Product-quotient surfaces: new invariants and algorithms

Abstract

In this article we suggest a new approach to the systematic, computer-aided construction and to the classification of product-quotient surfaces, introducing a new invariant, the integer \gamma , which depends only on the singularities of the quotient model X = (C_1 \times C_2)/G . It turns out that \gamma is related to the codimension of the subspace of H^{1,1} generated by algebraic curves coming from the construction (i.e., the classes of the two fibers and the Hirzebruch-Jung strings arising from the minimal resolution of singularities of X ). Profiting from this new insight we developed and implemented an algorithm in the computer algebra program MAGMA which constructs all regular product-quotient surfaces with given values of \gamma and geometric genus. Being far better than the previous algorithms, we are able to construct a substantial number of new regular product-quotient surfaces of geometric genus zero. We prove that only two of these are of general type, raising the number of known families of product-quotient surfaces of general type with genus zero to 75. This gives evidence to the conjecture that there is an effective bound \Gamma (p_g,q) \geq \gamma (cf. Conjecture 4.5). Finally we introduce a duality among product-quotient surfaces and prove that the dual surface of a surface of geometric genus zero has maximal Picard number, thus providing several new examples of surfaces with maximal Picard number.