Finite subsets of projective space, and their ideals

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