Rationality of rationally connected three folds admitting non-isomorphic endomorphisms

Abstract

We prove a structure theorem for non-isomorphic endomorphisms of weak ℚ-Fano three folds or, more generally, for three folds with a big anticanonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be rational. As a consequence, we show (without using the classification) that every smooth Fano threefold having a non-isomorphic surjective endomorphism is rational. © 2012 American Mathematical Society.