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

A -homology quadric is a normal projective algebraic surface with the same Betti numbers as the smooth quadric in 3. A smooth -homology quadric is either rational or of general type with vanishing geometric genus. Smooth minimal -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 PSL2() × PSL2() 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 XG and of its minimal desingularization Z. In this way we provide examples of singular -homology quadrics and minimal surfaces Z of general type with q = pg = 0 and K2 = 4 or 2 which contain the maximal number of disjoint (2)-curves. Conversely, we also show that if a smooth minimal surface of general type has the same invariant as Z and same number of (2)-curves, then we can construct geometrically a surface of general type with c12 = 8, c2 = 4.

$\mathbb{Q}$-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
Received: 18 January 2012
Revised: 24 May 2013
Accepted: 11 July 2013
Published: 22 December 2013
Amir Džambić
Institut für Mathematik
Johann Wolfgang Goethe Universität
Robert-Mayer-Str. 6–8
D-60325 Frankfurt am Main
Xavier Roulleau
Laboratoire de Mathématiques et Applications
Université de Poitiers
Téléport 2 - BP 30179
86962 Futuroscope Chasseneuil