Free curves and fundamental groups

2025/10/30 by Jovinelly, Eric, Lehmann, Brian, Riedl, Eric
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.27031

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.

Citations

• Optimal bounds in Bend-and-Break
• Campana rational connectedness and weak approximation
• On generic nefness of tangent sheaves
• Characterization of varieties of Fano type via singularities of Cox rings
• Schematic Harder-Narasimhan Stratification
• Foliations and Rational Connectedness in Positive Characteristic
• On different notions of tameness in arithmetic geometry
• Existence of minimal models for varieties of log general type
• Shokurov's Rational Connectedness Conjecture
• Rational Curves on Quasi-Projective Surfaces

Related

• Free Curves on Varieties
• Free quotients of fundamental groups of smooth quasi-projective\n varieties
• On complete families of curves with a given fundamental group in positive characteristic
• Restricted tangent bundles for general free rational curves
• The tame fundamental group schemes of curves in positive characteristic