Enriques surfaces with normal K3-like coverings
- 23 April 2021
- journal article
- research article
- Published by Mathematical Society of Japan (Project Euclid) in Journal of the Mathematical Society of Japan
- Vol. 73 (2), 433-496
- https://doi.org/10.2969/jmsj/83728372
Abstract
We analyze the structure of simply-connected Enriques surfaces in characteristic two whose K3-like coverings are 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.Keywords
This publication has 37 references indexed in Scilit:
- Arithmetic moduli and lifting of Enriques surfacesJournal für die reine und angewandte Mathematik (Crelles Journal), 2013
- Singularities appearing on generic fibers of morphisms between smooth schemesThe Michigan Mathematical Journal, 2008
- On actions of infinitesimal group schemesJournal of Algebra, 2003
- The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic $2$Tohoku Mathematical Journal, 1994
- Conductor, discriminant, and the Noether formula of arithmetic surfacesDuke Mathematical Journal, 1988
- On Minimally Elliptic SingularitiesAmerican Journal of Mathematics, 1977
- Coverings of the Rational Double Points in Characteristic pPublished by Cambridge University Press (CUP) ,1977
- Enriques' classification of surfaces in char.p, IIIInventiones Mathematicae, 1976
- Elliptic Curves and Wild RamificationAmerican Journal of Mathematics, 1967
- Vollständige Durchschnitte undp-BasenArchiv der Mathematik, 1965