Speaker: Martijn Kool (Imperial)
Title: Reduced classes and curve counting on surfaces

Abstract: Let S be a smooth projective surface with sufficiently ample line bundle L. The Göttsche conjecture (proved by several people) states that the number of δ-nodal curves in a general δ-dimensional linear subsystem of |L| is given by a universal polynomial in c_1(L)^2, c_1(L).c_1(S), c_1(S)^2, c_2(S). We define a perfect obstruction theory for stable pairs on S by describing their moduli space as cut out by a section of a vector bundle on a smooth space. The associated invariants are topological and generalise Göttsche's invariants to the non-ample case. Next, we define quite general reduced Gromov-Witten and stable pair invariants of the canonical bundle K_S. In various cases, we relate (some of) these three invariants. In the sufficiently ample case, we prove a version of the MNOP conjecture. This is joint work with Richard Thomas.