Christian Liedtke

Algebraic Surfaces in Positive Characteristic

These notes grew out of a series of lectures given at Sogang University, Seoul, in October 2009. They were meant for complex geometers, who are not familiar with characteristic-p-geometry but who would like to see similarities, as well as differences, to complex geometry. More precisely, these notes are on algebraic surfaces in positive characteristic and assume familiarity with the complex side of this theory, say, on the level of Beauville’s book [9].

GOOD REDUCTION OF K3 SURFACES

Let

Fundamental groups of Galois closures of generic projections

K

Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem

be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second

Morphisms to Brauer–Severi Varieties, with Applications to Del Pezzo Surfaces

\ell

Rational curves on K3 surfaces

-adic cohomology of a K3 surface over

Supersingular K3 surfaces are unirational

K

Uniruled surfaces of general type

implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study group actions and their quotients on models of varieties.

Arithmetic moduli and lifting of Enriques surfaces

For the Galois closure X gal of a generic projection from a surface X, it is believed that π 1 (X gal ) gives rise to new invariants of X. However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of π 1 (X gal ) that depends on π 1 (X) and data from the generic projection only. In all known examples, this quotient is in fact isomorphic to π 1 (X gal ). As a byproduct, we simplify the computations of Moishezon, Teicher and others.

ON THE BIRATIONAL NATURE OF LIFTING

We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus’ crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.