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On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves. (English) Zbl 1188.14021

Consider the defining ideal \(p_{K}(a,b,c)\) of the space of monomial curves \((t^a,t^b,t^c)\) in \(K^3\) for pairwise coprime integers \(a,b,c\) with \((a,b,c)\neq (1,1,1)\) and the symbolic Rees ring \(R=R_s(p_{K}(a,b,c))\). In the paper under review it is shown that if \(R\) is not finitely generated over \(K\), then it is a counterexample to the Hilbert’s fourteenth problem that can be formulated as follows. Let \(P\) be a polynomial finitely generated ring over \(K\) and \(L\) be a field satisfying \(K\subset L\subset Q(P)\). Hilbert asked whether the \(K\)-algebra \(L\cap P\) is finitely generated or not. The counterexample to this problem was discovered by M. Nagata [Proc. Int. Congr. Math. 1958, 459–462 (1960; Zbl 0127.26302)].
Finite generation of \(R\) is closely related to existence of negative curves on certain normal projective surfaces. In fact, in case of positive characteristic existence of a negative curve implies finite generation and in case \(\sqrt{abc}\not\in \mathbb{Z}\) the converse statement holds. In the paper under review a sufficient condition for existence of negative curve is studied. Applying it, the authors proved that if \((a+b+c)^2>abc\), then a negative curve exists. An example is given that the converse statement does not hold.

MSC:

14H50 Plane and space curves
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A50 Actions of groups on commutative rings; invariant theory

Citations:

Zbl 0127.26302
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References:

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