On the canonical map of surfaces with q>=6

TL;DR

This paper analyzes the canonical map of minimal complex surfaces with high irregularity, sharpening inequalities and proving birationality of the canonical map for certain surfaces with q>=6.

Contribution

It refines Castelnuovo's inequality for surfaces with q>0 and proves the birationality of the canonical map for surfaces with p_g=2q-3 and q>=6.

Findings

Sharpened Castelnuovo inequality: K^2 >= 3p_g + q - 7.

Proved canonical map is birational for q>=6 surfaces with p_g=2q-3.

Established inequality: K^2 >= 7χ + 2 for these surfaces.

Abstract

We carry out an analysis of the canonical system of a minimal complex surface of general type with irregularity q>0. Using this analysis we are able to sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7. Then we turn to the study of surfaces with p_g=2q-3 and no fibration onto a curve of genus >1. We prove that for q>=6 the canonical map is birational. Combining this result with the analysis of the canonical system, we also prove the inequality: K^2>=7\chi+2. This improves an earlier result of the first and second author [M.Mendes Lopes and R.Pardini, On surfaces with p_g=2q-3, Adv. in Geom. 10 (3) (2010), 549-555].