Graded algebras in algebraic geometry (Q2054459)

In this memoir, the authors report on several research areas of algebraic geometry where graded algebras play an important role. The materials were sourced from various texts of the first author that have appeared in informal proceedings of research meetings for more than a decade. These texts were expanded and updated to include accounts of recent published (and unpublished) work from both authors. Familiarity with many advanced topics of commutative algebra and algebraic geometry is assumed throughout. \medskip The book consists of 12 chapters, each followed by an exercises section. After an introduction, chapter 2 describes, from first principles, some of the most important algebras featuring in the remaining of the book, namely the symmetric algebra, the Rees algebra and the fiber cone algebra. The algebraic setup of rational maps is laid out in chapters 3 and 4. Chapter 3 includes sections on a characteristic-free criterion of birationality, on rational maps defined by monomials and on the degree of rational maps. Chapter 4 includes a section on Newton complementary duality (and the relation to rational maps) and a section on rational maps defined over a ground ring (a Noetherian domain). Chapter 5 is dedicated to the Jacobian ideal of a polynomial, describing applications to hyperplane arrangements and to families of singular plane curves. Chapters 6 and 7 deal with Cremona transformations; the first focusing on plane Cremona transformations and the interplay between them, their base ideals and their associated fat ideals. Chapter 7 contains sections on generalized Jonquières transformations, on the graph of a Cremona map and on monomial Cremona transformations. In chapter 8, several instances of the use of elimination algebras are described; including applications to Cremona transformations. Chapter 9 focuses on the Gauss map; including characterizations of the image of the Gauss map in several cases. Chapter 10 deals with tangent algebras and features a proof of the Fulton-Hansen connectedness theorem. Chapter 11 addresses join and secant algebras. The final chapter is devoted to Aluffi algebras. Section 12.3 gives the definition of the Aluffi algebra of a finitely generated module endowed with a fixed embedding into a free module. Earlier parts of this chapter deal with the embedded versions of this construct, with a section dedicated to the embedded Aluffi algebra of an ideal.