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arXiv · 2506.05248

Degree functions of graded families of ideals

Steven Dale Cutkosky·Jonathan Montaño
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Abstract

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.

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Steven Dale Cutkosky, Jonathan Montaño. 2025-06-05. Degree functions of graded families of ideals. https://arxiv.org/abs/2506.05248

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