Seminar
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| Location: | MSRI: Online/Virtual |
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Normal Reduction of Numbers, Normal Hilbert Coefficients and Elliptic Ideals in Normal 2-Dimensional Local Domains
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This is a joiint work with T. Okuma (Yamagata Univ.), M.E. Rossi (Univ. Genova) and K. Yoshida (Nihon Univ.).
Let (A, m) be an excellent two-dimensional normal local domain and let I be an inte grally closed m-primary ideal and Q be a minimal reduction of I (a parameter ideal with Ir+1 = QI rfor some r ≥ 1).
Then the reduction numbers
nr(I) = min{n | In+1 = QIn}, r¯(I) = min{n | IN+1 = QIN , ∀N ≥ n} are important invariants of the ideal and the singularity.
Also the normal Hilbert coefficients ¯ei(I) (i = 0, 1, 2) are defined by
)
+ ¯e2(I)
for n ≫ 0.
ℓA(A/In+1) = ¯e0(I)
(n + 2 2
)
− e¯1(I)
(n + 1 1
We can characterize certain class of singularities by these invariants. Namely, A is a rational singularity if and only if ¯r(A) = 1, or equivalently, ¯e2(I) = 0 for every I. We defined a pg ideal by the property ¯r(I) = 1 and in this language, A is a rational singularity if and only if every integrally closed m primary ideal is a pg ideal.
Our aim is to know the behavior of these invariants for every integrally closed m primary ideal I of a given ring A.
If A is an elliptic singularity, then it is shown by Okuma that ¯r(I) ≤ 2 for every I. Inspired by these facts we define I to be an elliptic ideal if ¯r(I) = 2 and strongly elliptic ideal if ¯e2 = 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 Spec(A). Let I be an m primary integrally closed ideal in A. We can take f : X → Spec(A) a resolution of A such that IOX = OX(−Z) is invertible. In particular pg(A) := h1(X, OX) and q(I) := h1(X, OX(−Z)) play important role in our theory.
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Notes
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