An orbifold approach to Severi Inequality

TL;DR

This paper proves the Severi inequality for certain algebraic surfaces and characterizes the surfaces attaining equality as double covers over Abelian surfaces, extending previous results to the canonical model.

Contribution

It extends the proof of Severi inequality to the canonical model of surfaces with additional assumptions and characterizes surfaces with equality as double covers over Abelian surfaces.

Findings

Proved Severi inequality for the canonical model of certain surfaces.

Characterized surfaces with $K_S^2 = 4\chi(S)$ as double covers over Abelian surfaces.

Provided a characterization of the ramification divisor for these double covers.

Abstract

For a smooth minimal surface of general type $S$ with $Albdim(S) = 2$, Severi inequality says that $K_S^2 \geq 4\chi(S)$, which was proved by Pardini. It is expected that when the equality is attained, $S$ is birational to a double cover over an Abelian surface branched along a divisor having at most negligible singularities. This was proved when $K_S$ is ample by Manetti. In this paper, we applied Manetti's method to the canonical model of $S$, with some additional assumptions we proved Severi inequality and characterized the surfaces with $K_S^2 = 4\chi(S)$.In addition, we gave a characterization of the double cover over an Abelian surface via the ramification divisor.