Convexity of multiplicities of filtrations on local rings
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913275036303360 |
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| author | Blum, Harold Liu, Yuchen Qi, Lu |
| author_facet | Blum, Harold Liu, Yuchen Qi, Lu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04902 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convexity of multiplicities of filtrations on local rings Blum, Harold Liu, Yuchen Qi, Lu Algebraic Geometry Commutative Algebra 14B05, 13H15 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. |
| title | Convexity of multiplicities of filtrations on local rings |
| topic | Algebraic Geometry Commutative Algebra 14B05, 13H15 |
| url | https://arxiv.org/abs/2208.04902 |