Surfaces close to the Severi lines

TL;DR

This thesis classifies surfaces of general type near the Severi lines, focusing on those with maximal Albanese dimension, and provides explicit descriptions of their canonical models as double covers of Abelian or elliptic surfaces.

Contribution

It offers a complete classification of surfaces where the key inequality becomes equality, detailing their structure as double covers over specific surfaces, extending results to various characteristics.

Findings

Classifies surfaces with equality in the key inequality as double covers of Abelian or elliptic surfaces.

Establishes lower bounds for $K_X^2$ when the inequality is not an equality.

Extends classification results to algebraically closed fields of characteristic not 2.

Abstract

In this Thesis we study surfaces of general type with maximal Albanese dimension for which the quantity $K_X^2-4\chi(\mathcal{O}_X)-4(q-2)$ vanishes or is "small", that is surfaces close to the Severi lines. Over the complex numbers, it is known that a surface $X$, provided that $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, has to satisfy the inequality $K_X^2-4\chi(\mathcal{O}_X)-4(q-2)\geq 0$. We give a constructive and complete classification of surfaces for which equality holds: these are surfaces whose canonical model is a double cover of an Abelian surface ($q=2$) or of a product elliptic surface ($q\geq 3$) branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice in the latter case. 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 fibration branched over an ample divisor with at most negligible singularities whose intersection with the elliptic fibre is $4$. Because these results are intimately related to theory of double covers, we see that their proof extend almost step by step to the case of any algebraically closed field of characteristic different from $2$. We also give some partial results over algebraically closed fields of characteristic $2$ after a study of double covers in that case.