Noether's inequality for non-complete algebraic surfaces of general type
type: Item
instance of: scholarly article
title: Noether's inequality for non-complete algebraic surfaces of general type
zbMATH Open document ID: 0824.14035
author: Shuichiro Tsunoda
author: De-Qi Zhang
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-05-03T00:00:00Z
review text: Let \(V\) be a nonsingular projective surface. M. Noether's inequality says that \(\dim H^ 0 (V,K_ V) \leq 1/2 (K_ V^ 2) + 2\) provided that \(V\) is a minimal surface of general type. Let \(D\) be a reduced effective divisor on \(V\) with only simple normal crossings. In the paper, the authors extend Noether's inequality to the case of open algebraic surfaces \(V - D\) of general type. In this case, there is a Zariski decomposition for the adjoint divisor: \(K_ V + D = P + N\). The authors prove that \(\dim H^ 0 (V,K_ V + D) \leq 9/8 (P^ 2) + 2\) and several other inequalities. The pairs \((V,D)\) such that the inequality above becomes an equality are precisely described. The case that \(D\) is semistable has been treated by \textit{F. Sakai} [Math. Ann. 254, 89-120 (1980; Zbl 0437.14017)].
reviewed by: M. Miyanishi
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 548920
zbMATH Keywords: open surfaces of general type
zbMATH Keywords: Noether's inequality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168854
OpenAlex ID: W2109384124
cites work: Q4100195
cites work: Q3784181
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
DOI: 10.2977/PRIMS/1195168854
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description: scientific article; zbMATH DE number 548920