Even canonical surfaces with small \(K^ 2\). II
type: Item
instance of: scholarly article
title: Even canonical surfaces with small \(K^ 2\). II
zbMATH Open document ID: 0865.14021
author: Kazuhiro Konno
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1997-07-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108357
review text: [For part I of this paper see Nagoya Math. J. 129, 115-146 (1993; Zbl 0780.14020).] A surface \(S\) is called canonical if it is minimal and its canonical divisor defines a birational map onto its image \(X\) in \(\mathbb{P}^{p_g-1}\) (where \(p_g\) is the geometric genus of \(S)\). Let \(Q(X)\) be the quadric hull of \(X\), that is, the intersection of all quadrics containing \(X\). It was conjectured by Reid that a (complex) canonical surface with \(K^2< 4p_g-12\) has a three-dimensional quadric hull \(Q(X)\). In part I (loc. cit.), \textit{K. Konno} gave a partial positive answer to Reid's conjecture. More precisely, if \(S\) is even (i.e. the canonical divisor can be written as \(K=2L)\) and \(q\) is the irregularity of \(S\), he proved that the conjecture is true when \(q=0\). He also proved that if \(S\) is even, \(K^2\leq 4p_g-12+q\) and \(S\) contains a pencil of trigonal or plane quintic curves, then \(Q(X)\) has dimension three. In the paper under review the author improves his previous work. In particular, he classifies even canonical surfaces with \(K^2=4p_g-12\) and \(Q(X) =X\), showing that they must be one of the following: -- Some explicit weighted complete intersection. -- A hyperquartic section of a normal Del Pezzo surface. -- A surface with a pencil of non-hyperelliptic curves of genus 7 whose general member has a \(g^2_6\). -- A double covering of a \(K3\) surface. -- A surface with a pencil of non-hyperelliptic, non-trigonal curves of genus 5. He also proves that, if \(q>0\), then \(S\) has a pencil of trigonal curves and the irreducible component of \(Q(X)\) containing \(X\) has dimension three. The above classification is done by studying the possible behavior of the map defined by the line bundle \(L\).
Mathematics Subject Classification ID: 14J29
zbMATH DE Number: 826767
zbMATH Keywords: canonical surfaces
zbMATH Keywords: canonical map
zbMATH Keywords: canonical divisor
zbMATH Keywords: birational map
zbMATH Keywords: Reid's conjecture
zbMATH Keywords: Del Pezzo surface
reviewed by: Enrique Arrondo
MaRDI profile type: Publication
cites work: L'application canonique pour les surfaces de type général
cites work: Q4100195
cites work: On the structure of polarized manifolds with total deficiency one. II
cites work: On polarized varieties of small Delta-genera
cites work: Q4283562
cites work: Q4749943
cites work: Algebraic surfaces of general type with small \(c^2_1\). I
cites work: Notes on canonical surfaces
cites work: Algebraic surfaces of general type with \(c_ 1^ 2=3p_ g-6\)
cites work: Q4006365
cites work: Even canonical surfaces with small K2, I
cites work: On rational surfaces I. Irreducible curves of arithmetic genus $0$ or $1$
cites work: Q3856877
cites work: Fibered algebraic surfaces with low slope
description: scientific article; zbMATH DE number 826767