Finite subsets of projective space, and their ideals

Abstract

Let $\mathcal{A}$ be a finite set of closed rational points in projective space, let $\mathcal{I}$ be the vanishing ideal of $\mathcal{A}$, and let $\mathcal{D}(\mathcal{A})$ be the set of exponents of those monomials which do not occur as leading monomials of elements of $\mathcal{I}$. We show that the size of $\mathcal{A}$ equals the number of axes contained in $\mathcal{D}(\mathcal{A})$. Furthermore, we present an algorithm for the construction of the Gröbner basis of $\mathcal{I}(\mathcal{A})$, hence also of $\mathcal{D}(\mathcal{A})$.

25 pages

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