Irregularity of surfaces with a linear pencil
type: Item
instance of: scholarly article
title: Irregularity of surfaces with a linear pencil
zbMATH Open document ID: 0651.14021
DOI: 10.1215/S0012-7094-87-05529-3
author: Gang Xiao
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let S be a smooth complex projective surface. Consider a nontrivial fibration \(f: S\to P^ 1.\) Suppose that F, a general fiber of f, is a smooth curve of genus g. The author shows that \(2q-1\leq g\) where \(q=h\) \(1(\omega_ S)\). Furthermore, he gives examples that this inequality is the best possible. The author first reduces the proof to the case when the Albanese map of S is generically finite to one. Then he observes that \(\dim (Im(H^ 0(\omega_ S)\to H^ 0(\omega_ F)))\leq g-q.\) Let A be a general 1- form. The author shows that the rank of the composition map \(H^ 0(\Omega^ 1_ S)^{\wedge}\to^{A}H^ 0(\omega_ S)\to H^ 0(\omega_ F)\) is \(q-1\). Hence \(2q-1\leq g\).
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4061395
zbMATH Keywords: irregularity
zbMATH Keywords: Albanese map
reviewed by: Lawrence Ein
MaRDI profile type: Publication
cites work: Q5749404
cites work: L'application canonique pour les surfaces de type général
cites work: Inégalités numériques pour les surfaces de type général. Appendice : «L'inégalité $p_g \ge 2q-4$ pour les surfaces de type général» par A. Beauville
cites work: Surfaces fibrées en courbes de genre deux
cites work: Q3724476
cites work: Algebraic surfaces with high canonical degree
cites work: Fibered algebraic surfaces with low slope
description: scientific article; zbMATH DE number 4061395