527 elliptic fibrations on Enriques surfaces

Simon Brandhorst; Víctor González-Alonso

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arXiv · 2408.00306

527 elliptic fibrations on Enriques surfaces

Simon Brandhorst·Víctor González-Alonso

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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.

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Simon Brandhorst, Víctor González-Alonso. 2024-08-01. 527 elliptic fibrations on Enriques surfaces. https://arxiv.org/abs/2408.00306

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