Stefan Schreieder (LMU, Munich):  The rationality problem for quadric bundles

Abstract: We study the rationality problem for quadric bundles X over rational bases S. By a theorem of Lang, such bundles are rational if r>2^n-2, where r denotes the fibre dimension and n=dim(S) denotes the dimension of the base. We show that this result is sharp. In fact, for any r at most 2^n-2, we show that many smooth r-fold quadric bundles over rational n-folds are not even stably rational. Our result is based on a generalization of the specialization method of Voisin and Colliot-Thélène—Pirutka.

Claire Voisin (College de France):  Segre numbers of tautological bundles on Hilbert schemes

Abstract:   We establish geometric vanishings in certain ranges for the top Segre classes of tautological bundles of punctual Hilbert schemes of K3 surfaces (which had been first obtained by Marian-Oprea-Pandharipande  by different methods) and also K3 surfaces blown-up at one point. We show how all the Segre numbers for any surface and any polarization are formally determined by these vanishings and we thus reduce the Lehn conjecture to showing that Lehn's function also has these vanishing properties. Marian-Oprea-Pandharipande in turn used the present results to complete recently the proof of Lehn's conjecture.