Related papers: Exceptional collections on $\Sigma _{ 2 }$

We investigate the commutativity of global products of functions on the two-sphere from the point of view of a construction started in [RT] and named the skewed product. We complete the construction of the skewed product of functions on the…

Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $\Gamma \lhd \Gamma_0 < \mathrm{PGL}_2(K)$ is a Schottky group normally contained in a larger group which is generated by order-$p$ elements each…

We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a…

We study points and 0-cycles on del Pezzo surfaces defined over a field K of characteristic 0, with emphasis on cubic surfaces. We prove that a cubic surface that admits a point defined over a field extension of K of degree coprime to 3…

In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In…

We equip the type $A$ diagrammatic Hecke category with a special derivation, so that after specialization to characteristic $p$ it becomes a $p$-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the…

We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf…

For a closed oriented surface $ \Sigma $ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let $X_{\Sigma,n}$ be the set of isomorphism classes of orientation preserving $n$-fold branched…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

Veech groups are discrete subgroups of SL(2, R) which play an important role in the theory of translation surfaces. For a special class of translation surfaces called origamis or square-tiled surfaces their Veech groups are subgroups of…

We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove…

We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting…

We study obstructions to a direct limit preserving right exact functor $F$ between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if $F$ is exact, all…

We generalize a classical result about the genus of curves in projective space by Gruson and Peskine to principally polarized abelian threefolds of Picard rank one. The proof is based on wall-crossing techniques for ideal sheaves of curves…

Working over a perfect field, I classify normal del Pezzo surfaces with base number one that contain a nonrational singularity. They form a huge infinite hierarchy; contractions of ruled surfaces lie on top of it. Descending the hierarchy…

The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an $E_2$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne…

We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories ${\mathcal D}_{2n}$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element…

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(\theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element…

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair $({\mathcal O},V)$ on any del Pezzo surface of degree 4, such that $V$ is a bundle of rank $r$ and degree $d$. As an application,…