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

Abstract

The defining ideal I X of a set of points X in P n 1 × . . . × P n k 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. ν(I X ) 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 ν(I X ) v(s; n 1 , . . . , n k ) always holds for s points in generic position in P n 1 × . . . × P n k . When k = 1, v(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 ) > v(s; n 1 , . . . , n k ). However, computational evidence suggests that in many cases ν(I X ) = v(s; n 1 , . . . , n k ).

Symbolic Powers Versus Regular Powers of Ideals of General Points in ℙ1× ℙ1

“…For s > 4, the ideal of s points of P 1 × P 1 is the ideal of s lines in P 3 , but the lines are never general, even if the points are.) Moving to P 1 × P 1 makes available to us the vast array of work done on products of projective spaces and surfaces in general, and on P 1 × P 1 in particular; see, for example [12,14,18,25,28,30].…”

“…It is well known that points with generic Hilbert function are general; i.e, for each s ≥ 1, there is a non-empty open subset of U s ⊂ (P 1 × P 1 ) s consisting of distinct ordered sets of s points of P 1 × P 1 with generic Hilbert function (see, for example, [30]). In particular, subschemes Z = P 1 + • • • + P s consisting of s distinct points for which every subset of the points has generic Hilbert function are general.…”