Surfaces with K^2<3\chi and finite fundamental group

TL;DR

This paper classifies minimal surfaces of general type with specific invariants, showing that certain fundamental groups imply the surface is a Campedelli surface, and provides new classification results for related surfaces.

Contribution

It establishes classification results for minimal surfaces with K^2=3p_g-5 and canonical maps as birational morphisms, and links fundamental group order to surface type.

Findings

Surfaces with K^2=3hi-1 and fundamental group of order 8 are Campedelli surfaces.

Fundamental groups of surfaces with K^2<3hi and no irregular etale cover have order at most 9.

Classification results for surfaces with K^2=3p_g-5 and birational canonical maps.

Abstract

In this paper we continue the study of algebraic fundamentale group of minimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, if K_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then S is a Campedelli surface. In view of the results of math.AG/0512483 and math.AG/0605733, this implies that the fundamental group of a surface with K^2<3\chi that has no irregular etale cover has order at most 9, and if it has order 8 or 9, then S is a Campedelli surface. To obtain this result we establish some classification results for minimal surfaces of general type such that K^2=3p_g-5 and such that the canonical map is a birational morphism. We also study rational surfaces with a Z_2^3-action.