Jay Kopper : Brill-Noether-Petri without degenerations

Posted by Gregory Taylor , part of the Graduate Algebraic Geometry Seminar.

At

Feb. 6, 2019, 3 p.m.

In

712 SEO

Abstract

The Brill-Noether theorem describes the number of maps a general smooth curve admits to projective space of given degree. A complete proof was given by Griffiths-Harris in 1980, building on work of Kleiman-Laksov and others. In 1982, Gieseker gave a proof of Petri's conjecture, a strictly stronger result. The proofs of these theorems require very delicate manipulations of transverse intersections and degenerations to special (reducible!) curves in the boundary of the Deligne-Mumford compactification of the moduli space. In 1986, Lazarsfeld gave a comparitively simple proof of the Gieseker-Petri theorem by explicitly constructing curves on K3 surfaces. In this talk I will describe the relevant background and give an overview of Lazarsfeld's proof.