Free curves on varieties

  • Frank Gounelas

    Institut für Mathematik Humboldt Universität zu Berlin Unter den Linden 6 10099 Berlin
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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:CXf:C\to X with large unobstructed deformation space as originally defined by Kollár, but we also give definitions and basic properties of varieties XX covered by a family of curves of a fixed genus gg so that through any two general points of XX there passes the image of a curve in the family. We prove that the existence of a free curve of genus g1g\geq1 implies the variety is rationally connected in characteristic zero and initiate a study of the problem in positive characteristic.

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Frank Gounelas, Free curves on varieties. Doc. Math. 21 (2016), pp. 287–308

DOI 10.4171/DM/534