Computing epsilon multiplicities in graded algebras
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914684997730304 |
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| author | Das, Suprajo Dubey, Saipriya Roy, Sudeshna Verma, Jugal K. |
| author_facet | Das, Suprajo Dubey, Saipriya Roy, Sudeshna Verma, Jugal K. |
| contents | This article investigates the computational aspects of the $\varepsilon$-multiplicity. Primarily, we show that the $\varepsilon$-multiplicity of a homogeneous ideal $I$ in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the $\varepsilon$-multiplicity of $I$ in terms of certain mixed multiplicities associated to $I$. In any dimension, under the assumptions that the saturated Rees algebra of $I$ is finitely generated, we give a different expression of the $\varepsilon$-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of $\varepsilon$-multiplicities. We further write a Macaulay2 algorithm to compute $\varepsilon$-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing epsilon multiplicities in graded algebras Das, Suprajo Dubey, Saipriya Roy, Sudeshna Verma, Jugal K. Commutative Algebra Algebraic Geometry Primary 13H15, 13D45, 13A02, 13A30, Secondary 14C20 This article investigates the computational aspects of the $\varepsilon$-multiplicity. Primarily, we show that the $\varepsilon$-multiplicity of a homogeneous ideal $I$ in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the $\varepsilon$-multiplicity of $I$ in terms of certain mixed multiplicities associated to $I$. In any dimension, under the assumptions that the saturated Rees algebra of $I$ is finitely generated, we give a different expression of the $\varepsilon$-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of $\varepsilon$-multiplicities. We further write a Macaulay2 algorithm to compute $\varepsilon$-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded. |
| title | Computing epsilon multiplicities in graded algebras |
| topic | Commutative Algebra Algebraic Geometry Primary 13H15, 13D45, 13A02, 13A30, Secondary 14C20 |
| url | https://arxiv.org/abs/2402.11935 |