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

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.