Related papers: Secant varieties of Segre-Veronese varieties P^m x…

We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension of their higher secant varieties. Via inverse systems (apolarity) and the study of…

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…

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…

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…

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the…

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 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…

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the…

For any irreducible non-degenerate variety $X\subset \mathbb{P}^r$, we give a criterion for the $(k,s)$-identifiability of $X$. If $k\leq s-1 <r$, then the $(k,s)$-identifiability holds for $X$ if and only if the $s$-identifiability holds…

Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the…

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…

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric…

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…

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet…

The $k$-th secant variety of a projective variety $X \subset \mathbb{P}^N$, denoted by $\sigma_k(X)$, is defined to be the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. In this paper, we examine the $k$-th secant…

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $\Pi$. It is known…

We prove that Segre-Veronese varieties are never secant defective if each degree is at least three. The proof is by induction on the number of factors, degree and dimension. As a corollary, we give an almost optimal non-defectivity result…

We explore injective morphisms from complex projective varieties $X$ to projective spaces $\mathbb{P}^s$ of small dimension. Based on connectedness theorems, we prove that the ambient dimension $s$ needs to be at least $2 \dim X$ for all…

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…

Let $X\subset \P^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset\P^N$ be its secant variety. $X\subset\P^N$ is called `secant defective' if $\dim(SX)$ is strictly…