Let PBn(Sg,p) be the pure braid group of a genus g>1 surface with p
punctures. In this paper we prove that any surjective homomorphism
PBn(Sg,p)→PBm(Sg,p) factors through one of the forgetful
homomorphisms. We then compute the automorphism group of PBn(Sg,p),
extending Irmak, Ivanov and McCarthy's result to the punctured case.
Surprisingly, in contrast to the n=1 case, any automorphism of
PBn(Sg,p), n>1 is geometric.Comment: 12 pages, changes on writin