Enriques surfaces with normal K3-like coverings

Abstract

We analyze the structure of simply-connected Enriques surface in characteristic two whose K3-like covering is normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct such surfaces and the resulting twistor lines in the moduli stack of Enriques surfaces, including the case that the K3-like covering is a normal rational surface rather then a normal K3 surface. Among other things, we show that elliptic double points indeed do occur. In this case, there is only one singularity. The main idea is to apply flops to Frobenius pullbacks of rational elliptic surfaces, to get the desired K3-like covering. Our results hinge on Lang's classification of rational elliptic surfaces, the determination of their Mordell-Weil lattices by Shioda and Oguiso, and the behavior of unstable fibers under Frobenius pullback via Ogg's Formula. Along the way, we develop a general theory of Zariski singularities in arbitrary dimension, which is tightly interwoven with the theory of height-one group schemes actions and restricted Lie algebras. Furthermore, we determine under what conditions tangent sheaves are locally free, and introduce a theory of canonical coverings for arbitrary proper algebraic schemes.

Del Pezzo surfaces and Mori fiber spaces in positive characteristic

“…The goal of this section is to study the complete local rings O ∧ Y,y at points y ∈ Y of codimension ≤ 2 contained in the conductor scheme C, and the schematic structure of the locus of non-smoothness N = Sing(Y /K). To avoid technical problems with respect to Kähler differentials of complete local rings, we also assume that the ground field K has finite p-degree, compare the discussion in [61], Section 1. Let us start with the case that the ramification locus is regular.…”

“…7, Theorem 5 the vector space Ω 1 κ/K is free of rank n−2, thus the B-module Ω 1 κ/K ⊗ κ B is free of the same rank. Since K has finite p-degree, so does κ, which ensures that Ω 1 B/κ is generated by the differentials da, db, dc modulo the relation b 2 dc = 0 (compare the discussion in [61], Section 1). Thus Ω 1 B/K has n + 1 generators and one relation, and we may use them to compute Fitting ideals.…”

The Deligne-Illusie Theorem and exceptional Enriques surfaces

“…Note that the canonical covering is a finite flat universal homeomorphism of degree deg(X/Y ) = p, and that the scheme X usually contains singularities, and easily may become non-normal. For more details, see for example [47], Section 4.…”

The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

“…The trichotomy of Enriques surfaces for p = 2 was developed by Bombieri and Mumford [12]. For more information on K3-like coverings, see for example [54].…”