Surfaces close to the Severi lines

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 $K_X^2\ge 4\chi ({\mathcal{O}}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for $q(X)\ge 3$: 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 $K_X^2\ne 4\chi ({\mathcal{O}}_X)+4(q-2)$ satisfies $K_X^2\ge 4\chi ({\mathcal{O}}_X)+8(q-2)$ 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 $4$.

References in corpus (4)

• Revêtements étales et groupe fondamental (SGA 1)
• The Severi inequality $K^2\ge 4\chi$ for surfaces of maximal Albanese dimension
• Slope inequalities and a Miyaoka-Yau type inequality
• Surfaces on the Severi line in positive characteristics

Cited by in corpus (1)

• Surfaces close to the Severi lines in positive characteristic