On the defining ideal of a set of points in multi-projective space

We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, we determine the degrees of the generators of the associated ideal I_X. Letting \nu(I_X) denote the minimal number of generators of I_X, we use this description of the degrees to construct a function v(s;n_1,...,n_k) with the property that \nu(\Ix) >= v(s;n_1,...,n_k) always holds for s points in generic position in P^{n_1} x ... x P^{n_k}. When k=1, v(s;n_1) equals the expected value for \nu(I_X) as predicted by the Ideal Generation Conjecture. If k >= 2, we show that there are cases with \nu(\Ix) > v(s;n_1,...,n_k). However, computational evidence suggests that in many cases \nu(\Ix) = v(s;n_1,...,n_k).