On some invariants of hypersurface singularities
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917347743236096 |
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| author | Mustaţă, Mircea |
| author_facet | Mustaţă, Mircea |
| contents | Given a hypersurface defined by $f$ in a smooth complex algebraic variety $X$, and a point $P$ on this hypersurface, we consider the invariant $β_P(f)$ given by the log canonical threshold at $P$ of ${\mathfrak m}_P\cdot J_f$, where ${\mathfrak m}_P$ is the ideal defining $P$ and $J_f$ is the Jacobian ideal of $f$. We show that this invariant satisfies most of the formal properties of the log canonical threshold of $f$ and give some examples. Dano Kim asked whether this invariant always gives an upper bound for the minimal exponent of $f$ at $P$. Motivated by this, we raise another question about minimal exponents, give a positive answer to a weaker version, and discuss some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On some invariants of hypersurface singularities Mustaţă, Mircea Algebraic Geometry Commutative Algebra 14B05, 14J17, 32S25 Given a hypersurface defined by $f$ in a smooth complex algebraic variety $X$, and a point $P$ on this hypersurface, we consider the invariant $β_P(f)$ given by the log canonical threshold at $P$ of ${\mathfrak m}_P\cdot J_f$, where ${\mathfrak m}_P$ is the ideal defining $P$ and $J_f$ is the Jacobian ideal of $f$. We show that this invariant satisfies most of the formal properties of the log canonical threshold of $f$ and give some examples. Dano Kim asked whether this invariant always gives an upper bound for the minimal exponent of $f$ at $P$. Motivated by this, we raise another question about minimal exponents, give a positive answer to a weaker version, and discuss some examples. |
| title | On some invariants of hypersurface singularities |
| topic | Algebraic Geometry Commutative Algebra 14B05, 14J17, 32S25 |
| url | https://arxiv.org/abs/2603.15579 |