On dominant rational maps from products of curves to surfaces of general type (Q2865927)

The \textit{Iitaka-Severi} set of a non-singular complex variety of general type \(X\) is the set of equivalence classes of generically finite dominant rational maps from \(X\) onto a variety of general type. It has been recently proved that this set is always finite.NEWLINENEWLINEIt is well known that, for a general curve of general type, this set has cardinality \(1\): in other words, a general curve of genus \(g \geq 2\) does not dominate any other curve of general type.NEWLINENEWLINEThe authors prove that the same holds for a product of two very general curves (of general type), if one of the two has genus at least \(7\). The author conjecture that the same result should hold substituting in the assumption the number \(7\) by the weaker \(3\). Indeed they also show (Remark 4.2) that when both curves are very general of genus \(2\), the Iitaka-Severi set of their product has cardinality two: more precisely is given by the class of the identity and the class of the involution obtained composing both hyperelliptic involutions. In the latter case the quotient is a regular surface (\(q:=h^1({\mathcal O})=0\)) of geometric genus (\(h^2({\mathcal O})\)) equal to \(4\).