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arXiv · 0801.3896

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

Kazuhiko Kurano·Naoyuki Matsuoka
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Abstract

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 $(a,b,c) \neq (1,1,1)$. If such a ring is not finitely generated over a base field, then it is a counterexample 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.

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Kazuhiko Kurano, Naoyuki Matsuoka. 2009-09-03. On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves. https://arxiv.org/abs/0801.3896

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