Igor Burban

Derived categories of irreducible projective curves of arithmetic genus one

We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all

Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves

t

Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraß cubics

-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder–Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.

Coherent sheaves on rational curves with simple double points and transversal intersections

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

TILTING ON NON-COMMUTATIVE RATIONAL PROJECTIVE CURVES

Abstract We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we compute explicitly. We show that the twist functors, which are associated to the structure sheaf and the structure sheaf k(p 0) of a smooth point p 0, generate an SL(2, ℤ)-action (up to shifts) on the bounded derived category of coherent sheaves on any Weierstraß cubic.

Vector Bundles on Singular Projective Curves

We study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems.

Maximal Cohen–Macaulay Modules Over Non–Isolated Surface Singularities and Matrix Problems

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy2 = x3 + x2z.

Maximal Cohen–Macaulay modules over surface singularities

In this survey artiele we report on reeent results known for vector bundles on singular projective curves (see (Drozd and Greuel; Drozd, Greuel and Kashuba; Yudin). We recall the description of vector bundles on tame and finite configurations of projective lines using the combinatorics of matrix problems. We also show that this combinatorics allows us to compute the cohomology groups of a vector bundle, the dual bundle of a vector bundle, the tensor product of two vector bundles, the dimension of the homomorphism spaces between two vector bundles, and finally to classify simple vector bundles.

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of

The composition Hall algebra of a weighted projective line

k\llbracket x,y,z\rrbracket/(xyz)