Free Curves in Singular Varieties

Rational curves are intricately linked to the birational geometry of varieties containing them.  Certain curves, called free curves, have the nicest deformation properties.  However, it is unknown whether mildly singular Fano varieties contain free rational curves in their smooth locus.  In this talk, we discuss free curves of higher genus.  Using recent results about tangent bundles, we prove that any klt Fano variety has higher genus free curves.  We then use the existence of such free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds; study the fundamental group of the smooth locus of a Fano variety; and obtain a new characterization of projective space via lengths of extremal rays.  This is joint work with Brian Lehmann, Eric Riedl, and Osamu Fujino.