The Defining Ideal of a Set of Points in Multi‐Projective Space

The defining ideal I X of a set of points X inℙ n 1× ... × ℙ n 1is investigated with a special emphasis on the case when X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, the degrees of the generators of the associated ideal I X are determined. ν( I X ) denotes the minimal number of generators of I X , and this description of the degrees is used to construct a function υ( s ; n 1 ,…, n k ) with the property that ν( I X )⩾ υ( s ; n 1 ,…, n k ) always holds for s points in generic position inℙ n 1× ... × ℙ n 1. When k = 1, υ( s ; n 1 ) equals the expected value for ν( I X ) as predicted by the ideal generation conjecture. If k ⩾ 2, it is shown that there are cases with ν( I X ) > υ( s ; n 1 , …, n k ). However, computational evidence suggests that in many cases ν( I X ) = υ( s ; n 1 , …, n k ).