Algebraic Preliminaries

This chapter brings in the ground algebraic material employed in the book. Based on elementary facts from matrices, deeper results are extracted from the point of view of commutative algebra and geometry, with an emphasis on the role of the adjugate matrix, cofactors, and other determinantal minors. The latter are a tool to approach various questions concerning matrices with homogeneous linear entries, such as the structure of the linear syzygies of ideals of cofactors, features of Hessian matrices, and inverse maps to birational maps. Additional emphasis is on linear sections over polynomial rings and the ideal theoretic notions intertwining with matrix theory along other parts of the book. For the reader’s convenience, definitions are given ab initio whenever possible. In particular, we develop the role of the partial derivatives of a determinant f and explain the tendency of \(2\times 2\) -minors to belong to the homogeneous defining ideal of the dual variety to f. In that thread, from the geometric side, some deeper results are collected that will be of importance throughout subsequent chapters, such as those concerning the algebraic properties of rational maps, the dual variety, and symbolic blowup theory. As before, the standing reference for basic commutative algebra is Simis (2023. Commutative algebra, 2nd edn. De Gruyter Graduate, Berlin).