# Waring Identifiability for powers of forms via degenerations

Alex Casarotti  Dipartimento di Matematica
Università degli Studi di Trento
via Sommarive 14 I-38123
Povo di Trento (TN), Italy<EMAIL_ADDRESS>and  Elisa Postinghel
Dipartimento di Matematica
Università degli Studi di Trento
via Sommarive 14 I-38123
Povo di Trento (TN), Italy<EMAIL_ADDRESS>
###### Abstract.

We discuss an approach to the secant non-defectivity of the varieties
parametrizing $k$-th powers of forms of degree $d$. It employs a Terracini
type argument along with certain degeneration arguments, some of which are
based on toric geometry. This implies a result on the identifiability of the
Waring decompositions of general forms of degree kd as a sum of $k$-th powers
of degree $d$ forms, for which an upper bound on the Waring rank was proposed
by Fröberg, Ottaviani and Shapiro.

###### Key words and phrases:

Identifiability, Waring problems, secant varieties, linear systems,
degenerations

###### 2020 Mathematics Subject Classification:

Primary: 14N07. Secondary: 14C20, 14D06, 14M25.

Both authors are members of INdAM-GNSAGA

##  1\. Introduction

Identifiability problems arise naturally in many fields of both applied and
classical algebraic geometry. A variety $X\subset\mathbb{P}^{N}$ is said to be
$h-$identifiable if the general point of its $h-$secant variety has a unique
decomposition as a sum of $h$ points of $X$. A classical application of
identifiability concerns particular polynomial decompositions. The Waring
problem for forms asks for a unique decomposition of a homogeneous polynomial
$F_{d}\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}$ as a sum of $d-$th powers of
linear forms, i.e.

(1.1) $F_{d}=L_{1}^{d}+\dots+L_{h}^{d},$

with $L_{i}\in\mathbb{C}[x_{0},\dots,x_{n}]_{1}$. A necessary condition for
identifiability is secant non-defectivity: a variety $X\subset\mathbb{P}^{N}$
of dimension $\dim(X)=n$ is said to be not $h-$(secant) defective if the
$h-$secant variety $\mathbb{S}ec_{h}(X)$, defined as the Zariski closure of
points in $\mathbb{P}^{N}$ lying in the span of $h$ points of $X$, has the
expected dimension $\min\\{N,h(n+1)-1\\}$. In [COV] the authors proved that,
for the Veronese case $X=\nu_{d}(\mathbb{P}^{n})$ and for all subgeneric ranks
$h$ (i.e. such that $\mathbb{S}ec_{h}(X)\subseteq\ {\mathbb{P}}^{N}$ does not
fill up the space), a general form $F$ of rank $h$ is identifiable, with a few
well known exceptions. In the case of generic rank, the situation is almost
the opposite: in [GM] it is proved that all forms of generic rank are not
identifiable with the following exceptions:
$(n,d,h)=(1,2k-1,k),(3,3,5),(2,5,7).$

In [FOS] the authors initiated the investigation of a generalization of the
classical Waring problems for forms. In particular they showed that a general
form $F_{kd}\in\mathbb{C}[x_{0},\dots,x_{n}]_{dk}$ can be written as a sum of
at most $k^{n}$ $k-$th powers of forms $G_{i}$’s of degree $d$

(1.2) $F_{kd}=G_{1}^{k}+\dots+G_{h}^{k},$

and that this bound is sharp, i.e. when $d$ is sufficiently large, $k^{n}$
computes the generic rank. The secant defectivity of the varieties
parametrizing $k-$th powers of forms of degree $d$ remains an open problem in
general.

In this paper we address both the secant defectivity and the identifiability
problems for such Waring decompositions. Denote with $V_{n,d}^{k}$ the variety
parametrizing $k-$th powers of homogeneous degree $d$ forms in $n+1$
variables:

$V_{n,d}^{k}:=\\{[F^{k}]|F\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}\\}.$

Our first main result about secant non-defectivity is the following.

######  Theorem 1.1.

The variety $V_{n,d}^{k}$ is not $h$-defective if $k\geq 3$ and
$h\leq\frac{1}{N+1}{N+k-3\choose N}$, where $N={n+d\choose d}-1$.

Our second result is about identifiability. A bridge from non-defectivity to
identifiability was built in [CM] first and then generalised in the recent
[MM]: whenever $X$ is a sufficiently regular variety (with non-degenerate
Gauss map), then if $X$ is not $h-$defective, then $X$ is
$(h-1)-$identifiable. Using this and Theorem 1.1 we obtain what follows.

######  Theorem 1.2.

A general form $F\in\mathbb{C}[x_{0},\dots,x_{n}]_{dk}$ of rank $h$ with
$k\geq 3$ is identifiable whenever

$h\leq\min\left\\{\frac{1}{N+1}{N+k-3\choose
N}-1,\left\lfloor\frac{{{n+kd}\choose{n}}}{N+1}\right\rfloor-1\right\\}$

We remark that in [Nen], the author showed that the secant defectivity of
$V^{k}_{n,d}$ can be bounded asymptotically, using a direct algebro-
computational argument, to $k^{n}-d^{n}$. In Section 6 we show that, for $d\gg
k$, our bound of Theorem 1.1 extends the latter.

In order to prove Theorem 1.1, we brought together a Terracini type argument
and several different degeneration techniques. By a classical application of
Terracini’s Lemma, non-defectivity problems for secant varieties translate
into the study of particular linear systems of hypersurfaces of projective
space with prescribed singularities. The first systematic study was used in
the proof of the celebrated Alexander and Hirschowitz Theorem for the case of
classical Waring problems (1.1), where secant varieties of Veronese embeddings
of ${\mathbb{P}}^{n}$ correspond to linear systems of hypersurfaces of
${\mathbb{P}}^{n}$ with prescribed double points in general position. In the
setting of the generalized Waring problem, as in (1.2) for $k\geq 2$, a direct
translation to linear systems of hypersurfaces with only double point
singularities is not possible. In order to prove secant non-defectivity in
this case it is necessary to impose a larger base locus to our linear systems.
In particular, we will be interested in studying the dimensions of linear
systems $\mathcal{L}:=\mathcal{L}_{N,k}(V,2^{h})$ of hyperurfaces of
$\mathbb{P}^{N}$ of degree $k$ that are singular at $h$ general points and
that contain the $d$-thVeronese embedding of ${\mathbb{P}}^{n}$,
$V\subset{\mathbb{P}}^{N}$. The study of such linear systems is carried out by
combining two types of degenerations introduced in [Po] and in [CDM] and [Pos-
SecDeg] respectively. We start by degenerating the ambient space
$\mathbb{P}^{N}$ to a scheme with two components and, in turn, the linear
system $\mathcal{L}$ to a fibered product of two linear systems, one on each
component, which are somewhat easier to deal with than the original one. In
fact one of them consists of hypersurfaces containing a linear subspace and a
collection of double points, the other one consists of hypersurfaces
containing $V$ and just one fat point of relatively large multiplicity with
support on $V$. Now, in order to study the latter, we perform a toric
degeneration of the Veronese $V$ to a union of $n$-dimensional linear spaces,
which will have the effect of reducing further the study of the limit linear
system.

It is worth mentioning that the study of Waring type problems and
identifiability of symmetric tensors has been implemented also in the applied
fields, from chemistry, biology to algebraic statistics. Recently in [BCMO],
the problem of identifiability for $k-$th powers of forms was linked to the
identifiability of centered Gaussian mixture models in applied statistics.

###  1.1. Organization of the paper

Section 2 contains all definitions and our Terracini type result that
translates non-defectivity of $V^{k}_{n,d}$ to the study of $\mathcal{L}$,
Proposition 2.16. In Section 3 we explain in detail the degenerations
techniques, both in the classical and in the toric setting. In Section 4 we
analyse two auxiliary linear systems arising from the degeneration of
$\mathcal{L}$, Proposition 4.3 and Corollary 4.8. Section 5 is devoted to the
proof of the main technical result, i.e. Theorem 5.2. Finally, in Section 6 we
explain to what extent our bounds are asymptotically better then the ones
known before in the literature.

###  1.2. Acknowledgments

The authors would like to thank Giorgio Ottaviani and Alessandro Oneto for
several useful discussions during the preparation of this article. We also
thank the referee for their useful comments.

##  2\. Powers of forms

In order to give a coherent and self-contained treatment of the subject, in
this section we recall some preliminary definitions and results.

We will work over the field of complex numbers $\mathbb{C}$.

###  2.1. Veronese embeddings

Let $W:=\mathbb{C}^{n+1}$ and $W^{\ast}$ the dual vector space. With
$\mathbb{P}^{n}=\mathbb{P}(W)$ we denote the projective space over
$\mathbb{C}$ of dimension $n$. We introduce the following integers

$N_{d}:={n+d\choose n}-1,\quad N_{d}^{k}:={N_{d}+k\choose N_{d}}-1.$

We will indicate $N_{d}$ simply by $N$, when no confusion may arise.

Notice that the following identities hold:

$\displaystyle\operatorname{h}^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(d))$
$\displaystyle=N_{d}+1$
$\displaystyle\operatorname{h}^{0}(\mathbb{P}^{N_{d}},\mathcal{O}_{\mathbb{P}^{N_{d}}}(k))$
$\displaystyle=N_{d}^{k}+1$

where $\operatorname{h}^{0}(\mathbb{P}^{a},\mathcal{O}_{\mathbb{P}^{a}}(b))$
denotes the number of global sections of the twisting sheaf
$\mathcal{O}_{\mathbb{P}^{a}}(b)$ on ${\mathbb{P}}^{a}$, for any integers
$a\geq 1,b\geq 0$. In other terms, ${{a+b}\choose b}$ is the dimension of the
linear systems of hypersurfaces of degree $b$ of ${\mathbb{P}}^{a}$ which, in
turn, is the projectivization of the complex vector spaces of forms of degree
$b$ in $a+1$ variables. With this in mind, we can make the following
identifications:

$\mathbb{P}^{N_{d}}=\mathbb{P}(\operatorname{Sym}^{d}(W^{*})),\quad\mathbb{P}^{N_{d}^{k}}=\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{*}))).$

Now we consider the following _Veronese embeddings_ :

$\displaystyle\nu_{d}:$ $\displaystyle\ \mathbb{P}^{n}\longrightarrow
V_{n}^{d}\subset\mathbb{P}(\operatorname{Sym}^{d}(W^{*}))$ $\displaystyle\
[L]\longmapsto[L^{d}]$ and $\displaystyle\nu_{k}:$ $\displaystyle\
\mathbb{P}^{N_{d}}\longrightarrow
V_{N_{d}}^{k}\subset\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{*})))$
$\displaystyle\ [F]\longmapsto[F^{k}]$

where $L\in\mathbb{C}[x_{0},\dots,x_{n}]_{1}$ is a linear form and
$F\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}$ is a form of degree $d$. The image of
the embeddings are called _Veronese varieties_.

######  Remark 2.1.

Note that both $\nu_{d}$ and $\nu_{k}$ are the maps corresponding to the
complete linear systems associated with the line bundles
$\mathcal{O}_{\mathbb{P}^{n}}(d)$ and $\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)$
respectively. As elements of $\mathbb{P}(\operatorname{Sym}^{d}(W^{\ast}))$
(respectively
$\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))$) the
image of $\nu_{d}(p)$ (respectively $\nu_{k}(p)$), with $p$ a point,
corresponds to the hyperplane parametrizing hypersurfaces of degree $d$ in
$\mathbb{P}^{n}$ (respectively of degree $k$ in $\mathbb{P}^{N_{d}}$) passing
through $p$.

We want to parametrize forms in $\mathbb{P}^{n}$ of degree $dk$, i.e. elements
in $\mathbb{C}[x_{0},\dots,x_{n}]_{dk}$, that can be written as $k-$th powers
of forms of degree $d$.

######  Remark 2.2.

Note that the Veronese varieties $\nu_{dk}(\mathbb{P}^{n})$ are always
contained in the set of all $k-$th powers of forms of degree $d$ because,
trivially, $L^{dk}=(L^{d})^{k}$.

Now, we let

$\displaystyle\phi_{dk}:$ $\displaystyle\
\mathbb{P}^{N_{d}}\longrightarrow\mathbb{P}^{N_{dk}}=\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))$
$\displaystyle\ \ [F]\longmapsto[F^{k}]$

be the map that assigns to each form $F\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}$
its $k-$th power.

######  Definition 2.3.

We call the scheme theoretic image

$V_{n,d}^{k}=\phi_{dk}({\mathbb{P}^{N_{d}}})\subseteq\mathbb{P}^{N_{dk}}$

the $(d,k)-$_Veronese variety_.

Under the previous identification, the classical Veronese varieties correspond
to the $(d,1)-$Veronese varieties. On the other hand, for $k>1$, $V_{n,d}^{k}$
is not a standard Veronese variety, indeed it is easy to see that the target
of $\phi_{dk}$ has dimension ${n+dk\choose dk}$, which is never equal to
${N_{d}+a\choose a}$ for any $a$. A priori we don’t know if the map
$\phi_{dk}$ is an isomorphism, as it happens for classical Veronese varieties,
see Lemma 2.11 below.

###  2.2. Secant varieties and identifiability

In this subsection we recall the definition of secant variety and the notion
of identifiability, following [CM].

Let $X\subset\mathbb{P}^{N}$ be a non degenerate reduced variety. Let
$X^{(h)}$ be the $h$-th symmetric product of $X$, that is the variety
parameterizing unordered sets of $h$ points of $X$. Let $U_{h}^{X}\subset
X^{(h)}$ be the smooth locus, given by sets of $h$ distinct smooth points.

######  Definition 2.4.

A point $z\in U^{X}_{h}$ represents a set of $h$ distinct points, say
$\\{z_{1},\ldots,z_{h}\\}$. We say that a point $p\in\mathbb{P}^{N}$ is in the
span of $z$, $p\in\langle z\rangle$, if it is a linear combination of the
$z_{i}$’s.

With this in mind, one can define the following object.

######  Definition 2.5.

The _abstract_ $h$-_secant variety_ is the $(hn+h-1)$-dimensional variety

$\textit{sec}_{h}(X):=\overline{\\{(z,p)\in
U_{h}^{X}\times\mathbb{P}^{N}|p\in\langle z\rangle\\}}\subset
X^{(h)}\times\mathbb{P}^{N}.$

Let $\pi:X^{(h)}\times\mathbb{P}^{N}\to\mathbb{P}^{N}$ be the projection onto
the second factor. The $h$-_secant variety_ is

$\mathbb{S}ec_{h}(X):=\pi(sec_{h}(X))\subset\mathbb{P}^{N},$

and $\pi_{h}^{X}:=\pi_{|sec_{h}(X)}:sec_{h}(X)\to\mathbb{P}^{N}$ is the
$h$-_secant map_ of $X$.

If the variety $X$ is irreducible and reduced we say that $X$ is $h$-defective
if

$\dim\mathbb{S}ec_{h}(X)<\min\\{\dim\textit{sec}_{h}(X),N\\}.$

The following is a classical result.

######  Theorem 2.6 (Terracini’s Lemma).

Let $X\subset\mathbb{P}^{N}$ be an irreducible variety. Then the follwing
holds.

  * •

For any $p_{1},\ldots,p_{k}\in X$ and $z\in\langle p_{1},\ldots,p_{k}\rangle$,
we have

$\langle T_{p_{1}}X,\ldots,T_{p_{k}}X\rangle\subseteq
T_{z}\mathbb{S}ec_{k}(X).$

  * •

There is a dense open set $U\subset X^{(k)}$ such that

$\langle T_{p_{1}}X,\ldots,T_{p_{k}}X\rangle=T_{z}\mathbb{S}ec_{k}(X),$

for any general point $z\in\langle p_{1},\ldots,p_{k}\rangle$ with
$(p_{1},\ldots,p_{k})\in U$.

The following notions are related to the notion of secant variety.

######  Definition 2.7.

Let $X\subset\mathbb{P}^{N}$ be a non degenerate subvariety. We say that a
point $z\in\mathbb{P}^{N}$ has _rank $h$ with respect to $X$_ if $z\in\langle
p\rangle$, for some $p\in U_{h}^{X}$ and $z\not\in\langle p^{\prime}\rangle$
for any $p^{\prime}\in U_{h^{\prime}}^{X}$, with $h^{\prime}<h$.

######  Definition 2.8.

A point $z\in\mathbb{P}^{N}$ is $h$-_identifiable with respect to_
$X\subset\mathbb{P}^{N}$ if $z$ is of rank $h$ and $(\pi^{X}_{h})^{-1}(z)$ is
a single point. The variety $X$ is said to be _$h$ -identifiable_ if the
$h$-secant map $\pi_{h}^{X}$ is birational, that is if the general point of
$\mathbb{S}ec_{h}(X)$ is $h$-identifiable.

It is clear, by Theorem 2.6, that when $X$ is $h$-defective, or more generally
when $\pi_{h}^{X}$ is of fiber type, then $X$ is not $h$-identifiable.

We now recall the recent result in [MM], in which the authors generalize the
approach in [CM] relating identifiability with the non defectivity of the
secant variety.

######  Theorem 2.9.

Let $X\subset\mathbb{P}^{N}$ be an irreducible and non-degenerate variety of
dimension $n$, $h\geq 1$ an integer, and assume that:

  * •

$(h+1)n+h\leq N$,

  * •

$X$ has non-degenerate Gauss map,

  * •

$X$ is not $(h+1)-$defective.

Then $X$ is $h-$identifiable.

In the next sections we will see how to rephrtase this theorem in the setting
of powers of forms in order to give identifiability results for $k-$th powers
of forms of degree $d$.

###  2.3. Geometric construction of $(d,k)-$Veronese varieties

Let us recall some facts from _apolarity theory_ ; the main reference is
[Ger].

######  Notation 2.10 (Apolarity).

We consider two polynomial rings in $n+1$ variables, both endowed with the
standard grading:

$\displaystyle R$
$\displaystyle=\mathbb{C}[x_{0},\dots,x_{n}]=\bigoplus_{i\in\mathbb{N}}R_{i}:=\mathbb{C}[x_{0},\dots,x_{n}]_{i}$
$\displaystyle S$
$\displaystyle=\mathbb{C}[y_{0},\dots,y_{n}]=\bigoplus_{i\in\mathbb{N}}S_{i}:=\mathbb{C}[y_{0},\dots,y_{n}]_{i}.$

Interpreting the elements of $S$ as partial derivatives in the $x_{i}$’s, the
pairing $S_{k}\times R_{l}\to\mathbb{C}$, sends $(F_{k},G_{l})$ to the
derivative $F_{k}\circ G_{l}\in R$ of $G_{l}$. If $k=l$ and if
$\mathcal{I}\subset R$ is a homogeneous ideal, the _orthogonal_
$\mathcal{I}_{k}^{\perp}\subset S_{k}$ is the following space of polynomials

$\mathcal{I}_{k}^{\perp}=\\{F\in S_{k}|F\circ G=0,\forall
G\in\mathcal{I}_{k}\\}.$

It is a standard fact of representation theory for the linear group $GL(W)$,
see for instance [Lan], that the space
$\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast}))$ can be decomposed
as direct sum of $GL(W)-$modules in the following way:

$\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast}))=\operatorname{Sym}^{dk}(W^{\ast})\oplus\mathcal{E}$

where

$\mathcal{E}=\operatorname{H}^{0}({\mathbb{P}}^{N},\mathcal{I}_{V_{n}^{d}}(k))$

is the $k-$th homogeneous part of the ideal of forms that vanish on the
Veronese variety $V_{n}^{d}=\nu_{d}(\mathbb{P}^{n})$, cf. notation of Section
2.1. In slightly different terms, we can describe
$\mathcal{E}=\operatorname{H}^{0}({\mathbb{P}}^{N},\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{V_{n}^{d}})$
as the space of global sections of the linear system of degree$-k$
hypersurfaces of ${\mathbb{P}}^{N}$ containing $V_{n}^{d}$. We have the
following exact sequence:

$0\rightarrow\mathcal{I}_{V_{n}^{d}}(k)\rightarrow\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\rightarrow\mathcal{O}_{V_{n}^{d}}(k)\rightarrow
0$

Considering the long exact sequence in cohomology and using the fact that
every Veronese variety is projectively normal, we have:

$0\rightarrow\operatorname{H}^{0}(\mathcal{I}_{V_{n}^{d}}(k))\rightarrow\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}}}(k))\rightarrow\operatorname{H}^{0}(\mathcal{O}_{V_{n}^{d}}(k))\rightarrow
0,$

where we omit the indication of the underlying space in the expressions
$H^{0}(\cdot)$, since it is clear from the context. This shows that

$\dim(\mathcal{E})={N_{d}+k\choose k}-{n+dk\choose dk}.$

It is easy to show that $GL(V)-$modules $\operatorname{Sym}^{dk}(W^{\ast})$
and $\mathcal{E}$ are apolar, i.e. for every pair of forms
$F,G\in\mathbb{C}[x_{0},\dots,x_{N_{d}}]$ with
$F\in\operatorname{Sym}^{dk}(W^{\ast})$ and $G\in\mathcal{E}$, one has that
$F\circ G=0$.

The constructions above fit into the following commutative diagram:

|
---|---
$\textstyle{\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{*})))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathbb{E}}}$$\textstyle{\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}(\operatorname{Sym}^{d}(W^{*}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{k}}$$\scriptstyle{\phi_{dk}}$$\textstyle{V_{n,d}^{k}\subset\mathbb{P}(\operatorname{Sym}^{dk}(W^{*}))}$

where the map $\pi_{\mathbb{E}}$ is the linear projection from the
(projective) linear space $\mathbb{E}=\mathbb{P}(\mathcal{E})$. We now prove
that the map $\phi_{dk}$ is in fact an isomorphism.

######  Lemma 2.11.

With the above notations, we have
$\mathbb{S}ec_{2}(V_{N_{d}}^{k})\cap\mathbb{E}=\emptyset$. In particular the
map $\phi_{dk}$ is an embedding.

###### Proof.

Note that an element $F\in\mathbb{S}ec_{2}(V_{N_{d}}^{k})$ is either of the
form $F=L^{k}$, $F=L^{k-1}M$ or $F=M^{k}+N^{k}$, where
$L,M,N\in\mathbb{C}[x_{0},\dots,x_{N_{d}}]_{1}$ are linear forms in
$\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))$, or
equivalently degree $d$ hypersurfaces in $\mathbb{P}^{n}$. In particular, if
there exist such an $F$ with
$F\in\mathcal{E}=\operatorname{H}^{0}(\mathcal{I}_{V_{n,d}}(k))$, then
$V_{n,d}$ would be contained in the vanishing locus of $F$. Moreover the zero
locus of $F$ is either a hyperplane ($\\{L=0\\}$), a union of two hyperplanes
($\\{L=0\\}\cup\\{M=0\\}$) or the union of a hyperplane and a subscheme $S$ of
degree $k-1$ ($\\{M+\zeta N=0\\}\cup S$), where $\zeta$ is a $k-$th root of
unity. Finally since $V_{n,d}$ is non-degenerate and irreducible the claim
follows.

∎

###  2.4. Identifiablity for $(d,k)-$Veronese varieties

From now on we will work with the projective notation, in particular
$\mathbb{E}$ has to be intended as the projectivization of the affine spave
$\mathcal{E}$ introduced in the previous section. Let us start by
characterizing hyperplanes in $\mathbb{E}$ as particular linear subsystems of
hypersurfaces in $\mathbb{P}^{N_{d}}$. Denote by $\pi_{dk}$ the linear
projection from the linear space
$\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))$ to $\mathbb{E}$, i.e.

$\pi_{dk}:\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))\dashrightarrow\mathbb{E}$

######  Lemma 2.12.

Let
$\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}^{k}}}(1)\otimes\mathcal{I}_{\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))})$
be the complete linear system of hyperplane sections of
$\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))$
containing the linear space $\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))$.
Then

$\nu_{k}^{*}(\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}^{k}}}(1)\otimes\mathcal{I}_{\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))}))\cong\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}})$

###### Proof.

Let
$H\in\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}})$
be a degree $k$ hypersurface in $\mathbb{P}^{N_{d}}$ that contains the
Veronese $V_{n,d}$. Then the linear span $\overline{H}$ of $(\nu_{k})_{*}(H)$
is a hyperplane in $\mathbb{P}^{N_{d}^{k}}$ that contains
$V_{n,dk}:=(\nu_{k})_{*}(V_{n,d})$. Since $\langle
V_{n,dk}\rangle=\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))$, we have that

$\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}})\subseteq\nu_{k}^{*}(\operatorname{H}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}^{k}}}(1)\otimes\mathcal{I}_{\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))})$

To conclude, it is enough to observe that

$\operatorname{h}^{0}(\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}})=\dim(\mathcal{E})=\operatorname{codim}(\mathbb{P}(\operatorname{Sym}^{dk}(W^{\ast}))),$

where $\operatorname{codim}$ here indicates the codimension in
$\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))$, and
$\nu_{k}^{*}$ induces an isomorphism of global sections. ∎

We need the following easy technical lemma:

######  Lemma 2.13.

The linear projection

$\pi_{dk}:\mathbb{P}(\operatorname{Sym}^{k}(\operatorname{Sym}^{d}(W^{\ast})))\dashrightarrow\mathbb{E}=\mathbb{P}(\operatorname{H}^{0}(\mathcal{I}_{V_{n,d}}(k)))$

is generically finite when restricted to
$V_{N_{d},k}=\nu_{k}(\mathbb{P}(\operatorname{Sym}^{d}(W^{\ast})))$.

###### Proof.

The map

$\pi_{dk_{|\nu_{k}(\mathbb{P}(\operatorname{Sym}^{d}(W^{\ast})))}}:V_{N_{d},k}\dashrightarrow\mathbb{E}$

is induced by a linear subsystem $\mathcal{F}$ of the line bundle
$\mathcal{L}=\mathcal{O}_{\mathbb{P}^{N_{d}^{k}}}(1)$ with

$\nu_{k}^{*}(\mathcal{F})=|\mathcal{O}_{\operatorname{Sym}^{d}(W^{\ast})}(1)\otimes\mathcal{I}_{V_{n,d}}(k)|.$

Now the claim follows easily from the fact that
$\operatorname{H}^{0}(\mathcal{I}_{V_{n,d}}(k))$ defines the Veronese variety
$V_{n,d}$ set-theoretically. ∎

Let $p_{1},\dots,p_{h}\in X\subset\mathbb{P}^{N}$ be general points. By Lemma
2.6, we have that $\mathbb{S}ec_{h}(X)$ has the expected dimension if and only
if
$\operatorname{H}^{0}(\mathcal{O}_{X}(1)\otimes\mathcal{I}_{p_{1}^{2},\dots,p_{h}^{2}})$
has the expected dimension, i.e.

$\dim\operatorname{H}^{0}(\mathcal{O}_{X}(1)\otimes\mathcal{I}_{p_{1}^{2},\dots,p_{h}^{2}})=\max\\{0,N+1-(n+1)h\\}.$

######  Notation 2.14.

If $p_{1},\dots,p_{h}\in X$ are general points, then we denote
$\mathcal{L}_{h,X}:=|\mathcal{O}_{X}(1)\otimes\mathcal{I}_{p_{1}^{2},\dots,p_{h}^{2}}|$.

Before moving on to the explicit description of the identifiability for
$V_{n,d}^{k}$, let us first prove a general proposition about linear systems
of projected varieties.

######  Proposition 2.15.

Let $X\subset\mathbb{P}^{N}$ be a smooth non-degenerate projective variety.
Moreover let $\mathbb{P}^{N}=\langle\mathbb{F},\mathbb{E}\rangle$ with
$\mathbb{F},\mathbb{E}$ skew linear subspaces. Let
$\pi_{\mathbb{E}}:{\mathbb{P}}^{N}\to\mathbb{F}$ and
$\pi_{\mathbb{F}}:{\mathbb{P}}^{N}\to\mathbb{E}$ be the natural projections.
If the projections restricted to $X$ are generically finite and
$\mathcal{L}_{h,X}$ has the expected dimension, then
$\mathcal{L}_{h,\pi_{\mathbb{F}}(X)}$ has the expected dimension if and only
if $\mathcal{L}_{h,\pi_{\mathbb{E}}(X)}$ has the expected dimension.

###### Proof.

Note that by symmetry of $\mathbb{E}$ and $\mathbb{F}$ it suffices to prove
only one of the implications. Let $q_{1},\dots,q_{h}$ be general points on
$\pi_{\mathbb{E}}(X)$, $x_{i}\in\pi_{\mathbb{E}}^{-1}(q_{i})$ a general point
in every fiber and $z_{i}=\pi_{\mathbb{F}}(p_{i})$. By the generality
assumption we have that $x_{1},\dots,x_{h}$ are general and so are
$z_{1},\dots,z_{h}$. Since $\pi_{\mathbb{F}}$ restricted to $X$ is generically
finite the space of hyperplanes
$\mathcal{L}_{h,\pi_{\mathbb{F}}(X)}=|\mathcal{O}_{\mathbb{E}}(1)\otimes\mathcal{I}_{z_{1}^{2},\dots,z_{h}^{2}}|$
correspond to $\mathcal{L}_{h,X}\otimes\mathcal{I}_{\mathbb{F}}$. Now the
splitting $\mathbb{P}^{N}=\langle\mathbb{F},\mathbb{E}\rangle$ induces the
linear projection

$\pi:\mathcal{L}_{h,X}\mapsto\mathcal{L}_{h,X}\otimes\mathcal{I}_{\mathbb{E}}$

such that $Ker(\pi)=\mathcal{L}_{h,\pi_{\mathbb{F}}(X)}$. Since, by
assumption, both the source and the kernel have the expected dimension by the
rank-nullity theorem the assertion follows. ∎

We are finally able to characterize the identifiability properties for the
case of powers of forms. Consider the following linear system of all
hypersurfaces of ${\mathbb{P}}^{N_{d}}$ of degree $k$ containing the Veronese
variety $V_{n,d}\subset{\mathbb{P}}^{N_{d}}$ and double at the points
$p_{1},\dots,p_{h}$ that lie in general position in ${\mathbb{P}}^{N_{d}}$:

$\mathcal{L}_{N_{d}}(V_{n,d},2^{h}):=\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}}\otimes\mathcal{I}_{p_{1}^{2},\dots,p_{h}^{2}}.$

######  Proposition 2.16.

In the above notation, let $p_{1},\dots,p_{h}$ be general points in
$\mathbb{P}^{N_{d}}=\mathbb{P}(\operatorname{Sym}^{d}(W^{\ast}))$. The linear
system $\mathcal{L}_{N_{d}}(V_{n,d},2^{h})$ has the expected dimension if and
only if $\mathbb{S}ec_{h}(V_{n,d}^{k})$ has the expected dimension.

###### Proof.

With the notations of Proposition 2.15 we have
$\mathbb{E}=\mathbb{P}(\operatorname{H}^{0}(\mathcal{I}_{V_{n,d}}(k)))$ and
$\mathbb{F}=\operatorname{Sym}^{dk}(W^{\ast})$. We have that
$\pi_{\mathbb{E}}$ restricted to $X=V_{N_{d}}^{k}$ is an isomorphism by Lemma
2.11, in particular it is generically finite. The same holds for
$\pi_{\mathbb{F}}=\pi_{dk}$ by Lemma 2.13. Now $\mathcal{L}_{h,V_{N_{d}}^{k}}$
has the expected dimension by Alexander-Hirschowitz (see Theorem 4.1 below)
and

$\mathcal{L}_{h,\pi_{dk}(V_{N_{d}}^{k})}=|\mathcal{O}_{\mathbb{P}^{N_{d}}}(k)\otimes\mathcal{I}_{V_{n,d}}\otimes\mathcal{I}_{p_{1}^{2},\dots,p_{h}^{2}}|$

by Lemma 2.12.

∎

##  3\. Degeneration techniques

In this section we will discuss two types of degenerations that will provide
main tools for the proofs of the results of this article.

###  3.1. The $\mathbb{F}\mathbb{P}-$degeneration

In this section we recall a degeneration procedure introduced in [Po], which
consists in degenerating the projective space $\mathbb{P}^{N}$ to a reducible
variety with two components, and then studying degenerations of line bundles
on the general fiber.

####  3.1.1. Degenerating the ambient space

Let $\Delta$ be a complex disc centred at the origin and consider the product
$\mathcal{Y}=\mathbb{P}^{N}\times\Delta$ with the natural projections
$\pi^{\mathcal{Y}}_{1}:\mathcal{Y}\to{\mathbb{P}}^{N}$ and
$\pi^{\mathcal{Y}}_{2}:\mathcal{Y}\to\Delta$. The second projection is a flat
morphism and we denote by $Y_{t}:=\mathbb{P}^{N}\times\\{t\\}$ the fiber over
$t\in\Delta$. We will refer to $Y_{0}$ and to $Y_{t}$, with $t\neq 0$, as the
_central fiber_ and the _general fiber_ respectively. Let
$f:\mathcal{X}\to\mathcal{Y}$ denote the blow-up of $\mathcal{Y}$ at a point
$(p,{0})\in Y_{0}$ in the central fiber. Consider the following diagram, where
$\pi^{\mathcal{X}}_{i}:=\pi^{\mathcal{Y}}_{i}\circ f$, for $i=1,2$:

|
---|---
$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi^{\mathcal{X}}_{2}}$$\scriptstyle{\pi^{\mathcal{X}}_{1}}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\mathcal{Y}}_{1}}$$\scriptstyle{\pi^{\mathcal{Y}}_{2}}$$\textstyle{\mathbb{P}^{N}}$$\textstyle{\Delta}$

The morphism $\pi^{\mathcal{X}}_{2}:\mathcal{X}\rightarrow\Delta$ is flat with
fibers denoted by $X_{t}$, $t\in\Delta$. For the general fiber we have
$X_{t}\cong Y_{t}=\mathbb{P}^{N}$, while the central fiber $X_{0}$ is the
reduced union of the strict transform of $Y_{0}$, that we shall denote with
${\mathbb{F}}$, and the exceptional divisor $\mathbb{P}\cong\mathbb{P}^{N}$ of
$f$. The two components $\mathbb{P}$ and ${\mathbb{F}}$ meet transversally and
we will denote by $R$ the intersection:
$R:={\mathbb{F}}\cap{\mathbb{P}}\cong{\mathbb{P}}^{N-1}$. We will say that
${\mathbb{P}}^{N}$ _degenerates_ to $X_{0}=\mathbb{P}\cup{\mathbb{F}}$.

We will now endow the general fiber $X_{t}$ with a line bundle and we will
describe its limits on $X_{0}$ via this degeneration. In order to do so, we
will give bases for the Picard groups of the components of $X_{0}$.

######  Notation 3.1.

We denote by $H_{\mathbb{P}}$ the hyperplane class of $\mathbb{P}$, so that
the Picard group of the exceptional component is generated by
$H_{\mathbb{P}}$. Moreover we denote with $H_{\mathbb{F}}$ the hyperplane
class of ${\mathbb{F}}$, pull-back of a general hyperplane of
$Y_{0}\cong{\mathbb{P}}^{N}$, and with $E:=\mathbb{P}|_{{\mathbb{F}}}$ the
exceptional class in ${\mathbb{F}}$: $H_{\mathbb{F}}$ and $E$ generate the
Picard group of ${\mathbb{F}}$.

In these bases, $R$ has class $H_{\mathbb{P}}$ in $\textrm{N}^{1}(\mathbb{P})$
and $E$ in $\textrm{N}^{1}({\mathbb{F}})$. A line bundle on $X_{0}$ will
correspond to a line bundle on $\mathbb{P}$ and a line bundle on
${\mathbb{F}}$, which match on the intersection $R$. In other terms, we can
describe the Picard group of $X_{0}$ as a fiber product

$\textrm{Pic}(X_{0})=\textrm{Pic}(\mathbb{P})\times_{\textrm{Pic}(R)}\textrm{Pic}({\mathbb{F}}).$

Consider the line bundle
$\mathcal{O}_{\mathcal{X}}(k)=(\pi^{\mathcal{X}}_{1})^{\ast}(\mathcal{O}_{\mathbb{P}^{N}}(k))$
and the following twist by a negative multiple of the exceptional divisor:

$\displaystyle\mathcal{M}_{\mathcal{X}}(k,a)$
$\displaystyle:=\mathcal{O}_{\mathcal{X}}(k)\otimes\mathcal{O}_{\mathcal{X}}(-a{\mathbb{P}}).$
The line bundle $\mathcal{M}_{\mathcal{X}}(k,a)$ will induce a line bundle on
each fiber $X_{t}$ by restriction: $\displaystyle\mathcal{M}_{t}(k,a)$
$\displaystyle:=\mathcal{M}_{\mathcal{X}}(k,a)|_{X_{t}},\ t\in\Delta.$ For
$t\neq 0$, since ${\mathbb{P}}\cap X_{t}=\emptyset$, we have
$\displaystyle\mathcal{M}_{t}(k,a)$ $\displaystyle=\mathcal{O}_{X_{t}}(k)$
while on the components of the central fiber we have
$\displaystyle\mathcal{M}_{\mathbb{P}}(k,a)$
$\displaystyle:=\mathcal{M}_{\mathcal{X}}(k,a)|_{\mathbb{P}}=\mathcal{O}_{\mathbb{P}}(aH_{\mathbb{P}}),$
$\displaystyle\mathcal{M}_{{\mathbb{F}}}(k,a)$
$\displaystyle:=\mathcal{M}_{\mathcal{X}}(k,a)|_{{\mathbb{F}}}=\mathcal{O}_{{\mathbb{F}}}(kH_{{\mathbb{F}}}-aE).$

The resulting line bundle on $X_{0}$ is a flat limit of the bundle
$\mathcal{O}_{X_{t}}(k)\cong\mathcal{O}_{\mathbb{P}^{n}}(k)$, for $t\to 0$.

####  3.1.2. Degenerating the Veronese variety

We will use the same notation as in Section 3.1.1. Let us set

$N:=N_{d}={{n+d}\choose n}-1,$

and let $V:=V_{n,d}=v_{d}(\mathbb{P}^{n})\subset\mathbb{P}^{N}$ denote the
$d$-th Veronese embedding of $\mathbb{P}^{n}$ in $\mathbb{P}^{N}$, and
consider the $1$-parameter family
$\mathcal{V}=V\times\Delta\subset\mathcal{Y}$ with the natural projections
${\pi^{\mathcal{Y}}_{1}}|_{\mathcal{V}}:\mathcal{V}\to{\mathbb{P}}^{N}$ and
${\pi^{\mathcal{Y}}_{2}}|_{\mathcal{V}}:\mathcal{V}\to\Delta$. The second
projection is a flat morphism and we denote by $V_{t}:=V\times\\{t\\}$ the
fiber over $t\in\Delta$. We pick a general point $(p,0)\in V_{0}\subset Y_{0}$
in the central fiber of $\pi_{2}^{\mathcal{Y}}:\mathcal{Y}\to\Delta$ supported
on the Veronese variety and we consider the blow-up
$f:\mathcal{X}\to\mathcal{Y}$ at $(p,0)$. This induces the blow-up
$f|_{\mathcal{V}}:\widetilde{\mathcal{V}}\to\mathcal{V}$ of $\mathcal{V}$ at
$(p,0)$ and the fibers of $(\pi_{2}^{\mathcal{Y}}\circ
f)|_{\mathcal{\widetilde{\mathcal{V}}}}$ are as follows: the general fiber is
a Veronese variety

$\displaystyle\widetilde{V}_{t}$ $\displaystyle\cong V,$ while the central
fiber is the reduced union of two components, $\displaystyle\widetilde{V}_{0}$
$\displaystyle=\widetilde{V}_{{\mathbb{F}}}\cup\Lambda,$ where
$\widetilde{V}_{{\mathbb{F}}}$ is the strict transform of $V_{0}$ under the
blow-up at $p$, while $\Lambda\cong\mathbb{P}^{n}$ is the exceptional divisor
on $\widetilde{V}$. Moreover we can write $\displaystyle\Lambda_{R}$
$\displaystyle:=\widetilde{V}_{{\mathbb{F}}}\cap\Lambda\subset\Lambda,$

and observe that $\Lambda_{R}\cong\mathbb{P}^{n-1}$.

Consider the line bundle $\mathcal{M}_{\mathcal{X}}(k,a)$ and twist it by the
ideal sheaf of $\widetilde{\mathcal{V}}$:

$\displaystyle\mathcal{M}_{\mathcal{X}}(k,a;\widetilde{\mathcal{V}})$
$\displaystyle:=\mathcal{M}_{\mathcal{X}}(k,a)\otimes\mathcal{I}_{\widetilde{\mathcal{V}}}=\mathcal{O}_{\mathcal{X}}(k)\otimes\mathcal{O}_{\mathcal{X}}(-a{\mathbb{P}})\otimes\mathcal{I}_{\widetilde{\mathcal{V}}}.$
This restricts to the following line bundles on the fibers $X_{t}$:
$\displaystyle\mathcal{M}_{t}(k,a;V)$
$\displaystyle=\mathcal{O}_{X_{t}}(k)\otimes\mathcal{I}_{V_{t}},\
t\in\Delta\setminus\\{0\\},$
$\displaystyle\mathcal{M}_{\mathbb{P}}(k,a;\Lambda)$
$\displaystyle=\mathcal{O}_{\mathbb{P}}(aH_{\mathbb{P}})\otimes\mathcal{I}_{\Lambda},$
$\displaystyle\mathcal{M}_{{\mathbb{F}}}(k,a;\widetilde{V})$
$\displaystyle=\mathcal{O}_{{\mathbb{F}}}(kH_{{\mathbb{F}}}-aE)\otimes\mathcal{I}_{\widetilde{V}}.$

The resulting line bundle on $X_{0}$ is a flat limit of the bundle
$\mathcal{O}_{\mathbb{P}^{n}}(k)\otimes\mathcal{I}_{V}$ on the general fiber.

####  3.1.3. Degenerating a collection of points in general position

We continue to use the notation of Sections 3.1.1-3.1.2. On the general fiber
of $\mathcal{Y}\to\Delta$, i.e. for $t\neq 0$, we consider a collection of
points $\\{x_{1,t}\dots,x_{h,t}\\}\subset Y_{t}$ in general position and that,
in particular, lie off the Veronese variety $V_{t}\subset Y_{t}$. After
choosing a point that lies generically on the Veronese variety in the central
fiber, $p\in V_{0}\subset Y_{0}$, we degenerate each point $x_{1,t}\in Y_{t}$
to an infinitely near point to $p\in V_{0}$ as follows. For every
$i\in\\{1,\dots,h\\}$, consider the curve
$(\pi^{\mathcal{Y}}_{1})^{-1}(x_{i})\subset\mathcal{Y}$ and its pull-back
$C_{i}$ on $\mathcal{X}$. The union $\bigcup_{i}C_{i}$ intersects each fiber
$X_{t}$ transversally in $h$ distinct points. For $t\neq 0$, the points
$\bigcup_{i}C_{i}\cap X_{t}$ are in general position. Moreover, by the
generality assumption, the curves
$(\pi^{\mathcal{Y}}_{1})^{-1}(p_{i})\subset\mathcal{Y}$ are not tangent to
$V_{0}\in Y_{0}$, therefore the intersection points $C_{i}\cap X_{0}$ lie on
${\mathbb{P}}$ but not inside $\Lambda$.

Consider the blow-up of $\mathcal{X}$ along the union of curves
$\bigcup_{i=1}^{h}C_{t}$, $g_{h}:\widetilde{\mathcal{X}}\to\mathcal{X}$, with
exceptional divisors $E_{C_{i}}$. Since these curves are disjoint, the result
does not depend of the order of blow-up. The general fiber, that we continue
to call $X_{t}$ by abuse of notation, is isomorphic to a ${\mathbb{P}}^{N}$
blown-up at $h$ points in general position and its Picard group will be
generated by the hyperplane class and by the classes of the exceptional
divisors:

$\operatorname{Pic}(X_{t})=\mathbb{Z}\langle H,E_{1,t},\dots,E_{h,t}\rangle.$

The central fiber has two components, the pull-back of ${\mathbb{F}}$ and the
strict transform of $\mathbb{P}\cong\mathbb{P}^{N}$, which is isomorphic to a
${\mathbb{P}}^{N}$ blown-up at $h$ points in general position: abusing
notation, we call ${\mathbb{F}}$ and $\mathbb{P}$ the two components, so that
$X_{0}={\mathbb{P}}\cup{\mathbb{F}}$. The Picard group of
$\widetilde{\mathbb{P}}$ is

$\operatorname{Pic}({\mathbb{P}})=\mathbb{Z}\langle
H_{\mathbb{P}},E_{1,t},\dots,E_{h,t}\rangle.$

For a vector ${\bf m}=(m_{1},\dots,m_{h})\in\mathbb{N}^{n}$, consider the
following sheaf on $\widetilde{\mathcal{X}}$:

$\displaystyle\mathcal{M}_{\widetilde{\mathcal{X}}}(k,a;\tilde{\mathcal{V}},{\bf
m})$
$\displaystyle:=\mathcal{O}(k)\otimes\mathcal{O}(-a{\mathbb{P}})\otimes\mathcal{O}(-(m_{1}E_{C_{1}}+\cdots+m_{h}E_{C_{h}}))\otimes\mathcal{I}_{\widetilde{V}}.$
It restricts to $\displaystyle\mathcal{M}_{t}(k,a;V,{\bf m})$
$\displaystyle=\mathcal{O}(k)\otimes\mathcal{O}(-(m_{1}E_{1,t}+\cdots+m_{h}E_{h,t}))\otimes\mathcal{I}_{V_{t}},\
t\neq 0,$ on the general fiber, and to
$\displaystyle\mathcal{M}_{\mathbb{P}}(k,a;\Lambda,{\bf m})$
$\displaystyle=\mathcal{O}_{\mathbb{P}}(aH_{\mathbb{P}}-(m_{1}E_{1}+\cdots+m_{h}E_{h}))\otimes\mathcal{I}_{\Lambda},$
$\displaystyle\mathcal{M}_{\mathbb{F}}(k,a;\widetilde{V},{\bf m})$
$\displaystyle=\mathcal{O}_{{\mathbb{F}}}(kH_{{\mathbb{F}}}-aE)\otimes\mathcal{I}_{\widetilde{V}}.$

on the components of the central fiber. The resulting line bundle on $X_{0}$
is a flat limit of the bundle on the general fiber.

####  3.1.4. Matching conditions

We will abbreviate the notation of the previous sections by setting

$\displaystyle\mathcal{M}_{\widetilde{\mathcal{X}}}$
$\displaystyle:=\mathcal{M}_{\widetilde{\mathcal{X}}}(k,a;\tilde{\mathcal{V}},{\bf
m})$ and $\displaystyle\mathcal{M}_{t}$
$\displaystyle:=\mathcal{M}_{t}(k,a;V,{\bf m})$
$\displaystyle\mathcal{M}_{\mathbb{P}}$
$\displaystyle:=\mathcal{M}_{\mathbb{P}}(k,a;\Lambda,{\bf m})$
$\displaystyle\mathcal{M}_{{\mathbb{F}}}$
$\displaystyle:=\mathcal{M}_{\mathbb{F}}(k,a;\tilde{V},{\bf m}).$

We are interested in computing the dimension of the space of global sections
of the line bundle on the central fiber which, by semicontinuity, gives an
upper bound to the dimension of the space of global sections of the line
bundle on the general fiber:

(3.1)
$\operatorname{h}^{0}(X_{0},\mathcal{M}_{0})\geq\operatorname{h}^{0}(X_{t},\mathcal{M}_{t}).$

In order to do so, we consider the natural restrictions to the intersection
$R=\mathbb{P}\cap{\mathbb{F}}$ of the central fiber:

$\displaystyle 0\to\hat{\mathcal{M}}_{\mathbb{P}}$
$\displaystyle\to{\mathcal{M}}_{\mathbb{P}}\to\mathcal{M}_{\mathbb{P}}|_{R}\to
0,$ $\displaystyle 0\to\hat{\mathcal{M}}_{\mathbb{F}}$
$\displaystyle\to{\mathcal{M}}_{\mathbb{F}}\to\mathcal{M}_{\mathbb{F}}|_{R}\to
0,$

where
$\hat{\mathcal{M}}_{\mathbb{P}}=\hat{\mathcal{M}}_{\mathbb{P}}(k,a;\Lambda,{\bf
m})$ and
$\hat{\mathcal{M}}_{\mathbb{F}}=\hat{\mathcal{M}}_{\mathbb{F}}(k,a;\tilde{V},{\bf
m})$ denote the kernels of the restriction maps. Since $R=H_{\mathbb{P}}$ on
$\mathbb{P}$ and $R=E$ on ${\mathbb{F}}$, we have

$\displaystyle\hat{\mathcal{M}}_{\mathbb{P}}$
$\displaystyle=\mathcal{O}_{\mathbb{P}}((a-1)H_{\mathbb{P}}-(m_{1}E_{1}+\cdots+m_{h}E_{h})\otimes\mathcal{I}_{\Lambda}$
$\displaystyle\ \hat{\mathcal{M}}_{\mathbb{F}}$
$\displaystyle=\mathcal{O}_{{\mathbb{F}}}(kH_{{\mathbb{F}}}-(a+1)E)\otimes\mathcal{I}_{\tilde{V}}.$

Consider the restriction maps of global sections:

$\displaystyle
r_{\mathbb{P}}:\operatorname{H}^{0}(\mathbb{P},\mathcal{M}_{\mathbb{P}})$
$\displaystyle\to\operatorname{H}^{0}(R,\mathcal{M}_{\mathbb{P}}|_{R}),$
$\displaystyle
r_{\mathbb{F}}:\operatorname{H}^{0}(\mathbb{P},\mathcal{M}_{\mathbb{F}})$
$\displaystyle\to\operatorname{H}^{0}(R,\mathcal{M}_{\mathbb{F}}|_{R}).$

We notice that the spaces of global sections of the restricted systems are
both subspaces of the space of global sections of the degree-$a$ line bundle
on $R\cong\mathbb{P}^{N-1}$:

$\operatorname{H}^{0}(R,\mathcal{M}_{\mathbb{P}}|_{R}),\operatorname{H}^{0}(R,\mathcal{M}_{\mathbb{F}}|_{R})\subseteq\operatorname{H}^{0}(R,\mathcal{O}_{R}(a)).$

A global sections of $\mathcal{M}_{0}$ consists of an element of
$\operatorname{H}^{0}(\mathbb{P},\mathcal{M}_{\mathbb{P}})$ and an element of
$\operatorname{H}^{0}({\mathbb{F}},\mathcal{M}_{\mathbb{F}})$ which match in
$\operatorname{H}^{0}(R,\mathcal{O}_{R}(a))$, i.e. the space of global
sections $H^{0}(X_{0},\mathcal{M}_{0})$ is described as a fiber product via
the following commutative diagram:

${\operatorname{H}^{0}(X_{0},\mathcal{M}_{0})}$${\operatorname{H}^{0}(\mathbb{P},\mathcal{M}_{\mathbb{P}})}$${\operatorname{H}^{0}({\mathbb{F}},\mathcal{M}_{{\mathbb{F}}})}$${\operatorname{H}^{0}(R,\mathcal{M}_{\mathbb{P}}|_{R}\cap\mathcal{M}_{\mathbb{F}}|_{R})}$$\scriptstyle{r_{\mathbb{P}}}$$\scriptstyle{r_{\mathbb{F}}}$

This yields the formula for the dimension of the spaces of global sections

$\operatorname{h}^{0}(X_{0},\mathcal{M}_{0})=\operatorname{h}^{0}(\mathbb{P},\hat{\mathcal{M}}_{\mathbb{P}})+\operatorname{h}^{0}({\mathbb{F}},\hat{\mathcal{M}}_{{\mathbb{F}}})+\operatorname{h}^{0}(R,\mathcal{M}_{\mathbb{P}}|_{R}\cap\mathcal{M}_{\mathbb{F}}|_{R}),$

which, in terms of dimensions of line bundles, reads as follows:

(3.2)
$\dim\mathcal{M}_{0}=\dim\hat{\mathcal{M}}_{\mathbb{P}}+\dim\hat{\mathcal{M}}_{{\mathbb{F}}}+\dim\mathcal{M}_{\mathbb{P}}|_{R}\cap\mathcal{M}_{\mathbb{F}}|_{R}+2.$

######  Remark 3.2.

There is an obvious isomorpshism between $\mathcal{M}_{t}(k,a;V,{\bf m})$,
line bundle on $X_{t}$, and the line bundle
$\mathcal{L}_{N,k}(V,{\bf{m}}):=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{V}\otimes\mathcal{I}_{Z}$
on ${\mathbb{P}}^{N}$, where $Z$ is a union of fat points in general position
in ${\mathbb{P}}^{N}$ with multiplicities respectively $m_{1},\dots,m_{h}$.
Such isomorphism is given by taking strict transforms of elements of
$\mathcal{L}_{N,k}(V,{\bf{m}})$.

Similarly, since ${\mathbb{P}}$ is the blow-up of ${\mathbb{P}}^{N}$ at $h$
points in general position and $\Lambda\subset{\mathbb{P}}$ is a general
linear subspace, then there is an isomorphisms
$\mathcal{M}_{\mathbb{P}}(k,a;\Lambda,{\bf
m})\cong\mathcal{L}_{N,a}(\Lambda,{\bf
m}):=\mathcal{O}_{{\mathbb{P}}^{N}}(a)\otimes\mathcal{I}_{\Lambda}\otimes\mathcal{I}_{Z}$.

Finally, since ${\mathbb{F}}$ is a ${\mathbb{P}}^{N}$ blown-up at a point $p$
on the Veronese variety $V\subset{\mathbb{P}}^{N}$ and since $\tilde{V}$ is
the strict transform of $V$ via this blow-up, then there is an isomorphisms
$\mathcal{M}_{\mathbb{F}}(k,a;\tilde{V},{\bf
m})\cong\mathcal{L}_{N,k}(V,a):=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{V}\otimes\mathcal{I}_{p^{a}}$.

###  3.2. Toric degeneration of the Veronese variety

We refer to [fulton-toric-book] for details on projective toric varieties
associated with convex lattice polytopes and to [GKZ] for details on coherent
triangulations. Let $\Delta_{n}\subset\mathbb{R}^{n}$ be the $n$-dimensional
simplex obtained as the convex hull of the points
$(0,\dots,0),(1,0,\dots,0),\dots,(0,\dots,0,1)$. Consider
$d\Delta_{n}=\Delta_{N}+\cdots+\Delta_{N}$, where $+$ denotes the Minkowski
sum of polytopes in $\mathbb{R}^{n}$. The polytope $d\Delta_{n}$ defines the
$d$-th Veronese embedding of ${\mathbb{P}}^{n}$ in ${\mathbb{P}}^{N}$, with
$N=N_{d}={{N+d}\choose N}-1$, that we shall call $V=V_{n,d}$ as in the
previous section. Consider the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$
and the set of lattice points $\mathcal{A}:=d\Delta_{n}\cap\mathbb{Z}^{n}$. We
have $\sharp\mathcal{A}=N+1$ and each such point corresponds to a coordinate
point of the ambient space ${\mathbb{P}}^{N}$.

####  3.2.1. Degenerating the Veronese to a union of linear spaces

Take a _regular triangulation_ of $d\Delta_{n}$, that is a decomposition of
$d\Delta_{n}$ into a finite union of simplices

$\bigcup_{i=1}^{d^{n}}S_{i},$

where

  * •

each $S_{i}$ is obtained as the convex hull of $n+1$ non-aligned points of
$\mathcal{A}$,

  * •

$\sharp S_{i}\cap\mathbb{Z}^{n}=n+1$,

  * •

for $i\neq j$, $S_{i}\cap S_{j}$ is a common faces of $S_{i}$ and $S_{j}$
(possibly empty),

  * •

there is a strictly convex piecewise linear function
$\lambda:\mathbb{R}^{n}\to\mathbb{R}$ whose domains of linearity are precisely
the $S_{i}$’s.

We can always assume that $S_{1}$ is the convex hull of the lattice points
$(0,\dots,0)$, $(1,0,\dots,0),\dots,(0,\dots,0,1)$, so that it lies at a
corner of $d\Delta_{n}$. Consider for example, for $n=2$ and $d=3$, the
subdivision into nine triangles and the piecewise linear function inducing it,
as shown in Figure 1.

Figure 1. A regular triangulation of $3\Delta_{2}$.

In this figure, $S_{1}$ is the triangle with vertices $(0,0),(1,0),(0,1)$.

Each $S_{i}$ defines a ${\mathbb{P}}^{n}$ as a toric variety, which we will
call $\Pi_{i}$, for $i=1,\dots,d^{n}$. Since a regular triangulation of
$d\Delta_{n}$ induces a 1-parameter embedded degeneration of
$V_{n,d}\subset{\mathbb{P}}^{N}$ to the union of toric varieties described by
the $S_{i}$’s (see [CDM] and [Pos-SecDeg] for details on the 2-dimensional
case), we have a degeneration of the $n$-dimensional Veronese variety
$V_{n,d}$ to a union of $d^{n}$ $n$-planes:

${\bf\Pi}:=\bigcup_{i=1}^{d^{n}}\Pi_{i}.$

The intersection table of these planes is encoded in the combinatorial data
described by the triangulation, that is: if $S_{i}\cap S_{j}$ is
$r$-dimensional, then $\Pi_{i}\cap\Pi_{j}\cong{\mathbb{P}}^{r}$, for $0\leq
r\leq n-1$.

######  Remark 3.3.

Because of the choice of $S_{1}$ made, we will say that $\Pi_{1}$ is a _sink_.
In practice this means that it is possible to choose a hyperplane of
${\mathbb{P}}^{N}$ that contains every $S_{i}$, $i>1$, but that does not
contain $S_{1}$.

Moreover, the union of planes ${\bf\Pi}\subset\mathbb{P}^{N}$ is a torus
invariant subscheme. In fact, consider the simplex $\Delta_{N}$, which defines
${\mathbb{P}}^{N}$ as a toric variety, with an action of the algebraic torus
$(\mathbb{C}^{\ast})^{N}$. Each $r$-dimensional face of $\Delta_{N}$
corresponds to a torus invariant linear subspace of dimension $r$ of
${\mathbb{P}}^{N}$. In particular vertices of $\Delta_{N}$ are in one-to-one
correspondence with $N+1$ linearly general points of ${\mathbb{P}}^{N}$, which
we may assume to be the coordinate points, after a suitable change of
coordinates. Each $r$-dimensional face of $\Delta_{N}$ corresponds to a
${\mathbb{P}}^{r}$ spanned by $r+1$ coordinate points. Since each $\Pi_{i}$ is
the linear span of $n+1$ coordinate points of ${\mathbb{P}}^{N}$, then the
union ${\bf\Pi}$ is embedded in a copy of ${\mathbb{P}}^{N}$ and it is
invariant under the action of the torus $(\mathbb{C}^{\ast})^{N}$. In
particular, each $\Pi_{i}$ will correspond to a _marked_ $n$-dimensional face
of $\Delta_{N}$ and we have $d^{n}$ such marked faces.

####  3.2.2. Degenerating a linear system intepolating the Veronese

We now consider the linear systems on ${\mathbb{P}}^{N}$ of degree$-k$
hypersurfaces containing the Veronese variety on the one hand, and the union
of $n-$planes ${\bf\Pi}$ on the other hand:

$\displaystyle\mathcal{L}_{N,k}(V_{n,d})$
$\displaystyle:=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{V_{n,d}},$
$\displaystyle\mathcal{L}_{N,k}({\bf\Pi})$
$\displaystyle:=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{\bf\Pi}$

######  Lemma 3.4.

In the above notation, we have

$\dim\mathcal{L}_{N,k}(V_{n,d})\leq\dim\mathcal{L}_{N,k}({\bf\Pi}).$

###### Proof.

Since ${\bf\Pi}$ is a flat degeneration of $V_{n,d}$, then the statement
follows by semi-continuity of the function $\dim$. ∎

##  4\. Some auxiliary linear systems

###  4.1. Hypersurfaces containing a linear subspace and $h$ double points in
general position

It is a well celebrated result of Alexander and Hirschowitz that if we impose
$h$ double points in general position to the hypersurfaces of degree $d$ of
${\mathbb{P}}^{N}$, there is only a finite list of cases where the dimension
is larger than that obtained via a parameter count, that is

$\operatorname{edim}\mathcal{L}_{N,d}(2^{h})=\max\left\\{-1,{{N+d}\choose
N}-h(N+1)-1\right\\}\leq\dim\mathcal{L}_{N,d}(2^{h}),$

where the expression on the left hand side is the _expected dimension_ of the
linear system.

######  Theorem 4.1 (Alexander-Hirschowitz Theorem).

The linear system $\mathcal{L}_{N,d}(2^{h})$ has the expected dimension,
except in the following cases:

  * •

$d=2$ and $N\geq 2$, $2\leq h\leq N$;

  * •

$d=3$ and $(N,h)=(4,7)$;

  * •

$d=4$ and $(N,h)=(2,5),(3,9),(4,14)$.

The interested reader may see [A],[AH1],[AH2],[AH3],[AH4]] for the original
proof based on specialisation of points (Horace method), and [BO] and
[Chandler] for a simplified proof. An alternative proof via a different
degeneration construction can be found here [PosPhD, Po]. This inspired the
degeneration approach developed in Section 3.1 that will be used to prove the
main result, Theorem 1.1.

In this section we want to present an analogous result about linear systems
with $h$ imposed double points in general position and a linear subspace. Let
$\Lambda\subset\mathbb{P}^{N}$ be a general linear subspace of dimension $n$
and let $Z\subset{\mathbb{P}}^{N}$ be a double point scheme with support a set
of points in general position. Let $\mathcal{I}_{\Lambda}$ be the ideal sheaf
of $\Lambda\subset\mathbb{P}^{N}$ and let $\mathcal{I}_{Z}$ be the ideal sheaf
of $Z\subset{\mathbb{P}}^{N}$. Consider the sheaf

$\mathcal{L}_{N,k}(\Lambda,2^{h}):=\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{\Lambda}\otimes\mathcal{I}_{Z}.$

Since the Hilbert polynomial of $\Lambda\subset\mathbb{P}^{N}$ at degree $k$
is

${{n+k}\choose n}$

or, in other terms,

(4.1)
$\operatorname{h}^{0}(\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{\Lambda})={{N+k}\choose
N}-{{n+k}\choose n},$

and since the scheme given by $h$ double points in general position of
${\mathbb{P}}^{N}$ has length $h(N+1)$, we can give the following definitions.

######  Definition 4.2.

The _virtual dimension_ of the linear system
$\mathcal{L}_{N,k,\Lambda}(2^{h})$ of hypersurfaces of ${\mathbb{P}}^{n}$ that
vanish along a linear subspace of dimension $n$,
$\Lambda\subset{\mathbb{P}}^{N}$, and double at $h$ points in general position
is the following integer:

$\operatorname{vdim}\mathcal{L}_{N,k}(\Lambda,2^{h})={{N+k}\choose
N}-{{n+k}\choose n}-h(N+1)-1.$

The _expected dimension_ of $\mathcal{L}_{N,k}(\Lambda,2^{h})$ is

$\operatorname{edim}\mathcal{L}_{N,k}(\Lambda,2^{h})=\max\left\\{-1,\operatorname{vdim}\mathcal{L}_{N,k}(\Lambda,2^{h})\right\\}.$

Since $\Lambda$ and the scheme of double points are disjoint, the virtual
dimension provides a lower bound to the actual dimension:

(4.2)
$\operatorname{edim}\mathcal{L}_{N,k}(\Lambda,2^{h})\leq\dim\mathcal{L}_{N,k}(\Lambda,2^{h}).$

######  Proposition 4.3.

Let $\Lambda\subset\mathbb{P}^{N}$ be a linear subspace of dimension $n$ and
let $Z_{\Lambda}\subset{\mathbb{P}}^{N}$ be a double point scheme supported on
a collection of points in general position in ${\mathbb{P}}^{N}$. Then if

(4.3) $h\leq\frac{1}{N+1}{{N+k-1}\choose N},$

and $(N,k-1,h)$ is not in the list of exceptions of Theorem 4.1, and $k\geq
2$, then

(4.4)
$\dim\mathcal{L}_{N,k}(\Lambda,2^{h})=\operatorname{edim}\mathcal{L}_{N,k}(\Lambda,2^{h}).$

###### Proof.

If $k=2$ and $h=0$, the conclusion follows from (4.1). If $k=2$ and $h=1$, it
is easy to see that all elements of ${\mathcal{L}}_{N,2}(\Lambda,2)$ are
pointed quadric cones containing $\Lambda$ and hence we have the isomorphism
${\mathcal{L}}_{N,2}(\Lambda,2)\cong{\mathcal{L}}_{N-1,2}(\Lambda)$. By (4.1),
we have that $\dim{\mathcal{L}}_{N-1,2}(\Lambda)={{N+1}\choose
2}-{{n+2}\choose 2}$. We conclude noticing that the latter equals the expected
dimension of ${\mathcal{L}}_{N,2}(\Lambda,2)$.

Now, assume $k\geq 4$ and consider the following exact sequence obtained by
restricting $\mathcal{L}_{N,k}(\Lambda,2^{h})$ to a general hyperplane
$H\subset\mathbb{P}^{N}$ such that $\Lambda\subseteq H$:

(4.5)
$0\to\mathcal{L}_{N,k-1}(2^{h})\to\mathcal{L}_{N,k}(\Lambda,2^{h})\to\mathcal{L}_{N,k}(\Lambda,2^{h})|_{H}\subseteq\mathcal{L}_{N-1,k}(\Lambda).$

Under the assumption (4.3) and using Theorem 4.1, the kernel system
$\mathcal{L}_{N,k-1}(2^{h})$ has dimension equal to its virtual dimension:

$\dim\mathcal{L}_{N,k-1}(2^{h})={{N+k-1}\choose N}-h(N+1)-1,$

and in particular $H^{1}({\mathbb{P}}^{N},\dim\mathcal{L}_{N,k-1}(2^{h}))=0$,
so that we have the following exact sequence in cohomology:

$0\to\operatorname{H}^{0}({\mathbb{P}}^{N},\mathcal{L}_{N,k-1}(2^{h}))\to
H^{0}({\mathbb{P}}^{N},\mathcal{L}_{N,k}(\Lambda,2^{h}))\to\operatorname{H}^{0}(H,\mathcal{L}_{N,k}(\Lambda,2^{h})|_{H})\to
0.$

Moreover by (4.1)

$\operatorname{h}^{0}({\mathbb{P}}^{N-1},\mathcal{L}_{N-1,k}(\Lambda))={{N-1+k}\choose
N-1}-{{n+k}\choose n},$

and so

$\operatorname{h}^{0}(H,\mathcal{L}_{N,k}(\Lambda,2^{h})|_{H})\leq{{N-1+k}\choose
N-1}-{{n+k}\choose n}.$

From the exact sequence of global sections we obtain:

$\displaystyle\operatorname{h}^{0}({\mathbb{P}}^{N},\mathcal{L}_{N,k}(\Lambda,2^{h}))$
$\displaystyle=\operatorname{h}^{0}({\mathbb{P}}^{N},\mathcal{L}_{N,k-1}(2^{h}))+\operatorname{h}^{0}(H,\mathcal{L}_{N,k}(\Lambda,2^{h})|_{H})$
$\displaystyle\leq{{N+k}\choose N}-{{n+k}\choose n}-h(N+1).$

We conclude the proof of this case using (4.2).

Finally, assume that $k=3$. In this case, the bound on the number of points is
$h\leq\frac{N}{2}+1$. We consider the restriction to a general hyperplane
containing $\Lambda$ as in (4.5). The kernel system is special by Theorem 4.1,
and one can easily check that it has dimension

$\dim\mathcal{L}_{N,2}(2^{h})={{N+2}\choose 2}-h(N+1)+{h\choose 2}-1,$

see for instance [PosPhD, Section 1.2.1]. Moreover, as a simple consequence of
Bézout’s Theorem, the linear system $\mathcal{L}_{N,3}(\Lambda,2^{h})$
contains in its base locus the lines spanned by pairs of points, each of which
intersects $H$ in a point. We claim that the base locus of
$\mathcal{L}_{N,3}(\Lambda,2^{h})$ is supported on the union of $\Lambda$ and
these lines. This implies that the restricted system is the complete linear
system of cubics containing $\Lambda$ and passing simply through the
${h\choose 2}$ trace points:

$\mathcal{L}_{N,3}(\Lambda,2^{h})|_{H}=\mathcal{L}_{N-1,3}(\Lambda,1^{{h\choose
2}}).$

We claim that the linear system on the right hand side of the above expression
is non-special, namely that the scheme given by $\Lambda$ and the simple
points impose independent conditions to the cubics of ${\mathbb{P}}^{N-1}$.
This shows that

$\displaystyle\dim\mathcal{L}_{N,3}(\Lambda,2^{h})$
$\displaystyle=\dim\mathcal{L}_{N,2}(2^{h})+\dim\mathcal{L}_{N-1,3}(\Lambda,1^{{h\choose
2}})+1$ $\displaystyle=\left({{N+2}\choose 2}-h(N+1)+{h\choose 2}-1\right)$
$\displaystyle\quad+\left({{N+2}\choose 3}-{{n+3}\choose 3}-{h\choose
2}-1\right)+1,$

which implies that $\mathcal{L}_{N,3}(\Lambda,2^{h})$ is non-special.

We are left to proving the two claims. For the second claim, first of all
notice that there is a hyperplane inside $H$, containing all ${h\choose 2}$
points. This can be taken to be the intersection with $H$ of a hyperplane of
${\mathbb{P}}^{N}$ containing the $h$ original points. Call $H_{1}\subset H$
the intersection and restrict the linear system
$\mathcal{L}_{N-1,3}(\Lambda,1^{{h\choose 2}})$ to it, giving rise to the
following exact sequence:

$0\to\mathcal{L}_{N-1,2}(\Lambda)\to\mathcal{L}_{N-1,3}(\Lambda,1^{{h\choose
2}})\to\mathcal{L}_{N-2,3}(1^{{h\choose 2}}).$

Both external linear systems are non-special (for the restricted one see for
instance [Po, Proposition 3.12]), then so is the middle one, concluding the
proof of the claim.

As for the first claim: we show that $\mathcal{L}_{N,3}(\Lambda,2^{h})$ has no
additional base locus other than $\Lambda$ and the lines spanned by pairs of
points. Let’s call $p_{1},\dots,p_{h}$ the $h$ assigned pints in general
position. Assume that $q$ is a point in ${\mathbb{P}}^{N}$ in linearly general
position with respect to $p_{1},\dots,p_{h}$. Since $h\leq\frac{N}{2}+1<N$,
there is a hyperplane $A$ containing $p_{1},\dots,p_{h}$ but not containing
$q$. Since $n<N$, there is a hyperplane $B$ containing $\Lambda$ but not
containing $q$. The cubic $2A+B$ belongs to $\mathcal{L}_{N,3}(\Lambda,2^{h})$
proving that $q$ cannot be a base point. Assume now that $q$ is a point in
${\mathbb{P}}^{N}$ not in linearly general position with respect to
$p_{1},\dots,p_{h}$, which means that there is a linear space spanned by some
of the $p_{i}$’s containing $q$, but such that $q$ does not belong to any of
the lines $\langle p_{i},p_{j}\rangle$. Let $\langle p_{i}:i\in I_{q}\rangle$
be the minimum such linear span and choose two distinct indices
$j_{1},j_{2}\in I_{q}$. Let $A_{1}$ be a hyperplane containing all $p_{i}$’s
with $i\neq j_{1}$ and let $A_{2}$ be a hyperplane containing all $p_{i}$’s
with $i\neq j_{2}$. Let $B$ be a hyperplane containing $\Lambda$, $p_{j_{1}}$
and $p_{j_{2}}$ and not containing $q$. The cubic $A_{1}+A_{2}+B$ belongs to
$\mathcal{L}_{N,3}(\Lambda,2^{h})$ proving that $q$ cannot be a base point.
Finally, since the multiplicity of the general element of the linear system of
cubics along the line $q\in\langle p_{i}:i\in I_{q}\rangle$ is exactly $1$,
then the above cases are exhaustive and this conclude the proof of the claim.
∎

###  4.2. Hypersurfaces containing the Veronese variety and a fat point

Let $V=V_{n,d}\subset\mathbb{P}^{n}$ be the $d$-th Veronese embedding of
$\mathbb{P}^{n}$ and let $\\{p^{a}\\}\subset V\subset{\mathbb{P}}^{n}$ be a
fat point scheme with support on $V$. Let $\mathcal{I}_{V}$ be the ideal sheaf
of $V\subset\mathbb{P}^{N}$ and let $\mathcal{I}_{p^{a}}$ be the ideal sheaf
of $Z\subset{\mathbb{P}}^{N}$. Consider the sheaf

$\mathcal{L}_{N,k}(V,a):=\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{V}\otimes\mathcal{I}_{p^{a}}.$

We are interested in computing the dimension of the space of global sections.
The Hilbert polynomial of $V\subset\mathbb{P}^{N}$ at degree $k$ is

${{n+kd}\choose n}$

or, equivalently, we have

$\dim\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{V}={{N+k}\choose
N}-{{n+kd}\choose n}-1.$

The scheme given by a point of multiplicity $a$ of ${\mathbb{P}}^{N}$ imposes

(4.6) ${{N+a-1}\choose N}$

conditions to the hypersurfaces of ${\mathbb{P}}^{N}$ of degree $k$. Therefore
the _virtual dimension_ of $\mathcal{L}_{N,k}(V,a)$, obtained by a parameter
count, is

${{N+k}\choose N}-{{n+kd}\choose n}-{{N+a-1}\choose N}-1.$

It does not yield a satisfactory notion of expected dimension for the linear
system $\mathcal{L}_{N,k}(V,a)$, due to the fact that the two subschemes $V$
and $\\{p^{a}\\}$ of ${\mathbb{P}}^{N}$ have nonempty intersection so that
some of the conditions imposed by them individually to the hypersurfaces of
degree $k$ of ${\mathbb{P}}^{N}$ will overlap. For instance, if we first
impose $V$ and then $\\{p\\}$, clearly the latter will not give any
independent condition, because, by the containment relation $p\in V$, $p$ is a
base point of the linear system
$\mathcal{L}_{N,k}(V)=\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{V}$.
When the support of a fat point subscheme
$Z=\\{p^{a}\\}\subset{\mathbb{P}}^{N}$, whose length is given in (4.6), lies
on the $n$-dimensional subvariety $V$, the restriction $Z|_{V}\subset V$ is a
subscheme of length

${{n+a-1}\choose n}.$

Therefore we may define the following notion of expected dimension.

####  4.2.1. A notion of expected dimension

We introduce the following refined parameter count.

######  Definition 4.4.

Let $V\subset\mathbb{P}^{N}$ be the $d$-th Veronese embedding of
$\mathbb{P}^{N}$ and let $\\{p^{a}\\}\subset{\mathbb{P}}^{N}$ be a fat point
scheme supported on $V$. The _expected dimension_ of $\mathcal{L}_{N,k}(V,a)$,
denoted by $\operatorname{edim}\mathcal{L}_{N,k}(V,a)$, is the following
integer:

$\max\left\\{-1,{{N+k}\choose N}-{{n+kd}\choose n}-\left[{{N+a-1}\choose
N}-{{n+a-1}\choose n}\right]-1\right\\}.$

That the integer of Definition 4.4 is a lower bound to the actual dimension of
$\mathcal{L}_{N,k}(V,a)$ is not an obvious statement. We will show that it
does when $a\leq k$.

######  Proposition 4.5.

Let $\nu_{d}:\mathbb{P}^{n}\to\mathbb{P}^{N}$ be the $d$-the Veronese
embedding. Let $V=V_{n,d}:=\nu_{d}(\mathbb{P}^{n})\subset\mathbb{P}^{N}$ and
let $Z_{V}=\\{p^{a}\\}\subset{\mathbb{P}}^{n}$ be an ordinary fat point of
multiplicity $a\leq k$ supported on $V$. Then

(4.7)
$\dim\mathcal{L}_{N,k}(V,a)\geq\operatorname{edim}\mathcal{L}_{N,k}(V,a).$

###### Proof.

We consider the linear system
$\mathcal{L}_{N,k}(a)=\mathcal{O}_{{\mathbb{P}}^{N}}\otimes\mathcal{I}_{Z_{V}}$
of the degree-$k$ hypersurfaces of ${\mathbb{P}}^{N}$ with a point of
multiplicity $a$ with support on $V$. Restriction to $V$ gives the following
Castelnuovo sequence:

$0\to\mathcal{L}_{N,k}(V,a)\to\mathcal{L}_{N,k}(a)\to\mathcal{L}_{N,k}(a)|_{V}.$

It is an easy observation that a fat point of multiplicity $a$ imposes
independent conditions to the hypersurfaces of fixed degree of
${\mathbb{P}}^{N}$, as long as the multiplicity does not exceed the degree.
Therefore we can obtain the dimension of the linear system
$\mathcal{L}_{N,k}(a)$ by a parameter count:

$\dim\mathcal{L}_{N,k,V}(a)={{N+k}\choose N}-{{N+a-1}\choose N}-1$

In particular $\operatorname{h}^{1}(\mathbb{P}^{N},\mathcal{L}_{N,k}(a))=0$,
so that we have the following sequence in cohomology:

$\displaystyle
0\to\operatorname{H}^{0}(\mathcal{L}_{N,k}(V,a))\to\operatorname{H}^{0}(\mathcal{L}_{N,k}(a))\to\operatorname{H}^{0}(\mathcal{L}_{N,k}(a)|_{V})\to\operatorname{H}^{1}(\mathcal{L}_{N,k}(V,a))\to
0.$

Since the Veronese morphism $\nu_{d}:\mathbb{P}^{n}\to{\mathbb{P}}^{N}$ gives
an isomorphism of ${\mathbb{P}}^{n}$ to its image $V$, then the pull-back of
$\mathcal{L}_{N,k}(a)|_{V}$ is a linear system of degree-$kd$ hypersurfaces of
$\mathbb{P}^{n}$:

$\nu_{d}^{\ast}(\mathcal{L}_{N,k}(a)|_{V})\subseteq\mathcal{O}_{\mathbb{P}^{n}}(kd)\otimes\mathcal{I}_{Z^{\prime}}=:\mathcal{L}_{n,kd}(a)$

where $Z^{\prime}\subset\mathbb{P}^{n}$ is a fat point of multiplicity $a$
with support a general point of $V$. Since $nk\geq a$ by the assumption, the
linear system $\mathcal{L}_{n,kd}(a)$ has dimension

$\dim\mathcal{L}_{n,kd}(a)={{n+kd}\choose n}-{{n+a-1}\choose n}-1.$

From this we obtain

$\dim\mathcal{L}_{N,k}(a)|_{V}\leq{{n+kd}\choose n}-{{n+a-1}\choose n}-1.$

Putting everything together gives the following inequalities

$\displaystyle\operatorname{h}^{0}(\mathcal{L}_{N,k}(V,a))$
$\displaystyle=\operatorname{h}^{0}(\mathcal{L}_{N,k}(a))-\operatorname{h}^{0}(\mathcal{L}_{N,k}(a)|_{V})+\operatorname{h}^{1}(\mathcal{L}_{N,k}(V,a))$
$\displaystyle\geq\operatorname{h}^{0}(\mathcal{L}_{N,k}(a))-\operatorname{h}^{0}(\mathcal{L}_{N,k}(a)|_{V})$
$\displaystyle\geq\left({{N+k}\choose N}-{{N+a-1}\choose
N}\right)-\left({{n+kd}\choose n}-{{n+a-1}\choose n}\right),$

which conclude the proof. ∎

####  4.2.2. Dimensionality via apolarity and toric geometry

Let $V=V_{n,d}\subset{\mathbb{P}}^{N}$ be the Veronese variety and let
${\bf\Pi}\subset{\mathbb{P}}^{N}$ be a union of $n$-planes, degeneration of
$V$, as in Section 3.2. Let $p\in V$ and $p_{0}\in\Pi_{1}\subset{\bf\Pi}$ be
general points. Consider the linear systems on ${\mathbb{P}}^{N}$

$\displaystyle\mathcal{L}_{N,k}(V,a)$
$\displaystyle:=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{V}\otimes\mathcal{I}_{p^{a}},$
$\displaystyle\mathcal{L}_{N,k}({\bf\Pi},a)$
$\displaystyle:=\mathcal{O}_{{\mathbb{P}}^{N}}(k)\otimes\mathcal{I}_{v}\otimes\mathcal{I}_{p_{0}^{a}}.$

Building on Lemma 3.4, we obtain the following result.

######  Proposition 4.6.

In the above notation, we have

$\dim\mathcal{L}_{N,k}(V,a)\leq\dim\mathcal{L}_{N,k}({\bf\Pi},a).$

###### Proof.

Since $p$ is a general point on $V$, we may assume that it degenerates to a
general point $p_{0}\in S_{1}$. Since ${\bf\Pi}\cup\\{p_{0}^{a}\\}$ is a flat
degeneration of the scheme $V\cup\\{p^{a}\\}$, then the Hilbert function of
the former is at most that of latter, by semi-continuity. This concludes the
proof. ∎

######  Proposition 4.7.

In the above notation and for any $1\leq a\leq k$, then

$\dim\mathcal{L}_{N,k}({\bf\Pi},a)={{N+k}\choose N}-{{n+kd}\choose
n}-{{N+a-1}\choose N}+{{n+a-1}\choose n}-1.$

###### Proof.

Given the union ${\bf\Pi}:=\bigcup_{i=1}^{d^{n}}\Pi_{i}$ of torus invariant
$n$-planes of ${\mathbb{P}}^{n}$, with $\Pi_{1}$ a sink and $p_{0}$ supported
generically on $\Pi_{1}$, there is a torus invariant hyperplane $H$ such that
$\Pi_{1}\cap H$ is an $(n-1)$-plane and $\Pi_{i}\subset H$ for $2\leq i\leq
d^{n}$ (cf. Remark 3.3). We can always assume that $p_{0}$ is a coordinate
point of ${\mathbb{P}}^{N}$ and we can call $p_{1},\dots,p_{N}$ the other
coordinate (torus invariant) points of ${\mathbb{P}}^{N}$. Hence we can
choose, without loss of generality, that $\Pi_{1}=\langle
p_{0},\dots,p_{n}\rangle$ and $H=\langle p_{1},\dots,p_{N}\rangle$, so that
$p_{0}\in\Pi_{1}$ and $p_{i}\notin\Pi_{i}$ for $i\geq 2$.

Let $R=\mathbb{C}[x_{0},\dots,x_{N}]$ be the homogeneous polynomial ring of
${\mathbb{P}}^{N}$ and consider the ideals
$\mathcal{I}_{p_{0}}\subset\mathbb{C}[x_{0},\dots,x_{N}]$ and
$\mathcal{I}_{\Pi_{i}}\subset\mathbb{C}[x_{0},\dots,x_{N}]$

$\displaystyle\mathcal{I}_{p_{0}}$ $\displaystyle=\langle
x_{1},\dots,n_{N}\rangle,$ $\displaystyle\mathcal{I}_{\Pi_{1}}$
$\displaystyle=\langle x_{n+1},\dots,n_{N}\rangle,$
$\displaystyle\mathcal{I}_{\Pi_{i}}$ $\displaystyle=\langle
x_{i_{n+1}},\dots,n_{i_{N}}\rangle,\ i\geq 2,$ $\displaystyle\mathcal{I}_{H}$
$\displaystyle=\langle x_{0}\rangle.$

By construction, for $i\geq 2$ , we have $0\in\\{{i_{n+1}},\dots,{i_{N}}\\}$.
Using Notation 2.10, we compute:

$\displaystyle\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}$
$\displaystyle=\\{y_{0}^{k-l}F_{l}(y_{1},\dots,y_{N}):F_{l}\in S_{l},0\leq
l\leq a-1\\},$ $\displaystyle\left[\mathcal{I}_{\Pi_{1}}^{-1}\right]_{k}$
$\displaystyle=\\{F_{k}(y_{0},\dots,y_{n}):F_{k}\in S_{k}\\},$
$\displaystyle\left[\mathcal{I}_{\Pi_{i}}^{-1}\right]_{k}$
$\displaystyle=\\{F_{k}(y_{i_{0}},\dots,y_{i_{n}}):F_{k}\in S_{k}\\},i\geq 2,$

where for $i\geq 2$, the index set $\\{i_{0},\dots,i_{n}\\}$ is the complement
of $\\{i_{n+1},\dots,{i_{N}}\\}\subset\\{0,\dots,N\\}$. We have the following
intersections

$\displaystyle\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}\cap\left[\mathcal{I}_{\Pi_{i}}^{-1}\right]_{k}$
$\displaystyle=\emptyset,i\geq 2,$
$\displaystyle\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}\cap\left[\mathcal{I}_{\Pi_{1}}^{-1}\right]_{k}$
$\displaystyle=\\{y_{0}^{k-l}F_{l}(y_{1},\dots,y_{n}):F_{l}\in S_{l}\\}.$

We compute the dimension of the latter intersection:

$\displaystyle\dim\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}\cap\left[\mathcal{I}_{\Pi_{1}}^{-1}\right]_{k}$
$\displaystyle=\dim\\{y_{0}^{k-l}F_{l}(y_{1},\dots,y_{n}):F_{l}\in S_{l},0\leq
l\leq a-1\\}$
$\displaystyle=\sum_{l=0}^{a-1}\dim\\{F_{l}(y_{1},\dots,y_{n}):F_{l}\in
S_{l}\\}$ $\displaystyle=\sum_{l=0}^{a-1}{{n-1+l}\choose{n-1}}$
$\displaystyle={{n+a-1}\choose{n}},$

where the last equality follows a standard relation of Newton coefficients,
commonly known as the _hockey stick identity_.

The number of conditions imposed to the linear system of degree-$k$
hypersurfaces of ${\mathbb{P}}^{N}$ by the scheme
$\\{p_{0}^{a}\\}\cup{\bf\Pi}$ is the dimension of the linear span of
$\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}$ and
$\left[\mathcal{I}_{\Pi_{i}}^{-1}\right]_{k}$, for $i=1\dots,d^{n}$, which is
the following integer:

$\displaystyle\quad\dim\left[\mathcal{I}_{p_{0}^{a}}^{-1}\right]_{k}+\dim\left\langle\left[\mathcal{I}_{\Pi_{i}}^{-1}\right]_{k},i=1\dots,d^{n}\right\rangle-\dim\left[\mathcal{I}_{q_{0}^{a}}^{-1}\right]_{k}\cap\left[\mathcal{I}_{\Pi_{1}}^{-1}\right]_{k}$
$\displaystyle={{N+a-1}\choose{N}}+{{n+kd}\choose{n}}-{{n+a-1}\choose{n}}.$

∎

######  Corollary 4.8.

The linear system $\dim\mathcal{L}_{N,k}(V,a)$ has the expected dimension
according to Definition 4.4.

###### Proof.

It follows from Propositions 4.5 and 4.7. ∎

##  5\. Proof of the main theorem

We are ready to prove our main theorem, Theorem 1.1. Thanks to Proposition
2.16, computing the dimension of the $h$-secant varieties of the
$(d,k)$-Veronese variety $V^{k}_{n,d}\subset\mathbb{P}^{N_{dk}}$ is equivalent
to computing the dimension of the linear system in $\mathbb{P}^{N}$ of
hypersurfaces containing the Veronese variety and double at $h$ general
points.

Let $V_{n,d}\subset\mathbb{P}^{N}$ be the $d$-th Veronese embedding of
$\mathbb{P}^{n}$ and let $Z\subset{\mathbb{P}}^{n}$ be a double point scheme
with support a set of points in general position. Let $\mathcal{I}_{V}$ be the
ideal sheaf of $V\subset\mathbb{P}^{N}$ and let $\mathcal{I}_{Z}$ be the ideal
sheaf of $Z\subset{\mathbb{P}}^{N}$. Consider the sheaf

$\mathcal{L}_{N,k}(V,2^{h}):=\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{V}\otimes\mathcal{I}_{Z}.$

Since the Hilbert polynomial of $V\subset\mathbb{P}^{N}$ in degree $k$ is
${{n+kd}\choose n}$ or, in other terms,

(5.1) $\dim\mathcal{O}_{\mathbb{P}^{N}}(k)\otimes\mathcal{I}_{V}={{N+k}\choose
N}-{{n+kd}\choose n}-1,$

and since $h$ double points in general position of ${\mathbb{P}}^{N}$ impose
$h(N+1)$ conditions to the hypersurfaces of ${\mathbb{P}}^{N}$ of degree $k$,
we can give the following definitions.

######  Definition 5.1.

The _virtual dimension_ of the linear system $\mathcal{L}_{N,k,V}(2^{h})$ of
hypersurfaces of ${\mathbb{P}}^{n}$ that vanish along the Veronese variety
$V=V_{n,d}\subset\mathbb{P}^{N}$ and double at $h$ points in general position
is the following integer:

$\operatorname{vdim}\mathcal{L}_{N,k}(V,2^{h})={{N+k}\choose N}-{{n+kd}\choose
n}-h(N+1)-1.$

The _expected dimension_ is

$\operatorname{edim}\mathcal{L}_{N,k}(V,2^{h})=\max\left\\{-1,\operatorname{vdim}\mathcal{L}_{N,k}(V,2^{h})\right\\}.$

Since $V$ and the scheme of double points are disjoint, the virtual dimension
provides a lower bound to the actual dimension:

(5.2)
$\dim\mathcal{L}_{N,k}(V,2^{h})\geq\operatorname{edim}\mathcal{L}_{N,k}(V,2^{h}).$

Using a degeneration argument, we shall show that if the number of points $h$
is not too large, then the linear system $\mathcal{L}_{N,k,V}(V,2^{h})$ has
dimension equal to the expected dimension.

######  Theorem 5.2.

Let $\nu_{d}:\mathbb{P}^{n}\to\mathbb{P}^{N}$ be the $d$-the Veronese
embedding. Let $V=V_{n,d}:=\nu_{d}(\mathbb{P}^{n})\subset\mathbb{P}^{N}$ and
let $Z_{V}\subset{\mathbb{P}}^{n}$ be a double point scheme supported on $h$
points in general position of ${\mathbb{P}}^{N}$. Then if $k\geq 3$ and

(5.3) $h\leq\frac{1}{N+1}{{N+k-3}\choose N}$

then

(5.4)
$\dim\mathcal{L}_{N,k}(V,2^{h})=\operatorname{edim}\mathcal{L}_{N,k}(V,2^{h}).$

###### Proof.

Using (5.2), it is enough to prove that the inequality
$\dim\mathcal{L}_{N,k}(V,2^{h})\leq\operatorname{edim}\mathcal{L}_{N,k}(V,2^{h})$
holds.

If $k=3$, then $h=0$ so the statement follows from equation (5.1). For $k\geq
4$, we will prove the statement by means of the $\mathbb{FP}-$degeneration
introduced in Section 3.1.2, applied to the line bundle

$\displaystyle\mathcal{L}_{\widetilde{\mathcal{X}}}:=$
$\displaystyle\mathcal{M}_{\widetilde{\mathcal{X}}}(k,k-1;\widetilde{\mathcal{V}},2,\dots,2).$
By Remark 3.2, the line bundle on the general fiber is isomorphic to
$\displaystyle\mathcal{L}_{t}:=$ $\displaystyle\mathcal{L}_{N,k}(V,2^{h}),$
while on the central fiber the linear systems on the two components are the
following: $\displaystyle\mathcal{L}_{\mathbb{P}}:=$
$\displaystyle{\mathcal{L}}_{N,k-1}(\Lambda,2^{h}),$
$\displaystyle\mathcal{L}_{\mathbb{F}}:=$
$\displaystyle{\mathcal{L}}_{N,k}(V,k-1).$ We consider the restriction to
$R=\mathbb{P}\cap{\mathbb{F}}$: the kernels on the two components are,
respectively: $\displaystyle\hat{\mathcal{L}}_{\mathbb{P}}:=$
$\displaystyle{\mathcal{L}}_{N,k-2}(\Lambda,2^{h}),$
$\displaystyle\hat{\mathcal{L}}_{\mathbb{F}}:=$
$\displaystyle{\mathcal{L}}_{N,k}(V,k).$ Since $R\cong\mathbb{P}^{N-1}$, the
two restricted systems satisfy the following:
$\displaystyle\mathcal{R}_{\mathbb{P}}:=$
$\displaystyle\mathcal{L}_{\mathbb{P}}|_{R}\subset\mathcal{L}_{N-1,k-1}(\Lambda_{R}),$
$\displaystyle\mathcal{R}_{\mathbb{F}}:=$
$\displaystyle\mathcal{L}_{\mathbb{F}}|_{R}\subset\mathcal{L}_{N-1,k-1}(\Lambda_{R}),$

where we recall that $\Lambda_{R}=\Lambda\cap R\cong\mathbb{P}^{n-1}$.

We first look at the exceptional component $\mathbb{P}$. By Proposition 4.3,
since $k-2\geq 2$ and

$h\leq\frac{1}{N+1}{{N+k-3}\choose{N}},$

both linear systems $\mathcal{L}_{\mathbb{P}}$ and
$\hat{\mathcal{L}}_{\mathbb{P}}$ have the expected dimension, that is

$\displaystyle\dim\mathcal{L}_{\mathbb{P}}$ $\displaystyle={{N+k-1}\choose
N}-{{n+k-1}\choose n}-h(N+1)-1$ (5.5)
$\displaystyle\dim\hat{\mathcal{L}}_{\mathbb{P}}$
$\displaystyle={{N+k-2}\choose N}-{{n+k-2}\choose n}-h(N+1)-1.$

Moreover, we have a short exact sequence of spaces of global sections:

$0\to\operatorname{H}^{0}(\mathbb{P},\hat{\mathcal{L}}_{\mathbb{P}})\to\operatorname{H}^{0}(\mathbb{P},{\mathcal{L}}_{\mathbb{P}})\to\operatorname{H}^{0}(R,\mathcal{R}_{\mathbb{P}})\to
0.$

In particular, we can compute

$\displaystyle\dim\mathcal{R}_{\mathbb{P}}$
$\displaystyle=\dim{\mathcal{L}}_{\mathbb{P}}-\dim\hat{\mathcal{L}}_{\mathbb{P}}+1$
$\displaystyle={{N+k-2}\choose{N-1}}-{{n+k-2}\choose{n-1}}-1$
$\displaystyle=\dim\mathcal{L}_{N-1,k-1}(\Lambda_{R}).$

We conclude that $\mathcal{R}_{\mathbb{P}}$ is the complete linear system

$\mathcal{R}_{\mathbb{P}}=\mathcal{L}_{N-1,k-1}(\Lambda_{R}).$

On the component ${\mathbb{F}}$, using Corollary 4.8, we have that both
$\mathcal{L}_{\mathbb{F}}$ and $\hat{\mathcal{L}}_{\mathbb{F}}$ have the
expected dimension, that is

$\displaystyle\dim\mathcal{L}_{\mathbb{F}}$ $\displaystyle={{N+k}\choose
N}-{{n+kd}\choose n}-\left({{N+k-2}\choose N}-{{n+k-2}\choose n}\right)-1$
(5.6) $\displaystyle\dim\hat{\mathcal{L}}_{\mathbb{F}}$
$\displaystyle={{N+k}\choose N}-{{n+kd}\choose n}-\left({{N+k-1}\choose
N}-{{n+k-1}\choose n}\right)-1.$

We claim that

$\mathcal{R}_{\mathbb{F}}=\mathcal{L}_{N-1,k-1}(\Lambda_{R}),$

so that, together with the above argument, we have

(5.7)
$\mathcal{R}_{\mathbb{P}}\cap\mathcal{R}_{\mathbb{F}}=\mathcal{O}_{\mathbb{P}^{N-1}}(k-1)\otimes\mathcal{I}_{\Lambda_{R}}.$

In order to prove the claim, we observe that by semicontinuity, and precisely
Formula (3.1), and by (5.2), we have

(5.8)
$\dim\mathcal{L}_{0}\geq\dim\mathcal{L}_{t}\geq\operatorname{edim}\mathcal{L}_{t}={{N+k}\choose
N}-{{n+kd}\choose n}-h(N+1)-1.$

Using Formula (3.2), i.e.,

(5.9) $\displaystyle\dim\mathcal{L}_{0}$
$\displaystyle=\dim\hat{\mathcal{L}}_{\mathbb{P}}+\dim\hat{\mathcal{L}}_{{\mathbb{F}}}+\mathcal{R}_{\mathbb{P}}\cap\mathcal{R}_{\mathbb{F}}+2$

and observing that
$\mathcal{R}_{\mathbb{P}}\cap\mathcal{R}_{\mathbb{F}}=\mathcal{R}_{\mathbb{F}}$,
we obtain

$\displaystyle\dim\mathcal{R}_{\mathbb{F}}$
$\displaystyle\geq\operatorname{edim}\mathcal{L}_{t}-\dim\hat{\mathcal{L}}_{\mathbb{P}}-\dim\hat{\mathcal{L}}_{\mathbb{F}}-2={{N+k-2}\choose{N-1}}-{{n+k-2}\choose{n-1}}-1;$

the proof of the latter equality is easy and left to the reader. Since

$\displaystyle\dim\mathcal{R}_{\mathbb{F}}\leq\dim\mathcal{O}_{\mathbb{P}^{N-1}}(k-1)\otimes\mathcal{I}_{\Lambda_{R}}={{N+k-2}\choose{N-1}}-{{n+k-2}\choose{n-1}}-1,$

the claim follows and we have

(5.10)
$\dim\mathcal{R}_{\mathbb{P}}\cap\mathcal{R}_{\mathbb{F}}={{N+k-2}\choose{N-1}}-{{n+k-2}\choose{n-1}}-1,$

Using (5.5), (5.6), (5.9) and (5.10), we obtain
$\dim\mathcal{L}_{0}=\operatorname{edim}\mathcal{L}_{t}$. We conclude using
(5.8).

∎

Theorem 1.1 is now just a corollary of what we just proved.

######  Corollary 5.3.

For $k\geq 3$, if (5.3) holds, then the $(d,k)-$Veronese variety
$V_{n,d}^{k}\subset\mathbb{P}^{N_{dk}}$ is non-defective.

###### Proof.

It follows from Theorem 5.2 and Proposition 2.16. ∎

Theorem 1.2 is an easy consequence of Theorem 1.1 and Theorem 2.9.

######  Corollary 5.4.

For $k\geq 3$, if

$h\leq\min\left\\{\frac{1}{N+1}{N+k-3\choose
N}-1,\left\lfloor\frac{{{n+kd}\choose{n}}}{N+1}\right\rfloor-1\right\\}$

then the $(d,k)-$Veronese variety $V_{n,d}^{k}\subset\mathbb{P}^{N_{dk}}$ is
$h$-identifiable.

###### Proof.

It follows from Corollary 5.3 and Theorem 2.9. ∎

######  Remark 5.5.

Note that the bound given in Theorem 1.1 can be strictly larger then the
expected generic rank for the $(d,k)-$Veronese $V_{n,d}^{k}$, see for example
the computations in Section 6 below. In this case Theorem 1.1 proves the non
defectivity of $V_{n,d}^{k}$ up to the generic rank. The identifiability
property instead is more subtle and in order to be able to use Theorem 2.9 we
must ensure that the embedded secant variety $\mathbb{S}ec_{h}(V_{n,d}^{k})$
has the same dimension as the abstract secant variety $sec_{h}(V_{n,d}^{k})$,
see definition 2.5. Equivalently we require that the $(h+1)-$secant map is
generically finite. Because of this assumption we are not able to prove
generic identifiability and we must restrict ourselves to the range given in
Theorem 1.2.

##  6\. Asymptotical Bound

In this section we relate our bound in Theorem 1.1 with the one given in
[Nen]. For the sake of completeness, we recall Nenashev’s result first.

######  Theorem 6.1.

[Nen, Theorem 1] Let $I$ be a homogeneous ideal generated by
$h\in\mathbb{N}\setminus\\{0\\}$ generic elements of some nonempty variety
$\mathcal{D}\subseteq\operatorname{Sym}^{r}(\mathbb{C}^{n})$ of $r$-forms that
is closed under linear transformations. Fix an integer $s\geq 0$. If

$h\leq\left({{r+s+n-1}\choose{n-1}}/{{s+n-1}\choose{n-1}}\right)-{{s+n-1}\choose{n-1}},$

then the dimension of $I$ in degree $(r+s)$ is maximal, i.e. it equals
$h{{s+n-1}\choose{n-1}}$.

Note that when $r=d(k-1)$ and $s=r$ the component of $I$ of degree $r+s=dk$
gives exactly the dimension of the $h-$ secant variety
$\mathbb{S}ec_{h}(V_{n,d}^{k})$, where $\mathcal{D}$ is the tangential variety
of $V_{n,d}^{k}$, i.e.

$\mathcal{D}=\\{F^{k-1}G|F,G\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}\\}.$

We have the following consequence.

######  Corollary 6.2.

The dimension of $\mathbb{S}ec_{h}(V_{n,d}^{k})$ is the expected one, i.e.

$\dim\mathbb{S}ec_{h}(V_{n,d}^{k})=h{{n+d}\choose{d}}-1$

for $h\leq\frac{{{n+dk}\choose{dk}}}{{{n+d}\choose{d}}}-{{n+d}\choose{d}}$.

Note that if we fix $k,n$ and let $d\gg 0$ we have that

$\frac{{{n+dk}\choose{dk}}}{{{n+d}\choose{d}}}-{{n+d}\choose{d}}\sim
k^{n}-d^{n}$

and if $d\gg k$ the bound is trivial. In Theorem 1.1 and under the same
assumptions we get

$\frac{1}{N+1}{{N+k-3}\choose{N}}\sim d^{n(k-4)}$

which gives non trivial bounds for $d\gg k$ when $k>4$.

The figure shows in red the bound given by Nenashev, in blue the bound of
Theorem 1.2 as a function of $d$ and in green the expected generic rank
${{n+kd}\choose{kd}}/{{n+d}\choose{d}}$ of $V_{n,d}^{k}$. In this case we have
set the values $k=5$ and $n=2$. For $d>3$ our bound overtakes Nenashev’s and
it continues to give informations also in the range $d>4$.

We also notice that for $d$ big enough our bound in Theorem 1.1 exceeds the
expected generic rank ${{n+kd}\choose{kd}}/{{n+d}\choose{d}}$ of
$V_{n,d}^{k}$, as predicted in Remark 5.5. This shows that under the
hypothesis of a very high degree reembedding of $\mathbb{P}^{n}$ the variety
$V_{n,d}^{k}$ is never defective for every $k\geq 3$.

## References


###### Abstract

The aim of this article is to introduce the concept of centrally-extended
Jordan endomorphisms and proving that if $R$ is a non-commutative prime ring
of characteristic not two, and $G$ is a CE- Jordan epimorphism such that
$[G(x),x]\in Z(R)$ ($[G(x),x^{*}]\in Z(R)$) for all $x\in R$, then $R$ is an
order in a central simple algebra of dimension at most $4$ over its center or
there is an element $\lambda$ in the extended of $R$ such that $G(x)=\lambda
x$ ($G(x)=\lambda^{*}x^{*}$) for all $x\in R$.

On centrally-extended Jordan endomorophisms in rings

Aziza Gouda and H. Nabiel

Department of Mathematics, Faculty of Science, Fayoum University, Fayoum,
Egypt

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City,
Cairo, Egypt

<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
Mathematics Subject Classification (2020): 16N60, 16W10, 16U10.
Keywords: Derivations, Centrally-extended derivations, Endomorphisms,
Centrally-extended endomorphisms, Centrally-extended Jordan endomorphisms,
prime rings.

##  1 Introduction

Throughout this paper, $R$ denotes an associative ring with center $Z(R)$,
$Q_{mr}(R)=Q$ denotes the maximal right ring of quotients of $R$ and the
center of $Q_{mr}(R)$ is called the extended centroid of $R$ and denoted by
$C$. By a ring with involution $*$, we mean a ring equipped with an involution
$*$, it is also called $*$-ring. For $x,y\in R$ we have $[x,y]=xy-yx$ and
$x\circ y=xy+yx$. In 2016, Bell and Daif [2] introduced the concept of
centrally-extended derivations and centrally-extended endomorphisms on a ring
$R$ with center $Z(R)$. A map $D$ on $R$ is called a centrally-extended
derivation (CE-derivation) if $D(x+y)-D(x)-D(y)\in Z(R)$ and
$D(xy)-D(x)y-xD(y)\in Z(R)$ for all $x,y\in R$. A map $T$ of $R$ is said to be
a centrally-extended endomorphism (CE-endomorphism) if for each $x,y\in R$,
$T(x+y)-T(x)-T(y)\in Z(R)$ and $T(xy)-T(x)T(y)\in Z(R)$. They showed that if
$R$ is a semiprime ring with no nonzero central ideals, then every CE-
derivation is a derivation and every CE-epimorphism is an epimorphism.

Through the last five years, the authors in [[13], [8],[7], [10], [11], [14]
[9], [12], [15]] introduced the notions of CE-generalized
$(\theta,\phi)$-derivations, CE- generalized *-derivations, CE-reverse
derivations, CE-generalized reverse derivations, CE-higher derivation, CE-
higher *derivation, CE-homoderivation, CE- $\alpha$-homoderivation,
multiplicative CE-derivation and multiplicative generalized reverse *CE
derivation and proved similar results.

In [3] Bhushan et al. gave the notion of a CE- Jordan derivation to be a
mapping $D$ of $R$ satisfying $D(x+y)-D(x)-D(y)\in Z(R)$ and $D(x\circ
y)-D(x)\circ y-x\circ D(y)\in Z(R)$ for all $x,y\in R$. Also, they gave the
notion of a CE- Jordan *-derivation to be a mapping $D$ of a ring $R$ with
involution $*$ satisfying $D(x+y)-D(x)-D(y)\in Z(R)$ and $D(x\circ
y)-D(x)y^{*}-xD(y)-D(y)x^{*}-yD(x)\in Z(R)$ for all $x,y\in R$. They proved
the following. Let $R$ be a non-commutative prime ring(with involution $*$) of
characteristic not 2. If $R$ admits a CE-Jordan derivation $d$ of $R$ such
that $[d(x),x]\in Z(R)$ ($[d(x),x^{*}]\in Z(R)$) for all $x\in R$, then either
$d=0$ or $R$ is an order in a central simple algebra of dimension at most $4$
over its center. Also they obtained the same results in the case of CE- Jordan
*-derivations.

Bhushan et al. in [4] introduced the notions of CE- generalized derivation
Jordan derivation is a mapping $F$ of $R$ satiesfying $F(x+y)-F(x)-F(y)\in
Z(R)$ and $F(x\circ y)-D(y)x-yD(x)\in Z(R)$ for all $x,y\in R$, where $D$ is a
CE-Jordan derivation of $R$. They proved the following: (i) Let $R$ be a non-
commutative prime ring(with involution $*$) of characteristic not 2. If $R$
admits a CE-generalized derivation $F$ constrained with a CE-Jordan derivation
$d$ of $R$ such that $[F(x),x]\in Z(R)$ ($[d(x),x^{*}]\in Z(R)$) for all $x\in
R$, then $R$ is an order in a central simple algebra of dimension at most $4$
over its center or $F(x)=\lambda x$ for all $x\in R$, where is $\lambda\in C$.
(ii) Let $R$ be a non-commutative prime ring with involution $*$ and
characteristic not 2. If $R$ admits a CE-generalized derivation $F$
constrained with a CE-Jordan derivation $d$ of $R$ such that $[F(x),x^{*}]\in
Z(R)$) for all $x\in R$, then $R$ is an order in a central simple algebra of
dimension at most $4$ over its center or $F=0$.

Our Objective in this article is to introduce the notion of CE- Jordan
endomorphisms and prove the following results:

(i) Let $R$ be a non-commutative prime ring of characteristic not two. If $R$
admits a CE- Jordan epimorphism $G$ such that $[G(x),x]\in Z(R)$ for all $x\in
R$, then $R$ is an order in a central simple algebra of dimension at most $4$
over its center or $G(x)=\lambda x$ for all $x\in R$, where is $\lambda\in C$.

(ii) Let $R$ be a non-commutative prime ring of characteristic not two with
involution $*$. If $R$ admits a CE- Jordan epimorphism $G$ such that
$[G(x),x^{*}]\in Z(R)$ for all $x\in R$, then $R$ is an order in a central
simple algebra of dimension at most $4$ over its center or
$G(x)=\lambda^{*}x^{*}$ for all $x\in R$, where is $\lambda\in C$.

##  2 Preliminary results

The following lemmas help us to prove our results.

######  Lemma 2.1.

[5, Theorem 3.2] Let $R$ be a prime ring. If $g:R\longrightarrow R$ is an
additive mapping such that $[g(x),x]=0$ for all $x\in R$, then there exists
$\lambda\in C$ and an additive mapping $\beta:R\longrightarrow C$, such that
$g(x)=\lambda x+\beta(x)$ for all $x\in R$.

######  Lemma 2.2.

[6, Lemma 1] Let $R$ be a prime ring with extended centroid $C$. Then the
following statements are equivalent: (i) $R$ satisfies $s_{4}$. (ii) $R$ is
commutative or $R$ embeds into $M_{2}(K)$, for a field $K$. (iii) $R$ is
algebraic of bounded degree $2$ over $C$. (iv) $R$ satisfies
$[[x^{2},y],[x,y]]$.

In the following lemma, we generalize [5, Proposition 3.1] to use it in our
results.

######  Lemma 2.3.

Let $R$ be a $2$-torsion free semiprime ring with a nonzero map $\alpha$:
$R\longrightarrow R$ and $\alpha(x+y)-\alpha(x)-\alpha(y)\in Z(R)$. If
$[\alpha(x),x]\in Z(R)$ for all $x\in R$, then $[\alpha(x),x]=0$ for all $x\in
R$.

###### Proof.

By our hypothesis, we have

$[\alpha(x),x]\in Z(R)\>\textit{ for all }\>x\in R.$ (2.1)

Linearizing (2.1) implies that

$[\alpha(x),y]+[\alpha(y),x]\in Z(R)\>\textit{ for all }\>x,y\in R.$ (2.2)

Putting $x^{2}$ instead of $y$ in (2.2) and using (2.1) yield

$2[\alpha(x),x]x+[\alpha(x^{2}),x]\in Z(R)\>\textit{ for all }\>x\in R.$ (2.3)

From (2.1) and (2.3), we obtain

$\begin{split}0=&[\alpha(x),2[\alpha(x),x]x+[\alpha(x^{2}),x]]=2[\alpha(x),x]^{2}+[\alpha(x),[\alpha(x^{2}),x]]\>\textit{
for all }\>x\in R.\end{split}$

From (2.1), we have
$[\alpha(x^{2}),x^{2}]=[\alpha(x^{2}),x]x+x[\alpha(x^{2}),x]\in Z(R)$ for all
$x\in Z(R)$ Thus,

$\begin{split}0=&[\alpha(x),[\alpha(x^{2}),x]x+x[\alpha(x^{2}),x]]=2[\alpha(x),x][\alpha(x^{2}),x]+[\alpha(x),[\alpha(x^{2}),x]]x\\\
&+x[\alpha(x),[\alpha(x^{2}),x]]=2[\alpha(x),x][\alpha(x^{2}),x]-4x[\alpha(x),x]^{2}\>\textit{
for all }\>x\in R.\end{split}$ (2.4)

So, we find

$0=[\alpha(x),2[\alpha(x),x][\alpha(x^{2}),x]-4x[\alpha(x),x]^{2}]=-8[\alpha(x),x]^{3}\>\textit{
for all }\>x\in R.$ (2.5)

The two torsion freeness of $R$ and the semiprimeness imply that
$[\alpha(x),x]=0$ for all $x\in R$. ∎

By the following lemma, we say [1, Proposition 2.2] is true when we take
$\alpha$: $R\longrightarrow R$ such that $\alpha(x+y)-\alpha(x)-\alpha(y)\in
Z(R)$ for all $x,y\in R$ instead of $\alpha$ is additive. Also, we use it in
proving of our results.

######  Lemma 2.4.

Let $R$ be a $2$-torsion free semiprime ring admitting an involution $*$. If
$\alpha$: $R\longrightarrow R$ a nonzero map,
$\alpha(x+y)-\alpha(x)-\alpha(y)\in Z(R)$ for all $x,y\in R$ and
$[\alpha(x),x^{*}]\in Z(R)$ for all $x\in R$, then $[\alpha(x),x^{*}]=0$ for
all $x\in R$.

###### Proof.

By our assumpation, we have $[\alpha(x),x^{*}]\in Z(R)$ for all $x\in R$.
Putting $x^{*}$ instead of $x$, we get $[\alpha(x^{*}),x]\in Z(R)$ for all
$x\in R$. Define a map $\gamma$: $R\longrightarrow R$ such that
$\gamma(x)=\alpha(x^{*})$ for all $x\in R$. We note
$\gamma(x+y)-\gamma(x)-\gamma(y)\in Z(R)$ for all $x\in R$ and
$[\gamma(x),x]\in Z(R)$ for all $x\in R$. Lemma (2.3) gives $[\gamma(x),x]=0$
for all $x\in R$. This implies that $[\alpha(x^{*}),x]=0$ for all $x\in R$,
hence $[\alpha(x),x^{*}]=0$ for all $x\in R$. ∎

##  3 Centrally-extended Jordan endomorphism

######  Definition 3.1.

A map $G$: $R\longrightarrow R$ is called a centrally-extended Jordan
endomorphism (CE-Jordan endomorphism) if $G(x+y)-G(x)-G(y)\in Z(R)$ and
$G(x\circ y)-G(x)\circ G(y)\in Z(R)$ for all $x,y\in R$.

In the following examples, we make sure the existance of CE-Jordan
endomorphisms.

######  Example 3.2.

Suppose R = $M_{2}(\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers. It
is obvious that the mapping $G$: $R\longrightarrow R$ defined by $G$
$(\begin{matrix}a&b\,\\\ c&t\,\\\ \end{matrix})$ = $(\begin{matrix}c&0\,\\\
0&c\,\\\ \end{matrix})$ is a CE-Jordan endomorphism but not a Jordan
endomorphism.

######  Example 3.3.

Let R = $M_{2}(\mathbb{Z})$. The mapping $G$: $R\longrightarrow R$ defined by
$G$$(\begin{matrix}a&b\,\\\ c&t\,\\\ \end{matrix})$ =
$(\begin{matrix}t&-b\,\\\ -c&a\,\\\ \end{matrix})$ is a CE-Jordan endomorphism
but not a CE-endomorphism.

Example 3.2 shows that Bell and Daif [2, , Theorem 2.8] can not be extended to
the case of CE-endomorphisms but fortunately, we proved the additive condition
in the following lemma.

######  Lemma 3.4.

Let $R$ be a $2$-torsion free ring with no nonzero central ideals. Then every
CE- Jordan epimorphism $G$ of $R$ is additive.

###### Proof.

Since $G$ is a CE-endomorphism, then we have for $x,y,w\in R$

$G(x+y)=G(x)+G(y)+a_{1},\>a_{1}\in Z(R).$ (3.1)

And,

$G(w\circ(x+y))=G(w)G(x+y)+G(x+y)G(w)+a_{2},\>a_{2}\in Z(R).$ (3.2)

By (3.1), we get

$G(w\circ(x+y))=G(w)(G(x)+G(y)+a_{1})+(G(x)+G(y)+a_{1})G(w)+a_{2}.$ (3.3)

Also, we have

$\begin{split}G(w\circ(x+y))&=G((w\circ x)+(w\circ y))=G(w\circ x)+G(w\circ
y)+a_{3}\\\ &=G(w)G(x)+G(x)G(w)+a_{4}+G(w)G(y)+G(y)G(w)\\\
&+a_{5}+a_{3},\>a_{3},a_{4},a_{5}\in Z(R).\end{split}$ (3.4)

Equation (3.3) and (3.4) imply that $2a_{1}G(w)+a_{2}=a_{3}+a_{4}+a_{5}$,
i.e., $2a_{1}G(w)\in Z(R)$ for all $w\in R$. Since $R$ is $2$-torsion free,
then $a_{1}G(w)\in Z(R)$ for all $w\in R$. Using $G$ is surjective gives
$aw\in Z(R)$ for all $w\in R$. Thus, $a_{1}R$ is a central ideal, hence
$a_{1}R=(0)$. Therefore, letting $A(R)$ be the two-sided annihilator of $R$,
we have $a_{1}\in A(R)$. But $A(R)$ is a central ideal, so $a_{1}=0$ and by
(3.1), we get our proof. ∎

######  Corollary 3.5.

Let $R$ be a non-commutative prime ring of characteristic not two. If $G$ is a
CE-Jordan epimorphism of $R$, then $G$ is additive.

Now, we are ready to prove our first theorem.

######  Theorem 3.6.

Let $R$ be a non-commutative prime ring of characteristic not two. If $R$
admits a CE- Jordan epimorphism $G$ such that $[G(x),x]\in Z(R)$ for all $x\in
R$, then $R$ is an order in a central simple algebra of dimension at most $4$
over its center or $G(x)=\lambda x$ for all $x\in R$, where is $\lambda\in C$.

###### Proof.

By our hypothsis and Lemma (2.3), we obtain

$[G(x),x]=0\>\textit{ for all }x\in R.$ (3.5)

Linearizing (3.5) gives

$[G(x),y]+[G(y),x]=0\>\textit{ for all }x,y\in R.$ (3.6)

Substituting $x\circ y$ for $y$ in (3.6) gives

$[G(x),x\circ y]+[G(x\circ y),x]=0\>\textit{ for all }x,y\in R.$

Thus,

$\begin{split}0&=[G(x),x\circ y]+[G(x)G(y)+G(y)G(x),x]\\\ &=[G(x),x\circ
y]+G(x)[G(y),x]+[G(y),x]G(x)\>\textit{ for all }x,y\in R.\end{split}$ (3.7)

Corollary (3.5) implies that $G$ is additive, then (3.5) and Lemma (2.1) give
$G(x)=\lambda x+\beta(x)$ for all $x\in R$ where $\beta:R\longrightarrow C$ is
an additive mapping. By (3.7), we arrive at

$\begin{split}0&=[\lambda x+\beta(x),x\circ y]+(\lambda x+\beta(x))[\lambda
y+\beta(y),x]+[\lambda y+\beta(y),x](\lambda x+\beta(x))\\\ &\>\textit{ for
all }x,y\in R.\end{split}$ (3.8)

Since $\lambda$ and $\beta(x)\in C$ for all $x\in R$, then

$\begin{split}0&=\lambda[x,x\circ
y]+\lambda^{2}x[y,x]+\lambda^{2}[y,x]x+2\lambda\beta(x)[y,x]\\\
&=\lambda[x^{2},y]+\lambda^{2}[y,x^{2}]+2\lambda\beta(x)[y,x]\\\
&=(\lambda^{2}-\lambda)[y,x^{2}]+2\lambda\beta(x)[y,x]\>\textit{ for all
}x,y\in R.\end{split}$ (3.9)

So, we find

$0=(\lambda^{2}-\lambda)[[x^{2},y],[y,x]]\>\textit{ for all }x,y\in R.$

The primeness of $R$ leads to the primeness of $Q$, then we get either
$\lambda^{2}=\lambda$ or $[[x^{2},y],[x,y]]=0$ for all $x,y\in R$. By Lemma
(2.2), the second case is equivalent to the $s_{4}$ identity and $R$ is
assumed to be non-commutative, therefore $R$ is an order in a central simple
algebra of dimension at most $4$ over $Z(R)$.
On the other hand, let us assume that $\lambda^{2}=\lambda$. Then from (3.9),
we have $2\lambda\beta(x)[y,x]=0$ for all $x,y\in R$. Since characteristic $R$
is not two, then characteristic $Q$ is not two and so, we find
$\lambda\beta(x)[y,x]=0$ for all $x,y\in R$. The primeness of $Q$ gives
$\lambda=0$ or $\beta(x)[y,x]=0$ for all $x,y\in R$. Assume $\lambda=0$. Then
$G(x)=\beta(x)\in C$ for all $x\in R$. Since $G$ is surjective, then
$R\subseteq C$, i.e., $R$ is a commutative ring, a contradiction. If
$\beta(x)[y,x]=0$ for all $x,y\in R$, then the primeness of $Q$ implies that
for each $x\in R$ either $\beta(x)=0$ or $[y,x]=0$ for all $y\in R$. Write
$E_{1}=\\{x\in R:\beta(x)=0\\}$ and $E_{2}=\\{x\in R:[y,x]=0\>\textit{ for all
}y\in R\\}$. Therefore, we note that $E_{1}$ and $E_{2}$ are proper additive
subgroups of $R$ and it is union equls $R$ which is impossible. Thus, either
$R=E_{1}$ or $R=E_{2}$. It implies that either $\beta(x)=0$ for all $x\in R$
or $[y,x]=0$ for all $x,y\in R$. If $[y,x]=0$ for all $x,y\in R$, then $R$ is
commutative, a contradiction. If $\beta(x)=0$ for all $x\in R$, then
$G(x)=\lambda x$ for all $x\in R$. ∎

Now, we prove the second theorem that dealing with ring with involution.

######  Theorem 3.7.

Let $R$ be a non-commutative prime ring of characteristic not two with
involution $*$. If $R$ admits a CE- Jordan epimorphism $G$ such that
$[G(x),x^{*}]\in Z(R)$ for all $x\in R$, then $R$ is an order in a central
simple algebra of dimension at most $4$ over its center or
$G(x)=\lambda^{*}x^{*}$ for all $x\in R$, where is $\lambda\in C$.

###### Proof.

Lemma (2.4) gives

$[G(x),x^{*}]=0\>\textit{ for all }x\in R.$ (3.10)

Applying involution in (3.10), we get

$[G(x)^{*},x]=0\>\textit{ for all }x\in R.$ (3.11)

Linearizing (3.11) gives

$[G(x)^{*},y]+[G(y)^{*},x]=0\>\textit{ for all }x,y\in R.$ (3.12)

Substituting $x\circ y$ for $y$ in (3.12) gives

$[G(x)^{*},x\circ y]+[G(x\circ y)^{*},x]=0\>\textit{ for all }x,y\in R.$

Thus,

$\begin{split}0&=[G(x)^{*},x\circ y]+[(G(x)G(y)+G(y)G(x))^{*},x]\\\
&=[G(x)^{*},x\circ y]+G(x)^{*}[G(y)^{*},x]+[G(y)^{*},x]G(x)^{*}\>\textit{ for
all }x,y\in R.\end{split}$ (3.13)

Since $*G$ is additive, then by (3.11) and Lemma (2.1), we find
$G(x)^{*}=\lambda x+\beta(x)$ for all $x\in R$ where $\beta:R\longrightarrow
C$ is an additive mapping. So, (3.13) becomes

$\begin{split}0&=[\lambda x+\beta(x),x\circ y]+(\lambda x+\beta(x))[\lambda
y+\beta(y),x]+[\lambda y+\beta(y),x](\lambda x+\beta(x))\\\ &\>\textit{ for
all }x,y\in R.\end{split}$ (3.14)

We have $\lambda$ and $\beta(x)$ for all $x\in R$ belong to $C$, then

$(\lambda^{2}-\lambda)[y,x^{2}]+2\lambda\beta(x)[y,x]=0\>\textit{ for all
}x,y\in R.$ (3.15)

So, we get $(\lambda^{2}-\lambda)[[x^{2},y],[y,x]]=0$ for all $x,y\in R$. As
above, we find either $\lambda^{2}=\lambda$ or $R$ is an order in a central
simple algebra of dimension at most $4$ over its center. If
$\lambda^{2}=\lambda$, then eqution (3.15) implies that $\beta(x)[y,x]=0$ for
all $x,y\in R$. By the same manner as in Theorem (3.6) we arrive at
$\beta(x)=0$ for all $x\in R$. Therefore $G(x)^{*}=\lambda x$ for all $x\in
R$. The involution leads to $G(x)=\lambda^{*}x^{*}$ for all $x\in R$, where
$\lambda\in C$. ∎

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