Algebraic Geometry and Combinatorics Seminar: Free curves on Fano varieties

Abstract: Rational curves play a pivotal 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 an improvement in Mori's Bend-and-Break that achieves the optimal degree bound and allows us to improve our understanding of sweeping families of rational curves in singular varieties. We then move on to discussing free curves of higher genus.  Using some ideas on stability of vector bundles, we show that any klt Fano variety has these higher-genus free curves. We then use the existence of these free curves to get some applications, including the existence of free rational curves in terminal Fano threefolds, the lengths of extremal rays of the cone of curves, and studying the fundamental group of the smooth locus of a terminal variety. This is joint work with Eric Jovinelly and Brian Lehmann.

Host: Roya Beheshti