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The fiber cone of a monomial ideal in two variables. (English) Zbl 1430.13047

Let \(S=K[x_1,\dots,x_n]\) be the polynomial ring and \(I\subset S\) be a graded ideal. Let \(\mu(I^k)=\dim_K F (I)_k\), where \(F (I)_k\) is the kth graded component of the fiber cone \(F(I)\) of \(I\) .
In this paper under review the authors by using Gröbner bases determine in an explicit way the depth of the fiber cone and its relation ideal for classes of monomial ideals in two variables. These classes include concave and convex ideals as well as symmetric ideals.

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05E40 Combinatorial aspects of commutative algebra
13A02 Graded rings
13C14 Cohen-Macaulay modules
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
68W30 Symbolic computation and algebraic computation
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References:

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