Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities. (English) Zbl 1353.14004
In this paper, the authors study the multiplicity of the jumping numbers of an \(\mathfrak{m}\)-primary ideal \(\mathfrak{a}\) in a two-dimensional local ring with a rational singularity. Recall the multiplicity \(m(c)\) of a jumping number \(c\) is defined as the dimension of \(J(\mathfrak{a}^{c-\epsilon})/J(\mathfrak{a}^c)\). The main result is that the Poincare series \(P_{\mathfrak{a}}(t)=\sum_{c\in \mathbb{R}_{>0}}m(c)t^c\) is rational, i.e, it belongs to the field of fractional funtions \(\mathbb{C}(z)\) where the indeterminate \(z\) corresponds to a fractional power \(t^{1/e}\) for suitable interger \(e\). This generalizes previous result in [C. Galindo and F. Monserrat, Adv. Math. 225, No. 2, 1046–1068 (2010; Zbl 1206.14011)]. The authors actually obtained explicit formulas of \(P_{\mathfrak{a}}(t)\) using the intersection numbers of the maximal jumping divisors. The results on multiplicities also leads to a simple method to detect jumping numbers.
Reviewer: Linquan Ma (Salt Lake City)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
Citations:
Zbl 1206.14011Software:
MultiplierIdealsDim2References:
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