Automorphisms and quotients of quaternionic fake quadrics

Amir Džambić and Xavier Roulleau

Vol. 267 (2014), No. 1, 91–120

DOI: 10.2140/pjm.2014.267.91

Abstract

A $\mathbb{Q}$-homology quadric is a normal projective algebraic surface with the same Betti numbers as the smooth quadric in ${\mathbb{P}}^3$. A smooth $\mathbb{Q}$-homology quadric is either rational or of general type with vanishing geometric genus. Smooth minimal $\mathbb{Q}$-homology quadrics of general type are called fake quadrics. Here we study quaternionic fake quadrics, that is, fake quadrics whose fundamental group is an irreducible lattice in ${PSL}_2\left(ℝ\right)\times {PSL}_2\left(ℝ\right)$ derived from a division quaternion algebra over a real number field. We provide examples of quaternionic fake quadrics $X$ with a nontrivial automorphism group $G$ and compute the invariants of the quotient $X∕G$ and of its minimal desingularization $Z$. In this way we provide examples of singular $\mathbb{Q}$-homology quadrics and minimal surfaces $Z$ of general type with $q=p_g=0$ and $K^2=4$ or $2$ which contain the maximal number of disjoint $\left(-2\right)$-curves. Conversely, we also show that if a smooth minimal surface of general type has the same invariant as $Z$ and same number of $\left(-2\right)$-curves, then we can construct geometrically a surface of general type with $c_1^2=8$, $c_2=4$.

Keywords

$\mathbb{Q}$-homology quadrics, surfaces with $q=p_{g}=0$, fake quadrics, surfaces of general type, automorphisms

Mathematical Subject Classification 2010

Primary: 14G35, 14J10, 14J29

Secondary: 14J50, 11F06, 11R52

Milestones

Received: 18 January 2012

Revised: 24 May 2013

Accepted: 11 July 2013

Published: 22 December 2013