The Determinantal Varieties

Abstract

This is the first of a series of chapters where we apply the techniques of chapter 5 to concrete examples. We consider the determinantal varieties for the generic, generic symmetric, and generic skew symmetric matrices. We describe explicitly the terms of their minimal free resolutions over fields of characteristic 0. We also show that in characteristic p > 0 the resolution can be different than in characteristic 0. In section 6.1 we deal with ideals of minors of generic matrices over a field of characteristic 0. We prove Lascoux's result providing the description of terms in minimal free resolutions of these ideals. We also treat in more detail the special cases of Eagon—Northcott and Gulliksen—Negard complexes. Section 6.2 is devoted to determinantal ideals in positive characteristic. We prove Hashimoto's result that the resolution of 2 × 2 minors of a 5 × 5 generic matrix over a field of characteristic 3 is different than the corresponding resolution over a field of characteristic 0. Here we make use of theory of Schur complexes developed in section 2.4. Section 6.3 deals with the ideals of minors of a generic symmetric matrix. Again we calculate the terms of a minimal free resolutions of such ideals over a field of characteristic 0.