Mathematics Department Colloquium: Hilbert schemes and enumerative geometry.

Colloquium | February 14 | 4:10-5 p.m. | 60 Evans Hall

 Claire Voisin, Collège de France

 Department of Mathematics

Consider an algebraic curve in 3-space; when projected generically to a plane, it will acquire a number of double points. This number depends only on the degree and the genus of the curve. Computing similar numbers when the curve is replaced by a surface arbitrarily embedded will be the subject of the lecture. One key difference with the curve case is the fact that we have to work with the Hilbert scheme of k points, instead of the k-th symmetric product, and I will spend some time on the construction of the Hilbert scheme. The main result I will present is the Lehn conjecture, now a theorem, computing all these numbers for all surfaces in terms of their numerical (complex cobordism) invariants.