Some algebraic surfaces with canonical map of degree 10, 12, 14

TL;DR

This paper constructs algebraic surfaces with specific degrees of their canonical maps and various irregularities, demonstrating the possible combinations and expanding understanding of their geometric properties.

Contribution

It proves the existence of surfaces with canonical map degrees 10, 12, 14 and all relevant irregularities, using $\

Findings

Surfaces with d=10 and all irregularities 0,1,2 exist.

Surfaces with d=12 and irregularities 1,2 exist.

Surfaces with d=14 and irregularities 0,1 exist.

Abstract

Surfaces of general type with canonical map of degree d bigger than 8 have bounded geometric genus and irregularity. In particular the irregularity is at most 2 if d>= 10. In the present paper, the existence of surfaces with d=10 and all possible irregularities, surfaces with d = 12 and irregularity 1 and 2, and surfaces with d = 14 and irregularity 0 and 1 is proven, by constructing these surfaces as $ \mathbb{Z}_2^3 $-covers of certain rational surfaces. These results together with the construction by C. Rito of a surface with d=12 and irregularity 0 show that all the possibilities for the irregularity in the cases d=10, d=12 can occur, whilst the existence of a surface with d=14 and irregularity 2 is still an open problem.