相关论文: On secant defective varieties, in particular of di…

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…

We study the dimensions of secant varieties of the Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for…

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality…

We study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties…

Let V_n be the Segre embedding of (P^1) x ... X (P^1) (n times). We prove that the higher secant varieties, \sigma_s(V_n), always have the expected dimension, except for \sigma_3(V_4), which is of dimension 1 less than expected.

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical…

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…

In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space…

Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant varieties use only isolated reduced points. In particular, any secant variety is…

We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb P^{n+a}$ with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is…

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in $\PP^r$ using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety $\Sec X_P$. We also give explicit formulas in…

The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We…

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…

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kaehler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of…

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic…

We study the $A$-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We prove a Gale dual characterization of…

We prove that for any $m\geq3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we…

We give a classification, up to finite cover, of flat compact complete Hermite-Lorentz manifolds up to complex dimension 4.

A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant…

For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we…