Free curves and fundamental groups
Eric Jovinelly, Brian Lehmann, Eric Riedl

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Geometric and Algebraic Topology
