Related papers: Rank Two Sheaves on K3 Surfaces: A Special Constru…

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is…

This paper classifies rank two vector bundles on a del Pezzo threefold $X$ of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on $X$,…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In…

Let X be an irreducible 2n-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular, if its Azumaya algebra End(F) deforms with X to every Kahler deformation of X. We show that if F is a slope-stable reflexive sheaf…

In this paper we prove some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer $N$, there is a K3 surface with Picard number 2 and at…

We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every…

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector…

In 2017, Catanese--Perroni gave a natural correspondence between the Picard group of a double cover and a set of pairs of a vector bundle of rank two and a certain morphism of vector bundles on the base space. In this paper, we describe the…

The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves…

We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the…

We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space.…

Let $X$ be a general cubic hypersurface in $\mathbb P^4$. If $x\in X$ is a general point there are exactly six distinct lines in $X$ passing through $x$, that lie on the rank 3 quadric cone with vertex $x$ of lines that have intersection…

A generic K3 surface of degree 2t is a general complex projective K3 surface whose Picard group is generated by the class of an ample divisor whose with respect to the intersection form is 2t. We show that if X is the Hilbert square of a…

We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely $H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}$, $H \oplus E_8(-1) \oplus D_4(-1)$,…

We prove that a double cover of $\mathbb{P}^2$ ramified along a general smooth curve B of degree $2s$, for $s \geq 3$, supports a rank 2 special Ulrich bundle.

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is:…

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of…

Let X be a general complex Fano threefold of genus 8. We prove that the moduli space of rank two semistable sheaves on X with Chern numbers c_1=1, c_2=6 and c_3=0 is isomorphic to the Fano surface F(X) of conics on X. Inside F(X), the…