Automorphisms and quotients of quaternionic fake quadrics

TL;DR

This paper investigates quaternionic fake quadrics, constructing examples with automorphisms, analyzing their quotients, and deriving new surfaces with specific invariants, advancing understanding of their geometric and automorphic properties.

Contribution

It introduces new examples of quaternionic fake quadrics with automorphisms and computes invariants of their quotients, linking these to the construction of surfaces with particular invariants.

Findings

Examples of quaternionic fake quadrics with non-trivial automorphism groups.

Computed invariants of minimal desingularisations of quotient surfaces.

Established a method to construct surfaces with K^2=8 from quotients with specific invariants.

Abstract

A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some quaternion algebras over real number fields. We provide examples of quaternionic fake quadrics X with a non-trivial automorphism group and compute the invariants of the minimal desingularisation of the quotient of X by this group. In that way we obtain minimal surfaces of general type Z with q=p_g=0 and K^2=4,2 which contain the maximal number of disjoint nodal curves. We then prove that if a surface of general type has the same invariant as Z and same number of nodal curves, we can construct geometrically a surface of general type with K^2=2c_2=8.