Surfaces close to the Severi lines in positive characteristic

TL;DR

This paper investigates the numerical inequalities of algebraic surfaces in positive characteristic, classifies those near the Severi lines, and describes their geometric structures, especially focusing on surfaces with maximal Albanese dimension and specific invariants.

Contribution

It establishes new bounds relating invariants of surfaces in positive characteristic and classifies surfaces achieving equality, revealing their structure as double covers of elliptic surfaces.

Findings

Proves inequality: if $K_X^2<rac{9}{2}\chi(\mathcal{O}_X)$, then $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$.

Classifies surfaces where equality holds for $q(X)\geq 3$, as double covers of elliptic surfaces.

Shows a similar partial result in characteristic two, extending the classification.

Abstract

Let $X$ be a surface of general type with maximal Albanese dimension over an algebraically closed field of characteristic greater than two: we prove that if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. Moreover 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. In addition we expose a similar partial result over algebraically closed fields of characteristic two. 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 $4$.