Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove th...

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Bibliographic Details
Published inJournal of algebra Vol. 590; pp. 413 - 438
Main Author Kurano, Kazuhiko
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.01.2022
Subjects
Cox ring
Ehrhart ring
Finite generation
Symbolic Rees ring
Toric variety
Ehrhart ring
Finite generation
Symbolic Rees ring
Cox ring
Toric variety
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Summary:Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,…,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety (Remark 2.7).
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2021.10.016