On the classification of product-quotient surfaces with $q=0$, $p_g=3$ and their canonical map

Abstract

In this work, we present new results to produce an algorithm that returns, for any fixed pair of positive integers $K^2$ and $\chi$, all regular surfaces $S$ of general type with self-intersection of the canonical class $K_S^2=K^2$ and Euler characteristic $\chi ({\mathcal{O}}_S)=\chi$, which are product-quotient surfaces. The key result we obtain is an algebraic characterization of all families of regular product-quotients surfaces, up to isomorphism, arising from a pair of $G$-coverings of ${\mathbb{P}}^1$. As a consequence of our work, we provide a classification of all regular product-quotient surfaces $S$ of general type with $23\le K_S^2\le 32$ and $\chi ({\mathcal{O}}_S)=4$. Furthermore, we study their canonical map and present several new examples of surfaces of general type with a high degree of the canonical map.