Related papers: Constructing Maximal Cohen-Macaulay Sheaves on Sym…
We study reflexive modules over one dimensional Cohen-Macaulay rings. Our key technique exploits the concept of $I$-Ulrich modules.
Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection…
In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of…
Let $R$ be a commutative Noetherian local ring. We characterize when its completion has an isolated singularity, thereby strengthening the Dao-Takahashi refinement of the Auslander-Huneke-Leuschke-Wiegand theorem. We investigate the ascent…
Via the BGG-correspondence a simplicial complex D on [n] is transformed into a complex of coherent sheaves L(D) on the projective space n-1-space. In general we compute the support of each of its cohomology sheaves. When the Alexander dual…
Let $R$ be a Cohen-Macaulay local ring. In this paper, we first describe the radicals of annihilators of stable categories of maximal Cohen-Macaulay $R$-modules. We then prove that the Alexandrov topology of the stable category of maximal…
We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois…
We determine, up to isomorphism, the indecomposable maximal Cohen-Macaulay modules over certain complete one-dimensional local rings of finite Cohen-Macaulay type. We then investigate the direct sum relations of maximal Cohen-Macaulay…
We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent…
The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety. We focus on the hypersurface case. Let $M$ be a complex manifold, $X\subset M$ a singular…
A commutative local Cohen-Macaulay ring R of finite Cohen-Macaulay type is known to be an isolated singularity; that is, Spec(R)-m is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded AS…
The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite…
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose…
Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of…
We describe, by matrix factorizations, all the rank two maximal Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$.
Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental…
Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting…
Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive…
Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime ideals P_1,...,P_s. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the…
We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq…