Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

New surfaces with canonical map of high degree

Pages: 1811 – 1823



Christian Gleissner (University of Bayreuth, Lehrstuhl Mathematik VIII, Bayreuth, Germany)

Roberto Pignatelli (Dipartimento di Matematica, Università di Trento, Italy)

Carlos Rito (Universidade de Trás-os-Montes e Alto Douro, UTAD Quinta de Prados, Vila Real, Portugal)


We give an algorithm that, for a given value of the geometric genus $p_g$, computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree $32$. We also construct a surface with $q = 1$ and canonical map of degree $24$. These are regular surfaces with $p_g = 3$ and base point free canonical system. We discuss the case of regular surfaces with $p_g = 4$ and base point free canonical system.

Received 11 October 2018

Accepted 17 March 2020

Published 13 July 2023