Vol. 267, No. 1, 2014

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Automorphisms and quotients of quaternionic fake quadrics

Amir Džambić and Xavier Roulleau

Vol. 267 (2014), No. 1, 91–120
Abstract

A $ℚ$-homology quadric is a normal projective algebraic surface with the same Betti numbers as the smooth quadric in ${ℙ}^{3}$. A smooth $ℚ$-homology quadric is either rational or of general type with vanishing geometric genus. Smooth minimal $ℚ$-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 ${PSL}_{2}\left(ℝ\right)×{PSL}_{2}\left(ℝ\right)$ 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 $ℚ$-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