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SUMMARY:Kei-ichi Watanabe (Nihon University and Meiji University)
DTSTART:20210319T000000Z
DTEND:20210319T013000Z
DTSTAMP:20260423T021353Z
UID:FOTR/43
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FOTR/43/">No
 rmal reduction numbers\, normal Hilbert coefficients and elliptic ideals i
 n normal 2-dimensional local domains</a>\nby Kei-ichi Watanabe (Nihon Univ
 ersity and Meiji University) as part of Fellowship of the Ring\n\n\nAbstra
 ct\nThis is a joint work with T. Okuma (Yamagata Univ.)\, M.E. Rossi (Univ
 . Genova) and K. Yoshida (Nihon Univ.).\n\nLet $(A\, \\mathfrak{m})$ be an
  excellent two-dimensional normal local domain and let $I$ be an integrall
 y 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 t
 he reduction numbers \n\\[\nnr(I) = \\min\\{ n \\mid \\overline{I^{n+1}} =
  Q \\overline{I^n} \\}\,  \n\\]\nand\n\\[\n\\overline{r}(I) = \\min\\{n \\
 mid \\overline{I^{N+1}} = Q\\overline{I^N}\, \\forall N \\ge n \\}\n\\]\na
 re important invariants of the ideal and the singularity. Also the normal 
 Hilbert coefficients $\\overline{e}_i(I)$\, for $i = 0\, 1\, 2$\, are defi
 ned by\n\\[\n\\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)\\\,.\n\\]\nfor 
 $n\\gg 0$. \n\nWe can characterize certain class of singularities by these
  invariants. Namely\, $A$ is a\nrational singularity if and only if $\\ove
 rline{r}(A) = 1$\, 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 integra
 lly closed $\\mathfrak{m}$ primary ideal is a$p_g$ ideal.\n\nOur aim is to
  know the behavior of these invariants for every integrally closed $\\math
 frak{m}$ primary ideal $I$ of a given ring $A$.\n\nIf $A$ is an elliptic s
 ingularity\, 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 i
 f $\\overline{r}(I) = 2$ and strongly elliptic ideal if $\\overline{e}_2 =
  1$.\n\nWe will show several nice equivalent properties for $I$ to be an e
 lliptic or a strongly elliptic ideal.\n\nOur tool is resolution of singula
 rities of $\\mathrm{Spec}(A)$. Let $I$ be an $\\mathfrak{m}$-primary integ
 rally 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.\n\nThis 
 talk is based on our joint work appeared in arXiv 2012.05530 and arXiv 190
 9.13190.\n
LOCATION:https://researchseminars.org/talk/FOTR/43/
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