Free Curves on Varieties

TL;DR

This paper explores generalizations of rationally connected varieties by studying free curves of higher genus, establishing their implications for the rational connectivity of varieties in characteristic zero, and initiating positive characteristic analysis.

Contribution

It introduces the concept of free curves of higher genus and demonstrates their role in characterizing rationally connected varieties over fields of characteristic zero.

Findings

Existence of free higher genus curves implies rational connectivity in characteristic zero.

Defined varieties covered by fixed genus curves with two general points connected.

Initiated study of free curves in positive characteristic.

Abstract

We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves $f:C\to X$ with large unobstructed deformation space as originally defined by Koll\'ar, but we also give definitions and basic properties of varieties $X$ covered by a family of curves of a fixed genus $g$ so that through any two general points of $X$ there passes the image of a curve in the family. We prove that the existence of a free curve of genus $g \geq 1$ implies the variety is rationally connected in characteristic zero and initiate a study of the problem in positive characteristic.