An elliptic surface covered by Mumford's fake projective plane (Q908983)

An algebraic surface M of general type with \(K_{M^ 2}=9\) and \(p_ g=q=0\) has been studied by Mumford. This surface is called Mumford's fake projective plane, because it has the same Betti numbers as the complex plane. By its construction M has an unramified Galois covering \(V\to M\) of order 8. Moreover, a simple group G of order 168 acts on V and M is the quotient of V by a 2-Sylow subgroup of G. The quotient surface \(Y=V/G\) and its minimal desingularization \(\tilde Y\) is studied. (Thus M is a branched covering over Y of order 21.) It is shown that Y has only 4 isolated singularities as singularites, 3 of them are rational double points of type \(A_ 2\) and the 4-th singularity is the quotient singularity of type (7,3) by the cyclic group of order 7. Moreover, it is shown that \(\tilde Y\) is an elliptic surface with the following data: the Kodaira dimension 1, the Euler number 12, \(p_ g=q=0\), the second pluri- genus 1, the third pluri-genus 1, the combination of singular fibers \(_ 2I_ 0+_ 3I_ 0+I_ 3+I_ 3+I_ 3+I_ 3\). Since Mumford's paper is based on the theory of the 2-adic unit ball due to Mustafin and Kurihara, the theory of schemes over the ring \({\mathbb{Z}}_ 2\) of 2-adic integers is the main tool in this paper.