≅ (/^2)^* (see, for example, <cit.>). The induced map on tangent spaces is given by
	
    f:(/^2)^*   →^1 _↪=_↪([t]/t^2)
    
    		t_ℓ^*   ↦θ_ℓ⊗ t.

	This map is an isomorphism, since θ_ℓ⊗ t forms a basis for ^1 _↪. 
	
	By lemma:obfinj (see Appendix <ref>), we have  that ob_f is an injective obstruction map for f. Thus by standardsmooth it follows that f is smooth.











§ TORIC VARIETIES



 §.§ Preliminaries

In this section, we will fix certain notation and recall relevant facts about toric varieties. We refer the reader to <cit.> for more details on toric geometry.

We will consider a lattice M with the dual lattice N=Hom(M,ℤ) and associated vector space N_ℝ=N⊗_ℤℝ.
Given a fan Σ in N_, there is an associated toric variety X_Σ with an action by the torus T_N=[M] (see <cit.>). The variety X_Σ has a T_N-invariant open affine cover ={U_σ}_σ∈Σ_max, where Σ_max is the set of maximal cones in Σ. Here, each   U_σ is defined as 
    U_σ= Spec [σ^∨∩ M];   σ^∨={∈ M⊗: v()≥0  for all  v∈σ}.


We will denote the regular function on T_N associated with ∈ M by χ^.
We denote the set rays of Σ by Σ(1). Given a ray ρ∈Σ(1), the  primitive lattice generator of ρ is denoted by n_ρ and the evaluation of n_ρ at ∈ M is denoted by ρ(). 
Recall that the support of the fan Σ is |Σ|=⋃_σ∈Σσ. 

Important aspects of the geometry of X_Σ can be seen from the combinatorics of the fan Σ. In particular:

	
  * The variety X_Σ is smooth if and only if Σ is smooth, that is every cone σ∈Σ is generated by part of a basis of N <cit.>;
	
  * X_Σ is -factorial if and only if Σ is simplicial <cit.>;
	
  * X_Σ is complete if and only if |Σ|=N_ <cit.>.


The variety X_Σ has a torus factor if it is equivariantly isomorphic to the product of a nontrivial torus and a toric variety of smaller dimension. We will frequently assume that the toric variety does not have a torus factor; this is equivalent to the condition  that N_ is spanned by the primitive ray generators n_ρ for  ρ∈Σ(1) <cit.>.





 §.§ Cohomology of the structure sheaf

Let Σ be a fan in N_ and X_Σ be the associated toric variety. The cohomology groups of the structure sheaf of X_Σ naturally have an M-grading, and each graded piece can be understood combinatorially using certain subsets of N_. For every ∈ M, we associate the subset of N_ given by

    V_:=⋃_σ∈Σ{n_ρ: ρ()<0}_ρ∈Σ(1)∩σ⊆ N_.





The following result establishes a relation between the sheaf cohomology of the structure sheaf _X_Σ and the reduced singular cohomology of V_.
 van1
	For ∈ M and k≥ 0, we have
	
    H^k(X,_X_Σ)_≅H^k-1(V_,).


When the set V_ is contractible to a point, its reduced singular cohomology vanishes. In fact, we will make use of the following vanishing result:
prop:O
	Suppose that |Σ| is convex. Then for all k>0,
	
    H^k(X_Σ,_X_Σ)=0.







 §.§ Cohomology of the torus invariant divisors

Torus invariant divisors on X_Σ and the cohomology groups of the associated reflexive sheaves can be understood combinatorially as well; we focus here on the prime torus invariant divisors which are in bijection with the rays of Σ <cit.>. We denote the divisor corresponding to ρ∈Σ(1) by D_ρ. 


Similar to above, the torus action on X_Σ induces a natural M-grading on the sections of the sheaf (D_ρ). The homogeneous part of H^0(U_σ,(D_ρ)) of degree  is denoted by H^0(U_σ,(D_ρ))_. By <cit.>, we have
H^0(U_σ,(D_ρ))_ is nonzero if and only if for all ρ'∈Σ(1)∩σ

    ρ'()≥
    	0     ρ'≠ρ,
    
    	-1      ρ'=ρ.


For ρ∈Σ(1), and ∈ M,  we define the subset

    V_ρ,:= ⋃ _σ∈Σ{ n_ρ' |[  ρ'()<0    ρ'≠ρ; ρ'()<-1    ρ'=ρ ]}_ρ'∈Σ(1)∩σ⊆ N_.

When Σ is simplicial, this is a (topological realization of a) simplicial complex.
If ρ()≠-1, it is immediate to see that V_ρ,=V_. For every σ∈Σ, there is a canonical exact sequence

    0 → H^0(U_σ,(D_ρ))_· H^0(V_ρ,∩σ,) → 0,

see <cit.>.
Since λ is either an isomorphism or the zero map, there is a unique -linear section s: H^0(V_ρ,∩σ,)→ of λ.

There is a natural closed cover 

    _ρ,={V_ρ,∩σ}_σ∈Σ_max

of the set V_ρ, indexed by elements of Σ_max, with all of its intersections being contractible. 
The above exact sequence of vector spaces leads to a short exact
sequence of alternating Čech complexes with respect to the covers ,Σ_max,_ρ, (cf. <cit.>,<cit.>):

    ⊕_ρ∈Σ(1)
    ∈ MČ^k(,(D_ρ))_[r,hook,"ι"]     ⊕_ρ∈Σ(1)
    ∈ MČ^k(Σ_max,) [r,twoheadrightarrow,"λ"below]    ⊕_ρ∈Σ(1)
    ∈ MČ^k(_ρ,,)  [l, bend right=20,"s"].

We will use d to refer to the differential of any one of these complexes. Note that although ι and λ are compatible with d, the section s is not.

	We will frequently use the notation χ^ and f_ρ to distinguish between the individual direct summands appearing in the terms of (<ref>), for example, a basis of ⊕_ρ∈Σ(1)
∈ MČ^k(Σ_max,) is given by {χ^· f_ρ}_ρ,.



For the middle and right Čech complexes in (<ref>), there exists a canonical isomorphism between Čech cohomology and singular cohomology (see <cit.>. In other words, we have 

    H^k(Č^∙(Σ_max,))≅ H^k(|Σ|,);    H^k(Č^∙(_ρ,,))≅ H^k(V_ρ,,) .

Since H^0(|Σ|,)= and H^k(|Σ|,)=0 for k≥1, the long exact sequence of cohomology implies that the boundary map induces an isomorphism

    H^k-1(V_ρ,,)≅ H^k(X_Σ,(D_ρ))_

for k≥1,
see <cit.>. 

cor:vanishdegree
	Suppose that H^k(X_Σ,_X_Σ)_=0. If ρ()≠-1, then
	
    H^k-1(V_ρ,,)=0.



	If ρ()≠-1, then V_ρ,=V_. Then the claim follows from van1



exactseqlocal
	Since T_N⊆ U_σ for every σ, we have the injection 
	
    ι: T_N↪⋂ U_σ.

	At this stage, we have not yet endowed (D_ρ) with the structure of a sheaf of Lie algebras. Nonetheless, we can imitate Construction <ref>, and obtain an exact sequence of quasi-coherent sheaves
	
    0 [r]    (D_ρ)  [r,"ι"]    ι_*((D_ρ)_|T_N) [r,"λ"]    ι_*((D_ρ)_|T_N)/ (D_ρ) [r]    0.

	Additionally, we observe that
	
    (D_ρ)(U_σ)    = ⊕_∈ M H^0(U_σ,(D_ρ))_,
    ι_*((D_ρ)_|T_N)(U_σ)    = (D_ρ)(T_N)
    		= ⊕_∈ M.

	Comparing with (<ref>),
	we can view the quotient sheaf as follows:
	
    ι_*((D_ρ)_|T_N)/ 	(D_ρ)(U_σ) = ⊕_∈ M H^0(V_ρ,,).

	Subsequently, the middle and right Čech complexes in (<ref>) can also be expressed as 
	
    ⊕_ρ∈Σ(1)
    ∈ MČ^k(Σ_max,)   =
    		⊕_ρ∈Σ(1)Č^k(,ι_*((D_ρ)_|T_N)), 
    ⊕_ρ∈Σ(1)
    ∈ MČ^k(_ρ,,)    = ⊕_ρ∈Σ(1)Č^k(,ι_*((D_ρ)_|T_N)/ 	(D_ρ) ).

	



















	We consider the toric threefold X_Σ whose fan Σ in ^3 may be described as follows.
The generators of its rays are given by the columns of the following matrix:
	code-for-first-row = 
	
	
    [first-row,first-col,margin]
    			   ρ_0   ρ_1   ρ_2   ρ_3   ρ_4   ρ_5   ρ_6   ρ_7   ρ_8   ρ_9   ρ_10   ρ_11   ρ_12
       0   0   -1   -1   -1    0   1    0   0    0   1    1    2 
       0   1    1    0   -1   -1   0    0   1   -1   0    0    0 
       1   1    1    1    1    1   1   -1   0    0   0   -1   -1 
    		.

A set of rays forms a cone in Σ if the corresponding set of vertices belong to a common simplex in Figure fig:fan2, where the ray ρ_7 corresponds to the point at infinity. Taking the lattice N=^3, it is straightforward to verify that Σ is smooth and complete.
	
It can be shown that H^1(X_Σ,(D_ρ))_≅H^0(V_ρ,,) is non-zero only for the (ρ,) pairs

    (ρ_3,(1,0,0)),(ρ_11,(0,0,1)),(ρ_10,(-1,0,-1)),(ρ_10,(-1,0,0)).

Similarly, H^2(X_Σ,(D_ρ))_ is zero except for the (ρ,) pair (ρ_0,(0,0,-1)). Although in principle there are infinitely many ray-degree pairs that needs to be checked, there are actually only finitely many different simplicial complexes that can occur, and each case can be verified individually. 

For the cases with non-vanishing cohomology, Figure fig:hyp shows the intersections of Σ with, and projections of the simplicial complexes V_ρ, onto, the hyperplane ⟨ -,⟩=-1.
The first four simplicial complexes have two connected components, hence have H^0(V_ρ,,)=1, while the final simplical complex has a single cycle, hence H^1(V_ρ,,)=1. We will see in Example <ref> that the toric threefold X_Σ is unobstructed. 
	 






 §.§ Euler sequence

Let X_Σ be a -factorial toric variety with no torus factors. To understand locally trivial deformations of X_Σ we will need control of its tangent sheaf. There is an exact sequence of sheaves 

    0 [r]   _((X_Σ),)⊗__X [r]   ⊕_ρ∈Σ(1)(D_ρ)  [r,"η"]    _X_Σ[r]    0

called the Euler sequence, see  <cit.> (and dualize). By <cit.>, the image of the local section χ^ of (D_ρ) is given by the derivation (ρ,) defined by 
    (ρ,)(χ^)=ρ()χ^+.

When (X_Σ) is trivial, the map η  is an isomorphism.

prop:isofromEulereq
Let X_Σ be a -factorial toric variety with no torus factors. Then for k≥0 
    η: ⊕_ρ∈Σ(1)H^k(X_Σ,(D_ρ))→ H^k(X_Σ,_X_Σ)

is:
	
		
  * injective if H^k(X_Σ,_X_Σ)=0 and k≥1;
		
  * surjective if H^k+1(X_Σ,_X_Σ)=0 and k≥0.
		
	


	The claims follow by applying the assumptions of cohomology vanishing to the long exact sequence induced by the Euler sequence, see also <cit.>,<cit.>.

Combining η with (<ref>) we obtain the following:
prop:cohom2
	Let X_Σ be a -factorial toric variety with no torus factors. Then for k≥1 
    ⊕_ρ∈Σ(1), ∈ M 
    ρ()=-1H^k-1(V_ρ,,) → H^k(X_Σ,_X_Σ)

	is:
	
		
  * injective if H^k(X_Σ,_X_Σ)=0;
		
  * surjective if H^k+1(X_Σ,_X_Σ)=0;
		
	


	The proof follows from combining the results of prop:isofromEulereq, Equation (<ref>), and cor:vanishdegree.






§ DEFORMATIONS OF TORIC VARIETIES





 §.§ The combinatorial deformation functor

Let X_Σ be a -factorial toric variety with no torus factors. In this section, we define the combinatorial deformation functor and show that under appropriate hypotheses it is isomorphic to '_X_Σ, the functor of locally trivial deformations of X_Σ. 


	We define a bilinear map 
	
    [-,-]:⊕_ρ∈Σ(1)(D_ρ)|_T_N×⊕_ρ∈Σ(1)(D_ρ)|_T_N→⊕_ρ∈Σ(1)(D_ρ)|_T_N

	by setting
	
    [χ^· f_ρ,χ^'· f_ρ']:=ρ(')χ^+'· f_ρ'-ρ'()χ^+'· f_ρ

	and extending linearly.
	Here f_ρ and f_ρ' denote the canonical sections of (D_ρ)|_T_N and (D_ρ')|_T_N respectively. 

It is straightforward to check that [-,-] is alternating and satisfies the Jacobi identity. Hence, it is a Lie bracket.

thm:liebracket
	Let X_Σ be a -factorial toric variety without any torus factors. Then the bracket of Definition <ref> extends to a Lie bracket on ⊕_ρ∈Σ(1)(D_ρ) such that the map
	
    η: ⊕_ρ∈Σ(1)(D_ρ) →_X_Σ
 of the Euler sequence (see <ref>) is a map of sheaves of Lie algebras.


	Suppose that χ^∈ H^0(U_σ,(D_ρ)) and χ^'∈ H^0(U_τ,(D_ρ')). By (<ref>), for all ρ”∈Σ(1) ∩σ∩τ we obtain the following system of inequalities:
	
    ρ'()≥0,   ρ'(')≥-1
       ρ”()≥0,   ρ”(')
    			≥0   if ρ”≠ρ,ρ'
       ρ()≥ -1,   ρ(')≥0.

	This means that for all ρ”∈Σ(1) ∩σ∩τ,
	
    ρ”(+')≥ 0
    	if ρ”≠ρ, ρ',
    if ρ”=ρ and ρ(')≠0
    if ρ”=ρ' and ρ'()≠0.

	Thus, using (<ref>) again we observe that
	
    ρ'()χ^+'∈ H^0(U_σ∩ U_τ, (D_ρ)),   ρ(')χ^+'∈ H^0(U_σ∩ U_τ, (D_ρ')).

	Therefore, the bracket of Definition <ref> extends to a Lie bracket on ⊕_ρ(D_ρ). 
	
	It is straightforward to verify that
	
    [(ρ,), (ρ',')]=ρ(')(ρ',+')-ρ'()(ρ,+').

	It follows that the Lie bracket on ⊕_ρ(D_ρ) is compatible with η and the Lie bracket on _X_Σ, hence η is a map of sheaves of Lie algebras.


Combining with exactseqlocal and Construction <ref>, the above result allows us to define the homotopy fiber analogue deformation functor (see Definition <ref>) for toric varieties. We make this explicit here.


	 Let X_Σ be a -factorial toric variety.
We define  to be the composition λ∘∘ s.
	 The combinatorial deformation functor _Σ:→ is defined on objects as follows:
	
    _Σ(A)   = {α∈⊕_ρ, Č^0(_ρ,,_A) : (α)=0 } /∼

	where α=β in _Σ(A)  if and only if there exists γ∈⊕_ρ,Č^0(,(D_ρ))⊗_A such that 
	
    ι(γ)⊙(s(α)) = (s(β)).

	It is defined on morphisms in the obvious way.


The following is our primary result:

thm:combiso
	Let X_Σ be a -factorial toric variety without any torus factors.
	Suppose that H^1(X_Σ,_X_Σ)=
H^2(X_Σ,_X_Σ)
=0. Then, _Σ is isomorphic to '_X_Σ. In fact, we have the following isomorphisms of deformation functors:

    _Σι^-1∘∘ s≅_⊕_ρ(D_ρ)η≅__X_Σexp≅'_X_Σ.
  


	According to  rem:applications<ref>, we have  __X_Σexp≅'_X_Σ. By Theorem <ref>, η induces  a morphism of Čech complexes of sheaves of Lie algebras:
	
    η:⊕_ρ∈Σ(1)Č^∙(,(D_ρ)) →Č^∙(,_X_Σ).

	Using the vanishing of H^k(X_Σ,_X_Σ) for k=1 and 2 and prop:isofromEulereq, we have that ⊕ H^k(X_Σ, (D_ρ)) → H^k(X_Σ, _X_Σ) is surjective for k=0, bijective for k=1 and injective for k=2.
	Hence, Theorem <ref> implies that  
	
    _⊕_ρ(D_ρ)η≅__X_Σ.

	
	Combining thm:liebracket and exactseqlocal, _Σ may be identified with the homotopy fiber analogue deformation functor 
	_↪ 
	(Definition <ref>) where =⊕_ρ(D_ρ) and =⊕_ρ(D_ρ)_|T_N. By compa2, prop:isosecfunctor, and rem:H1vanishing we obtain 
	
    _Σι^-1∘∘ s≅_⊕_ρ(D_ρ) ,

	completing the proof.


We are especially interested in understanding all deformations, not just locally trivial ones. The following corollary allows us to do this when the singularities of X_Σ are mild enough.

mildsing
	Let X_Σ be a complete toric variety. Assume that X_Σ is smooth in codimension 2, and -factorial in codimension 3. Let Σ be any simplicial subfan of Σ containing all three-dimensional cones of Σ. Then _X_Σ is isomorphic to _Σ.



	First, we observe that any toric variety associated with a fan is Cohen-Macaulay, see <cit.>. Since Σ is a subfan of Σ, the resulting toric variety X_Σ is an open subset of X_Σ. By the construction of Σ, the inequality (X_Σ∖ X_Σ)≥ 4 follows from the orbit-cone correspondence (<cit.>. This allow us to utilize  CMiso (see Appendix <ref>), thereby obtaining the isomorphism
	  
    _X_Σ≅_X_Σ.

	  
	  From the preceding discussion, our focus now shifts to _X_Σ. Observe that X_Σ is smooth in codimension 2 and -factorial. By <cit.> (cf. also <cit.>), the deformations of X_Σ are locally trivial, leading us to conclude that 
	
    _X_Σ≅'_X_Σ.
 
	
	

	We will now prove that 
    _Σ≅'_X_Σ

	by showing that X_Σ satisfies all the assumptions in thm:combiso; this will complete the proof. For ∈ M, recall the sets V_ defined in <ref>, and denote the corresponding set for Σ by V_.  Let V_^(2) denote the 2-skeleton of V_. Concretely,
	
    V_:   =⋃_σ∈Σ{n_ρ: ρ()<0}_ρ∈Σ(1)∩σ⊆ N_,
    
    			V_^(2):   =⋃_σ∈Σ(3){n_ρ: ρ()<0}_ρ∈Σ(1)∩σ⊆ N_,

	where Σ(3) denotes the 3-dimensional cones of Σ.
	It is straightforward to see that the singular cohomology groups H^k-1(V_,) for k=1,2 depend solely on V_^(2). Moreover, since Σ contains all three-dimensional cones of Σ, we have
	
    V_^(2)= V_^(2).

	
	
	 It follows from the preceding discussion combined with van1 that for k=1,2 and every ∈ M, 
	 
    H^k(X_Σ,_X_Σ)_≅ H^k-1(V_,)= H^k-1(V^(2)_,)≅ H^k(X_Σ,_X_Σ)_.
	
	Moreover, since Σ is a complete fan and hence |Σ| is convex, by prop:O, we have H^k(X_Σ,_X_Σ)=0 for k≥ 1. Therefore, by (<ref>) we obtain that
	
    H^1(X_Σ,_X_Σ)=  H^2(X_Σ,_X_Σ)=0.

	Additionally, as Σ is complete,  X_Σ does not have any torus factors. Since Σ(1)=Σ(1), the variety X_Σ also does not have any torus factors.
	
	By the above discussion, X_Σ satisfies all the assumptions in thm:combiso. Therefore, we conclude that
	_Σ and  '_X_Σ are isomorphic.


Let X=X_Σ be a -factorial toric variety with no torus factors.
	The Lie bracket on ⊕_ρ∈Σ(1)(D_ρ) may be interpreted as coming from the Lie bracket on the tangent sheaf of the affine space ^#Σ(1) associated to the Cox ring of X_Σ, as we now briefly explain. 
	The variety X=X_Σ arises as a geometric quotient π:X→ X of an open subset X of ^#Σ(1) under the action of the quasitorus G=((X), ^*), see <cit.>. We will call X the Cox torsor of X (although it is actually only a torsor when X is smooth).[In e.g. <cit.>, X is called the characteristic space of X.]

	The variety X is itself toric, given by a fan Σ whose cones are in dimension-preserving bijection with the cones of Σ (cf. <cit.>). In particular, for a cone σ∈Σ, denote by σ∈Σ the corresponding cone. Since affine space has trivial class group, the generalized Euler exact sequence for the Cox torsor gives a torus-equivariant isomorphism
	
    η: ⊕_ρ∈Σ(1)(D_ρ)  →_X_Σ

	inducing a G-equivariant Lie bracket on ⊕_ρ∈Σ(1)(D_ρ).  

Moreover, for any torus-invariant open subset U_σ⊆ X, π induces an isomorphism
	
    ⊕_ρ∈Σ(1)(D_ρ)(U_σ) π^*≅(⊕_ρ∈Σ(1)(D_ρ)^G(U_σ)).

The Lie bracket on  
⊕_ρ∈Σ(1)(D_ρ)
from Definition <ref> is exactly obtained by applying this isomorphism to the above bracket on ⊕_ρ∈Σ(1)(D_ρ). 


rem:cox2
	The discussion of Remark <ref> can be used to show that there is an isomorphism between _⊕(D_ρ)  and the functor _X^G of G-invariant deformations of the Cox torsor X.
	 Indeed, the sheaf _X^G≅⊕_ρ∈Σ(1)(D_ρ)^G controls G-invariant deformations of X (since X is smooth). By the above we have an isomorphism of Čech complexes
	 
    ⊕_ρ∈Σ(1)Č^∙({U_σ},(D_ρ)) π^*≅⊕_ρ∈Σ(1)Č^∙({U_σ},(D_ρ)^G)

	 and the claim follows.
	 Note that if H^1(X,_X)= H^2(X,_X)=0, thm:combiso implies that we actually have an isomorphism
	
    _X^G →'_X_Σ,

that is, G-invariant deformations of the Cox torsor are equivalent to locally trivial deformations of X_Σ.

This is very much related to the work of <cit.>, in which G-invariant deformations of some affine scheme Y are compared with deformations of the quotient X under the action of G of an invariant open subscheme U⊆ Y; here G is a linearly reductive group. In the toric setting, the natural affine scheme Y to consider is the spectrum of the Cox ring ^#Σ(1). However, as affine space is rigid, one will only obtain trivial deformations in this manner.

The above discussion can be generalized far beyond toric varieties. Let X be any normal -factorial variety with finitely generated class group and no non-trivial global invertible functions, and let X be the relative spectrum of its Cox sheaf <cit.>. We again call X the Cox torsor of X (although it is only a torsor if X is factorial). As before, π:X→ X is a geometric quotient by the group G=((X),^*).
If X is factorial, there is a generalized Euler sequence 

    0→_ ((X),)⊗_X→π_*(_X^G) →_X→ 0,

see e.g. <cit.> for even more general conditions guaranteeing such a sequence.
In any case, given such a generalized Euler sequence, if H^1(X,_X)=H^2(X,_X)=0, we may apply Theorem <ref> to conclude that _X' is isomorphic to the functor 
(_X')^G
of locally trivial G-invariant deformations of the Cox torsor X.










 §.§ The combinatorial deformation equation

In this section, we specialize the setup discussed in <ref> to the combinatorial deformation functor _Σ.
As discussed in the proof of thm:combiso, this functor may be identified with a homotopy fiber analogue deformation functor _↪ with respect to the open cover ={U_σ}_σ∈Σ_max of X_Σ, allowing us to apply the setup of <ref>. However, we will think about _Σ instead as in Definition <ref> with respect to the various closed covers _ρ,, which are also indexed by elements of Σ_max.

To start solving the deformation equation (<ref>), we need to fix bases for the tangent and obstruction spaces of _Σ.
We make this explicit here.
 By exactseqlocal and Equation (<ref>) we obtain the tangent space

    ^1 _Σ≅⊕_ρ∈Σ(1), ∈ M H^0(V_ρ,,).

To any connected component C of the simplicial complex V_ρ, we associate a zero cocycle  θ_C ∈Ž^0(_ρ,,) as follows:

    {θ_C}_σ=  
    
    	1    if σ∈Σ_max and  C∩σ≠∅,
    
    	0    otherwise.

The images of θ_C span H^0(V_ρ,,) and removing any one of these provides a basis.
Doing this for all ρ, with H^0(V_ρ,,)≠ 0, we obtain a basis 
    θ_1 ,…,θ_p
 of ^1 _Σ
where each θ_i is of the form θ_C·χ^· f_ρ for some ρ∈Σ(1), ∈ M, and connected component C of V_ρ, (see Convention <ref>).
This will also determine

    α^(1)= ∑_ℓ=1^p t_ℓ·θ_ℓ.





An obstruction space for _Σ is given by

    ⊕_ρ∈Σ(1), ∈ M H^1(V_ρ,,),

again by exactseqlocal and Equation (<ref>). Before choosing cocycles ω_1,…,ω_q whose images give a basis of the obstruction space, 
we fix any -linear map 

    ψ: ⊕_ρ,Č^1(_ρ,,) →⊕_ρ,Č^0(_ρ,,)

that is compatible with the direct sum decomposition and such that d∘ψ(ω)=ω if ω∈Č^1(_ρ,,) is a coboundary.

Here is one explicit way to do this: fix an ordering of the elements of Σ_max. For each connected component C of V_ρ,, this determines a unique cone σ_C that is minimal among all cones σ∈Σ_max intersecting C non-trivially.
For any cone σ intersecting C, there is a unique sequence τ_1=σ,τ_2,…,τ_k=σ_C such that for each i≥ 1, τ_i∩τ_i+1∩ C≠∅, k is minimal, and the sequence is minimal in the lexicographic order with respect to the previous properties. 
Given ω∈Č^1(_ρ,,)
we then define

    ψ(ω)_σ=∑_i=1^k-1ω_τ_i,τ_i+1

if σ∩ C≠∅ for some connected component C, and set ψ(ω)_σ=  0 otherwise.
Note that in particular, ψ(ω)_σ_C=0. It is straightforward to verify that ψ has the desired property.

We now choose one-cocyles 
    ω_1,…,ω_q∈⊕Ž^1(_ρ,,)
 whose images in 

    ⊕_ρ∈Σ(1), ∈ M H^1(V_ρ,,)

form a basis, such that each ω_i lies in a single direct summand, and such that d(ψ(ω))=0. From an arbitrary set of cocycles ω_1',…,ω_q' whose images form a basis, we may set

    ω_i=ω_i'-d(ψ(ω_i'))

to obtain that d(ψ(ω_i))=0.


In this situation, we refer to the corresponding deformation equation (<ref>) for _Σ as the combinatorial deformation equation. By prop:defeqsolving, we know that for each order r it has a solution 

    β^(r+1)∈⊕_ρ,Č^0(_ρ,,)⊗_r+1;    γ_ℓ^(r+1)∈_r+1.

In fact, we may obtain a solution using the map ψ:

	Let η∈⊕_ρ,Č^1(_ρ,,)⊗_r+1 be the normal form of (α^(r))-∑_ℓ=1^qg_ℓ^(r)ω_ℓ with respect to · J_r. Then β^(r+1)=ψ(η) and γ_ℓ^(r+1) 
give a solution to the combinatorial deformation equation, where γ_ℓ^(r+1) is determined by

    ∑_ℓ=1^q γ_ℓ^(r+1)·ω_ℓ=η-d(β^(r+1)).



	By prop:defeqsolving<ref>, there exists
	
    β' ∈⊕_ρ,Č^0(_ρ,,)⊗_r+1;   γ_ℓ'∈_r+1

	such that	
	
    η= d(β')+∑_ℓ=1^q γ_ℓ' ·ω_ℓ.

	Applying d∘ψ to both sides we obtain
	
    d(β^(r+1))= d(β').

	Since the ω_ℓ are linearly independent, this implies γ_ℓ^(r+1)=γ_ℓ'.

To summarize the contents of Proposition <ref>, in order to solve the combinatorial deformation equation, we only need to reduce (α^(r))-∑_ℓ=1^qg_ℓ^(r)·ω_ℓ to its normal form with respect to · J_r (an unavoidable algebraic step), and then apply the map ψ (a purely combinatorial step). 
By hull, iteratively solving the combinatorial deformation equation gives us a procedure for computing the hull of _Σ.
We will do this explicitly in several examples in <ref>.




 §.§ Higher order obstructions

To solve the combinatorial deformation equation discussed in <ref> we have to compute (s(α)). In this section, we will derive a general formula for (s(α)) which is of theoretical interest and present explicit formulas for lower order terms.

We first establish some notation. For w=(w_1,…,w_p)∈^p_≥0, we denote t_1^w_1… t_p^w_p by t^w. We choose θ_1,…,θ_p using the construction in <ref>.  Let φ∈(^p,M) be the map sending the ℓ-th basis vector of ^p to the degree in M of θ_ℓ. Consider 

    α=∑_ρ∈Σ(1)∑_w∈_≥ 0^p ∖{0} c^w_ρ· t^w·χ^φ(w)· f_ρ∈⊕_ρ∈Σ(1) 
    ∈ MČ^0(_ρ,,)⊗,

where c^w_ρ is a cochain in Č^0(_ρ,φ(w),) and χ^φ(w)· f_ρ specifies the summand in which c_ρ^w lies (see Convention <ref>). 


	For integers d≥ 1 and 1≤ k ≤ d we define the set Δ_d,k as follows:
	
    Δ_d,k={(a_1,…,a_d)∈_≥ 1^d : [ a_i <i or a_i=k-∑_j>i
     j<  a_j a_j-j                               if i<k;                                a_i=k                               if i=k;                               a_i< i                              if i> k ]}.

	Likewise, for a∈Δ_d,k, we set
	
    (a)=(-1)^#{i | a_i>i}.





	For an integer d≥ 1 and w ∈_≥ 0^p ∖{0} we define the set ∇_w,d as follows: 
	
    ∇_w,d={=(_1,…,_d)∈ (_≥ 0^p∖{0})^d | ∑_i=1^d _i=w}.




	Here we give some examples of Δ_d,k and ∇_w,d. For d=3 and k=1,2,3 we have the following sets:
	
    Δ_3,1={{1,1,1},{1,1,2}}
       Δ_3,2={{2,2,1},{2,2,2}}
       Δ_3,3={{3,1,3},{2,3,3}}.

	For w=(1,1,1) and d=2,3, we have the following sets:
	
    ∇_(1,1,1),3   ={(π(e_1),π(e_2),π(e_3)) :  π∈ S_3}
    ∇_(1,1,1),2   ={(π(e_1),π(e_2+e_3)) :  π∈ S_3}.

	Here, π(e_i) denotes the image of the standard basis vector e_i of ^3 under the action of π∈ S_3.
	These sets appear in the formula for the coefficient of 
	t_1t_2t_3 in Table Table:obspoly.




Consider any BCH formula

    x y=∑_d ≥ 1∑_σ(x,y)∈_d b_σ [σ(x,y)]

where b_σ∈, _d is some set of words σ(x,y) of length d  in x and y, and 
[σ(x,y)] denotes the iterated Lie bracket (see <ref>); we may for example take Dynkin's formula. Set  

    (σ)=(-1)^#{i | σ_i=y}.

We will use the notation ρ⃗=(ρ_1,…,ρ_d) for a d-tuple of rays of Σ. In this section, for compactness of notation and to avoid confusion with elements of _d, we will denote elements of Σ_max by i and j.



thm:obsformula
	 The coefficient of t^w in (s(α))_ij is the product of χ^φ(w)
	with
	
    ∑_d≥ 1
    ∈∇_w,d
    ρ⃗∈Σ(1)^d(∑_σ∈_d(σ) b_σ∏_k=1^d (c^_k_ρ_k)_σ(i,j)_d-k+1)
    		 (
    		∑_k=1… d
    a∈Δ_d,k(a)∏_ℓ≠ kρ_ℓ(φ(_a_ℓ)) · f_ρ_k).



We postpone the proof of thm:obsformula until the end of this section. We first proceed to describe the explicit formulas for lower order terms. Using the notation 



    α^(1)_i    = t_1·χ^_1· c^(1,0)_1,i·  f_1 + t_2·χ^_2· c^(0,1)_2,i·  f_2
    α^(2)_i   = α^(1)_i+
    	t_1t_2·χ^_1+_2· (c^(1,1)_1,i· f_1+ c^(1,1)_2,i· f_2)
    α^(3)_i   = α^(2)_i+ t_1^2t_2·χ^2_1+_2· (c^(2,1)_1,i· f_1+ c^(2,1)_2,i· f_2)+ t_1t_2^2·χ^_1+2_2· (c^(1,2)_1,i· f_1+ c^(1,2)_2,i· f_2)


we list the formulas for the coefficients of  t_1t_2,t_1t_2^2,t_1t_2^3 and t_1^2t_2^2 in Table <ref>. By applying λ to the coefficient of t_1t_2 in (s(α^(1)))_ij, we recover the combinatorial cup-product from <cit.>. Similarly, using the notation

    α^(2)_i    = t_1·χ^_1· c^e_1_1,i·  f_1 + t_2·χ^_2· c^e_2_2,i·  f_2+ t_3·χ^_3· c^e_3_3,i·  f_3+ 
       t_1t_2·χ^_1+_2· (c^e_1+e_2_1,i· f_1+ c^e_1+e_2_2,i· f_2) + t_1t_3·χ^_1+_3· (c^e_1+e_3_1,i· f_1+ c^e_1+e_3_3,i· f_3)
       + t_2t_3·χ^_2+_3· (c^e_2+e_3_2,i· f_2+ c^e_2+e_3_3,i· f_3)


we also list a formula for the coefficient of t_1t_2t_3 of (α^(2))_ij in Table <ref>.
We thus have explicit formulas for all obstructions of third order, and fourth order obstructions involving only two deformation directions. In theory, we could write down similar formulas for higher order obstructions using thm:obsformula, but they become increasingly large. We note that unlike for the cup product case, the formulas for obstructions of degree larger than two involve not only first order deformations but also higher order perturbation data.


















We now start on proving thm:obsformula.
Let ρ_1,…,ρ_m be not-necessarily distinct rays of Σ, and _1,…,_m∈ M. We will need an explicit formula for the iterated Lie bracket

    [χ^_m· f_ρ_m χ^_m-1· f_ρ_m-1 ⋯ χ^_1· f_ρ_1],

see <ref> for notation. 


prop:iteratedlie
	The iterated Lie bracket
	
    [χ^_m· f_ρ_m χ^_m-1· f_ρ_m-1 ⋯ χ^_1· f_ρ_1]

	is equal to 
	
    χ^_1+⋯+_m·∑_k=1^m(∑_a∈Δ_m,k(a)∏_i≠ kρ_i(_a_i) )· f_ρ_k.



	The proof is by induction. The base case m=1 is immediate. For proving the induction step, it is enough to show that
	
    ∑_a∈Δ_m-1,k∑_j=1^m-1(a) ∏_i≠ kρ_i(_a_i)ρ_m(_j)= ∑_a∈Δ_m,k(a) ∏_i≠ kρ_i(_a_i)

	for k=1,…,m-1 and
	
    ∑_k=1^m-1-ρ_k(_m)·∑_a∈Δ_m-1,k(a)∏_i≠ kρ_i(_a_i)· = ∑_a∈Δ_m,m(a) ∏_i≠ mρ_i(_a_i).


	
	To establish this, we use the following straightforward inductive relations on the sets Δ_m,k. For k≤ m-1 the map
	π_k: Δ_m,k→Δ_m-1,k
	defined by 
    π_k(a) =(a_1,…,a_m-1)

	is (m-1)-to-1 and satisfies (π_k(a))=(a). Additionally, for k≤ m-1 the map π̂_k: Δ_m-1,k→Δ_m,m defined by 	
    π̂_k(a)=(a_1,…,a_k-1,m,a_k+1,…,a_m-1,m)

is injective, (π̂_k(a))=-(a), and we have
	
    Δ_m,m= ⋃̇_k=1^m-1π̂_k ( Δ_m-1,k).

The proofs of (<ref>) and (<ref>) follow directly from these observations.




We have
	
    (α)_ij   = ∑_d ≥ 1∑_σ(x,y)∈_d b_σ [σ(α_i,-α_j)].

	Expanding the right-hand side, we obtain that the coefficient of  t^w is 
    ∑_d≥ 1
    ∈∇_w,d
    ρ⃗∈Σ(1)^d∑_σ∈_d(σ)· b_σ·∏_k=1^d (c^_k_ρ_k)_σ(i,j)_d-k+1·
    		[χ^φ(_d)· f_ρ_d ⋯ χ^φ(_1)· f_ρ_1].

	Applying prop:iteratedlie, we then obtain that this is equal to χ^φ(w)
	multiplied with the quantity in the statement of the theorem.









 §.§ Removing cones

In <ref>, we discussed how to compute the hull of _Σ using the combinatorial deformation equation. When X_Σ has mild singularities, this approach indeed yields the hull of _X_Σ, see mildsing. Although working with _Σ is less complex than _X_Σ, computations still involve dealing with every maximal cone of Σ and their pairwise intersections. In this section, we will explore methods to streamline the process of determining the hull of _Σ by reducing the number of maximal cones and intersections which we must consider. We will apply these techniques when we compute hull for examples in <ref>.

Throughout this section, Σ is any simplicial fan.

	Let 
	
    ={(ρ,)∈Σ(1)× M | H^0(V_ρ,,)≠0}.

	We let Γ⊆Σ(1)× M be the smallest set containing  that satisfies the following:
	
		
  * If (ρ,),(ρ',')∈Γ, ρ≠ρ', and ρ'()≠ 0 then (ρ,+')∈Γ;
		
  * If (ρ,),(ρ,')∈Γ and ρ(')≠ρ() then (ρ,+')∈Γ.
	
	We then define _Γ= ⊕_(ρ, )∈Γ(D_ρ)_.


It is straightforward to verify that the subsheaf _Γ is stable under the Lie bracket (Definition <ref>)  on 

    ⊕_ρ∈Σ(1)(D_ρ)= ⊕_ρ∈Σ(1)
    ∈ M(D_ρ)_.
 
Let Σ' be a subfan of Σ with Σ_max'⊆Σ_max. We say that Σ' covers Γ if for every (ρ,)∈Γ, V_ρ,⊆ |Σ'|. In this case, we obtain a cover '_ρ, of V_ρ, by Σ_max'.








	Fix the lattice N=^3. We consider a fan Σ with six rays, where the generator of the  ith ray ρ_i is given by the ith column of the following matrix:
	
    [  1  0 -1  0  0  0;  0  1  e -1  0  0;  0  0  a  b  1 -1 ].

	We assume that e,b ≥0 (the reason for this assumption will be explained  in <ref>).
Rays belong to a common cone of Σ if the corresponding set of vertices in Figure fig:cones belong to the same simplex, with the ray ρ_6 as a vertex at infinity.

	Suppose that we know that the ray-degree pairs (ρ,)∈  are of the form 
	
    ( ρ_2, (*,*,0) )   or  (ρ_5, (*,*,-1) )   or  (ρ_6, (*,*,1) ).
 
	We will establish this in lemma:SigmaD.
	It is then straightforward to verify that any element of Γ not in  must be of the form
		
    ( ρ_5, (*,*,-1) )   or  (ρ_6, (*,*,1) )   or  (ρ_5, (*,*,0) )   or  (ρ_6, (*,*,0) ).

	
	Given the above assumption on , we claim that the fan Σ' with maximal cones
	
    σ_1   = (ρ_1,ρ_4,ρ_5),   σ_2= (ρ_1,ρ_2,ρ_5), 
    σ_3   = (ρ_2,ρ_3,ρ_5),   σ_4= (ρ_3,ρ_4,ρ_5).

	covers Γ. Indeed, it is straightforward to see that for (ρ,)∈Γ, n_ρ_6∉ V_ρ, and the claim follows.
	




Suppose that Σ' covers Γ.
	We define the functor 
	
    _Σ',Γ(A)   = {α∈⊕_(ρ, )∈ΓČ^0('_ρ,,)⊗_A : (α)=0 } /∼

	where α=β in _Σ',Γ(A)  if and only if there exists 
    γ∈⊕_(ρ,)∈ΓČ^0(',(D_ρ))_⊗_A
 such that 
	
    ι(γ)⊙(s(α)) = (s(β)).

	This functor is defined on morphisms in the obvious way.



thm:samehull
	Let Σ be a simplicial fan and suppose that Σ' covers Γ.	
	Then there are smooth maps __Γ→_Σ and __Γ→_Σ' that induce isomorphisms on tangent spaces. In particular, if ^1 _Σ is finite dimensional, then _Σ and _Σ',Γ have the same hull.


	We have the natural map of functors f_1: __Γ,→_⊕(D_ρ) induced from the injection _Γ→⊕(D_ρ) and the cover . Consider the open cover '={U_σ}_σ∈Σ_max' of X_Σ'⊆ X_Σ. Similar to the construction of the deformation functor __Γ=__Γ,  using the Čech complex Č^∙(,_Γ), we can define the deformation functor __Γ,'  using the Čech complex Č^∙(',_Γ).  The natural map on Čech complexes induces a morphism of functors f_2:__Γ,→__Γ,'.
	
	
	To prove the theorem, we will show that f_1 and f_2 are smooth maps with isomorphisms on the tangent spaces, and there are isomorphisms g_1 and g_2 as in the following diagram:
	
	
    _⊕(D_ρ)[r,"g_1","≅"swap ]     _Σ
    __Γ,[ru,"f_1"] [rd,"f_2"]       
       __Γ,'[r,"g_2","≅"swap ]    _Σ',Γ.

	
	
	First, consider the map f_1. By construction
	
    ^1 __Γ= ⊕_(ρ,)∈Γ H^1(,(D_ρ))_=⊕_(ρ,) ∈Σ(1)× M H^1(,(D_ρ))_= ^1 _⊕(D_ρ)

	and 
	
    ⊕_(ρ,)∈ Q H^2(,(D_ρ))_→⊕_(ρ,) ∈Σ(1)× M H^2(,(D_ρ))_

	is an inclusion. Thus, by standardsmooth the map f_1 is smooth. 

	Now consider f_2.  The maps between the tangent and obstruction spaces are isomorphisms since Σ' covers Γ, so f_2 is also smooth by standardsmooth.

	We know that _Σ is the homotopy fiber analogue of _⊕(D_ρ).	Likewise, a straightforward adaptation of  exactseqlocal implies that _Σ',Γ is the homotopy fiber analogue of __Γ,'. The isomorphisms g_1 and g_2 thus follows from compa2, prop:isosecfunctor, and rem:H1vanishing.


From the above theorem, we can reduce the number of maximal cones that need to be considered. In the combinatorial deformation equation, we also need to compute (α)_στ for every pair σ,τ of maximal cones. The following cocycle property and prop:closure reduce the number of cases that need to be calculated. Let ⋀^2 Σ_max be the set consisting of size two subsets of  Σ_max.


	We say a set ⊆⋀^2 Σ_max has the cocycle property  if for {σ,κ}, {τ,κ}∈ with σ∩τ⊆κ, it follows that {σ,τ}∈. For ⊆⋀^2 Σ_max, we define  to be the smallest set containing  that has the cocycle property.
 


prop:closure
	Let Σ be a simplicial fan, and let ⊆⋀^2 Σ_max be such that = ⋀^2 Σ_max. Then a cocycle ω∈Ž^1(_ρ,,) is determined by {ω_στ | {σ,τ}∈}. 


	Let _0= and define
	
	
    _i+1=_i∪{{σ,τ} | {τ,κ},{σ,κ}∈_i with σ∩τ⊆κ}.

	Since ⋀^2 Σ_max is finite, after finitely many steps we obtain _m=. We will prove by induction that for every {σ,τ}∈_i, ω_στ is determined by 
    {ω_σ'τ' | {σ',τ'}∈}.

	Clearly, for i=0, this is true. Suppose that  {σ,τ}∈_i for some i≥ 1. If {σ,τ}∈_i-1 the statement is true by induction. If V_ρ,∩σ∩τ=∅, then ω_στ=0. Thus, we may assume that V_ρ,∩σ∩τ≠∅. Since ω∈Ž^1(_ρ,,) and 
	V_ρ,∩σ∩τ∩κ≠∅, we have ω_στ=-ω_τκ+ω_σκ. By the induction hypothesis, ω_τκ and ω_σκ are already determined. Thus, the statement is true by induction.
	 



In situations where Σ has reasonable geometry, there is a canonical choice of . For a cone τ∈Σ, define

    (τ,Σ)={σ∈Σ | τ⊆σ}.

We say that (τ,Σ) is connected in codimension one if for any two maximal cones σ,σ'∈(τ,Σ), there is a sequence of maximal cones τ_1=σ,τ_2,…,τ_k=σ' such that τ_i∈(τ,Σ) and τ_i,τ_i+1 intersect in a common facet.

prop:codimone
	Let Σ be a simplicial fan and let ⊆⋀^2Σ_max consist of those pairs of cones that intersect in a common facet. Suppose that for every τ∈Σ, (τ,Σ) is connected in codimension one. Then = ⋀^2 Σ_max. In particular, a cocycle ω∈Ž^1(_ρ,,) is determined by {ω_στ | {σ,τ}∈}. 


	We will show that = ⋀^2 Σ_max; the second claim then follows from prop:closure. More specifically, we will show that for {σ,σ'}∈⋀^2 Σ_max, {σ,σ'}∈. We will induct on the dimension of τ=σ∩σ'. If σ,σ' intersect in a common facet, then we are done by definition of . 
Otherwise, suppose that we have shown all pairs intersecting in a face of dimension larger than τ belong to .

Fixing τ, we now show that any pair of maximal cones σ,σ' from (τ,Σ) belongs to . For this, we induct on the length of a sequence τ_1=σ,τ_2,…,τ_k=σ' in (τ,Σ) connecting σ,σ' in codimension one. If k=2, then again we are done by the definition of . For k>2, by induction we have that {σ,τ_2} and {τ_2,σ'} belong to . Since σ∩σ'=τ⊆τ_2, it follows that {σ,σ'}∈. The claim now follows by induction.


	It is straightforward to verify that if a simplicial fan Σ is combinatorially equivalent to a fan with convex support, it satisfies the hypotheses of prop:codimone.

It follows from  thm:samehull and prop:closure that in order to compute the hull of _X_Σ, we can use the functor _Σ',Γ  for some Σ' covering Γ, and, when computing obstructions, choose ⊆⋀^2 Σ'_max such that =⋀^2 Σ'_max. We will apply these constructions in several examples in <ref>.



	Let Σ' be the fan from Example <ref>.
	Define the set 
    ={{σ_1,σ_2}, {σ_2,σ_3}, {σ_3,σ_4}, {σ_4,σ_1}}.

	We have {σ_i,σ_i+1}, {σ_i+1,σ_i+2}∈ with 
	σ_i∩σ_i+2⊆σ_i+1 for i=1,2. Thus, {σ_1,σ_3} and {σ_2,σ_4} are in .
	It follows that = ⋀^2 Σ'_max.                                       

In fact,  is the set of all pairs of maximal cones intersecting in a common facet. Since
Σ' has the property that (σ,Σ') is connected in codimension one for all σ, prop:codimone guarantees that 
= ⋀^2 Σ'_max. 





 §.§ Unobstructedness

In this section, we will
provide a sufficient criterion for a toric variety to have unobstructed deformations.

thm:unobstructed
	Let X=X_Σ be a complete toric variety that is smooth in codimension 2 and -factorial in codimension 3. Let
	
    ={(ρ,)∈Σ(1)× M | H^0(V_ρ,,)≠0}

and let Γ be as in Definition <ref>.
	If H^1(V_ρ,,)=0 for all pairs (ρ,)∈Γ satisfying ρ()=-1, then X_Σ is unobstructed.
In particular, setting 
	
    := { (ρ,+)∈Σ(1)× M |
    			(ρ,)∈𝒜; ∈∑_(ρ',')∈𝒜_≥0·' 
    		},

if H^1(V_ρ,,)=0 for all pairs (ρ,)∈ satisfying ρ()=-1, then X_Σ is unobstructed.
	

By mildsing, after possibly replacing Σ be any simplicial subfan of Σ containing all three-dimensional cones, we may assume that 
_X≅_Σ.
By thm:samehull, the functors _Σ and _Γ,Σ have the same hulls. But an obstruction space for _Γ,Σ is given by 

    ⊕_(ρ,)∈Γ H^1(V_ρ,,),

see showdeffun2. Moreover, by cor:vanishdegree, we know that for any (ρ,)∈Γ,  ρ()≠-1 implies H^1(V_ρ,u,)=0 and the first claim of the theorem follows.
To show the second claim, observe that  contains Γ.


To illustrate the significance of the above theorem, we provide an example of an unobstructed toric threefold whose unobstructedness does not follow by degree reasons alone.


	We consider the smooth toric threefold from Example <ref>. 
	The set  from thm:unobstructed consists of exactly

    (ρ_3,(1,0,0)),(ρ_11,(0,0,1)),(ρ_10,(-1,0,-1)),(ρ_10,(-1,0,0)).

As noted earlier, we also have H^1(V_ρ,,)=0 except for (ρ,)=(ρ_0,(0,0,-1)).

There are infinitely many positive integer combinations of degrees 
    (1,0,0),(0,0,1),(-1,0,-1),(-1,0,0)
 that sum to (0,0,-1). For example,

    (0,0,-1)=(-1,0,-1)+(1,0,0).

Hence, we cannot conclude by degree reasons alone that X_Σ is unobstructed. However, since ρ_0 does not appear in any element of , (ρ_0,(0,0,-1) is not in  and we conclude that X_Σ is in fact unobstructed.
	 









§ EXAMPLES



 §.§ Primitive collections and rigidity


Throughout <ref>, we will assume that Σ is a smooth complete fan. The data of a fan can be provided by specifying the ray generators and listing the maximal cones. Instead of specifying the maximal cones we can describe the fan using the notion of primitive collections:


	A subset ⊆Σ(1) is a primitive collection if the elements of  do not belong to a common cone in Σ, but the elements of every proper subset of  do.

	Let ={ρ_1,…,ρ_k} be a primitive collection and let σ∈Σ be the unique cone such that n_ρ_1+⋯+n_ρ_k lies in the relative interior of σ. Then there is a relation
	
    n_ρ_1+⋯+n_ρ_k= ∑_ρ∈σ∩Σ(1) c_ρn_ρ

	where c_ρ>0 for all ρ∈σ∩Σ(1). This is the primitive relation associated to , see <cit.>. The degree of  is the integer 
    ()= k-∑ c_ρ.




  
lemma:rigid
	Let Σ be a smooth complete fan. If ()>0 for every primitive collection of cardinality 2, then X_Σ is rigid.


	We will show that H^1(X_Σ,_X_Σ)=0 in this case. According to Proposition <ref>, we have
	
    H^1(X_Σ,_X_Σ)≅⊕_ρ∈Σ(1), ∈ M 
    ρ()=-1H^0(V_ρ,,).

	For any ρ∈Σ(1) and ∈ M, H^0(V_ρ,,)≠0 only when the simplicial complex V_ρ, is disconnected. Hence, it is enough to show that V_ρ, is connected for all ρ∈Σ(1) and ∈ M.

	Assuming that  V_ρ,≠∅, let ρ_min∈Σ(1) be any ray not equal to ρ such that ρ_min() is minimal. Then for any ρ'≠ρ with ρ'()<0, we claim that   ρ' shares a cone of Σ with ρ_min. Indeed, if not, ={ρ_min,ρ'} forms a primitive collection, and since 
()>0 we have only two possibilities:

    n_ρ_min+n_ρ'=0;    n_ρ_min+n_ρ'= n_ρ”.

The first case is impossible since both ρ_min() and ρ'() are less than zero. The second case is impossible since it would follow that ρ”()<ρ_min(), contradicting the choice of ρ_min. This implies the claim, and the connectedness of V_ρ, follows.	







	A smooth toric variety X_Σ is Fano (respectively weak Fano), if and only if ()>0 (respectively ()≥0) for every primitive collection (see <cit.>). As a corollary, we obtain the well-known result that every smooth toric Fano variety is rigid (see <cit.>).
	We also obtain the rigidity for smooth toric weak Fano varieties with no degree zero primitive collections of cardinality 2 (cf. <cit.> for a similar result).



It is well-known that ^n is the only smooth complete toric variety with Picard rank 1. Since it is Fano, it is rigid by the previous remark. Thus we may focus on smooth complete toric varieties with higher Picard rank.



	




 §.§ Picard rank two

In this section, we will prove that every smooth complete toric variety with Picard rank 2 is unobstructed. Any n-dimensional toric variety X of this type can be expressed as 

    X ≅(_^s⊕_^s(a_1)⊕⋯⊕_^s(a_r)),

for r,s≥ 1, r+s=n and 0≤ a_1≤⋯≤ a_r, see <cit.>, <cit.>.

A fan Σ with X=X_Σ may be described as follows, see also <cit.>. Fix the lattice N=^n, and consider the ray generators given by the columns of the following matrix:
code-for-first-col = ,code-for-first-row = 

	
    A=
    		
    			[small,first-row,first-col,margin]
    				   ρ_1   ⋯   ρ_s   ρ_s+1   ⋯   ρ_n   ρ_n+1   ρ_n+2
                               
       3-3<>I_s         3-3<>  0         0   -1
                         ⋮   ⋮
                         0   -1
    
       3-3<> 0         3-3<>I_r         -1   a_1
                         ⋮   ⋮
                         -1   a_r

		The primitive collections for Σ are {ρ_1,…,ρ_s,ρ_n+2} and {ρ_s+1,…,ρ_n,ρ_n+1}. 

	thm:ranktwo
	Let 
	
    X= (_^s⊕_^s(a_1)⊕⋯⊕_^s(a_r)).

	Then:
	
		
  * X is rigid if and only if s>1, or s=1 and a_r≤1. If s=1, then 
		
    _ H^1(X,_X)= ∑_j=1^rmax{a_j-1,0};
 
		
  * X is unobstructed.
		
	



	 We will make use of the isomorphism
	 
    H^k(X,_X)≅⊕_ρ∈Σ(1), ∈ M 
    ρ()=-1H^k-1(V_ρ,,),

	 for k=1,2 (see Proposition <ref>).
	 
	 We will first show claim <ref>.  
	 Suppose that H^1(X,_X)≠0. In this case, there must exist a ray ρ∈Σ(1) with ρ(𝐮)=-1 and a primitive collection {ρ',ρ”} with exactly two rays which are distinct from ρ and satisfy ρ'()<0, ρ”()<0. Consequently, either r or s must equal one. However, if r=1, then ρ'()=-ρ”(). Hence, it must be the case that s=1 and the rigidity for s>1 follows.

	 
	
	Assume that s=1.  From the primitive collections, it follows that the simplicial complex V_ρ, is disconnected only when it consists of the two vertices n_ρ_1 and n_ρ_n+2. This occurs if and only if the following conditions on the ray-degree pairings are satisfied:  there exists j ∈{1,…,r+1} such that ρ_s+j()=-1 and 
    ρ_1()<0,   ρ_n+2()<0,   ρ_s+i()≥ 0    for all  i ∈{1,…,r+1}∖{ℓ}.

	If a_r≥ 2, then for ρ=ρ_s+r=ρ_n and =(-1,0,…,0,-1), the above set of conditions is satisfied, and thus we obtain  V_ρ, is disconnected. In fact, all degree-ray pairs (ρ,) satisfying the above conditions are given by choosing  ρ=ρ_s+j for j∈{1,…,r} and setting 
	
    =d· e_1-e_s+j

	for d satisfying 1-a_j≤ d ≤ -1.
	Thus, we obtain
	
    ⊕_∈ MH^0(V_ρ_s+j,,)=max{a_j-1,0},

	for j∈{1,…,r}, and claim <ref> follows.

	To prove claim <ref>, we show that if X is not rigid, then H^2(X,_X)=0. Hence, we may again assume that s=1. It is suffices to show that for all ray-degree pairs (ρ,), every connected component of V_ρ, is contractible to a point.
	Given that
	
    n_ρ_s+1+…+n_ρ_n+1=0,

	for every ∈ M there exists at least one j∈{1,…,r+1} such that ρ_s+j()≥ 0. Let σ be the cone in Σ generated by {ρ_s+1, …, ρ_n+1}∖{ρ_s+j}. Then V_ρ, is the join of the simplicial complexes
	 ( V_ρ,∩{n_ρ_1,n_ρ_n+2} ) and (V_ρ,∩σ).
	Since V_ρ,∩σ is either empty or a simplex, it follows that every connected component of V_ρ, is contractible to a point. This completes the proof of claim <ref>.
























	




 §.§ Split P1-bundles over Hirzebruch surfaces

As shown in <cit.>, there exists a  ^1-bundle over the second Hirzebruch surface _2 that exhibits quadratic obstructions. In fact, Picard rank three toric threefolds represent the minimal cases in terms of both dimension and Picard rank where obstructions can occur. This is because smooth complete toric varieties of dimension at most 2 (<cit.>) and those with Picard rank at most 2 (thm:ranktwo) are unobstructed. 

In this section and the next, we examine toric threefolds that are ^1-bundles over the Hirzebruch surface 
    _e=(_^1⊕_^1(e)).
 Every toric threefold of Picard rank 3 is either of this form, or the blowup of a toric threefold of Picard rank 2 in a point or a ^1 (see <cit.>, <cit.>). 

	




 Any toric ^1-bundle over _e can be expressed as

    X≅(𝒪_𝔽_e⊕𝒪_𝔽_e(aF+bH)),

where _e=(_^1⊕_^1(e)) is the eth Hirzebruch surface, and F and H respectively represent the classes in (_e) of the fiber and _𝔽_e(1) in the ^1-bundle fibration of _e over ^1. 
In other words, 

    (𝔽_e)= ℤF⊕ℤH   with    F^2=0,  F· H=1,   H^2=e.

Since we are considering X up to isomorphism, we can take e,b≥0, see  e.g. <cit.>. Under this assumption, the fan Σ from Example <ref> describes X, that is, X=X_Σ. The primitive collections are given by {ρ_1,ρ_3}, {ρ_2,ρ_4}, and {ρ_5,ρ_6}. From this, we obtain the primitive relations:


    n_ρ_5+ n_ρ_6   =0;
    
    	n_ρ_2+ n_ρ_4   = b· n_ρ_5;
    
    	n_ρ_1+n_ρ_3   = 
    	
    		e · n_ρ_2 + a· n_ρ_5 if  a≥ 0 
    
    		e · n_ρ_2 + a·  n_ρ_6 if  a< 0.





As a first step for computing the hull for several examples of ^1-bundles over _e, we will describe H^1(X,_X) and H^2(X,_X) in the following two lemmas. 






lemma:H2deg
	Let X= (𝒪_𝔽_e⊕𝒪_𝔽_e(aF+bH)) with e,b≥0. Then
	
    H^2(X,_X)≅⊕_ρ∈Σ(1), ∈ M 
    ρ()=-1 H^1(V_ρ,,) is non-zero

	if and only if  b≥2 and (b-1)e+a≥ 2. Moreover, H^1(V_ρ,,)≠0 if and only if ρ=ρ_5 and =(x,y,-1) satisfies
	
    -b+1≤ y ≤ -1,   	ey-a+1≤ x ≤ -1.






	Clearly, when V_ρ, is the simple cycle formed by the vertices ρ_1,ρ_2,ρ_3,ρ_4, we have H^1(V_ρ,,)≠0. 
	We claim that for any other simplicial complex of the form V_ρ,, its connected components are contractible to a point. Indeed,  any such simplicial complex V_ρ, can be expressed as the join of the simplicial complexes:
	
    V_1= V_ρ,∩{n_ρ_1,n_ρ_3},    V_2=V_ρ,∩{n_ρ_2,n_ρ_4},    V_3=V_ρ,∩{n_ρ_5,n_ρ_6}.

	From (<ref>), V_3 is either empty or one of the vertices. In the latter case, V_ρ, is contractible to that vertex. Therefore, we may restrict to the case V_ρ, is the join of the simplicial complexes V_1 and V_2. In that case, the connected components of V_ρ, are contractible unless V_1 and V_2 each consist of two vertices. 
	  
	
	
	This scenario occurs if and only if the following conditions on ray-degree pairing are satisfied: either ρ_5()=-1 or ρ_6()=-1, and 
	
    ρ_i()<0   for all   i=1,2,3,4.
 
	There is no  satisfying the second case (this is immediate from primitive relations (<ref>)). Since ρ_5()=-1, we can represent =(x,y,-1). Under this representation, the inequalities can be expressed as:
	
    ey-a+1   ≤ x ≤ -1
    
    		-b+1   ≤ y ≤ -1.

This system of inequalities has an integral solution (namely (-1,-b+1,-1)) if and only if b≥ 2 and (b-1)e+a≥ 2. 






lemma:H1deg
	Let X≅(𝒪_𝔽_e⊕𝒪_𝔽_e(aF+bH)) with e,b≥0. Then
	
    H^0(V_ρ,,) ≠0

	in exactly the following cases:
	
		
  * The pair (ρ,) takes the form (ρ_2,(x,-1,0)), where x satisfies
		
    -e+1≤ x ≤ -1.

		Type 1 occurs if and only if e≥ 2;
		
		
  * The pair (ρ,) takes the form (ρ_6,(x,y,1)), where x,y satisfies
		 
    0≤ y≤ b,    	ey+a+1 ≤ x ≤ -1.

		 Type 2 occurs if and only if a≤ -2;
		
		
  * The pair (ρ,) takes the form (ρ_5,(x,y,-1)), where x,y satisfies
		
    -b+1≤ y≤ -1,    0 ≤ x ≤ ey-a.

		Type 3 occurs if and only if e+a≤0 and b≥ 2;
		
		
  * The pair (ρ,) takes the form (ρ_5,(x,0,-1)), where x satisfies
		
    b=0,    -a+1    ≤ x ≤ -1.

		Type 4 occurs if and only if
		a≥ 2 and b=0.
	

	















	Any simplicial complex V_ρ, can be expressed as the join of the simplicial complexes:
	
    V_1= V_ρ,∩{n_ρ_1,n_ρ_3},    V_2=V_ρ,∩{n_ρ_2,n_ρ_4},    V_3=V_ρ,∩{n_ρ_5,n_ρ_6}.

	If at least two of V_1,V_2 and V_3 are non-empty, then V_ρ, is connected. According to (<ref>), V_3 is either empty or consists of single vertex. Hence, V_ρ, can only have more than one connected component when it consists of exactly two vertices, either {n_ρ_1, n_ρ_3} or {n_ρ_2, n_ρ_4}. There are four distinct scenarios in which this occurs:
	
	Type 1:
	
    ρ_2()=-1,   ρ_1()<0,   ρ_3()<0,   ρ_4()≥ 0,   ρ_5()≥0 ,  ρ_6()≥ 0.

	Thus, we can represent =(x,-1,0), and the above 
	inequalities lead to
	
    -e+1≤ x ≤ -1.

	It is immediate to see that Type 1 occurs if and only if e≥ 2.
		
	Type 2:
	
    ρ_6()=-1,   ρ_1()<0,   ρ_3()<0,   ρ_2()≥ 0,   ρ_4()≥0 .

	Thus, we can represent =(x,y,1), and the above inequalities lead to
	
    0≤ y≤ b,    	ey+a+1 ≤ x ≤ -1.

	If Type 2 occurs, then =(-1,0,1) is always included, which requires a≤ -2.
	
	
	Type 3:
	
    ρ_5()=-1,   ρ_2()<0,   ρ_4()<0,   ρ_1()≥ 0,   ρ_3()≥0 .

	Thus, we can represent =(x,y,-1), and the above inequalities lead to
	
    -b+1≤ y≤ -1,    0 ≤ x ≤ ey-a.

	If Type 3 occurs, then =(0,-1,-1) is always included, which requires e+a≤0.
	
	Type 4:
	
    ρ_5()=-1,   ρ_1()<0,   ρ_3()<0,   ρ_2()≥ 0,   ρ_4()≥0 .

	Thus, we can represent =(x,0,-1), and the above inequalities lead to
	
    b=0,    -a+1    ≤ x ≤ -1.

	It is immediate to see that Type 4 occurs if and only if a≥ 2 and b=0.
	




Using lemma:H2deg and lemma:H1deg, we will apply the unobstructedness result (thm:unobstructed) based on the ray-degree pairs.

lemma:possibleobs
	Let X≅(𝒪_𝔽_e⊕𝒪_𝔽_e(aF+bH)) with e, b ≥ 0. Then X is unobstructed, except possibly in the following cases:
	
		
  * e=1, a≤ -2 and b≥ 3-a; 
		
  * e≥ 2, a≤ -e and b ≥  1+2-ae.  
	
	
	
	
 




	Suppose first that e=0. By lemma:H2deg H^2(X,_X)≠0 if and only if a≥ 2 and b≥ 2. In this case, lemma:H1deg implies that H^1(X,_X)=0. Therefore, when e=0,  X is always unobstructed.
	
	Now assume that e=1. Then by lemma:H2deg H^2(X,_X)≠0 if and only if 
	
    b ≥max{2, 3-a }.

	 In this case, Type 1 and Type 4 of lemma:H1deg do not occur.
	By thm:unobstructed, to have obstructions, we must have Type 2 and Type 3 elements of lemma:H1deg. Hence, we have a≤-2 and combining this with (<ref>) yields <ref>.
	
	
	Now assume that e≥ 2. By lemma:H2deg H^2(X,_X)≠0 if and only if  
    b ≥max{2, 1+2-ae}.

	In this case, Type 4 of lemma:H1deg does not occur and Type 1 does occur. By thm:unobstructed, Type 3 must occur. 
	However, the occurrence of both Type 1 and Type 3 together implies Type 2 also occurs. Hence, we have a≤-e; combining this with (<ref>) yields <ref>.







For the cases outlined in lemma:possibleobs, we will review the inequalities that define  the sets of  with H^1(X,_X)_≠0 and H^2(X,_X)_≠0,  as some of the inequalities are redundant.



Case: e=1, a≤ -2, and  b≥ 3-a. 
In Figure fig:type1, we depict the projections onto the xy-coordinates of the degrees  where H^1(X,_X)_≠0 (split into types 2 and 3) and H^2(X,_X)_≠0. 
In this situation, the degrees with H^2(X,_X)_≠0 satisfy the inequalities

    y≥ -b+1,    x≤ -1,    x≥ y-a+1.

This defines a triangular region, which degenerates to a single point if b=3-a. 

Type 1 and Type 4 of H^1(X,_X)_ do not occur. The region for Type 2 is defined by the inequalities

    y≥ 0,    x≤ -1,    x≥ y+a+1

resulting in a triangular region, which degenerates to a single point if a=-2. The region for Type 3 is defined by the inequalities

    y≤ -1,    x≥ 0,    x≤ y-a

resulting in a triangular region.
We always have the following relation among the degrees:

    (-1,a-2,-1)=(0,a,-1)+(0,-2,-1)+(-1,0,1).

For the degree on the left hand side, H^2(X,_X) is non-zero, while H^1(X,_X) is non-vanishing for each of the degrees on the right hand side.



Case: e≥2, a≤ -e, and  b≥ 1+2-ae. 
In Figure fig:type2, we depict the projections onto the xy-coordinates of the degrees  where H^1(X,_X)_≠0 (split into types 1, 2, and 3) and H^2(X,_X)_≠0. 
In this situation, the degrees with H^2(X,_X)_≠0 satisfy the inequalities

    y≥ -b+1,    x≤ -1,    x≥ ey-a+1.

If (a-2)/e is an integer, the convex hull of these degrees is a triangle, which degenerates to a single point when b= 1+(2-a)/e. If (a-2)/e is not an integer, we define 
    η=⌊a-2e⌋,
 and obtain a trapezoidal region (which degenerates to a line when η=-b+1) by eliminating the top portion of the triangle where there are no lattice points. 

Type 4 of H^1(X,_X)_ does not occur.
The region for Type 1 is defined by the (in)equalities

    y=-1,    x≤ -1,    x≥ -e+1,

and forms a line. The region for Type 2 is defined by the inequalities

    y≥ 0,    x≤ -1,    x≥ ey+a+1.

If (-a-2)/e is an integer, we obtain a triangle, which degenerates to a single point when a=-2. If (-a-2)/e is not an integer, we define

    μ= ⌊-a-2e⌋

and obtain a trapezoidal region (which degenerates to a line  when μ=0) by eliminating the top portion of the triangle where there are no lattice points. 

The region for Type 3 is defined by the inequalities

    y≤ -1,    x≥ 0,    x≤ ey-a.

If a/e is an integer, we obtain a triangle, which degenerates to a single point when a=-e. If a/e is not an integer, we define

    ξ=⌈ae⌉

and obtain a trapezoidal region (which degenerates to a line when ξ=-1) by eliminating the bottom portion of the triangle where there are no lattice points.

If a≢1  e, then ξ-η=1, and we have the following relation:

    (-1,η,-1)=(0,ξ,-1) + (-1,-1,0).

If a≡ 1  e, then ξ-η=2, and we have the following relation:

    (-1,η,-1)= (1,ξ,-1) + 2(-1,-1,0).

In both cases, for the degree on the left hand side, H^2(X,_X) is non-zero, while H^1(X,_X) is non-vanishing for each of the degrees on the right hand side.




 §.§ Split P1-bundles: obstruction computations

Our next goal is to compute the hull for several examples. Instead of directly working with _Σ, we will apply the results from <ref>. Specifically, we will identify Σ' and Γ such that _Σ',Γ has the same hull as _Σ. Additionally, we will determine a minimal  such that = ⋀^2 Σ'.


lemma:SigmaD
	Let X_Σ≅(𝒪_𝔽_e⊕𝒪_𝔽_e(aF+bH)) with e,b≥0. Then the fan Σ' with the maximal cones
	
    σ_1   = (ρ_1,ρ_4,ρ_5),   σ_2= (ρ_1,ρ_2,ρ_5), 
    σ_3   = (ρ_2,ρ_3,ρ_5),   σ_4= (ρ_3,ρ_4,ρ_5).

	 covers Γ. Moreover, the set
	 
	 
    ={{σ_1,σ_2}, {σ_2,σ_3}, {σ_3,σ_4}, {σ_4,σ_1}}

		satisfies = ⋀^2 Σ'_max.                                        
	


From lemma:H1deg, we observe that (ρ,)∈ must be of the form 

    ( ρ_2, (*,*,0) )   or  (ρ_5, (*,*,-1) )   or  (ρ_6, (*,*,1) ).
 
The proof then follows from Example <ref> and Example <ref>.

 
For notational simplicity, we will index cochains by the numbers 1,2,3,4 instead of the cones σ_1,…,σ_4, e.g. for 1≤ i,j≤ 4 we write α_i instead of α_σ_i and ω_ij instead of ω_σ_iσ_j.
Recall that in <ref>, we needed to make several choices while computing the hull of _Σ (or similarly _Σ',Γ). 
In the examples below, we will make these choices as follows:

	
  * We always use the graded lexicographic local monomial order.
	
  * For constructing the map ψ, we choose the ordering of cones in Σ' as σ_1<σ_2<σ_3<σ_4.
	
  * For Type 1 and Type 2 first order deformations, V_ρ, is given by two vertices n_ρ_1 and n_ρ_3. When choosing a basis of ^1 _Σ',Γ, we will always take the connected component n_ρ_3.
		
  * For Type 3 first order deformations, V_ρ, is given by two vertices n_ρ_2 and n_ρ_4. 
When choosing a basis of ^1 _Σ',Γ, we will always take the connected component n_ρ_2.
	
  * For ray-pairs such that H^1(V_ρ,,)≠0, we know V_ρ, is the simple cycle with vertices n_ρ_1,n_ρ_2, n_ρ_3,n_ρ_4 ordered cyclically.
	In these cases, we choose the cocycle
	ω∈Ž^1('_ρ,,) by setting
	
    ω_34=-1,   ω_12=ω_23=ω_41=0.

	Then ψ(ω)_i=0 for i=1,…,4, and thus
	d(ψ(ω))=0 as required.

With the above choices, it follows that α_1^(r) is always zero.  





 






[Case (e,a,b)=(1,-2,5)]
	Here, we explain how to use the setup in <ref> and <ref> to compute the hull of _X for 
	
    X=(𝒪_𝔽_1⊕𝒪_𝔽_1(-2F+5H)).

	This example possesses the minimal ^1 dimension among those ^1-bundles in lemma:possibleobs that have no quadratic obstructions, but may have third order obstructions. We will compute its hull, showing that it indeed has a third order obstruction.


	By lemma:H1deg, H^1(X_Σ,T_X_Σ) is non-zero only in the following degrees:
	
    _1=(0,-1,-1),   _2=(1,-1,-1),   _3=(0,-2,-1),   _4=(-1,0,1).

	By lemma:H2deg, H^2(X_Σ,T_X_Σ) is non-zero only in the degree 
	
    =(-1,-4,-1).
 
	See Figure fig:1,-2,5 for an illustration.
	The only non-negative integer combinations of the _i giving  are of the form
	
    = 2_3+_4=_1+_2+_3+2_4=2_1+2_2+3_4.

	Therefore, the hull must have the form
	
    [[t_1,…,t_4]]/J;    J=⟨ a_1t_3^2t_4 +a_2· t_1t_2t_3t_4^2+a_3  t_1^2t_2^2t_4^3 ⟩

	for some a_1,a_2,a_3∈. Here, t_i is the deformation parameter with degree _i.
	
	In Table table:1,-2,5 we summarize the deformation equation computations. See the ancillary files <cit.> for code carrying out these computations in Macaulay2 <cit.>. The monomials highlighted in blue are those for which obstructions are possible. The deformation data is interpreted as follows. For each α_i^(r), the coefficient of a monomial t^w is obtained by multiplying the entries in the σ_i row by the coefficient of t^w in the first column of the table. The expression for α_i^(r) is then the sum of these products, considering only monomials of total degree less than or equal to r. As explained in Example <ref>, every element of Γ must be of the form 
	
    ( ρ_5, (*,*,-1) )   or  (ρ_6, (*,*,1) )   or  (ρ_5, (*,*,0) )   or  (ρ_6, (*,*,0) ).

	This implies that only monomials of the form t_1^w_1t_2^w_2t_3^w_3t_4^w_4 with 
    |w_1+w_2+w_3-w_4|≤ 1

	 will appear. Additionally, we restrict our attention to relevant monomials, see Remark <ref>:
we only need to consider monomials dividing 	t_3^2t_4, t_1t_2t_3t_4^2, or t_1^2t_2^2t_4^3.
		
The obstruction data should be interpreted as follows. For the normal form of ((α^(r))-∑_ℓ=1^q g_ℓ^(r)·ω_ℓ)_ij with respect to · J_r,	the coefficient of t^w∈_r+1 is the product of the entries in the row labeled σ_iσ_j  with the coefficient of t^w found in the first column of the table. We also list the coefficients of the γ_ℓ^(r+1) in the rightmost column of the table.

After computing α^(7), we obtain the deformation equation
	
    (α^(7))   ≡ 0 ^8+ ⟨  t_3^2t_4 -2· t_1t_2t_3t_4^2+  t_1^2t_2^2t_4^3 ⟩.

	Consequently, we conclude that a_1=a_3=1 and a_2=-2.
	After applying the change of variables  t_3'=t_3-t_1t_2t_4, we observe that
	
    t_3'^2t_4= t_3^2t_4 -2· t_1t_2t_3t_4^2+  t_1^2t_2^2t_4^3.

	We obtain that the spectrum of the hull has two irreducible components, both of dimension three. One is smooth and the other generically non-reduced of multiplicity two.
	
	
	
	
	

	
	
	
	
	
	
	
	
	
	





[Case (e,a,b)=(2,-3,4)]
	Here, we compute the hull of _X for 
	
    X=(𝒪_𝔽_2⊕𝒪_𝔽_2(-3F+4H)).

	 We will show that the hull  does not have any quadratic obstructions, but does have third order obstructions. We will utilize this example in the proof of thm:obstructed. 
	
	By lemma:H1deg, H^1(X_Σ,T_X_Σ) is non-zero only in the following degrees:	 
	
    _1   =(0,-1,-1),    _2   =(1,-1,-1),    _3   =(-1,-1,0),
    _4   =(-1,0,1),    _5   =(-2,0,1).

	By lemma:H2deg, H^2(X_Σ,T_X_Σ) is non-zero only in the degrees 
	
    _1=(-1,-3,-1),  _2=(-2,-3,-1).
 
	See Figure fig:2,-3,4 for an illustration.
	The only non-negative integer combinations are of the form
	
    _1   = _2+2_3=_1+_2+_3+_4=2_2+_3+_5
       = 3_2+ 2_5=   _2+ 2_1+2_4= _1+2_2+_4+_5  
    _2   = _1+2_3=_1+_2+_3+_5=2_1+_3+_4
       =3_1+ 2_4= _1+2_2+2_5 =_2+ 2_1+_4+_5

	Therefore the hull must have the form [[t_1,…,t_5]]/ ⟨ g_1,g_2 ⟩, where
	
    g_1   =a_1t_2t_3^2+a_2t_1t_2t_3t_4+ a_3t_2^2t_3t_5+a_4t_2^3t_5^2+a_5t_1^2t_2t_4^2+a_6t_1t_2^2t_4t_5;
    
    		g_2   =b_1t_1t_3^2+b_2t_1t_2t_3t_5+ b_3t_1^2t_3t_4+b_4t_1^3t_4^2+b_5t_2^2t_1t_5^2+b_6t_2t_1^2t_4t_5.

	for some a_1,…,a_6,b_1,…,b_6∈.
	
	Doing a computation similar to Example <ref>, we obtain
	
    a_1=-b_1=a_4=-b_4=a_5=-b_5=1;
    
    		a_2=-b_2=a_3=-b_3=a_6=-b_6=2,

	see <cit.> for details.
	After applying the change of variables t_3'=t_3+t_1t_4+t_2t_5 we observe that
	
    t_2t_3'^2=g_1    	t_1t_3'^2=-g_2.

	As in Example <ref>, the spectrum of the hull has two irreducible components. One is smooth with dimension 3, and the other is a generically non-reduced component of multiplicity 2 and dimension 4.




[Case (e,a,b)=(3,-4,3)]
	Here we compute the hull of _X for 
	
    X=(𝒪_𝔽_3⊕𝒪_𝔽_3(-4F+3H)).

	We will see that the hull is irreducible and singular at the origin, and already determined by the quadratic obstructions. We note that these quadratic obstructions could have been computed using the methods of <cit.>.  

	By lemma:H1deg, H^1(X_Σ,T_X_Σ) is non-zero only in the following degrees:
	
    _1   =(0,-1,-1),    _2   =(1,-1,-1),    _3   =(-1,0,1),    _4   =(-1,-1,0),
    _5   =(-2,0,1),    _6   =(-2,-1,0),    _7   =(-3,0,1).

	By lemma:H2deg, H^2(X_Σ,T_X_Σ) is non-zero only in the degree 
	
    =(-1,-2,-1).
 
	See Figure fig:3,-4,3 for an illustration. The only non-negative integer combinations are of the form
	
    = _1+_4=_2+_6=2_1+_3=2_2+_7=_1+_2+_5.

	Therefore, the hull must have the form
	
    [[t_1,…,t_7]]/J;    J=⟨ a_1t_1t_4+ a_2t_2t_6+ a_3t_1^2t_3+a_4t_2^2t_7+a_5t_1t_2t_5 ⟩.

	
	By computing the quadratic obstructions, we obtain that a_1=a_2=1, see Table table:3,4,-3 or <cit.>. After applying the change of variables t_4'=t_4+a_3t_1t_3+a_5t_2t_5 and t_6'=t_6+a_4t_2t_7, we obtain 
	
    t_1t_4+ t_2t_6+ a_3t_1^2t_3+a_4t_2^2t_7+a_5t_1t_2t_5= t_1t_4'+t_2t_6'.

	Thus, the spectrum of the hull is irreducible and has dimension 6; it is the formal completion of the product of ^3 with the affine cone over a smooth quadric surface.








	














[Case e≥2, a=-e, b=3]
	Here, we compute the hull of _X for 
	
    X=(𝒪_𝔽_e⊕𝒪_𝔽_e(-eF+3H)),

	where e≥ 2. We will show that
	the spectrum of the hull consists of two irreducible components, with an arbitrarily large difference in their dimensions. The case e=2 was first analyzed in <cit.> and has minimal ^1 dimension among obstructed ^1-bundles.
	Similar to Example <ref>, the hull is already determined by the quadratic obstructions, which could have been computed using the methods of <cit.>.  

	By lemma:H1deg, H^1(X_Σ,T_X_Σ) has dimension 2e-1 and is non-zero in the degrees
	
    _1=(0,-1,-1),  _2k=(-k,-1,0),   _2k+1=(-k,0,1)

	for k=1,…,e-1. By lemma:H2deg, H^2(X_Σ,T_X_Σ) has dimension e-1  and is non-zero in the degrees 
	
    _k=(-k,-2,-1)
 
	for k=1,…,e-1. See Figure fig:e,-e,b. The only non-negative integer combinations are of the form
	
    _k= _1+_2k=2_1+_2k+1,

	for k=1,…,e-1. 
	
	Therefore, the hull must have the form
	
    [[t_1,…,t_2k+1]]/J;    J=⟨ a_1k· t_1t_2k+ a_2k· t_1^2t_2k+1:k=1,…,e-1 ⟩,

	for some a_1k,a_2k∈ where k=1,…,e-1.
	
	The deformation data restricted to the deformation parameters t_1,t_2k is shown in Table table:case: e,-e,3. Therefore, we obtain that a_1k=1 for k=1,…,e-1. After applying the change of variables
	
    t_2k'= t_2k+a_2k· t_1 t_2k+1,

	 we can express the hull as
	
    [[t_1,t_2',t_3, …, t'_2e-1 ]] /⟨ t_1 ⟩·⟨ t_2', t_4',…, t'_2e-2⟩.

	The spectrum of the hull has  two irreducible components, both smooth: one with dimension 2e-2, and the other with dimension e.
	
	 










	




















	We have carefully chosen the previous four examples so that degree constraints imply that the obstructions equations g_ℓ are polynomials instead of power series. Such constraints do not apply in general, such as in the case of (e,a,b)=(2,-4,4).



Using computations similar to those from the examples above, we now confirm that the cases in lemma:possibleobs do indeed yield obstructions, proving thm:obstructed.


	By lemma:possibleobs, the cases not listed in the theorem are unobstructed.
It remains to show that the cases of the theorem are indeed obstructed, with minimal degree of obstruction as claimed.
	Suppose first that e=1, a≤ -2 and b≥ 3-a. By Figure fig:type1 and the discussion following the proof of lemma:possibleobs, there is no degree of Type 1, so it is not possible to have relation among degrees that could provide a quadratic obstruction. However, we have
	
    (-1,a-2,-1)=(0,-a,-1)+(0,-2,-1)+(-1,0,1),

	 see (<ref>). If we restrict the computations to the deformation parameters associated with the ray-degree pairs 
	 
    (ρ_5,(0,-a,-1)),    (ρ_5,(0,-2,-1)),    (ρ_6,(-1,0,1))

	  we obtain results identical to the deformation parameters associated with the
	  ray-degree pairs 
	  
    (ρ_5,(0,-2,-1)),    (ρ_5,(0,-2,-1)),    (ρ_6,(-1,0,1))
 in Example <ref>. Indeed, the simplicial complexes V_ρ, and the rays involved are identical in these two situations.. Thus, we must have a third order obstruction, proving claim <ref>.
	  
	  Consider instead the case e≥ 2, a≤ -e and b ≥  1+(2-a)/e. We now  use Figure fig:type2 and the discussion following the proof of lemma:possibleobs. If  a≢1  e we have the relation
	  
    (-1,η,-1)=(0,ξ,-1)+(0,-1,-1)

	  see (<ref>). Restricting the computations to the deformation parameters associated with the ray-degree pairs 
	  
    (ρ_5,(0,ξ,-1)),    (ρ_2,(-1,-1,0))

	  we obtain results identical to the deformation parameters associated with the
	  ray-degree pairs 
	  
    (ρ_5,(0,-1,-1)),    (ρ_2,(-k,-1,0))

	   in Example <ref>. Thus, we must have a second order obstruction.
	  
	 If instead a≡ 1  e, it is not possible to have relations among degrees that could provide a quadratic obstruction since ξ-η=2. However, we have
	 
    (-1,η,-1)=(1,ξ,-1)  +  2· (-1,-1,0),

	 see (<ref>). Restricting the computations to the deformation parameters associated with the ray-degree pairs 
	 
    (ρ_5,(1,ξ,-1)),    (ρ_2,(-1,-1,0))

	 we obtain results identical to the deformation parameters associated with the
	 ray-degree pairs 
	 
    (ρ_5,(1,-1,-1)),    (ρ_2,(-1,-1,0))

	 in Example <ref>. Thus, we must have a third order obstruction. This completes the proof of claim <ref>.














	
equationsection



§ ISOMORPHISM OF DEFORMATION FUNCTORS

In this section, we will prove compa. Similar results to those in compa are known in some other situations; see <cit.> and <cit.>. Our proof of compa is an adaptation of these established results. We first establish a necessary lemma. 


	For x,y ∈Č^1(,)⊗_A, we define 
	
    _,(x,y)= {a ∈Č^0(,)⊗_A: a ⊙ x =y},

	where ⊙ is the action defined in action.


lifting lemma
	Let 
	
    0  → I → A'  A → 0

	be a small extension, and let x',y'∈Č^1(,)⊗_A' be representatives of elements in _,(A'). 
	Then: 
	
		
  * if a ∈_,(π(x'),π(y')), the obstruction to lifting a to  _,(x',y') is in Ȟ^1(,)⊗ I and a lifting exists if and only if the obstruction vanishes; 
		
  * given two lifts a',b'∈_,(x',y') of a∈_,(π(x'),π(y')), we have
		
    b' -a' ∈Ȟ^0(,)⊗ I.

	


	Let a'∈Č^0(,)⊗_A' be a lift of a. We set
	
    v' = (a'⊙ x')  -y' ∈Č^1(,)⊗_A'.

	It is immediate that v' ∈Č^1(,)⊗ I, since its image in Č^1(,)⊗_A is zero. We claim that v' ∈Ž^1(,)⊗ I. 
	Firstly, we observe the following facts:
	
		
  * Since v' ∈Č^1(,)⊗ I, by rem:obs<ref> we have (v')= d(v').
		
  * By applying  comm<ref>, we obtain  
    v'_jk:= a'_j x'_jk -a'_k  -y'_jk= y'_ij ( a'_j x'_jk -a'_k  -y'_jk) -y'_ij.

		
  * Since (x')=0=(y'), we have x'_ij x'_jk = x'_ik and -y'_jk -y'_ij = -y'_ik.
	
	By the first observation, it is enough to show that 
	
    v'_ij v'_jk= v'_ik.
 Using the second observation we obtain
	
    v'_ij v'_jk   = (a'_i x'_ij -a'_j  -y'_ij)  (y'_ij  a'_j x'_jk -a'_k  -y'_jk -y'_ij)  
       =  a'_i x'_ij x'_jk -a'_k  -y'_jk -y'_ij
       =  a'_i x'_ik -a'_k  -y'_ik
       = v'_ik

	Here, the penultimate equation follows from the third observation. Hence we have d(v')=0.

We next show that the class of v' is independent of the choice of lift a'. Let b'∈Č^0(,)⊗_A' be another lift of a and set w' = (b'⊙ x')  -y' similar to above.
Repeatedly using comm along with the fact that a'*-b'∈Č^0(,)⊗ I and rem:obs<ref>, one obtains that 

    v' -w'=(a' -b')=d(a' -b').

Since v',w'∈Č^1(,)⊗ I, comm<ref> implies that v' and w' differ by the coboundary d(a' -b') so they give the same cohomology class.

	If a'∈_,(x',y'), it is immediate that v'=0 hence its cohomology class is zero. Conversely,
	if the class of v' is zero in cohomology, then there exists c'∈Č^0(,)⊗ I such that 
	
    -w'= d(c')= (c').
 Since, c'∈Č^0(,)⊗ I we know c'  a' is also a lift of a.  We claim that
	
    (c' a' ⊙ x')  -y'=0.

	This can easily be verified by noticing
	
    0   = d(c')_ij+v'_ij
       = c'_i v'_ij -c'_j
       =c'_i (a'_i x'_ij -a'_j  -y'_ij)  -c'_j
       = c'_i (a'_i x'_ij -a'_j  -y'_ij)  (y'_ij -c'_j  -y'_ij)
       = (c'_i a'_i)  x'_ij -(c'_j a'_j)  -y'_ij.

	Here the second and fourth equations follow from an application of comm<ref>. This completes the proof of claim <ref>.
	
	Now we will show claim <ref>. Let a',b' be two lifts of a such that a'⊙ x'=y' and b'⊙ x'=y'. It is immediate that b' -a' ∈Č^0(,)⊗ I, since its image in Č^0(,)⊗_A is zero. We observe that
	
    y'   = b'⊙(-a'⊙ y')
       = (b' -a')⊙ x'
       = d(b' -a')+ y'.

	Here, the second equation is the multiplicative property of a group action, and the last equation follows from an application of comm<ref> with the observation b' -a'  ∈Č^0(,)⊗ I.
	Hence, we obtain d(b' -a')=0 and it follows that b' -a' ∈ H^0(,)⊗ I, proving claim <ref>.



	 By thm:deffunctor1, we have that Ȟ^1(,) represents the tangent space of _,, and Ȟ^2(,) is an obstruction space. Moreover, it is immediate to see that the map of cohomology groups Ȟ^1(,) →Ȟ^1(,) agrees with ^1_→^1_, and the map of cohomology groups Ȟ^2(,) →Ȟ^2(,) is an obstruction map for the morphism of functors f: _→_. Hence, using the standard smoothness criterion (standardsmooth), we obtain that f:_,→_, is smooth (cf. <cit.>). In fact, f:_,→_, is surjective.
	
	
	Now we proceed to show the injectivity of f:_,→_,. It is sufficient to show the following: for any A∈ and representatives x,y∈Č^1(,)⊗_A of elements in _,(A), if there exists γ∈_,(f(x),f(y)), then there also exists a ∈_,(x,y).
	
	
	
	By induction on _(A), we will prove the stronger fact that there exists a∈_,(x,y) such that f(a)=γ.
	During our proof of this claim the reader may refer to Figure figure:1, which illustrates the relationships between the elements under consideration.

	Clearly, the claim is true for A=. Let 
    0 → I → A'  A → 0
 be a small extension and assume the claim is true for A. Let x',y'∈Č^1(,)⊗_A' be representatives of two elements in _,(A') such that there exists γ'∈_,(f(x'),f(y')).
	We denote the images of x',y' in Č^1(,)⊗_A by x,y. Since 
	
    π(γ') ⊙ f(x)= f(y)
 by the induction hypothesis there exists a ∈_,(x,y) such that f(a)=π(γ'). 
	
	According to lifting lemma<ref>, the obstruction to lifting a to _,(x',y') lies in Ȟ^1(,)⊗ I. Moreover, this obstruction element maps to zero in Ȟ^1(,)⊗ I due to the existence of a lift for 
	f(a)=π(γ') in _,(f(x'),f(y')), provided by γ'. Therefore, by the injectivity of the map Ȟ^1(,)⊗ I →Ȟ^1(,)⊗ I, the obstruction element in Ȟ^1(,)⊗ I must vanish, ensuring the existence of a lifting b' ∈_,(x',y') of a. However, f(b') may not be equal to γ'. Nonetheless, according to lifting lemma<ref>, γ' -f(b') defines an element in Ȟ^0(,)⊗ I. Additionally, by the surjection Ȟ^0(,)→Ȟ^0(,) we may choose an element c'∈Ȟ^0(,)⊗ I that maps to γ' -f(b'). We set a'=c' b'. Clearly, a'∈_,(x',y') and π(a')=a.   Moreover, we have 
	
    f(a')   = f(c') f(b')
       =γ' -f(b') f(b')
       =γ'.

	This completes the proof of the inductive step, thus establishing the injectivity of the map f:_,→_,.
	
	
	
	
	 








	
	


§ COMPARISON THEOREM FOR OPEN SUBSCHEMES

Let X be a scheme over  and U⊆ X be an open subscheme. 
There is a natural map of functors
_X →_U obtained by restriction.
We make use of the following folklore result that was first brought to our attention by A. Petracci:
CMiso
	Suppose that X is a noetherian separated scheme over  and U⊂ X an open subscheme with complement Z=X∖ U.
Then the restriction map

    _X→_U

is injective if 
the depth of _X at every point of Z is at least two, and an isomorphism if the depth of _X at every point of Z is at least three. In particular, it is an isomorphism if Z has codimension at least three in X and X is Cohen-Macaulay.

The above theorem is stated and proved in the affine case (with slightly different hypotheses) by M. Artin in <cit.>. We now show how to globalize the argument of loc. cit. 
In the following, X will be a noetherian separated scheme over  and U an open subscheme with complement Z.

	Let k∈ and assume that the depth of _X at every point of Z is at least k. Then
	
    H^i(X,_X)=H^i(U,_X)

	for any i<k-1.


	By the depth condition, one obtains that ℋ^i_Z(_X)=0 for i<k, see <cit.>. Here, ℋ_Z^i is the sheaf of local cohomology with support in Z. By <cit.>, it follows that H^i(X,_X)=H^i(U,_U) for i<k-1.


Let X be affine and assume that the depth of _X at every point of Z is at least 2.
Let X' be any deformation of X over some A∈. Then

    H^0(X,_X')=H^0(U,_X').



	This is claim (*) from the proof of <cit.>, and follows from straightforward induction on the length of A. The base case A= follows from Lemma <ref>.



	Let X be affine and assume that the depth of _X at every point of Z is at least 3. Let U' be any deformation of U over some A∈. Then H^0(U,_U') is flat over A.


	This is the contents of <cit.>, see the final sentence of the proof. For the reader's convenience, we summarize the argument here.
The proof is by induction on the length of A; the case A= is trivial.
By Lemma <ref> we have that H^i(X,_X)=H^i(U,_U) for i=0,1, in particular this vanishes for i=1 since X is affine. 

	Realizing A as a small extension

    0→→ A → A_0→ 0

the flatness of U' implies the exactness of 

    0→_U →_U'→_U'⊗_A A_0→ 0.

By the isomorphisms of the previous paragraph, the exactness of 

    0→ H^0(X,_X) → H^0(U,_U') → H^0(U,_U'⊗_A A_0)→ 0

follows.
The ring H^0(U,_U'⊗_A A_0) is flat over A_0 by the induction hypothesis and the A-flatness of H^0(U,_U') follows by a version of the local criterion of flatness, see <cit.> or <cit.>.



We first show the injectivity of the map _X→_U when the depth of _X along Z is at least two. 
	Fix an affine open cover ={U_i}_i∈ I of X.
Consider two deformations X' and X” of X over A∈, and denote their restrictions to U by U' and U”. Assume that there is an isomorphism of deformations ϕ:U'→ U”. We thus have isomorphisms 

    ϕ_i^#:H^0(U_i∩ U, _U”)→ H^0(U_i∩ U, _U')

satisfying the obvious cocycle condition.
By Lemma <ref> 

    H^0(U_i, _X')=H^0(U_i∩ U, _U'),    
    	H^0(U_i, _X”)=H^0(U_i∩ U, _U”)

so we obtain isomorphisms ϕ_i:X'_|U_i→ X_|U_i”. By the cocycle condition, these glue to give an isomorphism X'→ X”. This shows that 
_X→_U is injective.

For the surjectivity when the depth is at least three, consider any deformation U' of U over A∈.
Let ι:U→ X denote the inclusion of U in X, and set _X':=ι_*(_U').
By Lemma <ref>, _X' is flat over A, hence defines a deformation X'. By construction, this restricts to the deformation U', hence _X→_U is surjective.



§ SOLVING THE DEFORMATION EQUATION

In this appendix, we will prove prop:defeqsolving and state and prove a lemma we used in proving hull.
We use notation as established in <ref> and <ref>.

	To solve (<ref>) we consider the small extension
	
    0→ J_r/(· J_r) → S/(·  J_r) → S/J_r→ 0.

	It follows from (<ref>) (modulo · J_r-1) that
	
    λ((s(α^(r)))) ≡λ((s(α^(r))))-∑_ℓ=1^q g_ℓ^(r)·ω_ℓ≡ 
    	0    J_r.



	From Proposition <ref> we obtain that the image of 
    ξ=λ((s(α^(r))))-∑_ℓ=1^q g_ℓ^(r)·ω_ℓ
 in Č^1(,/)⊗ S/(· J_r) is a cocycle. 
Because of this, the normal form of ξ with respect to · J_r is also a cocycle. In fact, the normal form belongs to 

    Ž^1(,/)⊗_r+1

since 

    ξ≡ 0 · J_r-1

and · J_r-1= J_r+^r+1. This latter equality follows from the assumption that J_r≡ J_r-1^r. Since the the images of the ω_ℓ span H^1(X,/), there exists

    β^(r+1)∈Č^0(,/)⊗_r+1,   γ_1^(r+1),…,γ_q^(r+1)∈_r+1
 satisfying (<ref>).
This implies claim <ref>.

We now prove claim <ref>.
By rem:obs<ref>,

    λ((s(α^(r+1))))≡λ((s(α^(r))))-d(β^(r+1))    ^r+2.

Equation (<ref>) then follows directly from (<ref>). Likewise, J_r+1≡ J_r^r+1 follows from the fact that γ_ℓ^(r+1) belongs to _r+1.


Finally, with notation as in <ref>, the following lemma is used in proving hull:

lemma:obfinj
		Consider the map of functors  f: (R,-)→_↪. 
There exists a natural injective obstruction map ob_f defined by
	 
    ob_f: (J/ J)^*    → H^1(,/)
    φ   ↦∑_ℓ=1^qφ(g_ℓ)·ω_ℓ,

	where g_ℓ denotes the image of g_ℓ in J/· J.


	By construction, J⊆^2.
	It is well-known that (J/ J)^* is an obstruction space for (R,-) (see <cit.>). An obstruction space for _↪ is given by Ȟ^1(,/) (see showdeffun2<ref>, prop:isosecfunctor). It is immediate that ob_f is injective since the ω_ℓ form a basis for H^1(,/) and the g_ℓ generate J. We claim that ob_f is an obstruction map for f.
	
	To prove that ob_f is an obstruction map for f, we first claim that we can choose n≫ 0 such that 
	
    (J+ ^n)/(· J+ ^n )≅ J/· J.

	We will show this using the ideas from the proof of <cit.>.
	According to the Artin-Rees lemma <cit.>, we have J∩^n ⊆· J for n≫0. 
	This implies that
	
    (· J+^n)∩ J=· J,

	which leads to the isomorphism
	
    (J+ ^n)/(· J+ ^n ) ≅ J/(· J +^n)∩ J = J/· J

	as desired.
	
	
	Now we will show that ob_f is an obstruction map for f. Let ζ∈(R,A) and consider a small extension
	
    0→ I → A' → A → 0.

	We will show that 
	
    ob_f(ϕ(ζ, A'))= ϕ(f(ζ),A')

	where we use ϕ to denote the map taking a small extension to its obstruction class for both functors (R,-) and _↪.
	
	We define a local morphism of -algebras η: S→ A' by mapping each variable t_ℓ to any lifting of its image under the map S→ R→ A. Because A,A' are Artin rings and by the discussion above, there exists r≫0 such that η,ζ factor respectively through S/^r+1 and R_r, and 
	
    (J+ ^r+1)/(· J+ ^r+1 ) ≅ J/· J.
	
	
	
	The image of  · J under η is zero, because _A'· I=0.	Consequently, η factors through S/· J.
	Furthermore, since g_ℓ-g_ℓ^(r)∈^r+1, it follows that J+^r+1= J_r and · J +^r+1= · J_r +^r+1. 
	From the above discussion, we thus have the following commutative diagram:
	
    [column sep=small]
    		0[r]    J [r] [d]    S [r] [d]     R [d,"𝕀"] [r]     0  
    
    		0[r]     J/ J [r] [d, "≅"]    S/(· J)[r] [d]    R[r] [d,"π_r"]    0 
    	
    		0[r]     J_r/(· J_r+^r+1) [r] [d, "η_r"]    S/(· J_r+^r+1)[r] [d,"η_r"]    R_r[r] [d,"ζ_r"]    0 
    
    		0[r]    I [r]     A' [r]      A  [r]     0  
    	.

	Since J⊆^2, it follows that  
    η: J/· J  J_r/(· J_r+^r+1)  I
 remains unaffected by the choice of η.
	
	
	
	
	Applying obstruction theory to ζ∈(R,A) (see for example, <cit.>), we obtain
	
    ϕ(ζ, A')= η.

	By (<ref>)
	we have 
	
    ϕ(α^(r),S/(· J_r+^r+1))    = ∑_ℓ=1^qg_ℓ^(r)·ω_ℓ

	where g_ℓ^(r) is the image of g_ℓ^(r) in J_r/(· J_r+^r+1).
	By the functoriality of obstruction theory, we have
	
    ϕ(f(ζ),A')   = ∑_ℓ=1^qη_r(g_ℓ^(r)) ·ω_ℓ
       = ∑_ℓ=1^qη(g_ℓ) ·ω_ℓ
       = ob_f(η).

	Thus, we have shown that ob_f is an injective obstruction map for f.









alpha
 	

