The total coordinate ring of a smooth projective surface
type: Item
instance of: scholarly article
title: The total coordinate ring of a smooth projective surface
zbMATH Open document ID: 1108.14007
author: Francisco Monserrat
author: Carlos Galindo
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: For smooth projective varieties \(X\) with finitely generated, free Picard group, one can define the Cox (or total coordinate) ring TC\((X)\) which, roughly speaking, looks like \[ \text{TC}(X)=\bigoplus_{D\in\text{Pic} X}\text{H}^0(X,\mathcal O(D)). \] It is factorial, but the general problem is to find out under which conditions it is finitely generated. A necessary condition for this is that the entire Mori cone is polyhedral. The authors of the paper under review investigate the case of \(\dim X=2\). Here they show that \(\text{TC}(X)\) is finitely generated if and only if \(X\) has only finitely many divisors with negative self-intersection and \(\text{TC}(X)\) contains a sufficiently large (e.g. covering the nef part), finitely generated subalgebra. This is fulfilled for surfaces with polyhedral Mori cone where, additionally, ``nef'' implies ``semiample''. Finally, this result is applied to investigate the case of rational surfaces. Encoding them by a configuration of points (to be blown up in) on \(\mathbb P^2\) or the Hirzebruch surfaces, sufficient conditions for the finite generatedness of \(\text{TC}(X)\) are given.
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 2136965
zbMATH Keywords: total coordinate ring (``Cox ring'')
zbMATH Keywords: algebraic surfaces
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.10.004
OpenAlex ID: W2048788081
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DOI: 10.1016/J.JALGEBRA.2004.10.004
description: scientific article; zbMATH DE number 2136965