Automorphisms and quotients of quaternionic fake quadrics

Details Automorphisms and quotients of quaternionic fake quadrics Automorphisms and quotients of quaternionic fake quadrics

2012-01-24

arxiv.org/abs/1201.5051v1

Mathematics

Algebraic Geometry

22 pages, comments welcome

Scientific paper

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 or 1 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.

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