On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves

JOURNAL OF ALGEBRA

卷:322

Article

Kurano, Kazuhiko¹  Matsuoka, Naoyuki¹ 

[1] Meiji Univ, Dept Math, Fac Sci & Technol, Tama Ku, Kawasaki, Kanagawa 2148571, Japan

关键词: Symbolic Rees ring;    Space monomial curve;    Hilbert's fourteenth problem;    Cowsik's question;   

DOI  :  10.1016/j.jalgebra.2008.08.015

来源: Elsevier

【 摘 要 】

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves (t(a), t(b), t(c)) for pairwise coprime integers a, b, c such that (a, b, c) not equal (1, 1, 1). If such a ring is not finitely generated over a base field, then it is a counter example to the Hilbert's fourteenth problem. Finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces. We study a sufficient condition (Definition 3.6) for existence of a negative curve. Using it, we prove that, in the case of (a + b + c)(2) > abc, a negative curve exists. Using a computer, we shall show that there exist examples in which this sufficient condition is not satisfied. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free