Dimension of families of determinantal schemes

Abstract

A scheme X ⊂ P n + c X\subset \mathbb {P}^{n+c} of codimension c c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t × ( t + c − 1 ) t \times (t+c-1) matrix and X X is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers a 0 , a 1 , . . . , a t + c − 2 a_0,a_1,...,a_{t+c-2} and b 1 , . . . , b t b_1,...,b_t we denote by W ( b _ ; a _ ) ⊂ Hilb p ⁡ ( P n + c ) W(\underline {b};\underline {a})\subset \operatorname {Hilb} ^p(\mathbb {P}^{n+c}) (resp. W s ( b _ ; a _ ) W_s(\underline {b};\underline {a}) ) the locus of good (resp. standard) determinantal schemes X ⊂ P n + c X\subset \mathbb {P}^{n+c} of codimension c c defined by the maximal minors of a t × ( t + c − 1 ) t\times (t+c-1) matrix ( f i j ) j = 0 , . . . , t + c − 2 i = 1 , . . . , t (f_{ij})^{i=1,...,t}_{j=0,...,t+c-2} where f i j ∈ k [ x 0 , x 1 , . . . , x n + c ] f_{ij}\in k[x_0,x_1,...,x_{n+c}] is a homogeneous polynomial of degree a j − b i a_j-b_i . In this paper we address the following three fundamental problems: To determine (1) the dimension of W ( b _ ; a _ ) W(\underline {b};\underline {a}) (resp. W s ( b _ ; a _ ) W_s(\underline {b};\underline {a}) ) in terms of a j a_j and b i b_i , (2) whether the closure of W ( b _ ; a _ ) W(\underline {b};\underline {a}) is an irreducible component of Hilb p ⁡ ( P n + c ) \operatorname {Hilb} ^p(\mathbb {P}^{n+c}) , and (3) when Hilb p ⁡ ( P n + c ) \operatorname {Hilb} ^p(\mathbb {P}^{n+c}) is generically smooth along W ( b _ ; a _ ) W(\underline {b};\underline {a}) . Concerning question (1) we give an upper bound for the dimension of W ( b _ ; a _ ) W(\underline {b};\underline {a}) (resp. W s ( b _ ; a _ ) W_s(\underline {b};\underline {a}) ) which works for all integers a 0 , a 1 , . . . , a t + c − 2 a_0,a_1,...,a_{t+c-2} and b 1 , . . . , b t b_1,...,b_t , and we conjecture that this bound is sharp. The conjecture is proved for 2 ≤ c ≤ 5 2\le c\le 5 , and for c ≥ 6 c\ge 6 under some restriction on a 0 , a 1 , . . . , a t + c − 2 a_0,a_1,...,a_{t+c-2} and b 1 , . . . , b t b_1,...,b_t . For questions (2) and (3) we have an affirmative answer for 2 ≤ c ≤ 4 2\le c \le 4 and n ≥ 2 n\ge 2 , and for c ≥ 5 c\ge 5 under certain numerical assumptions.