Eric Riedl : Nonfree curves and Geometric Manin's Conjecture

Posted by Eric Jovinelly , part of the Algebraic Geometry Seminar.

At

April 22, 2024, 3 p.m.

In

636 SEO

Abstract

Given a smooth Fano variety and a smooth curve B, let Hom(B,X) be the moduli space of maps from B to X. Let M be a component of Hom(B,X). If a general curve parameterized by M is free, it means M has the expected dimension and good deformation behavior. Components M consisting entirely of curves that are not free are more mysterious. We give a geometric characterization of which curves can be nonfree, explaining that roughly they come from fibrations. We show how this result can be seen as the analogue of Manin's Conjecture, which predicts the number of rational on a variety of bounded height. This is joint work with Brian Lehmann and Sho Tanimoto.