The normal reduction number of two-dimensional cone-like singularities

Abstract

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: $\mathrm{nr}(I)=\min \{n|\overline{I^{n+1}}=Q\overline{I^n}\},\bar{r}(I)=\min \{n|\overline{I^{N+1}}=Q\overline{I^N},\forall N\ge n\}$, $\mathrm{nr}(A)$, 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 $g\ge 2$.

arXiv comment13 pages; revised version. To appear in Proceedings of the American Mathematical Society

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Cited by in corpus (2)

• Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities
• Gorensteinness for normal tangent cones of elliptic ideals