Eric Jovinelly : Free curves in Singular Varieties

Posted by Izzet Coskun , part of the Algebraic Geometry Seminar.

At

Feb. 17, 2025, 3 p.m.

In

636 SEO

Abstract

Rational curves play a critical role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss free curves of higher genus. Using some ideas on stability of vector bundles, we show that any klt Fano variety has higher-genus free curves. We then use the existence of these free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds, obtain an optimal upper bound for any klt pair on the length of extreme rays of its Mori cone of curves, and study the fundamental group of the smooth locus of a klt Fano variety. This is joint work with Brian Lehmann and Eric Riedl.