Curves on generic Kummer varieties (Q753883)

Let \(\pi:K\to A\) be the Kummer variety of a q-dimensional Abelian variety over \({\mathbb{C}}\). Let C be a smooth curve of genus \(g,\) and \(\phi:C\to K\) be a nonconstant morphism. - The author shows that: 1. \((\phi,C)\) is rigid if \(g\leq q-2\). (In particular: Kummer surfaces are not uniruled.) 2. \(g\geq q-2\) if A is general among the Abelian varieties carrying a polarization of any given type. 3. As a consequence, if \(q\geq 3\) a general Abelian variety does not contain any hyperelliptic curve.