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 $\Gamma (p_g,q)\ge \gamma$ (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.