New surfaces with canonical map of high degree (Q6172850)

Let \(S\) be a smooth surface of general type with irregularity \(q\) and geometric genus \(p_{g} \geq 3\). Denote by \(\phi\) the canonical map of \(S\) and let \(d:=\mathrm{deg}(\phi)\). It is known since Beauville that if the canonical image \(\phi(S)\) is a surface, then \[d \leq 36 - 9q \quad \text{ if } \quad g\leq 3, \quad \quad d\leq 8 \quad \text{ if } \quad g\geq 4.\] In the paper under review the authors consider the problem of finding product-quotient surfaces \((A\times B) / G\) with at most canonical singularities having a canonical map of maximum degree. For these surfaces, one has \(K^{2} \leq 8\chi\), equality holding if and only if the quotient model \((A \times B) / G\) is smooth, i.e., the action of \(G\) is free. Here we have \[d \leq 32 - 8q \quad \text{ if } \quad g\leq 3,\] with equality holding if and only if \(G\) acts freely, \(p_{q}=3\), and the canonical system is base-point free. In order to be able to understand this system, the authors restrict themselves to the study of abelian groups \(G\). The main added value of the paper is an algorithm that, for a given value of the geometric genus \(p_{q}\) and some \(n \in \mathbb{N}\), computes all regular product-quotient surfaces with abelian group \(G\) that have at most canonical singularities and have a canonical system with at most \(n\) base points. Applying it to the case \(K^{2}=32\), one gets exactly two families of surfaces with \(p_{q}=3\), \(q=0\), and the canonical map of degree \(K^{2}=32\) onto \(\mathbb{P}^{2}\). The authors describe these surfaces as \((\mathbb{Z}/2)^{4}\)-coverings of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\).