Product-Quotient Surfaces: new invariants and algorithms

TL;DR

This paper introduces a new invariant gamma for product-quotient surfaces, develops an efficient algorithm to classify them, and discovers many new surfaces, including most of genus zero, supporting a conjecture about bounded invariants.

Contribution

It proposes the invariant gamma, creates an improved classification algorithm, and significantly expands the known examples of product-quotient surfaces, especially of genus zero.

Findings

Developed an algorithm to construct all regular product-quotient surfaces with given gamma and p_g.

Constructed numerous new regular product-quotient surfaces of genus zero.

Found only two of these surfaces are of general type, increasing known families to 75.

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 x 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 developped and implemented an algorithm which constructs all regular product-quotient surfaces with given values of gamma and geometric genus in the computer algebra program MAGMA. 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 of the form gamma < Gamma(p_g,q).