527 elliptic fibrations on Enriques surfaces

TL;DR

This paper proves that every Enriques surface has exactly 527 elliptic fibrations when considering multiplicities, providing explicit calculations for singular fibers and exploring the moduli space and ramification behavior.

Contribution

It extends the known classification of elliptic fibrations on Enriques surfaces to all such surfaces over algebraically closed fields, including multiplicities and explicit fiber calculations.

Findings

Every Enriques surface has 527 elliptic fibrations with multiplicities.

Explicit formulas for reducible singular fibers and their multiplicities.

The ramification indices relate to hyperbolic volume in the Morrison-Kawamata cone conjecture.

Abstract

Barth and Peters showed that a general complex Enriques surface has exactly 527 isomorphism classes of elliptic fibrations. We show that every Enriques surface has precisely 527 isomorphism classes of elliptic fibrations when counted with the appropriate multiplicity. Their reducible singular fibers and the multiplicities can be calculated explicitly. The same statements hold over any algebraically closed field of characteristic not two. To explain these results, we construct a moduli space of complex elliptic Enriques surfaces and study the ramification behavior of the forgetful map to the moduli space of unpolarized Enriques surfaces. Curiously, the ramification indices of a similar map compute the hyperbolic volume of the rational polyhedral fundamental domain appearing in the Morrison-Kawamata cone conjecture.