Some examples of algebraic surfaces with canonical map of degree 20

Nguyen Bin

doi:10.5802/crmath.267

Abstract

In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.

Highlights

If X is a minimal smooth complex projective surface, we denote by φ|KX | : X Ppg (X )−1 the canonical map of X, where KX is the canonical divisor of X and pg (X ) = dim H 0 (X, KX ) is the geometric genus
It is interesting to know which positive integers d occur as the degree of such canonical maps for surfaces of general type
X is a minimal surface of general type with canonical map of degree 20 satisfying the following: pg (X ) = 3, K

Summary

Introduction

It is interesting to know which positive integers d occur as the degree of such canonical maps for surfaces of general type. This problem is motivated by the work of A. For surfaces of general type, the degree d of the canonical map is at most 36 [9, Proposition 5.7]. We present a way to construct surfaces with d = 20 as Z42-covers of the Del Pezzo surface Y4 of degree 5. X is a minimal surface of general type with canonical map of degree 20 satisfying the following: pg (X ) = 3, K. There exist minimal surfaces of general type X satisfying the following d

Z42-coverings

Constructions of the surfaces in Theorem 4