A note on Veronese varieties

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。

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.

Veronese 簇的 Ulrich 秩与等变瞬子

Ulrich ranks of Veronese varieties and equivariant instantons

该研究通过射影空间上等变瞬子丛对称平方的一般形变，在任意次数Veronese三重簇上构造Ulrich丛，分类其秩并证明了Costa和Miró-Roig的猜想。

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

We classify the tangential varieties of the Segre-Veronese varieties which are Cohen-Macaulay or Gorenstein.

New examples of defective secant varieties of Segre-Veronese varieties

We prove the existence of defective secant varieties of three-factor and four-factor Segre-Veronese varieties embedded in certain multi-degree. These defective secant varieties were previously unknown and are of importance in the classification of defective secant varieties of Segre-Veronese varieties with three or more factors.

On secant defectiveness and identifiability of Segre-Veronese varieties

We give an almost asymptotically sharp bound for the non secant defectiveness and identifiability of Segre-Veronese varieties. We also provide new examples of defective Segre-Veronese varieties.

Horrocks splitting on Segre-Veronese varieties

We prove an analogue of Horrocks' splitting theorem for Segre-Veronese varieties building upon the theory of Tate resolutions on products of projective spaces.

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high dimensional case to the computation of the dimensions of secant varieties in low dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective s-th S…

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.

Ulrich bundles of arbitrary rank on Segre-Veronese varieties

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.

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.

Secant varieties of Segre-Veronese embeddings of (P^1)^r

We use a double degeneration technique to calculate the dimension of the secant variety of any Segre-Veronese embedding of (P^1)^r

Secant varieties to osculating varieties of Veronese embeddings of $\mathbb{P}^n$

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\mathbb{P}^n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity…

Higher Gauss Maps of Veronese Varieties---a generalization of Boole's formula and degree bounds for higher Gauss map images---

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.

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.

Severi's theorem for d-uple Veronese varieties

We characterize $d$-uple Veronese embeddings of finite-dimensional projective spaces. The easiest non-trivial instance of our theorem is the embedding of the projective plane in 5-dimensional projective space, a result obtained in 1901 by Severi when the underlying field is complex.

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。

Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

We give a partial "quasi-stratification" of the secant varieties of the order $d$ Veronese variety $X_{m,d}$ of $\mathbb {P}^m$. It covers the set $σ_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of curvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. For low border rank two different "quasi-strata" are disjoint and we compute the symmetric rank of their elements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties. To get a stratification we attach to each $P\in σ_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratum containi…

Equations for secant varieties of Veronese and other varieties

New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homoge…

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。

Projections of Grassmannians of lines and characterization of Veronese varieties

We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.

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…

非零特征域上的 Veronese 簇：综述

Veronese Varieties over Fields with non-zero Characteristic: A Survey

本综述汇集了非零特征域上 Veronese 簇的核与正规有理曲线不变子空间的近期结果，并指出基域需足够大以避免退化。

Most secant varieties of tangential varieties to Veronese varieties are nondefective

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space $\mathbb{P}^n$ have the expected dimension, modulo a few well-known exceptions. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $n$ of the projective space via a specializati…

Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty…

A Terracini Lemma for osculating spaces with applications to Veronese surfaces

Here we present a partial generalization to higher order osculating spaces of the classical Lemma of Terracini on ordinary tangent spaces. As an application, we investigate the secant varieties to the osculating varieties to the Veronese embeddings of the projective plane.

Veronese簇上低秩Ulrich丛的不存在性

Non-existence of low rank Ulrich bundles on Veronese varieties

本文通过研究维数至少为4的完全交的Veronese嵌入，证明此类Veronese簇上不存在秩不超过3的Ulrich丛。

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 $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.

Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to p…

Sumsets and Veronese varieties

In this paper, to any subset $\mathcal{A} \subset \mathbb{Z}^{n}$ we explicitly associate a unique monomial projection $Y_{n,d_{\mathcal{A}}}$ of a Veronese variety, whose Hilbert function coincides with the cardinality of the $t$-fold sumsets $t\mathcal{A}$. This link allows us to tackle the classical problem of determining the polynomial $p_{\mathcal{A}} \in \mathbb{Q}[t]$ such that $|t\mathcal{A}| = p_{\mathcal{A}}(t)$ for all $t \geq t_0$ and the minimum integer $n_0(\mathcal{A}) \leq t_0$ for which this condition is satisfied, i.e. the so-called {\em phase transition} of $|t\mathcal{A}|$.…

关于 Veronese 簇的秩 3 二次方程

On rank 3 quadratic equations of Veronese varieties

本文研究了 Veronese 簇的秩 3 二次方程轨迹的几何结构，探究其极小不可约分解，并分析了各不可约分支的几何性质、奇点消解以及非奇异性与奇异性。

Veronese varieties contained in hypersurfaces

Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\geq N_1(r,d)$ where the "expected dimension" $f_1(n,r,d)$ is nonnegative, in which case $f_1(n,r,d)$ equals the (pure) dimension. Using work by Gleb Nenashev, we prove that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the parameter space of $r$-dimensional $e$-uple Veronese varieties contained in the hypersurface is nonempty of pur…

Some effects of Veronese map on syzygies of projective varieties

Let $X \subset ¶^r$ be a nondegenerate projective variety and let $ν_{\ell} : ¶^r \to ¶^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective embedding $ν_{\ell} (X) \subset ¶^N$. We obtain that for a very ample line bundle $L \in {Pic}X$ such that $X \subset ¶H^0 (X,L)$ is $m$-regular in the sense of Castelnuovo-Mumford, $(X,L^{\ell})$ satisfies property $N_{\ell}$ for all $\ell \geq m$ (Theorem \ref{thm:main1}). This is a generalization of M. Green's work that $(¶^r,\mathcal{O}_{¶^r} (\ell))$ satisfi…

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。

A note on the Intersection of Veronese Surfaces

Motivated by our study (elsewhere) of linear syzygies of homogeneous ideals generated by quadrics and their restrictions to subvarieties of the ambient projective space, we investigate in this note possible zero-dimensional intersections of two Veronese surfaces in P^5. The case of two Veronese surfaces in P^5 meeting in 10 simple points appears also in work of Coble, Conner and Reye in relation to the 10 nodes of a quartic symmetroid in P^3, and we provide here a modern account for some of their results.

Double Veronese cones with 28 nodes

We study nodal del Pezzo 3-folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all $G$-birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group $\mathfrak{S}_4$ into the space Cremona group.

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.

Veronese簇的单项式投影：新结果与猜想

Monomial projections of Veronese varieties: new results and conjectures

本文研究Veronese簇单项式投影的齐次坐标环，证明了特定变量支撑条件下的二次代数性质，并完全刻画了二维情形下群不变量环的二次性与Koszul等价性。

通过Veronese嵌入的几何Arveson-Douglas猜想

The Geometric Arveson-Douglas Conjecture through Veronese Embeddings

本文证明几何Arveson-Douglas猜想在Veronese嵌入下不变，提供奇异簇新例，并归结到二次Gröbner基情形，同时证明基本类猜想不变性并给出无穷反例。

On multiples of divisors associated to Veronese embeddings with defective secant variety

In this note we consider multiples aD, where D is a divisor of the blow-up of P^n along points in general position which appears in the Alexander and Hirschowitz list of Veronese embeddings having defective secant varieties. In particular we show that there is such a D with h^1(X,D) > 0 and h^1(X,2D) = 0.

3x3 Minors of Catalecticants

Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the 3x3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3x3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the…

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。

A note on the Kuznetsov component of the Veronese double cone

This note describes moduli spaces of complexes in the derived category of a Veronese double cone $Y$. Focusing on objects with the same class $κ_1$ as ideal sheaves of lines, we describe the moduli space of Gieseker stable sheaves and show that it has two components. Then, we study the moduli space of stable complexes in the Kuznetsov component of $Y$ of the same class, which also has two components. One parametrizes ideal sheaves of lines and it appears in both moduli spaces. The other components are not directly related by a wall-crossing: we show this by describing an intermediate moduli sp…

Projections of Veronese surface and Morphisms from projective plane to Grassmannian

In this note we describe the image of $\PP^2$ in $ Gr(2, \CC^{4})$ under a morphism given by a rank two vector bundle on $\PP^2$ with Chern classes $(2,2).$

Setting Hidden Symmetries Free by the Noncommutative Veronese Mapping

The short note is devoted to the setting free of hidden symmetries in Verma modules over sl(2,C) by the noncommutative Veronese mappings.

Splitting Varieties for Cup Products with $\mathbb Z/3$-Coefficients

We connect Veronese embeddings to splitting varieties of cup products. We then give an algorithm for constructing splitting varieties for cup products with $\mathbb Z/n$ coefficients, with an explicit calculation for $n=3$. An application to the automatic realization of Galois groups is given.

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…

关于到判别式及密切簇的距离

ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIES

本文从代数几何角度研究到奇异超曲面簇的距离问题，计算其欧氏距离度，并建立 Chebyshev 多项式的极值性质。

小次数 Veronese 嵌入的高阶割线簇的素理想

On the prime ideals of higher secant varieties of Veronese embeddings of small degrees

本文研究小次数 Veronese 嵌入的高阶割线簇的素理想，证明 $\sigma_4(v_3(\mathbb{P}^3))$ 的理想由 36 个 5 次多项式生成，并推广到一般 $k$-割线情形，提出计算首个非平凡次数片的程序。

射影空间的线性合系

Linear syzygies of projective space

研究线性合系q=1情形下Veronese簇合系的Ein-Lazarsfeld猜想，证明在最高和次高阶线性合系上与猜想一致的消失性命题。

不变割线簇的非缺陷性

Nondefectivity of invariant secant varieties

本文证明一大类不变割线簇的非缺陷性，解决大维数 Grassmannian 割线猜想的多数情形，并改进 Chow、Segre-Veronese、Gaussian moment 等簇的非缺陷性、可辨识性与一般秩界。

Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety $v_2(\mathbb{P}^3)\subset \mathbb{P}^9$ to $\mathbb{P}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in $\mathbb{A}^4$, Cremona transforms of $\mathbb{P}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in $\mathbb{P}^6$.

Duality, Bridgeland wall-crossing and flips of secant varieties

Let $v_d(\mathbb{P}^2)\subset |\mathcal{O}_{\mathbb{P}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d(\mathbb{P}^2)$ by embedding the blow-up $\mbox{bl}_{v_d(\mathbb{P}^2)}|\mathcal{O}_{\mathbb{P}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on $\mathbb{P}^2$.