Surfaces close to the Severi lines

Abstract

Let X be a surface of general type with maximal Albanese dimension: if the Albanese morphism is composed with an involution, one has K 2 X ≥ 4χ(OX ) + 4(q − 2). We give a complete classification of surfaces for which equality holds for q(X) ≥ 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 twice the elliptic fibre.We also prove, in the same hypothesis, that a surface X with K 2 X = 4χ(OX ) + 4(q − 2) satisfies K 2 X ≥ 4χ(OX ) + 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.