On the normal sheaf of determinantal varieties

Abstract

Let X be a standard determinantal scheme X \subset \PP^n of codimension c, i.e. a scheme defined by the maximal minors of a t \times (t+c-1) homogeneous polynomial matrix A. In this paper, we study the main features of its normal sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a line bundle \shL on X \setminus Sing(X) such that \shN_X \otimes \shL is arithmetically Cohen-Macaulay and, even more, it is Ulrich whenever the entries of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and, finally, (3) \shN_X is \mu-(semi)stable provided the entries of A are linear forms.