The normal reduction number of two-dimensional cone-like singularities
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| Format: | Preprint |
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2019
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| _version_ | 1866908709312004096 |
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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 $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} =
Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1909_13190 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The normal reduction number of two-dimensional cone-like singularities Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi Commutative Algebra Algebraic Geometry Primary 13B22, Secondary 14B05, 14J17 Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} = Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$. |
| title | The normal reduction number of two-dimensional cone-like singularities |
| topic | Commutative Algebra Algebraic Geometry Primary 13B22, Secondary 14B05, 14J17 |
| url | https://arxiv.org/abs/1909.13190 |