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Hilbert functions on Cohen-Macaulay ideals with assigned generators’ degrees. (English) Zbl 1167.13305
Summary: We give information on the Hilbert function of a Cohen-Macaulay ideal \(I\) of the polynomial ring \(R=k[x_0, x_1,\dots , x_r]\) which is minimally generated by \(t\) forms of degrees \(d_1,\dots,d_t\). Mainly we deal with the codimension two case in which we show that the Dubreil bound \(t\leq d_1+1\) is a necessary and sufficient condition to have such an ideal and we give a sharp upper bound and lower bound for the Hilbert function. In codimension greater than two we give a characterization for having such an ideal and in codimension 3 we find an Hilbert function which is maximal for these ideals with \(d_1=\dots d_t=a\) and we produce a scheme which realizes such a Hilbert function.
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: EuDML
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