Cohen-Macaulay fiber cones

TL;DR

This paper characterizes the Cohen-Macaulay property of fiber cones of ideals using Hilbert series, providing formulas and conditions for specific classes of ideals, with applications to algebraic geometry and combinatorics.

Contribution

It offers a new characterization of Cohen-Macaulay fiber cones via Hilbert series and computes these series for various classes of ideals, including those with minimal mixed multiplicity.

Findings

Fiber cone of an m-primary ideal with minimal mixed multiplicity is Cohen-Macaulay iff its reduction number is at most one.

Hilbert series of fiber cones of ideals generated by quadratic sequences are computed via deformation to face rings.

Applications include properties of monomial space curves, straightening-closed ideals, and Huckaba-Huneke ideals.

Abstract

Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the quadric xw-yz=0, straightening-closed ideals in graded algebras with straightening law and Huckaba-Huneke ideals of analytic spread 1 and 2.