Geometry and Physics Seminar: Martijn Kool

The Hilbert scheme of curves in class \beta on a smooth projective surface S carries a natural virtual cycle. In many cases this cycle is zero (often when S has a holomorphic 2-form and \beta is not sub-canonical). However, in these cases one can often remove part of the obstruction bundle and obtain a non-trivial reduced virtual cycle. Both cycles have interesting applications. (1) Both are related to Pandharipande-Thomas' stable pair invariants on the total space of the canonical bundle over S. (2) The reduced virtual cycle is related to Severi degrees and classical curve counting on S. (3) The non-reduced virtual cycle is related to the Seiberg-Witten invariants of S (by work of Duerr-Kabanov-Okonek and Chang-Kiem