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Contracted ideals and the Gröbner fan of the rational normal curve. (English) Zbl 1163.13002
Let \(R= K[x, y]\) be the polynomial ring in two indeterminates over a field \(K\), and let \(m= (x, y)\). A homogeneous \({\mathfrak m}\)-primary ideal \(I\) is called contracted if there exists a linear form \(z\in R\) such that \(I= IR[{\mathfrak m}/z]\cap R\). Contracted ideals have been introduced by O. Zariski in his studies on the unique factorization property of integrally closed domains [cf. O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton (1960; Zbl 0121.27801)]. One of the goals of the authors is to determine those contracted ideals \(I\) whose associated graded ring \(\text{gr}_1(R)\) is Cohen-Macaulay. By a result of the authors and A. V. Jayanthan (cf. [Graded rings associated with contracted ideals, J. Algebra 284, 593–626 (2005; Zbl 1098.13007)]), this amounts to characterizing the lex-segment ideals with Cohen-Macaulay associated graded ring.
This leads to a surprising equivalence of the stated task with the problem of describing the Cohen-Macaulay initial ideals of the defining ideal \(P\) of the rational normal curve in lid. The authors solve that problem by showing that \(P\) has exactly \(2^{d-1}\) Cohen-Macaulay initial monomial ideals. In terms of the Gröbner fan of \(P\), an initial monomial ideal \(\text{in}_a(P)\) is Cohen-Macaulay if and only if a belongs to a union of \(2^{d-1}\) maximal closed cones which are explicitly described by linear homogeneous inequalities.
The authors also show that the union of a certain subfamily consisting of \(f_d\) of these cones (where \(f_d\) is the \((d+1)\)st Fibonacci number) is itself a cone. They call it the big Cohen-Macaulay cone and present some explicitly computed examples with small \(d\).

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Software:
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