Related papers: On separable higher Gauss maps

Let $X$ be a complex projective variety defined over $\mathbb R$. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to $X$. This rank takes into account only decompositions that are stable under…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in…

Let $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X \to U$ be an abelian scheme. We consider the set $\mathcal{R}$ of rational points $m \in U(k)$ such that the Mordell-Weil rank of the fibre $U_{m}$ is strictly bigger…

A $k$-regular graph on $v$ vertices is a {\em divisible design graph} if there exist integers $\lambda_1,\lambda_2,m,n$ such that the vertex set can be partitioned into $m$ classes of size $n$ and any two different vertices from the same…

A generalized gauge invariant Ising model on random surfaces with non-trivial topology is proposed and investigated with the dual transformation. It is proved that the model is self-dual in case of a self-dual lattice. In special cases the…

A projective variety $X\subset\mathbb{P}^N$ is $h$-identifiable if the generic element in its $h$-secant variety uniquely determines $h$ points on $X$. In this paper we propose an entirely new approach to study identifiability, connecting…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of…

R. Hartshorne conjectured and F. Zak proved that any n-dimensional smooth non-degenerate complex algebraic variety X in a m-dimensional projective space P satisfies Sec(X)=P if m<3n/2+2. In this article, I deal with the limiting case of…

A graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion.…

We prove that a product of $m>5$ copies of $\PP^1$, embedded in the projective space $\PP^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant…

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a…

Let $\phi$ be a birational map of the complex projective plane. We know that $\phi$ can be written as a composition of automorphisms of $\mathbb{P}^2_\mathbb{C}$ and the standard quadratic birational map $\sigma$. This writing, that is…

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks and…

A linear system of real quadratic forms defines a real projective variety. The real non-singular locus of this variety (more precisely of the underlying scheme) has a highly connected double cover as long as each non-zero form in the system…

Let $X$ be the moduli space of semistable rank 2 vector bundles over a smooth curve C of genus $g \ge 2$ and $\theta : X \to PH^0(L)^*$ be the map associated to the generalized theta divisor L on X. We prove that for C not hyperelliptic,…

This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the…

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a…

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of…