A variant of R{ö}hr's vanishing theorem with an application to the normal reduction number for normal surface singularities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909962898243584 |
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| author | Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| author_facet | Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| contents | Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the cohomology group of invertible sheaves on $X$ coincides with a natural lower bound. Applying this theorem, we establish upper bounds for the normal reduction number $\bar{\mathrm{r}}(A)$ of $A$. For example, we prove the inequality $\bar{\mathrm{r}}(A) \le p_a(A)+1$, where $p_a(A)$ denotes the arithmetic genus, a fundamental combinatorial (topological) invariant. We introduce the notion of almost cone singularities and give a sharper inequality $\bar{\mathrm{r}}(A) \le p_f(A)+1$ for such singularities, where $p_f(A)$ denotes the fundamental genus. We also show that $\bar{\mathrm{r}}(A)$ is not a combinatorial invariant in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A variant of R{ö}hr's vanishing theorem with an application to the normal reduction number for normal surface singularities Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi Algebraic Geometry Commutative Algebra Primary: 14J17, Secondary: 14B05, 13B22, 13G05 Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the cohomology group of invertible sheaves on $X$ coincides with a natural lower bound. Applying this theorem, we establish upper bounds for the normal reduction number $\bar{\mathrm{r}}(A)$ of $A$. For example, we prove the inequality $\bar{\mathrm{r}}(A) \le p_a(A)+1$, where $p_a(A)$ denotes the arithmetic genus, a fundamental combinatorial (topological) invariant. We introduce the notion of almost cone singularities and give a sharper inequality $\bar{\mathrm{r}}(A) \le p_f(A)+1$ for such singularities, where $p_f(A)$ denotes the fundamental genus. We also show that $\bar{\mathrm{r}}(A)$ is not a combinatorial invariant in general. |
| title | A variant of R{ö}hr's vanishing theorem with an application to the normal reduction number for normal surface singularities |
| topic | Algebraic Geometry Commutative Algebra Primary: 14J17, Secondary: 14B05, 13B22, 13G05 |
| url | https://arxiv.org/abs/2512.13042 |