Normal reduction numbers, normal Hilbert coefficients and elliptic ideals in normal 2-dimensional local domains

Abstract: This is a joint work with T. Okuma (Yamagata Univ.), M.E. Rossi (Univ. Genova) and K. Yoshida (Nihon Univ.).

Let $(A, \mathfrak{m})$ be an excellent two-dimensional normal local domain and let $I$ be an integrally closed $\mathfrak{m}$-primary ideal and $Q$ be a minimal reduction of $I$ (a parameter ideal with $I^{r+1} = Q I^{r}$ for some $r ≥ 1$). Then the reduction numbers \[ nr(I) = \min\{ n \mid \overline{I^{n+1}} = Q \overline{I^n} \}, \] and \[ \overline{r}(I) = \min\{n \mid \overline{I^{N+1}} = Q\overline{I^N}, \forall N \ge n \} \] are important invariants of the ideal and the singularity. Also the normal Hilbert coefficients $\overline{e}_i(I)$, for $i = 0, 1, 2$, are defined by \[ \ell_A(A/\overline{I^{n+1}}) = \overline{e}_0(I)\binom{n+2}2 - \overline{e}_1(I)\binom{n+1}1 + \overline{e}_2(I)\,. \] for $n\gg 0$.

We can characterize certain class of singularities by these invariants. Namely, $A$ is a rational singularity if and only if $\overline{r}(A) = 1$, or equivalently, $\overline{e}_2(I) = 0$ for every $I$. We defined a $p_g$ ideal by the property $\overline{r}(I) = 1$ and in this language, $A$ is a rational singularity if and only if every integrally closed $\mathfrak{m}$ primary ideal is a$p_g$ ideal.

Our aim is to know the behavior of these invariants for every integrally closed $\mathfrak{m}$ primary ideal $I$ of a given ring $A$.

If $A$ is an elliptic singularity, then it is shown by Okuma that $\overline{r}(I) \le 2$ for every $I$. Inspired by these facts we define $I$ to be an elliptic ideal if $\overline{r}(I) = 2$ and strongly elliptic ideal if $\overline{e}_2 = 1$.

We will show several nice equivalent properties for $I$ to be an elliptic or a strongly elliptic ideal.

Our tool is resolution of singularities of $\mathrm{Spec}(A)$. Let $I$ be an $\mathfrak{m}$-primary integrally closed ideal in $A$. We can take $f\colon X \to \mathrm{Spec}(A)$ a resolution of $A$ such that $I\mathcal{O}_X = \mathcal{O}_X(−Z)$ is invertible. In particular $p_g(A) := h^1(X,\mathcal{O}_X)$ and $q(I) := h^1(X,\mathcal{O}_X(−Z))$ play important role in our theory.

This talk is based on our joint work appeared in arXiv 2012.05530 and arXiv 1909.13190.

commutative algebra

Audience: researchers in the topic

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