相关论文: The $b$-secant variety of a smooth curve has a cod…

In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of…

In this paper we prove that if the r-th tensor power of the tangent bundle of a smooth projective variety X contains the determinant of an ample vector bundle of rank at least r, then X is isomorphic either to a projective space or to a…

Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this…

Let X be a smooth curve of genus g. When pi>=3g and d>=pi-2g+1 we show the existence of a double covering gamma:C-->X where C a smooth curve of genus pi with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing…

For a weighted projective line X, a wide subcategory of the category coh-X of coherent sheaves over X is called c-invariant if it is closed under the grading shift of the canonical element c. We proved that a c-invariant wide subcategory of…

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard…

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the first etale cohomology vector space H^1(X,Z/pZ). We study the representations of finite groups G of automorphisms of X…

We show that if a family of complex varieties over a base B admits a section when restricted to a very general curve in B, then the family must contain a subfamily of rationally connected varieties dominating B. As an application, we deduce…

We give a new method of computation of the degree of the third secant variety of a smooth curve C in P^(d-2) of genus 2 and degree d>=8, using the presentation of the third secant variety as the union of all scrolls that are defined via a…

Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if…

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point,…

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 $\sigma_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of curvilinear…

Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree…

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…

A non-singular connected algebraic curve $A$ in a simply connected algebraic surface $X$ can be knotted so that its homology class and the fundamental group of its complement in $X$ is preserved, provided $A$ is sufficiently complex (not…

Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More…

We show the geometric syzygy conjecture in positive characteristic. Specifically, if C is a general smooth curve of genus g defined over an algebraically closed field of characteristic p, then all linear syzygy spaces are spanned by…

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge…

In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = \pi_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset…