EVENT DETAILS AND ABSTRACT

Title: Nodal curves, old and new
Speaker: Richard Thomas
Speaker Info: Imperial College, London
Brief Description:
Special Note:
Abstract:

I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon, ...) and a solution using simple classical techniques. I will also explain how the motivation for these techniques would never have been dreamed up without modern ideas from string theory (Gromov-Witten invariants, BPS states) and geometry (the MNOP conjecture of Maulik-Nekrasov-Okounkov-Pandharipande).

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 \(c_2(S)\). I will describe a simple proof and recent extensions in work of Göttsche, Kool, Li, Rennemo, Shende and Tszeng.