Hadamard Products and Varieties Which Are Strongly Concise for All Systems of Coordinates

Abstract

We fix the Hadamard product (coordinate-wise multiplication) of a projective space. The strongly concise embedded varieties are the embedded varieties X such that all the points of the projective space have finite Hadamard X-rank. We prove that for all $n\ge 2m+1$, every m-dimensional projective manifold Y may be embedded in a projective space of dimension n as a strongly concise variety for all changes of coordinates. We prove that Y may be embedded in such a way that for a certain coordinate system not only it is not strongly concise, but the general point of the projective space has infinite Hadamard X-rank. We use previous works by D. Antolini, G. Montúfar, and A. Oneto.

1. Introduction

We work over an algebraically closed field $\mathbb{K}$ such that $\mathrm{char}\left(\mathbb{K}\right)=0$, e.g., $\mathbb{C}$ (see Section 5 for the real case). We fix a system of homogeneous coordinates $x_0,\dots ,x_n$ of ${\mathbb{P}}^n$, i.e., we take ${\mathbb{K}}^{n+1}$ with its usual coordinates. For all vectors $(a_0,\dots ,a_n)$ and $(b_0,\dots ,b_n)$ of ${\mathbb{K}}^{n+1}$, their Hadamard product is the vector $(a_0{b}_0,\dots ,a_n{b}_n)$ (coordinate-wise product).

If $a_i\ne 0$ for some i and $b_j\ne 0$ for some j, then the points $P=[a_0:\cdots :a_n]\in {\mathbb{P}}^n$ and $Q=[b_0:\cdots :b_n]\in {\mathbb{P}}^n$ are defined. If $a_h{b}_h\ne 0$ for some h, then the Hadamard product $P\star Q$ is defined. Let $X\subset {\mathbb{P}}^n$ and $Y\subset {\mathbb{P}}^n$ be integral projective varieties. In the classical case, one defines the join of X and Y, the secant varieties of X, and the X-ranks of the points of ${\mathbb{P}}^n$. These very classical notions have been heavily used and studied for their applications in multilinear algebra and complexity theory [1,2,3]. The interested reader may see in [4] the motivations for defining Hadamard product of two varieties $X,Y\subset {\mathbb{P}}^n$ ([5,6,7,8]). After [4], and stimulated by this book, many real-life applications were presented (n-body quantum potential scattering [9], neural networks [10], peptides [11]). Packages have been built for practical use in commutative algebra [12].

However, there are deep differences with respect to the classical case of secant varieties.

Let $X\subset {\mathbb{P}}^n$ be the projective variety. Taking iterated Hadamard products $X^{\star x}$, $x\ge 2$, one would like to send (or receive) the entire screen ${\mathbb{P}}^n$. This is not always possible, because there are varieties X such that $X^{\star x}=X$ for all $x\ge 2$ and hence the Hadamard powers of X only cover a very small part, X, of the screen ${\mathbb{P}}^n$. Among these non-coverings X, there are those used in multilinear algebra: all toric varieties and in particular the multiprojective spaces used to define the tensor rank (see Section 5). D. Antolini, G. Montúfar, and A. Oneto classified all such non-covering varieties X ([13], Cor. 3.40) and proved that the classical secant varieties of toric varieties cover the screen ${\mathbb{P}}^n$ taking Hadamard powers ([13], Cor. 3.9).

In the classical case, the secant varieties of X may be defined using the X-ranks of the points of ${\mathbb{P}}^n$ ([1], Ch. 2). In the Hadamard setup, the Hadamard rank of the variety X was only defined in 2025, by D. Antolini, G. Montúfar, and A. Oneto. The Hadamard rank of $P\in {\mathbb{P}}^n$ is the minimal positive integer x such that P is the Hadamard product of x elements of X, with the convention that it is ∞ if there is no such x ([13], Def. 2.1). They also classified the varieties X such that a general $P\in {\mathbb{P}}^n$ has Hadamard rank ∞ ([13], Cor. 3.40). In [14], Th. 1.1, there is a characterization of the varieties X such that all $P\in {\mathbb{P}}^n$ have finite Hadamard rank with respect to X. They are called strongly concise. By definition, an embedded variety $X\subset {\mathbb{P}}^n$ is said to be strongly concise if for all $i=0,\dots ,n$ there is $[p_0:\cdots :p_n]\in X$ such that $p_i=0$ and $p_j\ne 0$ for all $j\ne i$. Thus, strongly concise varieties are good for describing all $P\in {\mathbb{P}}^n$ using Hadamard products of points of X. Many example of strongly concise varieties are known ([14]).

However, the notion of strong conciseness depends on the coordinate system. What happens if a different coordinate system is used? It is easy to see that often a strongly concise variety in not strongly concise in another system (Examples 1–3). Let $PGL(n+1)$ or $\mathrm{Aut}\left({\mathbb{P}}^n\right)$ denote the group of all linear automorphisms of ${\mathbb{P}}^n$. We say that X is invariantly strongly concise if $g\left(X\right)$ is strongly concise for all $g\in PGL(n+1)$. This seems to be a geometrically important notion ([15]). On the applied side, only if X is invariantly strongly concise may we use X for all possible choices of coordinates to fill the screens. One person has X and then somebody else chooses the coordinates. If X is invariantly strongly concise, no enemy can choose bad coordinates. Also, the notion of Hadamard rank is only well behaved for small modifications of the point P and the variety X if P has finite Hadamard rank with respect to X. The invariantly strongly concise varieties have all these good properties. Invariance for any coordinate system may be useful in practical application to allow the choice of another, better, coordinate system.

Do they exist? Here, we construct many of them. We show that all smooth projective varieties X may be embedded in ${\mathbb{P}}^n$, $n\ge 2m+1$ as an invariantly strongly concise variety (Theorem 1). However, we also prove that it may be embedded as a variety which is not strongly concise (Theorem 2 and Remark 1).

Theorem 1.

Fix a smooth and connected projective variety Y of dimension m. Take an integer $n\ge 2m+1$. Then, there is an embedding of Y in ${\mathbb{P}}^n$ whose image X has the property that $g\left(X\right)$ is strongly concise for all $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$.

See Theorem 3 for an extension of Theorem 1 to the case of singular varieties Y such that $\dim \mathrm{Sing}\left(Y\right)\le m-2$.

Theorem 2.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Let $d_0$ be a positive integer such that X is scheme-theoretically cut out by forms of degree at most $d_0$. Fix an integer $d\ge d_0$. Let ${\nu }_{d,r}:{\mathbb{P}}^r\to {\mathbb{P}}^M$, $M=-1+\left({\displaystyle \genfrac{}{}{0pt}{}{r+d}{r}}\right)$ be the order-d Veronese embedding of ${\mathbb{P}}^r$. Let ${\mathbb{P}}^N$ denote the linear span of ${\nu }_d\left(X\right)$. Then, there is $g\in \mathrm{Aut}\left({\mathbb{P}}^N\right)$ such that $g\left({\nu }_d\left(X\right)\right)$ is contained in a binomial hypersurface.

Remark 1.

Take $g\left({\nu }_d\left(X\right)\right)$ as in Theorem 2. By [13] (Cor. 3.40), the generic $Q\in {\mathbb{P}}^N$ has infinite Hadamard rank with respect to $g\left({\nu }_d\left(X\right)\right)$ and hence $g\left({\nu }_d\left(X\right)\right)$ is not strongly concise.

Section 5 is divided into four parts. First, we discuss the classical varieties, e.g., the toric variety, and give three open questions. Then, we describe the modifications needed to handle Hadamard products over the real numbers and give an open question. Then, as in [13,14], we suggest tackling what happens in the Hadamard products without taking the Zariski closure. Then, we discuss the decomposition of an embedded variety as the Hadamard product of two lower-dimensional varieties (Remark 2). This is an important topic for the Hadamard products of matrices ([16]).

In the last section, we discuss how our paper is related to the previous literature (Section 6).

We thank the referees for crucial observations.

2. Preliminaries

Set $H_i:=\{x_i=0\}$. The hyperplanes $H_0,\dots ,H_n$ are the coordinate hyperplanes of ${\mathbb{P}}^n$. A coordinate linear space $V\subset {\mathbb{P}}^n$ is a finite intersection of coordinate hyperplanes. For $0\le i\le n-1$, let ${\Delta }_i\subset {\mathbb{P}}^n$ denote the union of all i-dimensional coordinate linear subspaces. Set $U_n:={\mathbb{P}}^n\setminus {\Delta }_{n-1}$. Let $G\subset PGL(n+1)$ denote the image of the diagonal subgroup of $GL(n+1)$. The abelian group G acts simply and transitively on $U_n$.

Let $X\subset {\mathbb{P}}^n$ be an integral projective variety. We recall that X is said to be concise if $X\cap U_n\ne \varnothing$, i.e., $X⊈{\Delta }_{n-1}$. The variety X is said to be strongly concise if for each $i\in \{0,\dots ,n\}$ there is $p=[p_0:\cdots :p_n]\in X$ such that $p_i=0$ and $p_j\ne 0$ for all $j\ne i$ ([14]). The variety X is strongly concise if and only if each point of ${\mathbb{P}}^N$ has finite Hadamard X-rank (and also in that case each point of ${\mathbb{P}}^n$ has Hadamard X-rank at most $3n$ by [14], Theorem A). For a general $g\in \mathrm{Aut}\left(X\right)$, the variety $g\left(X\right)$ is strongly concise ([14], Remark 4.7). It is easy to check that $g\left(X\right)$ is concise for all $g\in \mathrm{Aut}\left(X\right)$ if and only if X is non-degenerate, i.e., X spans ${\mathbb{P}}^n$ (Remark 3).

3. Examples

We give some easy examples to show that the notion of strong conciseness is non-trivial and that for some strongly concise $X\subset {\mathbb{P}}^n$ there is $g\in PGL(n+1)$ such that $g\left(X\right)$ is not strongly concise. In Section 5 we recall the classical known examples. Then, we discuss existence, non-existence, and uniqueness of the decomposition of a variety as the Hadamard product of other varieties (Remark 2).

Example 1.

Let $L\subset {\mathbb{P}}^n$ be a general line. Since L is general, $V\cap L=\varnothing$ for all codimension-2 linear subspace and it intersects each coordinate hyperplane transversally. Hence, for all $i=0,\dots ,n$ there is a unique $p_i\in L\cap H_i$ and $p_i\notin H_i\cap H_j$ if $j\ne i$. Thus, L is strongly concise. For every $g\in PGL(n+1)$ such that $g\left(L\right)\subset H_0$, the variety $g\left(L\right)$ is not concise and hence it is not strongly concise.

The next example is non-degenerate.

Example 2.

Let $C\subset {\mathbb{P}}^n$ be a rational normal curve. Since C is non-degenerate, $g\left(C\right)$ is concise for all $g\in PGL(n+1)$. We claim that there are $g,h\in PGL\left(n\right)$ such that $g\left(C\right)$ is strongly concise, while $h\left(C\right)$ is not strongly concise. Take a general $g\in PGL(n+1)$. The theorem of Bertini gives that each $H_i\cap g\left(C\right)$ is formed by n distinct points and that $g\left(C\right)\cap H_i\cap H_j=\varnothing$ for all $i\ne j$. Hence, for each $i=0,\dots ,n$ the curve $g\left(C\right)$ has n points, with the i-th coordinate equal to 0 and the other ones $\ne 0$. Take $p=[p_0:\cdots :p_n]$ such that $p_0=p_1=0$ and $p_j\ne 0$ for $j\ge 2$. Let H be a general hyperplane containing p. Fix $q\in C$ and let M be the osculating hyperplane of C at q. We have $\mathrm{deg}(M\cap C)=n$ and q is the unique point of C contained in M. There is $h\in PGL(n+1)$ such that $h\left(q\right)=p$ and $h\left(M\right)=H$. The point p is the unique point of C contained in $H_0$. Since $p_1=0$, $h\left(C\right)$ is not strongly concise.

Example 3.

Let $X\subset {\mathbb{P}}^n$ be an integral variety. The variety $g\left(X\right)$ is strongly concise for a general $g\in PGL(n+1)$. There are many such varieties X which are not strongly concise. Antolini, Montúfar, and Oneto classified the varieties X such that a general $p\in {\mathbb{P}}^n$ has infinite Hadamard rank with respect to X ([13], 3.12) (they are G-equivalent to those contained in a binomial hypersurface). They also proved that if a general $p\in {\mathbb{P}}^n$ has finite Hadamard rank, then the same is true for all $q\in U_n$ ([13], Prop. 3.11).

Remark 2.

There are smooth surfaces $W\subset {\mathbb{P}}^3$ that are not the Hadamard product of two lower-dimensional varieties ([17]). Many surfaces of ${\mathbb{P}}^3$ are the Hadamard product of two curves, but even in that case no uniqueness is expected. There is a classification of the quadrics $Q\subset {\mathbb{P}}^3$ that are the Hadamard product of two lines (they must be smooth) and the possible pairs of lines with Q as their Hadamard product ([18]).

4. Proof of Theorems 1 and 2

In this section we prove Theorems 1 and 2 and the following extension of it (Theorem 3) for all normal varieties and, more generally, for all varieties non-singular in codimension 1.

Theorem 3.

Let Y be an integral projective variety. Set $m:=\dim X$ and assume the inequality $\dim \mathrm{Sing}\left(Y\right)\le m-2$. For each $x>m$, let $T\left(x\right)$ denote the set of all $p\in \mathrm{Sing}\left(Y\right)$ such that the Zariski tangent space of X at p has dimension x. Let κ be the maximum of all integers $x+\dim T\left(x\right)$ for some $x>m$ such that $T\left(x\right)\ne \varnothing$. Fix an integer $n\ge \max \{2m+1,\kappa \}$. Then, there is an embedding of Y in ${\mathbb{P}}^n$ whose image X has the property that $g\left(X\right)$ is strongly concise for all $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$.

Remark 3.

Let $X\subset {\mathbb{P}}^n$ be an integral projective variety. Since X is integral, X is not concise if and only if $X\subset H_i$ for some $i\in \{0,\dots ,n\}$. Since $\mathrm{Aut}\left({\mathbb{P}}^n\right)$ acts transitively on the set of all hyperplanes of ${\mathbb{P}}^n$, $g\left(X\right)$ is concise for all $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$ if and only if X spans ${\mathbb{P}}^n$.

Lemma 1.

Let $X\subset {\mathbb{P}}^n$ be an integral projective variety. There is $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$ such that $g\left(X\right)$ is not strongly concise if and only if there is a hyperplane $H\subset {\mathbb{P}}^n$ such that H is not spanned by the set ${(H\cap X)}_{\mathrm{red}}$.

Proof. 

First, assume the existence of $H\subset {\mathbb{P}}^n$ such that H is not spanned by the set ${(H\cap X)}_{\mathrm{red}}$. Take a hyperplane $M\subset {\mathbb{P}}^n$ such that ${(H\cap X)}_{\mathrm{red}}\subset H\cap M$. Since $\mathrm{Aut}\left({\mathbb{P}}^n\right)$ acts transitively on the set of all complete flags of ${\mathbb{P}}^n$, there is $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$ such that $g\left(H\right)=H_0$ and $H\cap M=H_0\cap H_1$.

Conversely, assume ${(H_i\cap g\left(X\right))}_{\mathrm{red}}\subset H_i\cap H_j$ for some $i\ne j$. Take $H:=g^{-1}\left(H_i\right)$. □

Remark 4.

For all integers $N>0$ and $a\ge 0$, let $G(a+1,N+1)$ denote the Grassmannian of all a-dimensional linear subspaces of ${\mathbb{P}}^N$. The set $G(a+1,N+1)$ is a connected and smooth projective variety of dimension $(a+1)(N-a)$. The linear groups $GL(N+1)$ and $\mathrm{Aut}\left({\mathbb{P}}^N\right)$ act transitively on $G(a+1,N+1)$. We use the convention $\dim \varnothing =-1$. Fix integers $b\ge r\ge 0$ and $B\in G(b+1,N+1)$. Let $G(a+1,N+1)(B,r)$ denote the set of all $A\in G(a+1,N+1)$ such that $\dim A\cap B\ge r$. If $a<r$, then $G(a+1,N+1)(B,r)=\varnothing$. Now, assume that $a\ge r$. The set $G(a+1,N+1)(B,r)$ is an irreducible subvariety of $G(a+1,N+1)$ of dimension $(r+1)(b-r)+(a-r)(N-a)$ ([19] (p. 66), [20]). The set $G(a+1,N+1)(B,r)$ is called a Schubert cycle. Take another $B_1\in G(a+1,N+1)$. Since $\mathrm{Aut}\left({\mathbb{P}}^N\right)$ acts transitively on $G(b+1,N+1)$, $G(a+1,N+1)(B,r)\cong G(a+1,N+1)(B_1,r)$.

Remark 5.

Let $v_d:{\mathbb{P}}^r\to {\mathbb{P}}^M$, $M=-1+\left({\displaystyle \genfrac{}{}{0pt}{}{r+d}{r}}\right)$ be the order-d Veronese embedding of ${\mathbb{P}}^r$. Let $Z\subset {\mathbb{P}}^r$ be a zero-dimensional scheme. Set $z:=\mathrm{deg}\left(Z\right)$. If $d\ge z-1$, then the scheme ${\nu }_d\left(Z\right)$ is linearly independent.

Lemma 2.

Let $C\subset {\mathbb{P}}^r$ be a smooth, connected, and non-degenerate projective curve. Fix an integer $n\ge 3$ and an integer $d>2n^3$. Let ${\mathbb{P}}^N$ denote the linear span of $Y:={\nu }_d\left(C\right)$. We have $N>n$. Let $V\subset G(N-n-1,N+1)$ be a general linear subspace of dimension $N-n-1$. We have $V\cap C=\varnothing$. Let $\mathsf{\ell }:{\mathbb{P}}^N\setminus V\to {\mathbb{P}}^n$ denote the linear projection from V. Set $X:=\mathsf{\ell }\left(Y\right)$. The map ${\mathsf{\ell }}_{|Y}:Y\to X$ is an isomorphism and $g\left(X\right)$ is strongly concise for all $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$.

Proof. 

We have $\mathrm{deg}\left(Y\right)=d\mathrm{deg}\left(C\right)\ge d$. Hence, $N\ge d>2n^3$. We have $V\cap C=\varnothing$, because $\dim C=1<n$ and V is general. The assumption on n gives that ${\mathsf{\ell }}_{|Y}:Y\to X$ is an isomorphism. Every zero-dimensional scheme $Z\subset Y$ of degree at most $d+1$ is linearly independent. Since we are in characteristic 0, for each integer $i=1,\dots ,n-1$ the variety X has order of contact $i+1$ with the i-th osculating space of X at a general $p\in X$. For each $o\in X_{\mathrm{reg}}$ and $i\in \{1,\dots ,n-1\}$, let $A_i\left(o\right)\in G(i+1,n+1)$ denote the osculating space of X at o. Let $a_i\left(o\right)$ be the order of contact of X with $A_i\left(o\right)$ at o.

Claim 1: We have $a_i\left(o\right)=i+1$ for all $i\le n-2$ and $n\le a_{n-1}\left(o\right)\le n+1$.

Proof of Claim 1: By the definition of osculating space, we have $a_i\left(o\right)\ge i+1$. Let u be the smooth point of Y such that $\mathsf{\ell }\left(u\right)=o$. Set $B(u,x):=\langle (xu,Y)\rangle$, where x is a positive integer. Since $d\ge n+1$, we have $\dim B(u,x)=x-1$ for all $x\le n+2$. We have $a_i\left(o\right)=i+1$ if and only if $V\cap B(u,i)=\varnothing$. Since $B(u,i+1)$ has dimension at most $n-2$, $dimY=1$ and V is general, $V\cap B(u,i)=\varnothing$ for all $u\in Y_{\mathrm{reg}}$. We have $a_n\left(o\right)\le n+1$ if and only if either $V\cap B(u,n+1)=\varnothing$ (in this case $a_n\left(o\right)=n$) or $V\cap B(u,n+1)$ is a point. Since C has dimension 1 and V is general, we get Claim 1.

Claim 2: Every codimension-2 linear subspace of ${\mathbb{P}}^n$ contains at most $3n$ points of X.

Proof of Claim 2: Take $S\in S(Y,3n+1)$. Since $d\ge 3n$, we have $\dim \langle S\rangle =3n$.

The set $\mathsf{\ell }\left(S\right)$ is contained in a codimension-2 linear subspace of ${\mathbb{P}}^n$ if and only if $\dim (\langle S\rangle \cap V)\ge 2n+1$. Since $\dim Y=1$, $S(Y,3n+1)$ has dimension $3n+1$. We consider the Schubert cycle $G(N-n,N+1)(\langle S\rangle ,2n+1)$ described in Remark 4. For this Schubert cycle we have $a=N-n-1$, $b=3n$ and $r=2n+1<a$. By Remark 4, the Schubert cycle $G(N-n,N+1)(\langle S\rangle ,n+1)$ has dimension

$$(2n+2)(n-1)+(N-3n-2)(n+1)=(N-n)(n+1)-4n-4.$$

Since $\dim S(Y,3n+1)=3n+1<4n+4$ and V is general, we get Claim 2.

Assume that $g\left(X\right)$ is not strongly concise for some $g\in \mathrm{Aut}\left({\mathbb{P}}^n\right)$. By Lemma 1, there are $n+1$ linearly independent hyperplanes $M_0,\dots ,M_n$ such that $A\subset {\cup }_{i=1}^n(M_0\cap M_i)$, where $A:={(M_0\cap X)}_{\mathrm{red}}$. By Claim 2, we have $\#(A\cap H_i)\le 3n$ for all $i=1,\dots ,n$. By Claim 1, each connected component of the scheme-theoretic intersection $M_0\cap X$ has at most degree $n+1$. Since $\mathrm{deg}(M_0\cap X)=\mathrm{deg}\left(X\right)=d\mathrm{deg}\left(C\right)>2n^3$, we get a contradiction and conclude the proof of the lemma. □

Proof of Theorem 1.

The case $m=1$ is true by Lemma 2.

Assume $m\ge 2$. We fix a non-degenerate embedding $Y\subset {\mathbb{P}}^r$ for some r and an integer $d>2n^3$. Let ${\nu }_d:{\mathbb{P}}^r\to {\mathbb{P}}^M$, $M=-1+\left({\displaystyle \genfrac{}{}{0pt}{}{r+d}{r}}\right)$ denote the order-d Veronese embedding of ${\mathbb{P}}^r$. Let $C\subset {\mathbb{P}}^r$ be the intersection of Y with $m-1$ general hypersurfaces of degree $d+1$. The theorem of Bertini gives that $C\subset {\mathbb{P}}^r$ is a smooth and connected non-degenerate curve. Let ${\mathbb{P}}^N\subset {\mathbb{P}}^M$ denote the linear span of ${\nu }_d\left(Y\right)$. Since C is cut out by Y and hypersurfaces of degree $>d$, ${\nu }_d\left(C\right)$ spans ${\mathbb{P}}^N$. Note that $N>2n+2$ and in particular $N\ge n+m+2$. Let $V\subset {\mathbb{P}}^N$ be a general linear subspace of dimension $N-n-1$. Since $N-n-1>m=\dim \nu \left(Y\right)$, we have $V\cap {\nu }_d\left(Y\right)=\varnothing$. Let $\mathsf{\ell }:{\mathbb{P}}^N\setminus V\to {\mathbb{P}}^n$ denote the linear projection from V. Since $V\cap {\nu }_d\left(Y\right)=\varnothing$, ${\mathsf{\ell }}_{|{\nu }_d\left(Y\right)}$ is a morphism. Set $X:=\mathsf{\ell }({\nu }_d\left(Y\right)$). Since $n\ge 2m+1$, and V is general, ℓ induces an embedding of ${\nu }_d\left(Y\right)$. Hence, $X\cong Y$. By Lemma 2, $\mathsf{\ell }\left({\nu }_d\left(C\right)\right)$ is invariantly strongly concise in ${\mathbb{P}}^n$. Since $X\supseteq \mathsf{\ell }\left({\nu }_d\left(C\right)\right)$, X is invariantly strongly concise in ${\mathbb{P}}^n$. □

Proof of Theorem 3.

Since the case Y is smooth is true by Theorem 1, we may assume $\mathrm{Sing}\left(Y\right)\ne \varnothing$. Since we assumed that Y is smooth in codimension 1, we have $m\ge 2$. We fix a non-degenerate embedding $Y\subset {\mathbb{P}}^r$ for some r. Fix an integer $d>2n^3$. Let ${\nu }_d:{\mathbb{P}}^r\to {\mathbb{P}}^M$, $M=-1+\left({\displaystyle \genfrac{}{}{0pt}{}{r+d}{r}}\right)$ denote the order-d Veronese embedding of ${\mathbb{P}}^r$. Let ${\mathbb{P}}^N\subset {\mathbb{P}}^M$ denote the linear of ${\nu }_d\left(Y\right)$. Let $C\subset {\mathbb{P}}^r$ be the intersection of Y with $m-1$ general hypersurfaces of degree $d+1$. Since $\dim \mathrm{Sing}\left(Y\right)\le m-2$, the theorem of Bertini gives that $C\subset {\mathbb{P}}^r$ is a smooth and connected non-degenerate curve. We take d, N, and ℓ as in the proof of Theorem 1. Since $n\ge \{2m+1,\kappa \}$, ℓ induces an embedding. Since $\mathsf{\ell }\left({\nu }_d\left(C\right)\right)$ is invariantly strongly concise and $X\supset \mathsf{\ell }\left({\nu }_d\left(C\right)\right)$, X is invariantly strongly concise. □

Proof of Theorem 2.

Set $Y:={\nu }_d\left(X\right)$. Since X is scheme-theoretically cut out by forms of degree at most $d_0$, it is scheme-theoretically cut out by forms of degree d. Since ${\mathbb{P}}^N$ is the linear span $\langle Y\rangle$ of Y in ${\mathbb{P}}^M$, Y is scheme-theoretically cut out by ${\mathbb{P}}^N$ and a system of generators of ${\nu }_d\left(X\right)$ in ${\mathbb{P}}^M$.

We fix homogeneous coordinates $y_0,\dots ,y_r$ of ${\mathbb{P}}^r$. For each $\alpha =(a_0,\dots ,a_r)\in {\mathbb{N}}^{r+1}$, let $y^{\alpha }$ denote the monomial $y_0^{a_0}\cdots y_r^{a_r}$. The integer $\left|\alpha \right|:=a_0+\cdots +a_r$ is the degree of the monomial $y^{\alpha }$. Let E denote the set of all $\alpha \in {\mathbb{N}}^{r+1}$ such that $\left|\alpha \right|=d$. We have $\#E=M+1$ and the elements $z_{\alpha }:=y^{\alpha }$, $\alpha \in E$ are the variables of ${\mathbb{P}}^M$. Let F be the set of all $(\alpha ,\beta ,\gamma ,\delta )\in E^4$ such that $\alpha +\beta =\gamma +\delta$ and $\#\{\alpha ,\beta ,\gamma ,\delta \}>2$.

Take $(\alpha ,\beta ,\gamma ,\delta )\in F$. Since we work over a commutative field, $z_{\alpha }{z}_{\beta }-z_{\gamma }{z}_{\delta }$ vanishes identically on $v_d\left({\mathbb{P}}^r\right)$. The homogeneous ideal of ${\nu }_d\left({\mathbb{P}}^r\right)$ in ${\mathbb{P}}^M$ is generated by the quadratic forms $z_{\alpha }{z}_{\beta }-z_{\gamma }{z}_{\delta }$ for all $(\alpha ,\beta ,\gamma ,\delta )\in F$ ([21], (p. 52)). The rank of this quadratic form is the integer $\#\{\alpha ,\beta ,\gamma ,\delta \}$. Hence, Y is scheme-theoretically cut out by the restriction to ${\mathbb{P}}^N$ of the quadric hypersurfaces $z_{\alpha }{z}_{\beta }-z_{\gamma }{z}_{\delta }=0$, $(\alpha ,\beta ,\gamma ,\delta )\in F$. Since ${\nu }_d\left({\mathbb{P}}^r\right)$ spans ${\mathbb{P}}^M$ and it is scheme-theoretically defined by these quadric hypersurfaces, there is one such quadric hypersurface $Q^{\prime }$ such that $Q:=Q_{|{\mathbb{P}}^N}^{\prime }\ne 0$. The rank does not increase when restricting a quadric hypersurface to a linear subspace, because the rank of a quadratic form does not depend on the choice of coordinate system and we may extend a base of a linear subspace to a base of the ambient space. Thus, Q has rank at most 4. By construction $\{Q=0\}\supset Y$. Since Q has rank at most 4, there are homogeneous coordinates $x_0,\dots ,x_N$ of ${\mathbb{P}}^N$ for which Q is binomial in the sense of [4], Ch. 7. Choosing such a coordinate system, we obtain that $W^{\star x}=W$ for all $x\ge 2$ ([4], Proposition 7.1, [22], Proposition 4.7). Hence, in this coordinate system a general $p\in {\mathbb{P}}^N$ has infinite Hadamard rank with respect to W. Since $Y\subset W$, a general $p\in {\mathbb{P}}^N$ has infinite Hadamard rank with respect to Y. If ${\mathbb{P}}^N$ is equipped with homogeneous coordinates $w_0,\dots ,w_N$ with $w_i\ne x_i$ for some i, then this is equivalent to using $g\left(Y\right)$ instead of Y for some $g\in GL(N+1)$ such that $g{w}_i=x_i$ for all i, concluding the proof of the theorem. □

5. Concluding Remarks and Open Questions

We divide this section into three subsections. The first one describes classical embedded varieties, e.g., toric varieties, and their secant varieties. We explain why their Hadamard cousins have very different properties.

In the second subsection we explain the case of real varieties.

Then, we ask what happens before we take the Zariski closure to get the Hadamard product of two varieties.

5.1. Classical Examples and Their Secant Varieties

A very rough analogy is non-degenerate varieties (classical side) and concise varieties (the Hadamard side), joins and secant varieties (classical side) and Hadamard products and powers, X-ranks (classical) and Hadamard ranks ([13]). The classical case inspired several topics, results, and techniques on the Hadamard side. However, there are key differences. First of all, there are concise varieties $X\subset {\mathbb{P}}^n$ such that iterating the Hadamard powers we do not get all ${\mathbb{P}}^n$, and for them the general Hadamard rank is infinite. These examples are not pathological: all toric varieties have this bad property ([4], Proposition 7.1, [22], Proposition 4.7). Antolini, Montúfar, and Oneto proved that for all $r\ge 2$ the classical r-secant varieties of a toric variety have finite Hadamard ranks ([13], Example 3.8) and applied it to the Segre–Veronese embeddings ([13], Cor. 3.9). The case of toric varieties inspired the statement of Theorem 2 and its proof.

Question 1: Are all Hadamard ranks between the minimum and the maximum (or ∞) possible? Is it true at least if X is strongly concise?

If the generic Hadamard rank is finite, say z, the definition of Hadamard powers gives that all integers between 1 and z are Hadamard ranks of some $p\in {\mathbb{P}}^n$.

In the classical case, Question 1 is true ([23], Theorem 3.1). Note that [23], Th. 3.1, says more: if x is the generic X-rank, y is the maximum X-rank, and $x<c<y$, then it gives that the set of all $p\in {\mathbb{P}}^n$ with X-rank c has larger dimension than the set of all $p\in {\mathbb{P}}^n$ with X-rank $c+1$.

Question 2: Is this true for the Hadamard ranks of strongly concise varieties? For a non-degenerate $X\subset {\mathbb{P}}^n$ is it true for all $g\left(X\right)$ with g general in $PGL(n+1)$?

It would be nice to compute in interesting cases the maximum Hadamard rank. Blekherman and Titler proved an upper bound of the maximum rank in terms of the generic rank [24]. Antolini, Montúfar, and Oneto adapted the ideas in [24] to the Hadamard ranks of all $p\in U_n$ ([13], Prop. 3.11). Is their bound (twice the generic Hadamard rank) true for all $p\in {\mathbb{P}}^n$ if X is strongly concise?

Let $X\subset {\mathbb{P}}^n$ be integral and non-degenerate. Let F be the set of all $g\in PGL(n+1)$. The algebraic group $PGL(n+1)$ is irreducible. Hence, saying that $g\left(X\right)$ is strongly concise for a general $g\in PGL(n+1)$ means that E contains a nonempty Zariski-open subset of $PGL(n+1)$. Since in the definition of strict conciseness we only need to check finitely many hyperplanes, the coordinate hyperplanes, it is easy to see that E is Zariski-open in $PGL(n+1)$. Compute $PGL(n+1)\setminus E$ in interesting cases. The set $PGL(n+1)\setminus E$ is sent into itself by the action of the abelian group G.

Question 3: Let $X\subset {\mathbb{P}}^n$ and $Y\subset {\mathbb{P}}^n$ be non-degenerate and strongly concise subvarieties. Thus, for all $i\in \{0,\dots ,n\}$ there are $P_i:=\in X_i\cap H_i$ and $Q_i\in X_i$ with all coordinates, except the i-th one, non-zero. The points $P_i\star Q_i$, $0\le i\le n$, show that $X\star Y$ is strongly concise. Now, assume that X and Y are invariantly strongly concise. Under what assumptions is $X\star Y$ invariantly strongly concise?

5.2. Hadamard Products over the Real

The book [4] considers the complex numbers, but of course many real-life applications use and need the real numbers. Let $X\subset {\mathbb{P}}^n$ be an integral projective variety defined over $\mathbb{R}$ and whose embedding is defined over $\mathbb{R}$, i.e., X is scheme-theoretically cut out by homogeneous polynomials with real coefficients. We use both the Euclidean and the Zariski topology on $X\left(\mathbb{C}\right)$, $X\left(\mathbb{R}\right)$, ${\mathbb{P}}^n\left(\mathbb{C}\right)$, and ${\mathbb{P}}^n\left(\mathbb{R}\right)$. Since the embedding of X is defined over $\mathbb{R}$, we have $X\left(\mathbb{R}\right)={\mathbb{P}}^n\left(\mathbb{R}\right)\cap X\left(\mathbb{C}\right)$. For several reasons we assume that the smooth locus $X_{\mathrm{reg}}$ of X contains at least one real point, i.e., $X_{\mathrm{reg}}\left(\mathbb{C}\right)\cap X\left(\mathbb{R}\right)\ne \varnothing$. This assumption is equivalent to the assumption that the real algebraic set $X\left(\mathbb{R}\right)$ has dimension $\dim X$. It is also equivalent to the reasonable assumption that $X\left(\mathbb{R}\right)$ is Zariski-dense in $X\left(\mathbb{C}\right)$, and hence it is sufficient to detect the equations defining $X\left(\mathbb{C}\right)$ inside ${\mathbb{P}}^n\left(\mathbb{C}\right)$. With this assumption for each $p\in {\mathbb{P}}^r\left(\mathbb{R}\right)$, the real X-rank of p is defined ([24]), which is finite for all p and lower-bounded by the $X\left(\mathbb{C}\right)$-rank of p. As in [13], Def. 2.2, we may define the real Hadamard rank of p as the minimum positive integer x such that $p=p_1\star \cdots \star p_x$ with $p_i\in X\left(\mathbb{R}\right)$ for all i, or ∞ if there is no such integer x. We say that X is strongly concise over $\mathbb{R}$ if for each $i=0,\dots ,n$ there is $[p_0:\cdots :p_n]\in X_{\mathrm{reg}}\left(\mathbb{R}\right)$ such that $p_i=0$ and $p_j\ne 0$ for all $j\ne 0$. We say that X is s-concise over $\mathbb{R}$ if for each $i=0,\dots ,n$ there is $[p_0:\cdots :p_n]\in X\left(\mathbb{R}\right)$ such that $p_i=0$ and $p_j\ne 0$ for all $j\ne 0$. We add invariantly to these two definitions if they are true for the real varieties $g\left(X\right)$ for all $g\in GL(n+1,\mathbb{R})$. Since the Veronese embedding $v_d$ is defined over $\mathbb{R}$, our proofs in this paper work for invariantly strong conciseness over $\mathbb{R}$.

However, the classical side over $\mathbb{R}$ has many differences with the classical case over $\mathbb{C}$, and we now explain a crucial one. In the classical case, over $\mathbb{C}$ there is a generic X-rank, say z, and this is true also for Hadamard powers, although it may be ∞. In the classical case over $\mathbb{R}$ there is the notion of typical ranks ([25,26,27]). They are the integers t with the property that there is a nonempty Euclidean open subset U of ${\mathbb{P}}^n\left(\mathbb{R}\right)$ such that all $p\in U$ have real rank t. Easy examples (even the rational normal curves) give that X may have several different typical ranks [25,26]. If the generic Hadamard rank is finite, say w, then this definition with $X\left(\mathbb{R}\right)$ instead of $X\left(\mathbb{C}\right)$ gives finitely many integers and, perhaps, even ∞ with w as the smallest of these integers.

Question 4: Assume that the generic complex Hadamard rank of X is finite, w. Let z be the maximum typical rank. Are all integers t such that $w<t<z$ typical?

In the classical case for real ranks, the answer is yes ([27], Th. 1.1).

In [28], we considered and handled (again, inspired by classical constructions) a completely different problem: why, when using Hadamard products of elements of $X\left(\mathbb{C}\right)\setminus X\left(\mathrm{R}\right)$, we may get large subsets of ${\mathbb{P}}^n\left(\mathbb{R}\right)$ not covered by Hadamard products of elements of $X\left(\mathbb{R}\right)$.

5.3. The True Image of the Hadamard Products

Take subvarieties X and Y of ${\mathbb{P}}^n$. Even in the classical case the definition of the join of X and Y involves a Zariski closure. The definition of the Hadamard product involves a Zariski closure, i.e., we add to a well-known quasi-projective variety $A\subset {\mathbb{P}}^n$ all unknown points of $\overline{A}\setminus A$. Ref. [14] gives several examples in which $\overline{A}=A$, and this equality is easily a priori checked. An earlier example is [4], Th. 2.1, which was used to prove [13], Prop. 3.11. These cases were used to construct important examples and handle some steps in the proof of general theorems. It would be very important to describe $\overline{A}\setminus A$ in many other cases. For m powers of the same variety X, this is equivalent (with the notation of [13]) between ${\eta }_m\left(X\right)$ and the set of all $p\in {\mathbb{P}}^n$ with Hadamard rank at most m. This set is constructible, i.e., a finite union of locally closed subsets for the Zariski topology of ${\mathbb{P}}^n$ ([29], Ex. II.3.18 and Ex.3.19). Thus, the integer $\dim (\overline{A}\setminus A)$ is well defined.

6. Discussion

The Hadamard product of a projective space depends on the choice of a fixed coordinate system, while most definitions and results in algebraic geometry and projective algebraic geometry do not depend on it, and hence they allow a change of coordinate to put a system in a simpler form. Since the beginning of the field ([5,6,7,8]), the challenge has been not only to guess and prove theorems on the Hadamard products of varieties, but also to use them. Since the beginning of the field, classical algebraic geometry has been used to define the Hadamard product of two subvarieties, as explained in [4]. Let $X\subset {\mathbb{P}}^n$ be an irreducible variety. Earlier this year, the necessary and sufficient condition for the finiteness of all Hadamard X-rank ([14]) was obtained, and X was called strongly concise if it satisfies this condition. We say that X is invariantly strong concise if X is strongly concise with respect to all changes of coordinates.

In this paper we prove that for each $n\ge 2n+1$, every m-dimensional smooth projective variety may be embedded in ${\mathbb{P}}^n$ as an invariantly strongly concise variety. We also prove this for varieties with mild singularities. We also prove that there are many embeddings of Y which are not invariantly strongly concise, and we provide a way to produce these embeddings.

In the last section we outline a few open problems, for instance, on the maximum Hadamard rank and on the case over the real numbers. We also give suggestions for further research.