Related papers: Tangential projections and secant defective variet…

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another, fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly…

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…

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.

The paper concerns the affine varieties that are homogeneous with respect to a (non-standard) graduation over the group Z^m. Among the other properties it is shown that every such a variety can be embedded in its Zariski tangent space at…

We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its…

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…

We study degenerations of complex projective spaces $\mathbb P^n$ into normal projective klt varieties $X$. If the tangent sheaf of $X$ is semi-stable, we show that $X$ itself is a projective space. If $X$ is a threefold with canonical…

To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to…

We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety…

Let $Y \subset \P^r$ be a normal nondegenerate m-dimensional subvariety and let $\sigma(Y)$ denote the maximum dimension of a subvariety $Z \subset Y_{smooth}$ such that $Z$ contains a generic point of some divisor on $Y$ and the tangent…

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a…

We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois…

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.

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant…

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of…

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…

Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives…

For a smooth projective variety $X$, we consider when the diagonal $\Delta_X$ is nef as a cycle on $X\times X$. In particular, we give a classification of complete intersections and smooth del Pezzo varieties where the diagonal is nef. We…

The tangent degree $\tau(X)$ of a projective variety $X^n\subset\mathbb P^N$ is the number of tangent spaces to $X$ at smooth points passing through a general point of the tangent variety $Tan(X)\subseteq\mathbb P^N$, if positive and…

We make explicit the equations of any projective $PGL_2(\C)$-variety defined by quadrics. We study their zero-locus and their relationship with the geometry of the Veronese curve.