Surfaces on the Severi line

TL;DR

This paper characterizes minimal complex surfaces of general type with maximal Albanese dimension that achieve the Severi equality, showing they are double covers of their Albanese surfaces with specific branch divisors.

Contribution

It provides a precise geometric description of surfaces on the Severi line, linking the equality case to double covers of Albanese surfaces with particular branch divisors.

Findings

Surfaces with $K^2_S=4\,\chi( ext{O}_S)$ have $q(S)=2$.

Canonical models are double covers of Albanese surfaces.

Branch divisors are ample with negligible singularities.

Abstract

Let S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has $K^2_S\geq 4\chi(\mathcal O_S)$. We prove that the equality $K^2_S=4\chi(\mathcal O_S)$ holds if and only if $q(S):= h^1(\mathcal O_S)=2$ and the canonical model of $S$ is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities.