Rational endomorphisms of Fano hypersurfaces

TL;DR

This paper investigates the degrees of rational endomorphisms of general complex Fano and Calabi-Yau hypersurfaces, establishing congruence conditions and implications for birational properties, using specialization to characteristic p.

Contribution

It introduces new congruence conditions on endomorphism degrees and shows certain hypersurfaces are not birational to elliptic fibrations, advancing understanding of their birational geometry.

Findings

Degrees of rational endomorphisms satisfy specific congruences

Very general hypersurfaces of certain degrees are not birational to elliptic fibrations

Resolution of singularities in mixed characteristic is developed

Abstract

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree $d\ge 5\lceil (n+3)/6\rceil$ are not birational to elliptic fibrations. A key part of the argument is to resolve singularities of general p-cyclic covers in mixed characteristic p.