Geodesic
Free curves on varieties
Documenta mathematica, Tome 21 (2016), pp. 287-308
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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→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≥1 implies the variety is rationally connected in characteristic zero and initiate a study of the problem in positive characteristic.
DOI : 10.4171/dm/534
Classification : 14H10, 14M20, 14M22 
@article{10_4171_dm_534,
     author = {Frank Gounelas},
     title = {Free curves on varieties},
     journal = {Documenta mathematica},
     pages = {287--308},
     year = {2016},
     volume = {21},
     doi = {10.4171/dm/534},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/534/}
}
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Frank Gounelas. Free curves on varieties. Documenta mathematica, Tome 21 (2016), pp. 287-308. doi: 10.4171/dm/534

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