## 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

### Citations:

Zbl 0121.27801; Zbl 1098.13007

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