Background Steps in Determinantal Rings

This chapter takes a general overview of determinantal rings. Our general reference for the material discussed is Bruns and Vetter (1988, Determinantal rings, LNM No. 1327. Springer, Berlin) and Simis (2023, Commutative algebra, 2nd edn. De Gruyter Graduate, Berlin), where full proofs can be found. Of these, we kept a few explicit proofs that seemed to us relevant to the general style of the book, e.g., some of the early theorems of Eagon and Northcott. In addition, we look at some important rational determinantal varieties, such as the Veronese variety, the Segre variety, and the rational normal scrolls. In these, both the algebraic and the geometric sides are emphasized. The second part of the chapter is devoted to an account of linear sections of a generic matrix in the case where these sections are sufficiently general. Thus, first one focuses on the case where the sections are general linear forms and then surveys a collection of results assuming that the section is t-generic in the sense of Eisenbud and Harris. Both approaches are useful and will be called upon in the later chapters. The chapter assumes familiarity with first principles of commutative algebra, such as contained in Simis (2023, Commutative algebra, 2nd edn. De Gruyter Graduate, Berlin), and the basic notation of algebraic geometry.