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Characterization of the Hilbert-Samuel polynomials of curve singularities. (English) Zbl 0724.13021
Let A be a one-dimensional Cohen-Macaulay local ring, (b,e,\(\rho\)) a triplet of integers. Let \(\rho_{0,b,e}=(r+1)e-\left( \begin{matrix} r+b\\ r\end{matrix} \right)\), where r is the integer such that \(\left( \begin{matrix} b+r-1\\ r\end{matrix} \right)\leq e<\left( \begin{matrix} b+r\\ r+1\end{matrix} \right)\) and \(\rho_{1,b,e}=e(e-1)/2-(b-1)(b-2)/2\). The main result of the paper is that there exists a one-dimensional Cohen-Macaulay local ring A with embedding dimension b, multiplicity e and reduction number \(\rho\) iff \(b=e=1\) and \(\rho =0\) or \(2\leq b\leq e\) and \(\rho_{0,b,e}\leq \rho \leq \rho_{1,h,e}\). In this last case, one can choose A to be the quotient of a power series ring over an algebraically closed field of characteristic zero. Using this, the Hilbert-Samuel function of one- dimensional Cohen-Macaulay rings of small multiplicity is computed. Lastly, conditions for the Cohen-Macaulayness of gr(A) are established in terms of the reduction number.

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14H20 Singularities of curves, local rings
13C14 Cohen-Macaulay modules
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