Related papers: Some remarks on osculating self-dual varieties

Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$.…

Let $X\subset \mathbb P^d$ be a $m$-dimensional variety in $d$-dimensional projective space. Let $k$ be a positive integer such that $\binom{m+k}k \le d$. Consider the following interpolation problem: does there exist a variety $Y\subset…

We study varieties $X \subseteq \mathbb P^N$ of dimension $n$ such that $T_X(k)$ is an Ulrich vector bundle for some $k \in \mathbb Z$. First we give a sharp bound for $k$ in the case of curves. Then we show that $k \le n+1$ if $2 \le n \le…

Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a…

We show that every supersingular K3 surface is birational to a double cover of a projective plane.

Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P^2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of…

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general…

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…

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results…

For X a compact Riemann surface of positive genus, the strange duality conjecture predicts that the space of sections of certain theta bundle on moduli of bundles of rank r and level k is naturally dual to a similar space of sections of…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M…

A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs,…

Let $S \subset \mathbb{P}^g$ be a smooth $K3$ surface of degree $2g-2$, $g \geq 3$. We classify all the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) \neq 0$ and the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) <…

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also…

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that…

We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with…