Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

Abstract

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces). Comment: 24 pages; v3: journal version, correcting 20 root types to 19 and ruling out the remaining type 4A_2+A_1 (in new section 11)

Normal forms for quasi-elliptic Enriques surfaces and applications

“…One can also apply our results to the study of maximal root types supported on Enriques surfaces, i.e., rank 9 root lattices whose vertices correspond to smooth rational curves. In [Sch19] there is given a complete classification of the maximal root types for singular and classical Enriques surfaces. This can now be complemented for many types on supersingular Enriques surfaces.…”

Twelve Rational curves on Enriques surfaces

“…This splits on the K3 cover into two disjoint sections. Hence, the reasoning from [20][21][22] (or [15]) applies to show that independent of the characteristic, Y arises from Kondo's construction.…”