arXiv subjects

Lothar Göttsche

Publications and source records attributed to Lothar Göttsche.

25 records · Page 2Linked to original sources

arXiv2003

Hilbert schemes of points on surfaces

The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections.

math.AG↗

arXiv1997

Euler number of the compactified Jacobian and multiplicity of rational curves

We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $δ$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.

alg-geom↗

arXiv1996

Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b_+=1

We prove structure theorems for the Donaldson invariants of 4-manifolds with b_+=1, similar to those of Kronheimer and Mrowka in the case b_+>1: We show that for a 4-manifold with b_+=1 and two different period points F, G on the boundary of the positive cone, the difference of the Donaldson invariants at F and G satisfies the k^{th}-order simple type condition for a number k, explicitely given in terms of F and G. We give a formula for this difference in terms of a set of basic classes, expressed in a universal formula involving modular forms. The leading terms coincide with the formulas of Kronheimer and Mrowka in the simple type case. The basic classes are related to the Seiberg Witten invariants. The proof of these results is based on the conjecture of Kotschick and Morgan on the structure of the wall-crossing terms. We relate the problem to new theta functions for indefinite lattices.

alg-geom↗

arXiv1996

The quantum cohomology of blow-ups of P^2 and enumerative geometry

We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed multiplicities at fixed general points. We show that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2. In particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.

alg-geom↗

arXiv1995

Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$

We study the Donaldson invariants of simply connected $4$-manifolds with $b_+=1$, and in particular the change of the invariants under wall-crossing. We assume the conjecture of Kotschick and Morgan about the shape of the wall-crossing terms (which Oszvath and Morgan are now able to prove), and are determine a generating function for the wall-crossing terms in terms of modular forms. As an application we determine all the Donaldson invariants of the projective plane in terms of modular forms. The main tool are the blowup formulas, which are used to obtain recursive relations.

alg-geom↗

arXiv1995

Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces

We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for computing the Donaldson invariants of all rational surfaces in almost all chambers. In particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the projective plane of degree at most 50.

alg-geom↗

arXiv1994

Variation of moduli spaces and Donaldson invariants under change of polarization

The paper determines the change of moduli spaces of rank $2$ sheaves on surfaces with $p_g=0$ under change of polarization and the corresponding change of the Donaldson invariants. In this revised version we have made some minor stylistic changes in the previous text. In addition we have added a final chapter of about 20 pages (announced in the previous version), in which the six lowest order terms (three of them non-zero) of the change are computed explicitely using computations in the cohomology of Hilbert schemes of points.

alg-geom↗