On the Defining Ideal of a Set of Points in Multi-projective Space

We investigate the defining ideal IX of a set of points X in P n1 × · · · × Pk 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 IX. Letting ν(IX) denote the minimal number of generators of IX, we use this description of the degrees to construct a function v(s;n1, . . . , nk) with the property that ν(IX) ≥ v(s;n1, . . . , nk) always holds for s points in generic position in P1 ×· · ·×Pk . When k = 1, v(s; n1) equals the expected value for ν(IX) as predicted by the Ideal Generation Conjecture. If k ≥ 2, we show that there are cases with ν(IX) > v(s;n1, . . . , nk). However, computational evidence suggests that in many cases ν(IX) = v(s; n1, . . . , nk).