Related papers: On the irreducibility of secant cones, and an appl…

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…

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…

Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we…

For $n\geq 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let $\mathcal{O}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of…

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a…

For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain…

Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $\sigma_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus…

Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of $S$-integral points of varieties over number fields $k$, including many cases with geometrically irreducible boundary divisors. Our approach builds…

We present a slightly different formulation of Zak's theorem on tangencies as well as some applications. In particular, we obtain a better bound on the dimension of the dual variety of a manifold and we classify extremal and…

Let $X_{m,d}\subset \mathbb {P}^N$, $N:= \binom{m+d}{m}-1$, be the order $d$ Veronese embedding of $\mathbb {P}^m$. Let $\tau (X_{m,d})\subset \mathbb {P}^N$, be the tangent developable of $X_{m,d}$. For each integer $t \ge 2$ let $\tau…

In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y of A_g generically contained in the…

In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3--secant planes to a variety. Precisely we prove that if $X\subseteq \PP^r$ is an irreducible, non--degenerate, projective complex variety of dimension…

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of…

We study the $m$-th Gauss map in the sense of F.~L.~Zak of a projective variety $X \subset \mathbb{P}^N$ over an algebraically closed field in any characteristic. For all integer $m$ with $n:=\dim(X) \leq m < N$, we show that the contact…

The classic trisecant lemma states that if $X$ is an integral curve of $\PP^3$ then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2.…

Let $X \subset \mathbb{P}^r$ be smooth and irreducible and for $k \ge 0$ let $\nu_k(X)$ (resp., $\delta_k(X)$) be the $k$-th contact (resp., the $k$-th secant) defect of $X$. For all $k \ge 0$ we have the inequality $\nu_k(X) \ge…

Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta$ be a positive integer such that $\mathcal I_{Z,Y}(\delta)$ is generated by…

Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it…

Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set…

Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we…