Rationality of rationally connected threefolds admitting non-isomorphic endomorphisms

TL;DR

This paper establishes a structure theorem for certain threefolds with non-isomorphic endomorphisms and provides criteria for their rationality, concluding that smooth Fano threefolds with such endomorphisms are rational.

Contribution

It introduces a new structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds and offers a criterion for rationality of fibred rationally connected threefolds.

Findings

Every smooth Fano threefold with a non-isomorphic surjective endomorphism is rational.

Provides a criterion for rationality of fibred rationally connected threefolds.

Establishes a structure theorem for non-isomorphic endomorphisms of certain threefolds.

Abstract

We prove a structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds, or more generally for threefolds with big anti-canonical 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.