Algebraic Geometry / Moduli Seminar 

28/02/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Richard Thomas, Imperial College

Nodal curves old and new

I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques, but techniques that one would never really have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande 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 d-dimensional linear family of curves should contain a finite number of such $d-$nodal curves. The classical problem - at least in the case of $S$ being the projective plane - is to determine this number. The Göttsche 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 $c_2(S)$. There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler approach which is joint work with Martijn Kool and Vivek Shende.