Generators and relations for finitely generated graded normal rings of dimension two
type: Item
instance of: scholarly article
title: Generators and relations for finitely generated graded normal rings of dimension two
zbMATH Open document ID: 0658.13015
author: Frances Van Dyke
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: After Pinkham, a finitely generated graded normal ring \(R=\oplus R_ k \) of dimension 2 over \({\mathbb{C}}\) \((R_ 0={\mathbb{C}})\) is isomorphic to \({\mathcal L}(D)=\oplus^{\infty}_{n=0}L(nD), \) where D is a ``fractional'' divisor on a Riemann surface X of genus g, i.e. \(D=D_ 0+\sum^{k}_{i=1}\gamma_ iP_ i, \) \(D_ 0=\sum_{P\in X}n_ PP, \) \(n_ P\in {\mathbb{Z}}\), all but finitely many \(n_ P=0\), \(P_ i\in X\), \(\gamma_ i\in {\mathbb{Q}}\), \(0<\gamma_ i<1\) and L(nD) denotes the set of meromorphic functions f such that \(div(f)+nD\geq 0\). The present paper shows that in many cases a set of homogeneous generators of \({\mathcal L}(D)\) and their relations can be determined if a set of homogeneous generators and their relations are known for \({\mathcal L}(D_ 1)\), where \(D_ 1<D\) and \({\mathcal L}(D_ 1)\) has a much simpler structure than \({\mathcal L}(D)\). In particular the degrees of homogeneous generators and relations can be found for all D for which \(\deg (D_ 0)\geq 2g+1\) or \(D_ 0\) is the canonical divisor on a nonhyperelliptic curve of genus \(g>3\) (this extends some results of Mumford and Saint-Donat).
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4075247
zbMATH Keywords: coordinate ring of a normal affine surface
zbMATH Keywords: finitely generated graded normal ring
zbMATH Keywords: divisor on a Riemann surface
zbMATH Keywords: degrees of homogeneous generators
zbMATH Keywords: relations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4075247