# The Kodaira-Spencer map for minimal toric hypersurfaces

Julius Giesler
University of Tübingen

(August 27, 2024)

###### Abstract

In this article we study infinitesimal deformations of toric hypersurfaces. We
introduce a Kodaira-Spencer map and compute its kernel. By introducing some
new Laurent polynomials we make our computation as explicit as possible. This
widely generalizes results of Griffiths for projective hypersurfaces.

##  1 Introduction

We start with an $n$-dimensional lattice polytope $\Delta$ and a nondegenerate
Laurent polynomial $f$ with Newton polytope $\Delta$ and set

$\displaystyle Z_{f}:=\\{f=0\\}\subset T.$

By results from ([Bat22]) there is (under a mild condition
$F(\Delta)\neq\emptyset$) a projective toric variety $\mathbb{P}$ to a
simplicial fan $\Sigma$, such that the closure $Y_{f}$ of $Z_{f}$ in
$\mathbb{P}$ has at most terminal singularities and the canonical divisor
$K_{Y_{f}}$ is nef. We say that $Y_{f}$ is a minimal model of  $Z_{f}$.
The toric variety $\mathbb{P}$ does not depend on $f$. Therefore we vary $f$
over all such polynomials, write $f\in B:=B(\Delta)$ to obtain a family of
minimal toric hypersurfaces

$\mathcal{X}:=\\{(x,f)\in\mathbb{P}\times B\mid\,x\in
Y_{f}\\}\overset{pr_{2}}{\rightarrow}B.$

Next to a tangent vector

$\operatorname{Spec}\,\mathbb{C}[\epsilon]/(\epsilon^{2})\rightarrow B$

at $f\in B$ we associate a first-order infinitesimal deformation of $Y_{f}$
(in $\mathcal{X}$) by base change along $v$. These infinitesimal deformations
and all those in $\mathbb{P}$ are parametrized by Kodaira Spencer maps

$\displaystyle\kappa_{\mathbb{P},f}:H^{0}(Y,N_{Y/\mathbb{P}})\rightarrow\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y}),$
$\displaystyle\kappa_{f}:\,\,H^{0}(Y,N_{Y/\mathcal{X}})\rightarrow\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y}).$

In fact $H^{0}(Y,N_{Y/\mathcal{X}})\subset H^{0}(Y,N_{Y/\mathbb{P}})$ and
$\kappa_{f}$ equals the restriction of $\kappa_{\mathbb{P},f}$ (see section
9). In Theorem 6.1 we identify the kernel of $\kappa_{\mathbb{P},f}$ with the
Lie algebra of the automorphism group of $\mathbb{P}$:

######  Theorem 1.1.

Let $\Delta$ be an $n$-dimensional lattice polytope, where $n\geq 2$, with
$l^{*}(\Delta)>0$. If $n=2$ assume that $l^{*}(\Delta)\geq 2$. Then given
$f\in B$

$\displaystyle\ker(\kappa_{\mathbb{P},f})\cong\operatorname{Lie}\,\operatorname{Aut}(\mathbb{P})$
(1)

where $l^{*}(\Delta)$ denotes the number of interior lattice points of
$\Delta$. The proof is cohomological. Concerning the cokernel of
$\kappa_{\mathbb{P},f}$ we prove:

######  Corollary 1.2.

Given the conditions of the theorem if $n\geq 4$ then the following sequence
is exact

$\displaystyle 0\rightarrow
Im(\kappa_{\mathbb{P},f})\rightarrow\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})\rightarrow\operatorname{Ext}_{\mathcal{O}_{\mathbb{P}}}^{1}(\Omega_{\mathbb{P}}^{1},\mathcal{O}_{\mathbb{P}})\rightarrow
0$

that is all infinitesimal deformations of $Y_{f}$ arise from
$Im(\kappa_{\mathbb{P},f})$ or from infinitesimal deformations of
$\mathbb{P}$.

A more detailed description of the second term is a topic of current research
(see [IlTu18], [Pet20]) and ([Mav03]) for the case that $Y_{f}$ is Calabi-Yau.

To be more explicit on $\ker(\kappa_{\mathbb{P},f})$ we recall the definition
of the canonical closure $C(\Delta)$ of $\Delta$, a polytope that might be
slightly larger than $\Delta$. Given $P\subset M_{\mathbb{R}}$ let $L(P)$ be
the $\mathbb{C}$-vector space with basis $x^{m}$, $m\in P\cap M$. Then

$\displaystyle H^{0}(Y,N_{Y/\mathcal{X}})\cong L(\Delta)/\mathbb{C}\cdot f$
$\displaystyle H^{0}(Y,N_{Y/\mathbb{P}})\cong L(C(\Delta))/\mathbb{C}\cdot f.$

We apply results from ([BG99]) to identify a basis of
$\ker(\kappa_{\mathbb{P},f})$ explicitly with certain Laurent polynomials
having support on $C(\Delta)$ (Corollary 8.4):

######  Corollary 1.3.

Given the conditions of Theorem 1.1, let

$\displaystyle f=\sum\limits_{m\in M\cap\Delta}a_{m}x^{m}\in B$

Then

$\displaystyle\ker(\kappa_{\mathbb{P},f})\cong\Big{\langle}\,x_{i}\cdot\frac{\partial
f}{\partial x_{i}}\mid
i=1,...,n\,\Big{\rangle}\quad\oplus\bigoplus\limits_{\alpha\in
R(N,\Sigma)}w_{-\alpha}(f)\cdot\mathbb{C}$

where $R(N,\Sigma)\subset M$ denote the (Demazure) roots of $\Sigma$ and
$w_{-\alpha}(f)$ denote certain Laurent polynomials. Concerning the kernel of
$\kappa_{f}$ we prove

######  Theorem 1.4.

Given the conditions of Theorem 1.1 then

$\displaystyle\ker(\kappa_{f})\cong\Big{\langle}\,x_{i}\cdot\frac{\partial
f}{\partial x_{i}}\mid
i=1,...,n\,\Big{\rangle}\quad\oplus\bigoplus\limits_{\alpha\in
R(N,\Sigma_{\Delta})}w_{-\alpha}(f)\cdot\mathbb{C}$

Application: Let $n=3$. Then $Y$ is smooth and thus

$\displaystyle\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})\cong
H^{1}(Y,T_{Y}).$ (2)

There is a period map

$\displaystyle\mathcal{P}_{B,f}:f\mapsto[H^{2,0}(Y_{f})].$

This map is holomorphic and by a result of Griffiths the differential
$d\mathcal{P}_{B,f}$ factors as

${L(C(\Delta))/\mathbb{C}\cdot
f}$${H^{1}(Y,T_{Y})}$${\operatorname{Hom}(H^{2,0}(Y_{f}),H^{1,1}(Y_{f}))}$$\scriptstyle{\kappa_{\mathbb{P},f}}$$\scriptstyle{d\mathcal{P}_{B,f}}$$\scriptstyle{\Phi_{f}}$

The infinitesimal Torelli theorem for $Y_{f}$ asks if $\Phi_{f}$ is injective.
If we know $\ker(d\mathcal{P}_{B,f})$ we may check if

$\displaystyle\ker(d\mathcal{P}_{B,f})\supseteq\ker(\kappa_{f})$ (3)

is an equality, in which case we get

$\displaystyle\ker(\Phi_{f|{Im\,\kappa_{\mathbb{P},f}}})=0$

giving a partial result towards the ITT. A calculation of
$\ker(d\mathcal{P}_{B,f})$ should work with jacobian rings. In the
dissertation ([Gie23]) we worked this out for $n=3$. In fact these methods
should also apply to higher dimensions.

One might be also interested in subfamilies of $\mathcal{X}\rightarrow B$:
Given $B^{0}\subseteq B$ with restriction $\kappa_{f}^{0}$ of $\kappa_{f}$ it
is obvious that

$\displaystyle x_{1}\cdot\frac{\partial f}{\partial
x_{1}},...,x_{n}\cdot\frac{\partial f}{\partial x_{n}}\in\ker(\kappa_{f}^{0})$

but hard to say when $w_{-\alpha}(f)\in\ker(\kappa_{f}^{0})$. We show that if
$B^{0}$ equals the set of vertices of $\Delta$ then under some additional
condition on the polytope $\Delta$ none of the $w_{-\alpha}(f)$’s contribute
to $\ker(\kappa_{f})$ (see Lemma 10.3).

What is known in the direction of our results: For
$Y_{f}\subset\mathbb{P}^{n}$ a smooth degree $d$ hypersurface by ([Voi03,
Ch.6])

$\ker(\kappa_{f})\cong\ker(\kappa_{\mathbb{P},f})\cong J_{f,griff}^{d},$

where $J_{f,griff}^{d}$ denotes the $d$-th homogeneous component of

$J_{f,griff}:=\Big{(}\frac{\partial f}{\partial x_{0}},....,\frac{\partial
f}{\partial x_{n}}\Big{)}\unlhd\mathbb{C}[x_{0},...,x_{n}].$

In Example 8.5 we show that if $d\geq n+1$ then our results specialize to this
assertion. There are generalizations of this results to quasismooth
hypersurfaces in weighted projective spaces ([Dol82, Thm.4.3.2]). In the
dissertation ([Koe91]) it is dealt with families of nondegenerate curves in
toric surfaces. In ([Mav03]) both the kernel and the cokernel of the Kodaira-
Spencer map is dealt with for anticanonical hypersurfaces and in ([Ok87]) the
Kodaira-Spencer map has been studied for $\Delta$ an $n$-simplex.

The author declares no conflicts of interest. Data sharing is not applicable
to this article as no new data were created or analyzed in this study. The
data that support the findings of this study are openly available at
arXiv:2203.01092 [math.AG]

##  2 Notation and Background

Let $M$ and $N$ be dual lattices and
$T=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}\cong(\mathbb{C}^{*})^{n}$ the
$n$-dimensional torus. We fix an $n$-dimensional lattice polytope
$\Delta\subset M_{\mathbb{R}}$ and denote the normal fan of a lattice (or more
generally rational) polytope $F$ by $\Sigma_{F}$. We denote the projective
toric variety defined via $\Sigma_{F}$ by $\mathbb{P}_{F}$. $\Sigma[1]$ denote
the rays of the fan $\Sigma$ and $D_{i}$ (or $D_{\nu}$) the toric divisor to
the ray $\nu_{i}$ (or $\nu$).

Throughout this article we assume $f$ to be a Laurent polynomial with Newton
polytope $\Delta$, written

$f=\sum\limits_{m\in M\cap\Delta}a_{m}x^{m}.$

Set $\quad Z_{f}:=\\{f=0\\}\subset T$ and given another $n$-dimensional
rational polytope $F$ let $Z_{F,f}$ or just $Z_{F}$ be the closure of $Z_{f}$
in $\mathbb{P}_{F}$. We repeat some results and notions introduced in
([Bat22]):

######  Remark 2.1.

The linear equivalence class of $Z_{F,f}$ just depends on $\Delta$. More
precisely let

$\operatorname{Min}_{\Delta}(\nu):=\min\limits_{m\in\Delta}\langle
m,\nu\rangle$

for $\nu\in N\setminus\\{0\\}$. Then

$Z_{F,f}\sim_{lin}-\sum\limits_{\nu\in\Sigma_{F}[1]}\operatorname{Min}_{\Delta}(\nu)\cdot
D_{\nu}$

([Bat22, Prop.7.1]). Therefore we usually omit $f$ from the notation.

The nondegeneracy of $f$, written $f\in U_{reg}(\Delta)$, means that $Z_{f}$
is smooth and for every $k$-dimensional face $\Gamma$ of $\Delta$

$\displaystyle f_{|{\Gamma}},\,\frac{\partial f_{|{\Gamma}}}{\partial
x_{1}},...,\frac{\partial f_{|{\Gamma}}}{\partial x_{n}},$ (4)

where $f_{|{\Gamma}}:=\sum\limits_{m\in M\cap\Gamma}a_{m}x^{m}$, have no
common zero in the torus orbit
$(\mathbb{C}^{*})^{k}\subset\mathbb{P}_{\Delta}$ corresponding to $\Gamma$.
Write

$\displaystyle\Delta=\\{x\in M_{\mathbb{R}}\mid\langle
x,\nu\rangle\geq\operatorname{Min}_{\Delta}(\nu)\quad\forall\,\nu\in
N\setminus\\{0\\}\\}.$ (5)

we consider the Fine interior

$F(\Delta):=\\{x\in M_{\mathbb{R}}\mid\langle
x,\nu\rangle\geq\operatorname{Min}_{\Delta}(\nu)+1\quad\forall\,\nu\in
N\setminus\\{0\\}\\}.$

In general $F(\Delta)$ is a rational polytope contained in $\Delta$ and
contains all interior lattice points of $\Delta$. It is constructed by moving
every hyperplane touching $\Delta$ one step into the interior of $\Delta$. Let

$S_{F}(\Delta):=\\{\nu\in
N\setminus\\{0\\}\mid\operatorname{Min}_{F(\Delta)}(\nu)=\operatorname{Min}_{\Delta}(\nu)+1\\}$

denote the support of $F(\Delta)$ to $\Delta$ (those hyperplanes that touch
$\Delta$ and moved by one touch $F(\Delta)$).

Figure 1: Figure: On the left a polygon $\Delta$, on the right the rays of
$\Sigma_{\Delta}$. $F(\Delta)$ equals the interior lattice point of $\Delta$.
The illustrations shows that $(0,-1)\in S_{F}(\Delta)$. In general the support
vectors $S_{F}(\Delta)$ are contained in the convex span of the rays
$\Sigma_{\Delta}[1]$ (see [Bat22, Prop.3.11])

######  Definition 2.2.

([Bat22, Def.3.13])
Let $\Delta$ be a lattice polytope with presentation as in (5) and with
$F(\Delta)\neq\emptyset$. The polytope

$C(\Delta):=\\{x\in M_{\mathbb{R}}|\,\langle
x,\nu\rangle\geq\operatorname{Min}_{\Delta}(\nu)\quad\forall\,\nu\in
S_{F}(\Delta)\\}$

is called the canonical closure of $\Delta$.

######  Remark 2.3.

It is clear from the definition that $C(\Delta)$ is a rational polytope and
contains $\Delta$. The operators of taking the Fine interior $F$, the
canonical closure $C$ and the support $S_{F}$ could be defined analogously for
rational polytopes.

###### Proof.

The first formula follows from the exact sequence (9), the vanishing
$H^{1}(\mathbb{P},\mathcal{O}_{\mathbb{P}})=0$ due to Demazure and formula
(6). The second follows from Remark 3.10. ∎

######  Theorem 2.4.

([Bat22, Thm.8.2])
Assume $F(\Delta)\neq\emptyset$. Then there is a complete simplicial fan
$\Sigma$ with $\Sigma[1]=S_{F}(\Delta)$ and with associated toric variety
$\mathbb{P}$ such that with $Y=Y_{f}$ the closure of $Z_{f}$ in $\mathbb{P}$

  * •

$\mathbb{P}$ has $\mathbb{Q}$-factorial terminal singularities

  * •

the adjoint divisor $K_{\mathbb{P}}+Y$ is nef.

######  Corollary 2.5.

([Bat22, Thm.8.2])
$Y\subset\mathbb{P}$ has at most terminal singularities and $K_{Y}$ is nef. We
say that $Y=Y_{f}$ is a minimal model of $Z_{f}$.

######  Proposition 2.6.

([Bat22, Prop.7.4])
$Y\subset\mathbb{P}$ defines a nef and big $\mathbb{Q}$-Cartier divisor.

######  Lemma 2.7.

Let $F$ be a torus orbit of $\mathbb{P}$. If

$\emptyset\neq F\cap Y\subsetneq F$

then $Y$ intersects $F$ transversely in a subset of codimension $1$.

###### Proof.

Choose a common refinement $\tilde{\Sigma}$ of $\Sigma$ and $\Sigma_{\Delta}$
in $N$ such that the toric variety $\tilde{\mathbb{P}}$ to $\tilde{\Sigma}$ is
smooth. Let $\tilde{Y}$ denote the closure of $Z_{f}$ in $\tilde{\mathbb{P}}$.
Then for $\tilde{F}\subset\tilde{\mathbb{P}}$ a torus orbit either $\tilde{Y}$
is disjoint from $\tilde{F}$ (if $\tilde{F}$ contracts to a torus fixed point
on $\mathbb{P}_{\Delta}$) or intersects $\tilde{F}$ transversely in a subset
of codimension $1$ ([Bat94, Prop.3.2.1]).
There is a birational toric morphism $\tilde{\mathbb{P}}\rightarrow\mathbb{P}$
inducing a birational morphism $\tilde{Y}\rightarrow Y$. Assume that
$\tilde{F}$ contracts onto $F$. If $\tilde{Y}\cap\tilde{F}\neq\emptyset$ then
$Y\cap F\neq\emptyset$ and either $Y$ contains $F$ or intersects $F$
transversely in a subset of codimension $1$. ∎

######  Remark 2.8.

The singular loci both of $\mathbb{P}$ and $Y$ are of codimension $\geq 3$
since they are terminal. It will be important to choose smooth open subset
$V\subset\mathbb{P}$ and $U\subset Y$ such that $U=V\cap Y$ and $V$ is toric.
This works if we define $V$ as the union of all torus orbits of dimension
$\geq n-2$.

######  Remark 2.9.

To a toric divisor

$D=\sum\limits_{i=1}^{r}a_{i}D_{i}\subset\mathbb{P}$

we associate a polytope

$P_{D}:=\\{x\in M_{\mathbb{R}}\mid\langle x,\nu_{i}\rangle\geq-
a_{i}\quad\forall n_{i}\in\Sigma[1]\\}$

counting the global sections of $D$, that is
$H^{0}(\mathbb{P},\mathcal{O}_{\mathbb{P}}(D))\cong L(P_{D})$ (compare [CLS11,
Prop.4.3.2]). By the formula in Remark 2.1 and by ([Bat22, Prop.4.3]) to $Y$
is associated the polytope $C(\Delta)$ that is

$\displaystyle H^{0}(\mathbb{P},\mathcal{O}_{\mathbb{P}}(Y))\cong
L(C(\Delta)).$ (6)

It follows in the same way that to $Y+K_{\mathbb{P}}$ is associated
$F(\Delta)$.

######  Corollary 2.10.

$Y\subset\mathbb{P}$ is Cartier if and only if $C(\Delta)$ is a lattice
polytope.

###### Proof.

By Remark 2.1

$\displaystyle
Y\sim_{lin}-\sum\limits_{\nu\in\Sigma[1]}\operatorname{Min}_{\Delta}(\nu)D_{\nu}=-\sum\limits_{\nu\in\Sigma[1]}\operatorname{Min}_{C(\Delta)}(\nu)D_{\nu}.$

The function
$\operatorname{Min}_{C(\Delta)}:N_{\mathbb{R}}\rightarrow\mathbb{R}$ has the
advantage that it is linear on the cones of $\Sigma$ since $\Sigma$ refines
the normal fan of $C(\Delta)$. Thus $Y\subset\mathbb{P}$ is Cartier if and
only if $\operatorname{Min}_{C(\Delta)}$ is a support function for $Y$, that
is

$\operatorname{Min}_{C(\Delta)}(N)\subset\mathbb{Z}.$

Obviously this is the case if and only if $C(\Delta)$ is a lattice polytope. ∎

######  Remark 2.11.

In particular if $C(\Delta)=\Delta$ then $Y$ will be Cartier. In ([Bat22,
Ex.3.1.5]) there is mentioned an example of a $5$-dimensional lattice polytope
$\Delta$ for which $C(\Delta)$ is not a lattice polytope.

##  3 Reflexive sheaves and MCM-sheaves

Throughout this section $X$ denotes an $n$-dimensional normal irreducible
variety with $j:U\rightarrow X$ the inclusion of the smooth locus of $X$.

######  Definition 3.1.

A coherent sheaf $\mathcal{F}$ on $X$ is called reflexive if the natural map
$\mathcal{F}\rightarrow\mathcal{F}^{**}$ is an isomorphism, where
$\mathcal{F}^{**}$ denotes the double dual of the sheaf $\mathcal{F}$.
$\mathcal{F}^{**}$ is called the reflexive hull of $\mathcal{F}$.

######  Remark 3.2.

([CLS11, Prop.8.0.1, Thm.8.0.4])
Given an open subset $j:U\subset X$ with $\operatorname{codim}_{X}(X\setminus
U)\geq 2$ a reflexive sheaf is uniquely determined by its restriction to $U$,
that is

$\displaystyle\mathcal{F}\cong j_{*}(\mathcal{F}_{|{U}}).$ (7)

Furthermore if $\mathcal{F}$ is a coherent sheaf with $\mathcal{F}_{|{U}}$
locally free and $\operatorname{codim}_{X}(X\setminus U)\geq 2$ then
$j_{*}(\mathcal{F}_{|{U}})$ is reflexive ([Sch08, Prop.2.12]). The dual of a
coherent sheaf on a normal variety is always reflexive. In particular the
reflexive hull of a coherent sheaf is reflexive.

The tensor product of two reflexive is defined by

$\mathcal{F}\otimes_{r}\mathcal{G}:=(\mathcal{F}\otimes\mathcal{G})^{**}.$

Given two Weil divisors $D,D^{\prime}$ on $X$ we have
$\mathcal{O}(D+D^{\prime})\cong\mathcal{O}(D)\otimes_{r}\mathcal{O}(D^{\prime})$.

######  Definition 3.3.

Let

$\displaystyle\Omega_{X}^{p}:=j_{*}\Omega_{U}^{p}\quad 1\leq p\leq n,$
$\displaystyle T_{X}:=(\Omega_{X}^{1})^{*},$

be the sheaf of differential $p$-forms and the tangent sheaf. The sheaf

$N_{Y/\mathbb{P}}:=\iota_{*}\mathcal{O}_{U}(Y_{|{U}})$

is called the normal sheaf of $Y$ in $\mathbb{P}$.

######  Remark 3.4.

There is a different method for the construction of $T_{X}$: Let
$\Omega_{X,\textrm{K\"{a}hl}}^{p}$ denote the sheaf of Kähler
$p$-differentials on $X$ and

$T_{X,\textrm{K\"{a}hl}}:=(\Omega_{X,\textrm{K\"{a}hl}}^{1})^{*}.$

Then $\Omega_{X,\textrm{K\"{a}hl}}^{1}$ is coherent and thus by Remark 3.2 its
dual $T_{X,\textrm{K\"{a}hl}}$ is reflexive and coincides with $T_{X}$ on the
smooth locus $U$ of $X$. Since both sheaves are reflexive

$T_{X,\textrm{K\"{a}hl}}\cong T_{X}.$

By ([MuOd15, Ch.VI.1])

$\displaystyle
H^{0}(X,T_{X,\textrm{K\"{a}hl}})\cong\operatorname{Lie}(\operatorname{Aut}(X)),$
(8)

where $\operatorname{Lie}(\operatorname{Aut}(X))$ denotes the Lie algebra of
the automorphism group of $X$.

In order to apply Serre duality we need sheaves with the following property

######  Definition 3.5.

([CLS11, Def. before Thm.9.2.12], [Kol13, Def.5.2])
Let $\mathcal{F}$ be a coherent sheaf on $X$. Then $\mathcal{F}$ is called
maximal Cohen-Macaulay (for short:MCM) if for all $x\in X$ the stalk
$\mathcal{F}_{x}$ is a Cohen-Macaulay module over $\mathcal{O}_{X,x}$ of
dimension $n$.

######  Proposition 3.6.

[CLS11, Proof of Thm.9.2.10]
Let $V$ be an $n$-dimensional complete toric variety to a simplicial fan and
$D$ a $\mathbb{Q}$-Cartier divisor on $V$. Then the sheaves $\Omega_{V}^{p}$
for $p=0,...,n$ and $\mathcal{O}_{V}(D)$ are MCM.

Let $\mathcal{F}$ be a MCM-sheaf on $X$. Then Serre duality ([KM98, Thm.5.71,
Prop.5.75]) gives

$\displaystyle H^{k}(X,\mathcal{F})\cong
H^{n-k}(X,\operatorname{Hom}(\mathcal{F},\mathcal{O}_{X}(K_{X})))^{*}.$

For $Z\subset X$ a closed subset, $\mathcal{F}$ a coherent sheaf on $X$, we
work with the local cohomology groups $H_{Z}^{k}(X,\mathcal{F})$ on $X$. This
allows us to extend exact sequences which exist on the smooth locus $U$ of $X$
to all of $X$. First we recall a vanishing Theorem:

######  Theorem 3.7.

([HaKo04, Prop.3.3])
Let $X$ be an $n$-dimensional algebraic variety and $Z\subset X$ a closed
subset with $\operatorname{codim}_{X}(Z)\geq r$. Let $\mathcal{F}$ be a
coherent sheaf on $X$ which is MCM. Then

$\displaystyle H_{Z}^{k}(X,\mathcal{F})=0\quad k=0,...,r-1.$

######  Proposition 3.8.

The sequence of sheaves

$\displaystyle
0\rightarrow\mathcal{O}_{\mathbb{P}}\rightarrow\mathcal{O}_{\mathbb{P}}(Y)\rightarrow
N_{Y/\mathbb{P}}\rightarrow 0$ (9)

is exact.

###### Proof.

If $Y\subset\mathbb{P}$ is Cartier (see Corollary 2.10 for a criterion) then
there is nothing to show, but in general $Y$ just defines a
$\mathbb{Q}$-Cartier divisor.
We may assume that $n=\dim\,\mathbb{P}\geq 3$. Let
$\iota:V\rightarrow\mathbb{P}$ be the inclusion of the union of all torus
orbits of dimension $\geq n-2$ and $U:=V\cap Y$. If we take the pushforward
under $\iota_{*}$ of an ideal sheaf sequence for $U\subset V$ we get

$0\rightarrow\mathcal{O}_{\mathbb{P}}\rightarrow\mathcal{O}_{\mathbb{P}}(Y)\rightarrow
N_{Y/\mathbb{P}}\rightarrow R^{1}\iota_{*}(\mathcal{O}_{V}).$

$R^{1}\iota_{*}(\mathcal{O}_{V})$ is the sheaf associated to the presheaf

$W\mapsto H^{1}(W\cap V,\mathcal{O}_{W\cap V}).$

Assume that $W\subset\mathbb{P}$ is affine, $V\subset W$ an open subset such
that $\operatorname{codim}_{W}(Z)\geq 3$ for $Z:=W\setminus V$. We have to
show that $H^{1}(V,\mathcal{O}_{V})=0$. There is a local cohomology exact
sequence ([Gro67, Cor.1.9])

$...\rightarrow H^{1}(W,\mathcal{O}_{W})\rightarrow
H^{1}(V,\mathcal{O}_{V})\rightarrow H_{Z}^{2}(W,\mathcal{O}_{W})$

Since $W$ is affine $H^{1}(W,\mathcal{O}_{W})=0$ by Serre’s criterion and
$H_{Z}^{2}(W,\mathcal{O}_{W})=0$ by Theorem 3.7 since
$\mathcal{O}_{\mathbb{P}}$ is MCM and $\operatorname{codim}_{W}(Z)\geq 3$. ∎

We will use this exact sequence and the vanishing
$H^{1}(\mathbb{P},\mathcal{O}_{\mathbb{P}})=0$ in Remark 2.9 to compute
$H^{0}(Y,N_{Y/\mathbb{P}})$ by identifying $Y\subset\mathbb{P}$ with a torus
invariant divisor.

######  Corollary 3.9.

The normal sheaf $N_{Y/\mathbb{P}}$ is MCM.

###### Proof.

Consider the exact sequence (9). For $y\in Y$ the $\mathcal{O}_{Y,y}$-module
$(N_{Y/\mathbb{P}})_{y}$ has dimension $n-1$ and by ([Kol13, Cor.2.62(3)]) it
is Cohen-Macaulay since both $\mathcal{O}_{\mathbb{P}}$ and
$\mathcal{O}_{\mathbb{P}}(Y)$ are MCM sheaves. ∎

######  Remark 3.10.

The normal sheaf
$N_{Y_{f}/\mathcal{X}}:=(I_{Y_{f}}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{O}_{Y_{f}})^{*}$
is trivial of rank $l(\Delta)-1$, where $\mathcal{I}_{Y_{f}}$ denotes the
ideal sheaf of $Y_{f}\subset\mathcal{X}$.

######  Corollary 3.11.

$\displaystyle H^{0}(Y,N_{Y/\mathbb{P}})\cong L(C(\Delta))/\mathbb{C}\cdot f,$
$\displaystyle H^{0}(Y,N_{Y/\mathcal{X}})\cong L(\Delta)/\mathbb{C}\cdot f.$

##  4 Kodaira-Spencer maps

Let $B$ be the image of $U_{reg}(\Delta)$ in $\mathbb{P}L(\Delta)$. We
consider the following natural family

$\mathcal{X}:=\\{(x,f)\in\mathbb{P}\times B|\,x\in
Y_{f}\\}\overset{pr_{2}}{\rightarrow}B.$

######  Definition 4.1.

Given a full-dimensional rational polytope $P\subset M_{\mathbb{R}}$ let
$L(P)$ be the $\mathbb{C}$-vector space with basis $x^{m}$, $m\in\Delta\cap M$
and $l(P)=\\#(P\cap M)$ its dimension.

Let $D:=\operatorname{Spec}\,\mathbb{C}[\epsilon]/(\epsilon^{2})$ denote the
dual numbers. Given a tangent vector $\phi:D\rightarrow B$ of $B$ at $f$ the
pullback

$\mathcal{X}\times_{B}D\rightarrow D$

defines an infinitesimal deformation (of first order) of $Y_{f}$ in
$\mathcal{X}$. Passing from $\mathcal{X}\rightarrow B$ to
$\mathcal{X}\times_{\phi}D\rightarrow D$ some information gets lost. Still we
can extract interesting information out of the situation (compare Kodaira
[Kod86, Thm.4.3]).
There are isomorphisms

$\displaystyle H^{0}(Y,N_{Y/\mathcal{X}})\quad\,\,\,\cong\\{\textrm{Inf. def.
of }Y_{f}\textrm{ in }\mathcal{X}\\}/\textrm{iso.}$ $\displaystyle
H^{0}(Y,N_{Y/\mathbb{P}})\quad\,\,\,\,\cong\\{\textrm{Inf. def. of
}Y_{f}\textrm{ in }\mathbb{P}\\}/\textrm{iso.}\,$
$\displaystyle\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})\cong\\{\textrm{Inf.
def. of }Y_{f}\\}/\textrm{iso.}$

Let us derive two Kodaira-Spencer maps in this context: There is an
isomorphism

$H^{1}(U,T_{U})\cong\operatorname{Ext}_{\mathcal{O}_{U}}^{1}(\Omega_{U}^{1},\mathcal{O}_{U}).$

since $U$ is smooth. Besides by ([KM92, Lemma (12.5.6)])

$\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})\cong\operatorname{Ext}_{\mathcal{O}_{U}}^{1}(\Omega_{U}^{1},\mathcal{O}_{U})\cong
H^{1}(U,T_{U}).$

since $\operatorname{codim}_{Y}(Y\setminus U)\geq 3$. For $V\subset\mathbb{P}$
an open subset as in Remark (2.8) with $V\cap Y=U$ the Kodaira-Spencer map

$H^{0}(U,N_{U/V})\rightarrow H^{1}(U,T_{U})$

is the coboundary map in the tangent sheaf sequence for $U\subset V$. We
derive one Kodaira-Spencer map $\kappa_{\mathbb{P},f}$ from the commutative
diagram

${0}$${H^{0}(U,T_{U})}$${H^{0}(U,T_{V|{U}})}$${H^{0}(U,N_{U/V})}$${H^{1}(U,T_{U})}$${0}$${H^{0}(Y,T_{Y})}$${H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})}$${H^{0}(Y,N_{Y/\mathbb{P}})}$${\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\kappa_{\mathbb{P}}=\kappa_{\mathbb{P},f}}$

The other Kodaira-Spencer map
$\kappa_{f}:H^{0}(Y,N_{Y/\mathcal{X}})\rightarrow\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})$
is gotten similarly by working with $\mathcal{X}$ instead of $\mathbb{P}$.
These two maps associate two a deformation in $\mathbb{P}$ ($\mathcal{X}$) its
equivalence class.

##  5 Mavlyutov’s vanishing Theorem

######  Theorem 5.1.

([Mav08, Thm.2.4], [CLS11, Thm.9.3.3])
Let $V$ be an $n$-dimensional complete toric variety to a simplicial fan. If D
is a nef Cartier divisor on $V$, then

$H^{p}(V,\Omega_{V}^{q}\otimes\mathcal{O}(D))=0$

for $p>q$.

######  Construction 5.2.

(Multiplication morphism)
([Fuj06, 2.5, Prop.3.2], [CLS11, Lemma 9.2.6, Proof of Thm. 9.3.1])
Let $V$ be a normal (not necessarily complete) toric variety. There is a
useful construction to reduce computations of cohomology groups of
$\mathbb{Q}$-Cartier Weil divisors $D$ to cohomology of multiples $mD$ of $D$,
which are Cartier.

Namely for $l\in\mathbb{N}_{\geq 1}$ the map $\overline{\phi}_{l}:N\rightarrow
N$ given by

$n\mapsto l\cdot n$

yields a toric morphism $\phi_{l}:V\rightarrow V$. $\phi_{l}$ induces an
injection

$\displaystyle H^{p}(V,\Omega_{V}^{q}\otimes_{r}\mathcal{O}(D))$
$\displaystyle\rightarrow H^{p}(V,\Omega_{V}^{q}\otimes_{r}\mathcal{O}(lD))$
$\displaystyle\cong H^{p}(V,\Omega_{V}^{q}\otimes\mathcal{O}(lD)),\quad p\geq
0,$

where the last isomorphism follows since $\mathcal{O}(lD)$ is Cartier.
For us this result isr powerful in connection with Theorem 5.1 above.

##  6 Computation of $\ker(\kappa_{\mathbb{P},f})$

######  Theorem 6.1.

Let $\Delta$ be an $n$-dimensional lattice polytope, where $n\geq 2$, with
$l^{*}(\Delta)>0$. If $n=2$ assume that $l^{*}(\Delta)\geq 2$. Then given
$f\in B$

$\displaystyle\ker(\kappa_{\mathbb{P},f})\cong\operatorname{Lie}\,\operatorname{Aut}(\mathbb{P}).$
(10)

###### Proof.

We write $Y$ and $\kappa_{\mathbb{P}}$ for $Y_{f}$ and
$\kappa_{\mathbb{P},f}$. By the exact sequence from section 4

$\ker(\kappa_{\mathbb{P}})\cong
H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})/H^{0}(Y,T_{Y}).$

By Lemma (6.3) below if $n\geq 3$ then

$H^{0}(Y,T_{Y})=0.$

If $n=2$ this follows from $g(Y)\geq 2$ since the Fine interior is not a
point. By Remark 3.4 we are left to show that

$H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})\cong H^{0}(\mathbb{P},T_{\mathbb{P}}).$

######  Lemma 6.2.

There is an exact sequence

$\displaystyle 0\rightarrow
T_{\mathbb{P}}\otimes_{r}\mathcal{O}(-Y)\rightarrow T_{\mathbb{P}}\rightarrow
T_{\mathbb{P}|{Y}}^{**}\rightarrow 0.$

###### Proof.

Let $\iota:V\subset\mathbb{P}$ be the inclusion of the union of all torus
orbits of dimension $\geq n-2$ and $U=V\cap Y$. Consider the exact sequence

$0\rightarrow T_{V}\otimes\mathcal{O}(-Y_{|{V}})\rightarrow T_{V}\rightarrow
T_{V|{U}}\rightarrow 0$

and take the pushforward under $\iota$. Let

$\mathcal{F}:=\Omega_{\mathbb{P}}^{n-1}\otimes_{r}\mathcal{O}(-Y-K_{\mathbb{P}})\cong
T_{\mathbb{P}}\otimes_{r}\mathcal{O}(-Y).$

Then $R^{1}\iota_{*}(\mathcal{F}_{|{V}})=0$ since given $x\in V$ choose an
affine toric neighborhood $W$ of $x$ and (using Construction 5.2) replace
$\mathcal{O}_{W}(-Y-K_{\mathbb{P}})$ by an $m$-times multiple which is Cartier
in the definition of $\mathcal{F}$. Then by Proposition 3.6
$\mathcal{F}_{|{W}}$ is MCM and the vanishing

$H^{1}(V\cap W,\mathcal{F}_{|{V\cap W}})=0$

follows as in the proof of Proposition 3.8 from a local cohomology sequence. ∎

Take the long exact cohomology sequence

$\displaystyle 0$ $\displaystyle\rightarrow
H^{0}(\mathbb{P},T_{\mathbb{P}}\otimes_{r}\mathcal{O}(-Y))\rightarrow
H^{0}(\mathbb{P},T_{\mathbb{P}})\rightarrow H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})$
$\displaystyle\rightarrow
H^{1}(\mathbb{P},T_{\mathbb{P}}\otimes_{r}\mathcal{O}(-Y)).$

Write

$T_{\mathbb{P}}\otimes_{r}\mathcal{O}(-Y)\cong\Omega_{\mathbb{P}}^{n-1}\otimes_{r}\mathcal{O}(-Y-K_{\mathbb{P}}).$

By Construction 5.2 we have

$H^{k}(\mathbb{P},\Omega_{\mathbb{P}}^{n-1}\otimes_{r}\mathcal{O}(-Y-K_{\mathbb{P}}))\subset
H^{k}(\mathbb{P},\Omega_{\mathbb{P}}^{n-1}\otimes\mathcal{O}(-mY-
mK_{\mathbb{P}}))$

where $m\geq 1$ is such that $mY,mK_{\mathbb{P}}$ are Cartier. Now use Serre
duality and Theorem 5.1 to deduce

$\displaystyle
H^{k}(\mathbb{P},\Omega_{\mathbb{P}}^{n-1}\otimes\mathcal{O}(-mY-
mK_{\mathbb{P}}))\cong$ $\displaystyle
H^{n-k}(\mathbb{P},\Omega_{\mathbb{P}}^{1}\otimes\mathcal{O}(mY+mK_{\mathbb{P}}))=0$

for $k=0,1$ if $n\geq 3$. If $n=2$ the vanishing for $k=1$ follows from the
precise formula in ([Mav08, Cor.2.7]) (to $Y+K_{\mathbb{P}}$ is associated the
polytope $F(\Delta)$, see Remark 2.9).

∎

######  Lemma 6.3.

Let $\Delta$ be an $n$-dimensional lattice polytope with $n\geq 3$ and
$l^{*}(\Delta)>0$. Then

$H^{0}(Y,T_{Y})=0.$

###### Proof.

Using an ideal sheaf sequence we get

$\displaystyle 0\rightarrow T_{Y}\rightarrow\Omega_{Y}^{n-1}\rightarrow...$

where we have used $\Omega_{Y}^{n-1}\cong T_{Y}\otimes_{r}\mathcal{O}(K_{Y})$
and $h^{0}(Y,\mathcal{O}(K_{Y}))=l^{*}(\Delta)>0$. Thus

$\displaystyle h^{0}(Y,T_{Y})$ $\displaystyle\leq h^{0}(Y,\Omega_{Y}^{n-1})$
$\displaystyle=0$

For the last vanishing: Take a common toric resolution of singularities
$\mathbb{P}\overset{p}{\leftarrow}\tilde{\mathbb{P}}\rightarrow\mathbb{P}_{\Delta}$
with closure $\tilde{Y}$ of $Z_{f}$ in $\tilde{\mathbb{P}}$. Then
$h^{0}(\tilde{Y},\Omega_{\tilde{Y}}^{n-1})=0$ by ([DK86]) and
$p_{*}\Omega_{\tilde{Y}}^{n-1}=\Omega_{Y}^{n-1}$. ∎

We guess that the weaker assumption $F(\Delta)\neq\emptyset$ is sufficient for
the above Lemma.

##  7 The cokernel of $\kappa_{\mathbb{P},f}$

######  Corollary 7.1.

Given the conditions of the theorem if $n\geq 4$ then the following sequence
is exact

$\displaystyle 0\rightarrow
Im(\kappa_{\mathbb{P},f})\rightarrow\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})\rightarrow\operatorname{Ext}_{\mathcal{O}_{\mathbb{P}}}^{1}(\Omega_{\mathbb{P}}^{1},\mathcal{O}_{\mathbb{P}})\rightarrow
0$

that is all infinitesimal deformations of $Y_{f}$ arise from
$Im(\kappa_{\mathbb{P},f})$ or from infinitesimal deformations of
$\mathbb{P}$.

###### Proof.

Let $V$ denote the smooth locus of $\mathbb{P}$, $Z=\mathbb{P}\setminus V$ and
$U=V\cap Y$. Then $Z$ has codimension $\geq 3$ in $\mathbb{P}$ and $Y\setminus
U$ has codimension $\geq 2$ in $Y$. We could extend the upper exact sequence
in Section 4 to

$\displaystyle 0$ $\displaystyle\rightarrow H^{0}(U,T_{U})\rightarrow
H^{0}(U,T_{V|{U}})\rightarrow
H^{0}(U,N_{U/V})\overset{\kappa_{\mathbb{P}}}{\rightarrow}H^{1}(U,T_{U})$
$\displaystyle\rightarrow H^{1}(U,T_{V|{U}})\rightarrow H^{1}(U,N_{U/V})$

By the exact sequence (9) $H^{1}(Y,N_{Y/\mathbb{P}})=0$. Relate
$H^{1}(U,N_{U/V})$ and $H^{1}(Y,N_{Y/\mathbb{P}})$ via a local cohomology
sequence and use (Lemma 3.9, Theorem 3.7)

$H_{Z\cap Y}^{k}(Y,N_{Y/\mathbb{P}})=0\quad k<n-1.$

It follows $H^{1}(U,N_{U/V})=0$. For $T_{V|{U}}$ use an ideal sheaf sequence

$\displaystyle H^{1}(V,T_{V}\otimes\mathcal{O}(-Y_{|{V}}))\rightarrow
H^{1}(V,T_{V})\rightarrow H^{1}(U,T_{V|{U}})\rightarrow
H^{2}(V,T_{V}\otimes\mathcal{O}(-Y_{|{V}})).$

$V$ is a toric variety and by Construction 5.2

$\displaystyle H^{k}(V,T_{V}\otimes\mathcal{O}(-Y_{|{V}}))$
$\displaystyle=H^{k}(V,\Omega_{V}^{n-1}\otimes\mathcal{O}(-Y_{|{V}}-K_{\mathbb{P}|{V}}))$
$\displaystyle\subset
H^{k}(V,\Omega_{V}^{n-1}\otimes\mathcal{O}(-mY_{|{V}}-mK_{\mathbb{P}|{V}}))$

where $mY,mK_{\mathbb{P}}$ are Cartier. We show that these terms vanish for
$k=1,2$ and the result follows since

$H^{1}(V,T_{V})\cong\operatorname{Ext}_{\mathcal{O}_{\mathbb{P}}}^{1}(\Omega_{\mathbb{P}}^{1},\mathcal{O}_{\mathbb{P}})$

as in section 4. Use a local cohomology sequence

$H^{k}(\mathbb{P},\mathcal{F})\rightarrow
H^{k}(V,\mathcal{F}_{|{V}})\rightarrow H_{Z}^{k+1}(\mathbb{P},\mathcal{F})$

where

$\mathcal{F}:=\Omega_{\mathbb{P}}^{n-1}\otimes\mathcal{O}(-mY-
mK_{\mathbb{P}}).$

$\mathcal{F}$ is CM by Lemma 3.9. As in the Lemma above

$H^{k}(\mathbb{P},\mathcal{F})\cong
H^{n-k}(\mathbb{P},\Omega_{\mathbb{P}}^{1}\otimes\mathcal{O}(mY+mK_{\mathbb{P}}))=0\quad
k=1,2.$

Further by Theorem 3.7

$H_{Z}^{k}(\mathbb{P},\mathcal{F})=0\quad k<n-1.$

∎

######  Example 7.2.

If $\mathbb{P}=\mathbb{P}^{3}$, $\Delta=4\cdot\Delta_{3}$ then
$\dim\,Im(\kappa)=19$ and $H^{1}(\mathbb{P},T_{\mathbb{P}})=0$. But
$h^{1}(Y,T_{Y})=20$ since $Y$ is a K3 surface. The above Corollary does not
apply since $n=3$.

##  8 An explicit basis for $\ker(\kappa_{\mathbb{P},f})$

Let

$\displaystyle R(N,\Sigma):=\\{$ $\displaystyle\alpha\in
M\,|\,\langle\alpha,n(\alpha)\rangle=1\textrm{ for some
}n(\alpha)\in\Sigma[1]$ $\displaystyle\textrm{ and
}\langle\alpha,n_{j}\rangle\leq 0\textrm{ for
}n_{j}\in\Sigma[1]\setminus\\{n(\alpha)\\}\\}$

denote the roots of $\Sigma$. Likewise we define $R(N,\Sigma_{C(\Delta)})$ and
$R(N,\Sigma_{\Delta})$ by replacing $\Sigma$ by $\Sigma_{C(\Delta)}$ and
$\Sigma_{\Delta}$.

There are inclusions

$\displaystyle\Sigma_{C(\Delta)}[1]\underbrace{\subset}_{(\textrm{since
}\Sigma\textrm{ refines
}\Sigma_{C(\Delta)})}\Sigma[1]\underbrace{\subset}_{(\textrm{Figure
}\ref{figure_support_of_Fine_interior})}\operatorname{Convhull}(\Sigma_{\Delta}[1]).$
(11)

######  Lemma 8.1.

Let $\Delta$ be an $n$-dimensional lattice polytope with
$F(\Delta)\neq\emptyset$. Then

$R(N,\Sigma_{\Delta})\subset R(N,\Sigma)=R(N,\Sigma_{C(\Delta)}).$

###### Proof.

To the second equality: Let $\alpha\in R(N,\Sigma_{C(\Delta)})$, that is

$\langle\alpha,n(\alpha)\rangle=1,\quad\langle\alpha,n_{j}\rangle\leq
0\quad\textrm{ for }n_{j}\in\Sigma_{C(\Delta)}[1]\setminus\\{n(\alpha)\\}.$

$\Rightarrow\langle\alpha,n_{j}\rangle\leq 0$ for
$n_{j}\in\Sigma[1]\setminus\\{n(\alpha)\\}$, that is $\alpha\in R(N,\Sigma)$.
Conversely assume $\alpha\in R(N,\Sigma)$. If
$n_{i}\notin\Sigma_{C(\Delta)}[1]$ then $\alpha$ would have scalar product
$\leq 0$ with all vectors in $\Sigma_{C(\Delta)}[1]$ and thus would be zero
since $\Sigma$ refines $\Sigma_{C(\Delta)}$, a contradiction. The first
inclusion follows similarly by using (11). ∎

We ask for a basis of Laurent polynomials for

$\displaystyle\operatorname{Lie}\,\operatorname{Aut}(\mathbb{P})\subset
L(C(\Delta))/\mathbb{C}\cdot f.$

Remember the results from ([BG99]): Given $f\in B$ there is a map

$\displaystyle\phi_{f}:T\rightarrow B$
$\displaystyle(t_{1},t_{2},t_{3})\mapsto\Big{(}(x_{1},x_{2},x_{3})\mapsto
f(t_{1}x_{1},t_{2}x_{2},t_{3}x_{3})\Big{)}.$

By differentiating $\phi_{f}$ we get an injective homomorphism
$(d\phi_{f})_{e}:\operatorname{Lie}(T)\rightarrow T_{B,f}$ where $e=(1,1,1)$
with

$Im(d(\phi_{f})_{e})=\Big{\langle}x_{1}\cdot\frac{\partial f}{\partial
x_{1}},...,x_{3}\cdot\frac{\partial f}{\partial x_{3}}\Big{\rangle}.$

For $m\in M\cap C(\Delta)$ and $\alpha\in R(N,\Sigma_{C(\Delta)})$ define

$\displaystyle ht_{-\alpha}(m):=\max\\{k\in\mathbb{N}_{\geq
0}|\,m-k\cdot\alpha\in C(\Delta)\\}.$ (12)

Given $\alpha\in R(N,\Sigma_{C(\Delta)})$ we denote by $\Gamma_{-\alpha}\leq
C(\Delta)$ the facet to which $n(\alpha)$ is normal.

######  Remark 8.2.

Assuming

$\Gamma_{-\alpha}=\\{x\in M_{\mathbb{R}}|\,\langle
x,n_{\Gamma}\rangle=b_{\Gamma}\\}\cap C(\Delta)$

and $m\in M\cap C(\Delta)$ then

$\displaystyle ht_{-\alpha}(m)=\langle m,n_{\Gamma}\rangle-b_{\Gamma}.$ (13)

$\Gamma_{v}$$v$$m$ Figure 2: A lattice polytope with a column vector $v$ and
$ht_{v}(m)=2$.

Let $S_{C(\Delta)}$ denote the graded semigroup $\mathbb{C}$-algebra over

$\operatorname{Cone}(C(\Delta)\times\\{1\\})\cap(M\times\mathbb{Z})$

Then the function $ht_{-\alpha}$ continues to a map $S_{C(\Delta)}\rightarrow
S_{C(\Delta)}$ which respects the grading on $S_{C(\Delta)}$. For
$\lambda\in\mathbb{C}$ define a graded automorphism
$e_{-\alpha}^{\lambda}:S_{C(\Delta)}\rightarrow S_{C(\Delta)}$ by

$e_{-\alpha}^{\lambda}(x^{m}):=x^{m}\cdot(1+\lambda
x^{-\alpha})^{ht_{-\alpha}(m)}$

######  Corollary 8.3.

([BG99, Lemma 3.1, Thm.3.2b), Thm.5.4])
$\operatorname{Lie}\,\operatorname{Aut}(\mathbb{P})$ has a basis of
derivations, which act on $L(C(\Delta))$ as follows

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}:\quad x^{m}\mapsto
m_{i}\cdot x^{m},\quad i=1,...,n,$ $\displaystyle\frac{\partial
e_{-\alpha}^{\lambda}}{\partial\lambda}_{|{\lambda=0}}:\quad x^{m}\mapsto
ht_{-\alpha}(m)\cdot x^{m-\alpha},\quad\alpha\in R(N,\Sigma_{C(\Delta)}).$

∎

By definition of the tangent sheaf sequence the homomorphism

$j:H^{0}(\mathbb{P},T_{\mathbb{P}})\cong
H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})\rightarrow H^{0}(Y,N_{Y/\mathbb{P}})$

is given by applying the derivations from Corollary 8.3 to

$f=\sum\limits_{m\in\Delta\cap M}a_{m}x^{m}\in U_{reg}(\Delta)$

and restricting to $Y=Y_{f}$.

######  Corollary 8.4.

Given the conditions of Theorem 6.1 $\ker(\kappa_{\mathbb{P},f})$ has the
basis

$\displaystyle\quad\quad\quad\quad\quad\quad x_{1}\cdot\frac{\partial
f}{\partial x_{1}},...,x_{n}\cdot\frac{\partial f}{\partial x_{n}},$
$\displaystyle w_{-\alpha}(f):=\sum\limits_{m\in\Delta\cap
M}ht_{-\alpha}(m)\cdot a_{m}\cdot x^{m-\alpha},\quad\alpha\in
R(N,\Sigma_{C(\Delta)}).$

$1$$1$$1$$2$$2$$3$$\Gamma_{-\alpha}$$-\alpha$ Figure 3: On the left: The
column vector $-\alpha$ with facet $\Gamma_{-\alpha}$ and all lattice points
$m\in C(\Delta)$ with $ht_{-\alpha}(m)>0$. On the right: $w_{-\alpha}(f)$ has
support on the thick lattice points.

######  Example 8.5.

If

$\Delta=d\cdot\Delta_{n},\quad f\in U_{reg}(\Delta),$

then

$C(\Delta)=\Delta,\quad\Sigma=\Sigma_{\Delta},\quad\mathbb{P}=\mathbb{P}^{n}$

and $Y_{f}$ is a smooth degree $d$ hypersurface in $\mathbb{P}^{n}$. For such
an hypersurface it is shown in ([Voi03, Lemma 6.15]) that

$\displaystyle\ker(\kappa_{f})\cong J_{f,griff}^{d},$ (14)

if we work with the family $\mathcal{X}\rightarrow U_{reg}(\Delta)$ (if we
projectivize then we have to mod out $f$ from the kernel). Here
$J_{f,griff}^{d}$ denotes the $d$-th homogeneous component of Griffiths
Jacobian ideal

$J_{f,griff}:=(\frac{\partial f}{\partial x_{0}},...,\frac{\partial
f}{\partial x_{n}})\unlhd\mathbb{C}[x_{0},...,x_{n}].$

The roots of $\Sigma$ are given by

$\pm e_{i},\quad i=1,...,n,\quad\pm e_{i}\mp e_{j},\quad i,j=1,...,n,\quad
i\neq j$

and if $d\geq n+1$ then Theorem 8.4 restricts to the result (14) up to
homogenization.

##  9 An explicit basis for $\ker(\kappa_{f})$

######  Theorem 9.1.

Given the conditions of Theorem 6.1 $\ker(\kappa_{f})$ has the basis

$\displaystyle x_{i}\frac{\partial f}{\partial x_{i}},\quad i=1,...,n,\quad
w_{-\alpha}(f),\quad\alpha\in R(N,\Sigma_{\Delta}).$

We first prove the following Proposition which reduces the proof to a
combinatorial argument

######  Proposition 9.2.

Let $\Delta$ be an $n$-dimensional lattice polytope with
$F(\Delta)\neq\emptyset$. Then $\kappa_{f}$ equals the restriction of
$\kappa_{\mathbb{P},f}$ to $L(\Delta)/\mathbb{C}\cdot f$ and thus

$\displaystyle\ker(\kappa_{f})\cong\ker(\kappa_{\mathbb{P},f})\cap L(\Delta).$
(15)

###### Proof.

The following reduction step is similar to ([Koe91, Ch.2.1] and [Voi03, Lemma
6.15]): Let $V\subset\mathbb{P}$ be the union of all torus orbits of dimension
$\geq n-2$. Then $U=U_{f}:=V\cap Y_{f}$ is smooth and
$\operatorname{codim}_{Y_{f}}(Y_{f}\setminus U_{f})\geq 2$ for every $f\in B$.
Let

$W:=(V\times B)\cap\mathcal{X}.$

Consider the differential

$(pr_{1})_{*}:T_{W|{U}}\rightarrow T_{V|{U}}$

of the first projection. All the sheaves we consider are reflexive, therefore
there is no difference in working with $Y$, $\mathcal{X}$ and $\mathbb{P}$.
$pr_{1}$ restricts to an isomorphism

$Y_{f}\times\\{f\\}\rightarrow Y_{f},$

thus $(pr_{1})_{*}$ restricts to the identity on $T_{Y}$. The map

$(pr_{1})_{*}:N_{Y/\mathcal{X}}\cong
H^{0}(Y,N_{Y/\mathcal{X}})\otimes\mathcal{O}_{Y}\subset
H^{0}(Y,N_{Y/\mathbb{P}})\otimes\mathcal{O}_{Y}\rightarrow N_{Y/\mathbb{P}}$

is given by multiplication of sections. In effect we obtain a commutative
diagram

${0}$${H^{0}(Y,T_{Y})}$${H^{0}(Y,T_{\mathcal{X}|{Y}}^{**})}$${H^{0}(Y,N_{Y/\mathcal{X}})}$${\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})}$${0}$${H^{0}(Y,T_{Y})}$${H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})}$${H^{0}(Y,N_{Y/\mathbb{P}})}$${\operatorname{Ext}_{\mathcal{O}_{Y}}^{1}(\Omega_{Y}^{1},\mathcal{O}_{Y})}$$\scriptstyle{id}$$\scriptstyle{(pr_{1})_{*}}$$\scriptstyle{\kappa_{f}}$$\scriptstyle{id}$$\scriptstyle{\kappa_{\mathbb{P},f}}$

and

$\ker(\kappa_{f})\cong H^{0}(Y,T_{\mathcal{X}|{Y}}^{**})\cong
H^{0}(Y,T_{\mathbb{P}|{Y}}^{**})\cap
L(\Delta)\cong\ker(\kappa_{\mathbb{P},f})\cap L(\Delta).$

∎

The $x_{i}\frac{\partial f}{\partial x_{i}}$ obviously belong to $L(\Delta)$
but the $w_{-\alpha}(f)$ need not have support on $\Delta$ as the folowing
example shows:

######  Example 9.3.

Consider the polytope

$\displaystyle\Delta=\langle\begin{pmatrix}-1\\\ -1\\\
-1\end{pmatrix},\begin{pmatrix}5\\\ 1\\\ 3\end{pmatrix},\begin{pmatrix}-1\\\
10\\\ 0\end{pmatrix},\begin{pmatrix}-1\\\ -1\\\ 0\end{pmatrix}\rangle$

$\Delta$ has $3$ interior lattice points, $F(\Delta)$ is $1$-dimensional and
$C(\Delta)$ has the additional vertex $(1,-1,1)$. We obtain a family of
elliptic surfaces $\mathcal{X}\rightarrow B$. There are $7$ roots

$R(N,\Sigma)=\\{\begin{pmatrix}-3\\\ -1\\\
-2\end{pmatrix},\begin{pmatrix}-1\\\ -4\\\
-1\end{pmatrix},\begin{pmatrix}-1\\\ -3\\\
-1\end{pmatrix},\begin{pmatrix}-1\\\ -2\\\
-1\end{pmatrix},\begin{pmatrix}-1\\\ -1\\\
-1\end{pmatrix},\begin{pmatrix}-1\\\ 0\\\ -1\end{pmatrix},\begin{pmatrix}0\\\
-1\\\ 0\end{pmatrix}\\}$

and one root $(-1,0,-1)$ not belonging to $R(N,\Sigma_{\Delta})$. The column
vector $-\alpha=(1,0,1)$ belongs to the facet

$\Gamma_{-\alpha}=\langle\begin{pmatrix}5\\\ 1\\\
3\end{pmatrix},\begin{pmatrix}-1\\\ 10\\\ 0\end{pmatrix},\begin{pmatrix}-1\\\
-1\\\ 0\end{pmatrix},\begin{pmatrix}1\\\ -1\\\ 1\end{pmatrix}\rangle$

of $C(\Delta)$. The vertex $(-1,-1,-1)$ does not lie on $\Gamma_{-\alpha}$ and

$-\alpha+(-1,-1,-1)=(0,-1,0)\notin\Delta.$

Thus only $6$ of the roots in $R(N,\Sigma)$ reduce the number of moduli.

(Proof of Theorem 9.1)

The proof is rather technical. By Proposition 9.2

$\ker(\kappa_{f})=\ker(\kappa_{\mathbb{P},f})\cap L(\Delta)$

and $R(N,\Sigma_{\Delta})\subset R(N,\Sigma_{C(\Delta)})$ by Lemma 8.1. Let
$R:=R(N,\Sigma_{C(\Delta)})\setminus R(N,\Sigma_{\Delta})$. The Theorem is a
consequence of the three points below.

  * •

$\alpha\in R(N,\Sigma_{\Delta})\Rightarrow w_{-\alpha}(f)\in L(\Delta)$.

  * •

$\alpha\in R\Rightarrow w_{-\alpha}(f)\notin L(\Delta)$.

  * •

Varying $\alpha\in R$ the $w_{-\alpha}(f)$ are linearly independent in
$L(C(\Delta))/L(\Delta)$.

The necessity of the first two points is obvious and the last point assures
that no linear combination of the $w_{-\alpha}(f)$, where $\alpha\in R$, lies
in $\ker(\kappa_{f})$.

First point: To $\alpha\in R(N,\Sigma_{\Delta})$ is associated both
$\Gamma_{-\alpha}\leq\Delta$ and $\Gamma_{-\alpha}^{\prime}\leq C(\Delta)$. We
show

$\displaystyle\Gamma_{-\alpha}\subset\Gamma_{-\alpha}^{\prime},$ (16)

since then for $m\in M\cap\Delta$, $m\notin\Gamma_{-\alpha}$ we get
$m-\alpha\in\Delta$, that is $w_{-\alpha}(f)\in L(\Delta)$. Concerning (16):
Given $n_{i}\in\Sigma_{\Delta}[1]$ with $\langle\alpha,n_{i}\rangle=1$ and
$n_{j}\in\Sigma_{C(\Delta)}[1]$ with $\langle\alpha,n_{j}\rangle=1$ then
$n_{i}=n_{j}$ by (11). It follows
$\Gamma_{-\alpha}\subset\Gamma_{-\alpha}^{\prime}$ since

$\operatorname{Min}_{C(\Delta)}(n_{i})=\operatorname{Min}_{\Delta}(n_{i}).$

Thus $\Gamma_{-\alpha}\subset\Gamma_{-\alpha}^{\prime}$.

Second point: There is a facet $\Gamma_{-\alpha}$ of $C(\Delta)$ such that

$m-\alpha\in C(\Delta)\quad\textrm{for }m\in C(\Delta)\cap M,\quad
m\notin\Gamma_{-\alpha}.$

First assume that $\Gamma_{-\alpha}\cap\Delta$ is also a facet of $\Delta$.
There is $n_{j}\in\Sigma_{\Delta}[1]\setminus\\{n_{\Gamma_{-\alpha}}\\}$ with
$\langle\alpha,n_{j}\rangle>0$ since $\alpha\notin R(N,\Sigma_{\Delta})$.
Given $m\in Vert(\Gamma_{j})$, then $m\in\operatorname{Supp}(f)$ and
$m-\alpha\notin\Delta$ since

$\langle m-\alpha,n_{j}\rangle<\operatorname{Min}_{\Delta}(n_{j}).$

$\Rightarrow w_{-\alpha}(f)\notin L(\Delta)$. Assume that
$\Gamma_{-\alpha}\cap\Delta$ is a face of $\Delta$ of dimension $<n-1$. The
convex span

$\langle m\in Vert(\Delta)\,|\,m-\alpha\notin\Delta\rangle$

is of dimension $\geq n-1$. $\Rightarrow$ there is $m\in Vert(\Delta)$ with

$\quad m\notin\Gamma_{-\alpha},\quad m-\alpha\notin\Delta,$

that is $w_{-\alpha}(f)\notin L(\Delta)$.

Third point: Given a fixed facet $\Gamma=\Gamma_{-\alpha}$ of $C(\Delta)$ all

$\alpha\in R(N,\Sigma_{C(\Delta)})\setminus R(N,\Sigma_{\Delta})$

with $\Gamma_{-\alpha}=\Gamma$ build the lattice points on a lattice polytope
$P\subset M_{\mathbb{R}}$.
Given $\alpha\in Vert(P)$ there is $m\in\operatorname{Supp}(f)$ such that
$x^{m-\alpha}$ does not appear in the support of any other
$w_{-\alpha^{\prime}}(f)$. Thus $w_{-\alpha}(f)$ does not appear with nonzero
coefficient in any relation between the $w_{-\alpha^{\prime}}(f)$. We then
break down $P$ vertex by vertex.

Let $\Gamma_{1},\Gamma_{2}$ be two different facets of $C(\Delta)$ and
$\alpha_{1},\alpha_{2}\in R(N,\Sigma_{C(\Delta)})\setminus
R(N,\Sigma_{\Delta})$ roots to these facets. Given a relation in

$L(C(\Delta))/L(\Delta)$

in which both $w_{-\alpha_{1}}(f)$ and $w_{-\alpha_{2}}(f)$ appear with
nonzero coefficients there is $v\in\operatorname{Supp}(f)$ with

$\langle v-\alpha_{1},n_{1}\rangle<\operatorname{Min}_{\Delta}(n_{1}),\quad
v-\alpha_{1}+\alpha_{2}\in M\cap\Delta.$

Then

$\langle
v-\alpha_{1}+\alpha_{2},n_{1}\rangle\geq\operatorname{Min}_{\Delta}(n_{1}),$

but $\langle\alpha_{2},n_{1}\rangle\leq 0$ since $\alpha_{2}$ is a root for
$n_{2}\neq n_{1}$, a contradiction. ∎

######  Remark 9.4.

Given a common toric resolution of singularities

${\mathbb{P}_{\tilde{\Sigma}}}$${\mathbb{P}}$${\mathbb{P}_{\Delta}}$

there is a deformation of smooth toric hypersurfaces $\tilde{X}\rightarrow B$.
Take the Kodaira-Spencer map $\tilde{\kappa}_{f}$. Then
$R(N,\tilde{\Sigma})\subset R(N,\Sigma_{\Delta})$ and
$\ker(\tilde{\kappa}_{f})$ is gotten as $\ker(\kappa_{f})$ but with $\alpha\in
R(N,\tilde{\Sigma})$ instead of $\alpha\in R(N,\Sigma_{\Delta})$.

##  10 The number of moduli for subfamilies

######  Remark 10.1.

Let $\Delta$ be an $n$-dimensional polytope with $F(\Delta)\neq\emptyset$ and
$A\subset M\cap\Delta$ a subset containing all vertices of $\Delta$. Let

$f:=\sum\limits_{m\in A}a_{m}x^{m}.$

Then for $(a_{m})_{m\in A}$ generic $f$ is nondegenerate with respect to
$\Delta$ (see [GKZ94, Ch.10] and [Bat03, Prop.2.16]). Denote the resulting
open subset of $\mathbb{C}^{|A|}$ by $U_{A}$ and the restriction of
$\mathcal{X}$ to $U_{A}$ by $\mathcal{X}_{A}$. Taking the quotient by the
Laurent polynomials

$f,\quad x_{i}\frac{\partial f}{\partial x_{i}}\quad i=1,...,n$

will reduce the number of moduli of the subfamily $\mathcal{X}_{A}\rightarrow
U_{A}$ by $n+1$. Concerning the $w_{-\alpha}(f)$ it seems to be difficult to
decide in general if there are $c_{\alpha}\in\mathbb{C}$ with

$\sum\limits_{\alpha}c_{\alpha}w_{-\alpha}(f)\in T_{U_{A},f}\setminus\\{0\\}.$

Therefore we restrict to some special cases:

######  Example 10.2.

Assume that $\Delta$ is an $n$-dimensional simplex with
$F(\Delta)\neq\emptyset$ and $A$ equals the set of vertices of $\Delta$. Then
by varying the coefficients to $A$ we obtain a family with $\kappa=0$. This
generalizes:

######  Lemma 10.3.

Given the conditions of Theorem 6.1 let $A$ denote the set of vertices of
$\Delta$. Assume that for every facet

$\Gamma=\Delta\cap\\{x\in M_{\mathbb{R}}|\,\langle
x,n_{\Gamma}\rangle=b_{\Gamma}\\}$

with $n_{\Gamma}\in\Sigma_{\Delta}[1]$ there is no vertex of $\Delta$ lying in
the plane

$H_{\Gamma,+1}:=\\{x\in M_{\mathbb{R}}|\,\langle
x,n_{\Gamma}\rangle=b_{\Gamma}+1\\}$

Then the subfamily to $A$ has

$\\#\\{\textrm{vertices}\\}-n-1$

number of moduli.

###### Proof.

Given a root $\alpha$ and a lattice point $m$ on $\Delta$ then $m-\alpha$ lies
exactly one step closer to the facet $\Gamma_{-\alpha}$ (and not closer to any
other facet). Assume that $m$ and $m-\alpha$ are vertices of $\Delta$. Then
$m-\alpha$ lies on $\Gamma_{-\alpha}$ since else $m,m-2\alpha\in\Delta$ and
$m-\alpha$ would not be a vertex. But then $m\in H_{\Gamma_{-\alpha},+1}$
contradicting the assumption. ∎

## References

  * [Bat94] V. V. Batyrev, _Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties_ , Journal of Algebraic Geometry 3 (1994), no. 3, 493–535.
  * [Bat03] V. V. Batyrev, _Cohomology, Periods and the Hodge structure of toric hypersurfaces_ , (2003), https://www.math.arizona.edu/ swc/aws/2004/04BatyrevNotes.pdf.
  * [Bat22] V. V. Batyrev, _Canonical models of toric hypersurfaces_ , (2022), arXiv:2008.05814v1 [math.AG]
  * [BG99] W. Bruns, J. Gubeladze, _Polytopal Linear Groups_ , Journal of Algebra 218, 715-737 (1999).
  * [CLS11] D. A. Cox, J. B. Little and H. K. Schenck, _Toric varieties_ , Graduate Studies in Mathematics, 124, Amer. Math. Soc., Providence, RI, (2011).
  * [DK86] V. I. Danilov, A. G. Khovanskii, _Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers_ , Izv. Akad Nauk SSSR Ser Mat 50 (1986), 925–945.
  * [Dol82] I. Dolgachev, _Weighted projective varieties_ , Lecture Notes in Mathematics book series (LNM, volume 956), (1982).
  * [Fuj06] O. Fujino, _Multiplication maps and vanishing theorems for toric varieties_ , Math. Z. 257, 631–641, (2007).
  * [Gie23] J. Giesler, _Hodge theory of nondegenerate minimal toric hypersurfaces_ , (2023), unpublished.
  * [GKZ94] I. Gelfand, M. Kapranov, A. Zelevinsky, _Discriminants, Resultants and Multidimensional Determinants_ , Springer-Birkhäuser, (1994).
  * [Gro67] A. Grothendieck, _Local cohomology_ , Lecture Notes in Mathematics, Springer Verlag (1961).
  * [HaKo04] B. Hassett, S. J. Kovács, _Reflexive Pull-backs and base extension_ , Journal of Algebraic Geometry, 13, 233-247, (2004).
  * [IlTu18] N. Ilten, C. Turo, _Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product_ , (2018), arXiv:1812.09254 [math.AG].
  * [Kod86] K. Kodaira, _Complex Manifolds and Deformation of Complex Structures_ , First Edition A Series of Comprehensive Studies in Math. 283 Springer-Verlag, (1986).
  * [Koe91] R. J. Koelman, _The number of moduli of families of curves on toric surfaces_ , PhD Thesis, University of Nijmegen, (1991).
  * [Kol13] J. Kollár, _Singularities of the Minimal Model Program_ , Cambridge University Press, (2013).
  * [KM92] J. Kollár, S. Mori, _Classification of Three-Dimensional Flips_ , Journal of the American Mathematical Society , Vol. 5, (1992).
  * [KM98] J. Kollár and S. Mori, _Birational Geometry of Algebraic Varieties_ , Vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1998).
  * [Mav03] A. Mavlyutov, _Embedding of Calabi-Yau deformations into toric varieties_ , Mathematische Annalen volume 333, pages 45–65 (2005).
  * [Mav08] A. Mavlyutov, _Cohomology of rational forms and a vanishing theorem on toric varieties_ , Journal für die reine und angewandte Mathematik, (2008).
  * [MuOd15] D. Mumford, T. Oda, _Algebraic Geometry II_ , Texts and Readings in Mathematics, (2015).
  * [Ok87] M. Oka, _On the deformation of a certain type of algebraic varieties_ , Preprint, (1987).
  * [Pet20] A. Petracci, _Homogeneous deformations of toric pairs_ , Manuscripta Math. 166 (2021), no. 1-2, 37-72.
  * [Sch08] K. Schwede, _Generalized divisors and reflexive sheaves on normal varieties_ , (2008), https://www.math.utah.edu/ schwede/Notes/GeneralizedDivisors.pdf
  * [Voi03] C. Voisin, _Hodge Theory and Complex Algebraic Geometry II_ , Cambridge University Press, (2003).