28/02/2011, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática 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 on a complex surface , nodal curves - those with the simplest possible singularities - appear in codimension 1. More generally those with nodes occur in codimension . In particular a d-dimensional linear family of curves should contain a finite number of such nodal curves. The classical problem - at least in the case of being the projective plane - is to determine this number. The Göttsche conjecture states that the answer should be topological, given by a universal degree polynomial in the four numbers and . 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.