Abstract A Product-Quotient surface is the minimal resolution of singularities of a quotient of a product of curves by the action of a finite group of automorphisms. Product-Quotient surfaces are introduced by Catanese in a famous paper of 2000 and then extensively studied by several authors. Indeed, they are revealed to be a very useful tool for building new examples of algebraic surfaces and exploring their geometry in an accessible way. Hence a classification of them (by fixing some invariants, e.g. $K^2$ and $\chi$), is not only interesting by itself, but also also highly practical in various contexts.
During the talk, I will introduce one of the still open problems on the degree of the canonical map of surfaces of general type and I will show which is the role that Product-Quotient surfaces have been played on the results that I have given. Hence I will provide a brief overview on Product-Quotient surfaces and I will describe the most important tools that are developed by some authors to produce a classification of them via a computational algebra system (e.g. MAGMA).
I will introduce the results I have obtained to provide a more performant algorithm. The main result is a theorem that allows us to move from a database of $G$-coverings of the projective line (in pairs), that already exists and it has been produced in a recent work by Conti, Ghigi and Pignatelli, to a database of families of Product-Quotient surfaces.
I have used this to produce a huge list of families of Product-Quotient surfaces with $p_g=3$, $q=0$, and $K^2$ high. The classification is almost complete for $K^2$ equal to $32$, with very few exceptions. Finally, if time permits, I will give more details of what I did with this list to study the already mentioned open question on the degree of the canonical map of surfaces of general type