Date: Tuesday, April 16, 2013
Title: Algebraic geometry Spring lectures: The Gottsche conjecture
Abstract: I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques that one would never have thought of without ideas coming from string theory (GromovWitten invariants, BPS states) and modern geometry (the MaulikNekrasovOkounkovPandharipande conjecture).
In generic families of curves C on a complex surface S, nodal curves  those with the simplest possible singularities  appear in codimension 1. More generally those with d nodes occur in codimension d. In particular a ddimensional linear family of curves should contain a finite number of such dnodal curves. The classical problem  at least in the case of S being the projective plane  is to determine this number. The Gottsche conjecture states that the answer should be topological, given by a universal degree d polynomial in the four numbers C.C, c_1(S).C, c_1(S)^2 and c_2(S).
This was proved recently by YuJong Tzeng. I will explain a simpler proof which was joint work with Martijn Kool and Vivek Shende. The treatment will be very lowtech; I won't assume any prior knowledge. The main tool is Euler characteristics (which I will also explain).
Speaker: Richard Thomas
Institution: Imperial College
