#### DOCUMENTA MATHEMATICA, Vol. 21 (2016), 287-308

Frank Gounelas

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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