  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.
