Brauer groups of bielliptic surfaces and classification of irregular surfaces in positive characteristic

Abstract

In this work we tackle three problems about surfaces.

In Part I (Chapter 2) we study the Brauer groups of bielliptic surfaces in characteristic zero. More precisely, given a bielliptic surface X, we give explicit generators for the torsion of the second cohomology group H^2(X,Z) of each type of bielliptic surface, and we determine the injectivity (and possibly the triviality) of the Brauer maps arising from canonical covers and bielliptic covers. This part is based on

E. Ferrari, S. Tirabassi, M. Vodrup, On the Brauer Group of Bielliptic Surfaces, with an appendix The Homomorphism Lattice of Two Elliptic Curves by J. Bergström and S. Tirabassi, preprint, arXiv:1910.12537, 2019.

In Part II (Chapter 3 and Chapter 4) we deal with two problems of characterisation of surfaces in positive characteristic. In Chapter 3 we show that a smooth projective surface over an algebraically closed field of characteristic at least five is birational to an abelian surface if and only if P_1(S)=P_4(S)=1 and h^1(S,O_S)=2. This is based on

E. Ferrari, An Enriques Classification Theorem for Surfaces in Positive Characteristic, Manuscripta Mathematica 160, pp. 173–185, 2019.

Also, we discuss the fact that K3 surfaces are characterised by P_1(S)=P_2(S)=1 and h^1(S,O_S)=0. In Chapter 4 we study surfaces of general type with p_g(S)=h^1(S,O_S)=3 in positive characteristic. We compare our results to those in characteristic zero that were obtained in

C.D. Hacon, R. Pardini, Surfaces with p_g=q=3, Transactions of the American Mathematical Society 354, No.7, pp.2631–2638, 2002. G.P. Pirola, Surfaces with p_g=q=3, Manuscripta Mathematica 108, pp.163–170, 2002.