Cohen-Macaulay and Gorenstein tangential varieties of the Segre-Veronese Varieties

Segre-Veronese簇的非缺陷性

Non-defectivity of Segre-Veronese varieties

该研究通过对因子数量、次数和维数的归纳法，证明了次数均不小于3的Segre-Veronese簇无割线缺陷，并给出了单次数为1、其余次数不小于3的Segre-Veronese簇的近乎最优非缺陷性推论。

Higher secant varieties of Segre-Veronese varieties

We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by ${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it a {\it Segre-Veronese variety} and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for 3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the expected dimension except for a few cases. A list of examples of defective $X^s$'s is given.

Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the $s^{\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \bar{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.

Segre-Veronese流形的到达性

Reach of Segre-Veronese Manifolds

本文计算了与单位球面相交的Segre-Veronese簇的到达性、极值曲率及管状邻域体积，揭示了其几何性质。

Tangential varieties of Segre-Veronese surfaces are never defective

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.

Regularity and Segre-Veronese embeddings

This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. We then apply our results to study the Tate resolutions of these sheaves.

On non-secant defectivity of Segre-Veronese varieties

Let $SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese given as the image of the embedding induced by the line bundle $\mathcal{O}_{\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_r}}(d_1,\dots, d_r)$. We prove that asymptotically $SV^{\pmb n}_{\pmb d}$ is not $h$-defective for $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.

A note on the cactus rank for Segre-Veronese varieties

We give an upper bound for the cactus rank of any multi-homogeneous polynomial.

Secant Varieties of Segre--Veronese Varieties

We prove that the ideal of the variety of secant lines to a Segre--Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre--Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.

Segre-Veronese 簇的切簇

Tangential varieties of Segre-Veronese varieties

本文确定了 Segre-Veronese 簇切簇理想的极小生成元及齐次坐标环的 GL-表示分解，并证实了 Segre 簇情形下 Landsberg-Weyman 猜想。

关于0维完全交与Veronese簇的强Lefschetz性质

On Strong Lefschetz Property of 0-dimensional complete intersections and Veronese varieties

本文证明了齐次0维完全交在1次中的强Lefschetz性质成立，条件是其相应的关联形式（Macaulay逆系统）具有非零判别式。

On the ranks of the third secant variety of Segre-Veronese embeddings

We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $\dim V_i \geq 3$ for all $i$'s, then also all the ranks between $k$ and $2k-1$ arise.

Segre-Veronese 变种 $\mathbb{P}^m\times\mathbb{P}^n$ 通过 $\mathcal{O}(1,2)$ 嵌入的 secant 种子在 $n\gg m^3$，$m\geq3$ 时是非缺陷性的

Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$

该研究证明了在特定条件下，Segre-Veronese 变种的 secant 种子非缺陷性，通过推广 Brambilla 和 Ottaviani 的方法并利用计算机辅助证明验证基例。

Representation stability for syzygies of line bundles on Segre--Veronese varieties

The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology gro…

Chordal varieties of Veronese varieties and catalecticant matrices

It is proved that the chordal variety of the Veronese variety v_d(P^n) is projectively normal, arithmetically Cohen-Macaulay and its homogeneous ideal is generated by the 3 x 3 minors of two catalecticant matrices. These results are generalized to the catalecticant varieties Gor_{\leq}(T) with t_1 = 2. We also give a simplified proof of a theorem of O. Porras about the rank varieties of symmetric tensors.

Secant varieties of toric varieties arising from simplicial complexes

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $\mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.

Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.

A uniqueness result on the decompositions of a bi-homogeneous polynomial

In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial $p$ (i.e. a partially symmetric tensor of $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex, finite dimensional vector spaces) if its rank with respect to the Segre-Veronese variety $S_{d_1,d_2}(V_1,V_2)$ is at most $\min \{d_1,d_2\}$. Such a polynomial may not have a unique minimal decomposition as $p=\sum_{i=1}^rλ_i p_i$ with $p_i\in S_{d_1,d_2}(V_1,V_2)$ and $λ_i$ coefficients, but we can show that there exist unique $p_1, \ldots , p_{r'}$, $p_{1}', \…

Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?

When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format…

Tensor ranks on tangent developable of Segre varieties

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

On the representation type of a projectiva variety

Let X be a smooth arithmetically Cohen-Macaulay subvariety of Pn. We prove that the restriction to X of the Veronese 3-uple embedding of Pn embeds X as a variety of wild representation type.

A footnote to a footnote to a paper of B. Segre

The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is the unique case of a Veronese image of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable. The title originates from a paper of 1981, where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. We explore the apolar ideal of a sextic $F$ and the …

Curves in Segre threefolds

We study locally Cohen-Macaulay curves of low degree in the Segre threefold with Picard number three and investigate the irreducible and connected components respectively of the Hilbert scheme of them. We also discuss the irreducibility of some moduli spaces of purely one-dimensional stable sheaves and apply the similar argument to the Segre threefold with Picard number two.

Segre积与遗传表示型Cohen-Macaulay环的非交换消解

Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary representation type

本文利用倾斜理论与丛倾斜理论研究遗传表示型Cohen-Macaulay环，证明Segre积上丛倾斜模或非交换crepant消解的存在性定理，构造多个Gorenstein环例子，并对一个4维非Gorenstein环给出刚性Cohen-Macaulay模的完全分类。

From non Defectivity to Identifiability

A projective variety $X\subset\mathbb{P}^N$ is $h$-identifiable if the generic element in its $h$-secant variety uniquely determines $h$ points on $X$. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect. In this way we are able to improve all known bounds on identifiability. In particular we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.

Higher secant varieties of $\mathbb P^n \times\mathbb P^ 1$ embedded in bi-degree $(a,b)$

In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in \mathbb{Z}^+$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.

Cremona linearizations of some classical varieties

In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Non-Ulrich representation type

We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.

On the ideals of Secant Varieties to certain rational varieties

If $\X \subset ¶^n$ is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of $\X$. In this paper we find those equations in the following cases: $\X = ¶^{n_1}\times...\times¶^{n_t}\times¶^n$ is the Segre embedding of the product and $n$ is "large" with respect to the $n_i$ (Theorem 2.4); $\X$ is a Segre-Veronese embedding of some products with 2 or three factors; $\X$ is a Del Pezzo surface.

超胖点与相关张量

Superfat points and associated tensors

本研究聚焦n维空间中m对称的0维概形及其中的m-超胖点，重点分析n=2时的性质，并以平面2-超胖点为例，探究其在Veronese等簇上定义的簇及对应张量，给出此类概形的首个应用。

Higher secant varieties of $\mathbb{P}^n \times \mathbb{P}^m$ embedded in bi-degree $(1,d)$

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.

On the secant varieties of tangential varieties

Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Let $σ_{a,b}(X)\subseteq \mathbb{P}^r$, $(a,b)\in \mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $\dim σ_{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $\mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $\dim σ_{0,b}(X)$ is the expected o…

Real Rank with Respect to Varieties

We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of $X$ only. We show constructively that there exist varieties $X$ for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman-Teitler on the maximal $X$-rank is tight. We also give examples of varieties $X$ for which the gap between maximal complex and the maximal real rank is arbitra…

On tangential weak defectiveness and identifiability of projective varieties

A point $p\in\mathbb{P}^N$ of a projective space is $h$-identifiable, with respect to a variety $X\subset\mathbb{P}^N$, if it can be written as linear combination of $h$ elements of $X$ in a unique way. Identifiability is implied by conditions on the contact locus in $X$ of general linear spaces called non weak defectiveness and non tangential weak defectiveness. We give conditions ensuring non tangential weak defectiveness of an irreducible and non-degenerated projective variety $X\subset\mathbb{P}^N$, and we apply these results to Segre-Veronese varieties.

The Hitchhiker guide to: Secant Varieties and Tensor Decomposition

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most o…