# Surfaces close to the Severi lines

## TL;DR

This paper classifies surfaces of general type with maximal Albanese dimension near the Severi line, identifying specific double cover structures and establishing inequalities relating their invariants.

## Contribution

It provides a complete classification of surfaces achieving equality in a specific inequality, describing their canonical models as double covers of elliptic surfaces.

## Key findings

- Surfaces with $K_X^2= rac{9}{2}\chi( ext{O}_X)$ are classified as double covers of elliptic surfaces.
- Surfaces not satisfying the equality have $K_X^2 extgreater 4\chi( ext{O}_X)+8(q-2)$.
- Explicit geometric descriptions of the canonical models are given for equality cases.

## Abstract

Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. We also prove, in the same hypothesis, that a surface $X$ with $K_X^2\neq 4\chi(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq 4\chi(\mathcal{O}_X)+8(q-2)$ and we give a characterization of surfaces for which the equality holds. These are surfaces whose canonical model is a double cover of an isotrivial smooth elliptic surface branched over an ample divisor with at most negligible singularities whose intersection with the elliptic fibre is $4$.