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arXiv · 2208.04902

Convexity of multiplicities of filtrations on local rings

Harold Blum·Yuchen Liu·Lu Qi
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Abstract

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

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Harold Blum, Yuchen Liu, Lu Qi. 2024-03-20. Convexity of multiplicities of filtrations on local rings. https://arxiv.org/abs/2208.04902

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