Binomial vanishing ideals (Q307920)

The present paper under review deals with the study of the binomial vanishing ideals. Vanishing ideals has been a research topic for the decates in commutative algebra and algebraic geometry. The second author has been working on Cohen-Macaulayness, complete intersection and regularity of vanishing ideals as well as vanishing ideals over finite fields and graphs in recent years.NEWLINENEWLINELet \(S=K[x_1, \dots, x_n]\) be the standard polynomial ring over the field \(K\) and \(\mathbb P^{s-1}\) the projective space of dimension \(s-1\) over \(K\). Let \( \mathbb Y \subset \mathbb P^{s-1}\). The graded ideal \(I(\mathbb Y)\) generated by the homogeneous polynomials of \(S\) that vanish at all points of \(\mathbb Y\) is called the vanishing ideal of \(\mathbb Y\). For graded ideal \(I \subset S\), one defines \(V(I)\) the projective variety of \(I\) as the set of all \([\alpha]\) in \(\mathbb P^{s-1}\) such that \(f(\alpha)=0\) for all homogeneous polynomial \(f\) in \(I\). A binomial of \(S\) is an element of the form \(g=x^a-x^b\) for some \(a, b \in \mathbb N^n\) where \(x^a=x_1^{a_1}\dots x_n^{a_n}\). An ideal generated by binomials is called a binomial ideal. If \(x_i\) is not a zero-divisor of \(S/I\) for all \(i\), then the binonial ideal of \(I\) is called a lattice ideal.NEWLINENEWLINEIn this paper the authors classify binomial vanishing ideals as follows: \(I(\mathbb Y)\) is a binomial ideal if and only if \(V(I(\mathbb Y)) \cup\{[0]\}\) is a monoid under componentwise multiplication. As a consequence for small dimension, namely \(\dim(S/I(\mathbb Y))=1\), \(I(\mathbb Y)\) is a binomial ideal if and only if \(\mathbb Y \cup\{[0]\}\) is a submonoid of \(\mathbb P^{s-1}\cup\{[0]\}\).NEWLINENEWLINEThe set \(T =\{ [(x_1, \dots, x_s)]\in \mathbb P^{s-1}: x_i \in K\setminus \{0\} \text{ for all i} \}\) is called a projective torus in \(\mathbb P^{s-1}\). As a classification of graded lattice ideal the authors prove the following: Let \(K\) be an algebraically closed field of characteristic zero and let \(I\) be a graded ideal of \(S\) such that \(\dim S/I=1\). Then, \(I\) is a lattice ideal if and only if \(I\) is the vanishing ideal of a finite subgroup \(\mathbb Y\) of a projective torus \(T\).