Free Curves on Varieties

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 Kollár, 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 $g\geq1$ implies the variety is rationally connected in characteristic zero and initiate a study of the problem in positive characteristic.

2010 Mathematics Subject Classification: 14M20, 14M22, 14H10.

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