Free curves and fundamental groups

TL;DR

This work extends the landscape of rational curves on Fano-type varieties to free higher-genus curves, linking tangent-bundle positivity to the existence of r-free curves across characteristic 0, positive characteristic, and log settings. The authors develop a robust deformation and Frobenius-based framework to convert positive-curvature curves into r-free curves and then deduce strong consequences, notably finiteness of the fundamental group of the smooth locus. They establish a detailed equivalence web: absence of zero-quotients of the tangent bundle, existence of 1-free curves, and vanishing of log/differential tensor sections, with log-Fano and lc variants treated in parallel. The finite-fundamental-group results extend to the log open locus and rely on an appendix proving a Künneth-type formula for tame étale fundamental groups, enabling product arguments. The paper also provides a constructive program for representing nef classes by free curves and gives a terminal-Fano threefold case ensuring free rational curves in the smooth locus, offering both foundational theory and concrete geometric applications.

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 Künneth formula for tame étale fundamental groups.

Theorems & Definitions (83)

• Conjecture 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10