Ideals generated by submaximal minors

Jan O. Kleppe and R.M. Mir'o Roig

The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of Pⁿ defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^p(x)(Pⁿ), we show that Hilb^p(x)(Pⁿ) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf I_Y/I²_Y(s) where s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of Pⁿ defined by the maximal minors of the matrix obtained deleting a suitable row of A.