Related papers: Brill-Noether theorems and globally generated vect…

Using Bridgeland stability conditions we give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the…

In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of…

Let $X$ be a K3 surface with Picard group $\mathrm{Pic}(X)\cong\mathbb{Z} H$ such that $H^2=2n$. Let $M_{H}(\mathbf{v})$ be the moduli space of Gieseker semistable sheaves on $X$ with Mukai vector $\mathbf{v}$. We say that $\mathbf{v}$…

One describes, using a detailed analysis of Atiyah--Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This…

Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the…

Let M be a moduli space of stable vector bundles on a curve with rank and degree fixed and coprime. We give a simple proof that the rational cohomology of M is generated by the Kunneth components of the Chern classes of the universal…

Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the…

We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection…

We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general…

We give an explicit formula for Euler characteristics of line bundles on the Hilbert scheme of points on $\mathbb{P}^1\times\mathbb{P}^1$. Combined with structural results of Ellingsrud, G\"ottsche, and Lehn, this determines the Euler…

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and…

Let X be a ruled surface over a nonsingular curve C of genus $g\geq0$. Let $M_H:=M_{X,H}(2;c_1,c_2)$ be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $c_i:=c_i(E)$ for $i=1,2$. The main goal of this…

Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old…

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces…

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to…

We compute the stable cohomology of moduli spaces of hyperelliptic curves of a fixed genus embedded on a fixed Hirzebruch surface. We also describe these moduli spaces of embedded hyperelliptic curves in terms of moduli spaces of pointed…

Let (S,H) be a polarized K3 surface, $E$ be a coherent sheaf on S and W be a linear subspace in the space of global sections H^0(S,E). If we are lucky, there is an exact sequence 0 -> W tensor O -> E -> E' -> 0, which gives a correspondence…