I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me see the obvious.


Thank you very much.



Proof of "generic curve of genus at least 2 has no nontrivial maps to a positive genus curve"
OK, here's an answer. As I said in the comments, if the $C$ is a (smooth projective) curve with a simple Jacobian, then it can't map onto a curve of smaller genus. Since I was a bit curious myself, 
I found a reference for the next step: Koizumi "The ring of correspondences on a generic curve of genus g" Nagoya (1976).
The result is better than I expected! Koizumi proves that if $C$ is the geometric generic curve of $M\_g$ over the algebraic closure of the prime field, then $End(J( C) =\mathbb{Z}$; so in particular, it is simple. 


