2024/08/01 by Simon Brandhorst, Brandhorst, Simon, Víctor González‐Alonso +1 · 1 citation
Mathematics · #14J27 #14J28 #32G13 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2408.00306
openalex publication_date 2024/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.