Let B be a simply-connected projective variety such that the first cohomology groups of all line bundles on B are zero. Let E be a vector bundle over B and X = P(E). It is easily seen that a power of any endomorphism of X takes fibers to fibers. We prove that if X admits an endomorphism which is of degree greater than one on the fibers, then E splits into a direct sum of line bundles.
We study contraction of points on P 1 ( Q̄) with certain control on local ramification indices, with application to the unramified curve correspondence problem initiated by Bogomolov and Tschinkel.