An indispensable classification of monomial curves in $\mathbb{A}^4(\mathit{k})$

Anargyros Katsabekis and Ignacio Ojeda

Vol. 268 (2014), No. 1, 95–116

DOI: 10.2140/pjm.2014.268.95

Abstract

We give a new classification of monomial curves in ${\mathbb{A}}^4\left(k\right)$. It relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the defining ideal of a monomial curve in ${\mathbb{A}}^4\left(k\right)$ to have a unique minimal system of binomial generators. Furthermore, some other interesting results on more general classes of binomial ideals with unique minimal system of binomial generators are obtained.

Keywords

binomial ideal, toric ideal, monomial curve, minimal systems of generators, indispensable monomials, indispensable binomials

Mathematical Subject Classification 2010

Primary: 13F20

Secondary: 16W50, 13F55

Milestones

Received: 26 July 2012

Revised: 7 October 2013

Accepted: 11 October 2013

Published: 21 May 2014