Lothar Göttsche

Fundamental algebraic geometry : Grothendieck's FGA explained

Grothendieck topologies, fibered categories and descent theory: Introduction Preliminary notions Contravariant functors Fibered categories Stacks Construction of Hilbert and Quot schemes: Construction of Hilbert and Quot schemes Local properties and Hilbert schemes of points: Introduction Elementary deformation theory Hilbert schemes of points Grothendiecks existence theorem in formal geometry with a letter of Jean-Pierre Serre: Grothendiecks existence theorem in formal geometry The Picard scheme: The Picard scheme Bibliography Index.

Orbifold cohomology for global quotients

For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant part. In the case thar Y is S^n for a surface S with trivial canonical class we prove that (a small modification of) the orbifold cohomology of X is naturally isomorphic to the cohomology ring of the Hilbert scheme of n points on S computed in math.AG/0012166 by Lehn and Sorger.

A CONJECTURAL GENERATING FUNCTION FOR NUMBERS OF CURVES ON SURFACES

Abstract:I give a conjectural generating function for the numbers of δ-nodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V] for the case δ &\le; 6 and Kleiman–Piene [K-P] for the case δ &\le; 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau–Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso–Harris for the Severi degrees in ℙ2. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.

Hilbert schemes of zero-dimensional subschemes of smooth varieties

Fundamental facts.- Computation of the Betti numbers of Hilbert schemes.- The varieties of second and higher order data.- The Chow ring of relative Hilbert schemes of projective bundles.

Hodge numbers of moduli spaces of stable bundles on K3 surfaces

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is: Theorem: Let

Theta Functions and Hodge Numbers of Moduli Spaces of Sheaves on Rational Surfaces

X

Riemann-Roch theorems and elliptic genus for virtually smooth schemes

be a K3 surface,

REFINED CURVE COUNTING WITH TROPICAL GEOMETRY

L

Orbifold-Hodge numbers of Hilbert schemes

a primitive big and nef line bundle and

Rationality of moduli spaces of torsion free sheaves over rational surfaces

H