Noether's inequality for non-complete algebraic surfaces of general type. II
type: Item
instance of: scholarly article
title: Noether's inequality for non-complete algebraic surfaces of general type. II
zbMATH Open document ID: 0824.14036
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-11-13T00:00:00Z
review text: [For part I see \textit{S. Tsunoda} and the author, ibid. No. 1, 21-38 (1992; see the preceding review).] Let \(V\) be a non-ruled non-singular projective surface. Let \(D\) be a reduced, effective non-zero divisor on \(V\) with only simple normal crossings. The author considers here the case where the open surface \(V - D\) is ``minimal'' and of general type. On the one hand, \textit{F. Sakai} [Math. Ann. 254, 89-120 (1980; Zbl 0437.14017)] proved the Miyaoka-Yau type inequality: \((c_ 1^ 2) : = (K_ V + D)^ 2 \leq 3 c_ 2 : = 3c_ 2 (V) - 3e (D)\). Here the author will give a lower bound of \(c_ 1^ 2\) in terms of \(c_ 2\): \(c_ 1^ 2 \geq 1/9 c_ 2^ 2-2\) under the assumption that the rational map associated with the linear system \(| K_ V + D |\) maps \(V\) to a surface. When the above inequality becomes an equality, then \(p_ g (V - D) = 1/2c_ 1^ 2 + 2 = 3\), \(D\) is a smooth elliptic curve and \(V\) is Horikawa's canonical resolution of a double covering surface over \(\mathbb{P}^ 2\).
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 597470
zbMATH Keywords: Noether's inequality
zbMATH Keywords: open surface of general type
zbMATH Keywords: Chern class
zbMATH Keywords: Miyaoka- Yau type inequality
author: De-Qi Zhang
reviewed by: De-Qi Zhang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195167932
OpenAlex ID: W2063114317
cites work: L'application canonique pour les surfaces de type général
cites work: Q4100195
cites work: Normal Gorenstein surfaces with ample anti-canonical divisor
cites work: On deformations of quintic surfaces
cites work: Algebraic surfaces of general type with small \(c^2_1\). I
cites work: Varieties of special divisors on a curve. II.
cites work: On rational surfaces I. Irreducible curves of arithmetic genus $0$ or $1$
cites work: Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps
cites work: Noether's inequality for non-complete algebraic surfaces of general type
cites work: The structure of normal surfaces
DOI: 10.2977/PRIMS/1195167932
description: scientific article; zbMATH DE number 597470