The defining ideal of a set of points in multi-projective space

摘要

The defining ideal I-X of a set of points X in P-n1 x ... x P-nk is 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. v(IX) denotes the minimal number of generators of I-X, and this description of the degrees is used to construct a function v(s; n(1),...., n(k)) with the property that v(I-X) >= v(s; n(1),..., n(k)) always holds for s points in generic position in P-n1 x ... x P-nk. When k = 1, v(s; n(1)) equals the expected value for v(I-X) as predicted by the ideal generation conjecture. If k >= 2, it is shown that there are cases with v(I-X) > v(s; n(1),...,n(k)). However, computational evidence suggests that in many cases v(I-X) = v(s; n(1),..., n(k)).