Finite subsets of projective space, and their ideals

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