Free curves and fundamental groups

TL;DR

This paper demonstrates the existence of free higher-genus curves on certain Fano varieties and uses these curves to prove finiteness of the fundamental group of their smooth loci, extending results to positive characteristic and quasiprojective varieties.

Contribution

It introduces methods to find free curves on Fano varieties in various settings and links these to fundamental group finiteness, including an appendix on the K"unneth formula for tame étale fundamental groups.

Findings

Existence of free higher-genus curves on klt and lc Fano varieties.

Finiteness of the fundamental group of the smooth locus in these varieties.

Extension of methods to positive characteristic and quasiprojective varieties.

Abstract

We show that klt Fano varieties and certain lc Fano varieties contain free higher-genus curves in their smooth loci. Our methods also allow us to find free curves on varieties in positive characteristic and on quasiprojective varieties, under a natural positivity condition on the tangent bundle. We then use the existence of free curves to deduce finiteness of the fundamental group of the smooth locus in these settings. The paper includes an appendix by de Jong that establishes the K\"unneth formula for tame \'etale fundamental groups.