Related papers: Splitting criterion for reflexive sheaves

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

We investigate the positivity and extension of invertible sheaves on group homogeneous spaces over coherent bases. Bypassing the failure of standard limit arguments and the classical Weil--Cartier correspondence, we develop a valuative…

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of…

A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence…

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…

Inspired by Mukai's work on K3 surfaces, we introduce and study a notion of semi-rigidity for stable sheaves on smooth polarised varieties, designed to capture the existence of stable deformations of direct sums. We show that semi-rigidity…

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds:…

A procedure resolving a torsion-free coherent sheaf on a nonsingular $N$-dimensional projective algebraic variety into a locally free sheaf on a projective scheme of certain class is proposed. This is a higher-dimensional analog of the…

We study the geometry of the moduli stack of torsion-free sheaves on ribbons. We introduce a stratification of the stack by the complete type of the sheaves, and we investigate the geometric properties of the strata and their closure…

In this paper we study the cohomological criterion for the splitting of vector bundles on multiprojective spaces $\mathbb{P}^{n_1}\times\ldots\times\mathbb{P}^{n_s}$. We also give a generalization of vanishing cohomological criteria for…

We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on P(1,1,1,3) using a tilting bundle.

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite…

We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.

Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also…

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…

We introduce a notion of a connection on a coherent sheaf on a weighted projective line (in the sense of Geigle and Lenzing). Using a theorem of Huebner and Lenzing we show, under a mild hypothesis, that if one considers coherent sheaves…

We find locally free resolutions of length one for all semi-stable sheaves supported on curves of multiplicity five in the complex projective plane. In some cases we also find geometric descriptions of these sheaves by means of extensions.…

For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $\P^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of…

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough. Actually we prove more: given a line bundle…