as possible, namely ${{\mathbb Z}}$. Lefschetz [@lefschetz] proved in 1928 that this is still true if we restrict ourselves to hyperelliptic curves and we refer to Ciliberto’s survey [@Ciliberto] for what other results of that type were known before 1989. As of then the theorem of Lefschetz has been refined and generalized in several directions. For example, Ciliberto, van der Geer and Teixidor i Bigas [@CGT] studied (and to some extent characterized) the locus of curves for which the endomorphism ring of the Jacobian is strictly larger than ${{\mathbb Z}}$, while Zarhin addressed this problem over other base fields and also considered in [@zahrin2004] and [@zahrin2012] curves with a specified automorphism of a particular type other than a hyperelliptic involution: he proved that for a general such curve the endomorphism ring of its Jacobian is indeed as small as it could possibly be.

In the present paper we obtain a natural generalization of these results, at least over the base field ${{\mathbb C}}$, by proving such a minimality property for curves endowed with an action of a given (but arbitrary) finite group. More precisely, we prove that for a very general complex smooth projective curve $C$ with automorphism group $G$ 