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

TL;DR

This paper investigates the finite generation of symbolic Rees rings of space monomial curves, exploring conditions for negative curve existence and providing examples where these conditions do not hold, contributing to understanding Hilbert's fourteenth problem.

Contribution

It introduces a sufficient condition for negative curve existence and proves its validity for certain parameter ranges, while also identifying cases where the condition fails.

Findings

Negative curves exist when (a+b+c)^2 > abc.

Counterexamples show the sufficient condition is not always satisfied.

Finite generation relates to negative curve existence and Hilbert's fourteenth problem.

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.