Linear Sections of the Generic Square Matrix

This chapter captures quite a bit of the spirit of the book in that one views the generic square matrix as a primeval model for many other matrices. One deals with ladder ideals that are Gorenstein, and in the codimension three case, a full discussion of its properties is given, including an explicit alternating matrix for the corresponding maximal Pfaffians and the structure of its lower Pfaffians. In the sequel, based on such Gorenstein ideals, a family of codimension four Gorenstein ideals is discussed that does not seem to be immediately available as such in the known literature. Structural properties of this family are surveyed, some of which are still conjectured. The next topic concerns homological and differential minded discussions of various types of linear sections of the generic square matrix, some tightly related to the generic model, while others more qualified as sparse sections of certain strategic shape. For all these, the corresponding gradient ideal, polar map, and ideals of minors are thoroughly discussed. Throughout the chapter quite a heavy use is made of previous chapters.