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

الأصل · EN

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 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() >= v(s;n₁,...,nₖ) always holds for s points in generic position in Pⁿ¹ x... x Pⁿᵏ. 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() > v(s;n₁,...,nₖ). However, computational evidence suggests that in many cases ν() = v(s;n₁,...,nₖ).