Geir Ellingsrud

Bott’s formula and enumerative geometry

We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry compu- tations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons. MATHEMATICAL INSTITUTE, UNIVERSITY OF OSLO, P. 0. Box 1053, N-0316 OSLO, NORWAY E-mail address: ellingsr0math.uio .no MATHEMATICAL INSTITUTE, UNIVERSITY OF BERGEN, ALLEG 55, N-5007 BERGEN, NORWAY E-mail address: strommeti i.uib.no This content downloaded from 157.55.39.29 on Tue, 12 Apr 2016 10:23:14 UTC All use subject to http://about.jstor.org/terms

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles A and B of complementary dimensions in the Hilbert scheme H parameterizingr subschemes of given finite length n of a smooth projective surface S. The (n + 1)cycle A corresponds to the set of finite closed subschemes the support of which has cardinality 1. The (n-1)-cycle B consists of the closed subschemes the support of which is one given point of the surface. Since B is contained in A, indirect methods are needed. The intersection number is A.B = (--1)n-1n, answering a question by H. Nakajima.

Irreducibility of the punctual quotient scheme of a surface

AbstractIt is shown that the punctual quotient schemeQlr parametrizing all zero-dimensional quotients

Calabi-Yau Manifolds and Related Geometries

\mathcal{O}_{A^2 }^{ \oplus ^r } \to T

Variation of moduli spaces and Donaldson invariants under change of polarization

of lengthl and supported at some fixed point O∈A2 in the plane is irreducible.