2012/01/31 by Amir Džambić, Amir Dzambic, Xavier Roulleau
Mathematics · #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #Automorphism #Disjoint sets #Geometry and complex manifolds #Quadric #Quotient #Quotient algebra #Type (biology) #math.AG #msc:11F06 #msc:14G35 #msc:14J29 #msc:14J50
paper · pdf · doi:10.2140/pjm.2014.267.91
published as Pacific J. Math. 267 (2014) 91-120 · 22 pages, acknowledgements completed
arxiv created 2013/03/07 · openalex publication_date 2014/01/01 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P3, i.e. K2=2c2=8 and q=pg=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=pg=0 and K2=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 K2=2c2=8.