Dimension of Families of Determinantal Schemes

الأصل · EN

A scheme X⊂ ⁿ⁺ᶜ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t × (t+c-1) matrix and X is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers a₀,a₁,...,aₜ₊c₋₂ and b₁,...,bₜ we denote by W(b;a)⊂ ᵖ(ⁿ⁺ᶜ) (resp. Wₛ(b;a)) the locus of good (resp. standard) determinantal schemes X⊂ ⁿ⁺ᶜ of codimension c defined by the maximal minors of a t× (t+c-1) matrix (fij)ⁱ⁼¹,...,ᵗⱼ₌₀,...,ₜ₊c₋₂ where fij∈ k[x₀,x₁,...,xₙ₊c] is a homogeneous polynomial of degree aⱼ-bᵢ. In this paper we address the following three fundamental problems: To determine (1) the dimension of W(b;a) (resp. Wₛ(b;a)) in terms of aⱼ and bᵢ, (2) whether the closure of W(b;a) is an irreducible component of ᵖ(ⁿ⁺ᶜ), and (3) when ᵖ(ⁿ⁺ᶜ) is generically smooth along W(b;a). Concerning question (1) we give an upper bound for the dimension of W(b;a) (resp. Wₛ(b;a)) which works for all integers a₀,a₁,...,aₜ₊c₋₂ and b₁,...,bₜ, and we conjecture that this bound is sharp. The conjecture is proved for 2≤ c≤ 5, and for c≥ 6 under some restriction on a₀,a₁,...,aₜ₊c₋₂ and b₁,...,bₜ. For questions (2) and (3) we have an affirmative answer for 2≤ c ≤ 4 and n≥ 2, and for c≥ 5 under certain numerical assumptions.