Graded algebras in algebraic geometry
type: Item
instance of: scholarly article
title: Graded algebras in algebraic geometry
zbMATH Open document ID: 1487.13001
DOI: 10.1515/9783110640694
author: Zaqueu Ramos
author: Aron Simis
published in: De Gruyter Expositions in Mathematics
publication date: 2021-12-02T00:00:00Z
review text: 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 These texts were expanded and updated to include accounts of recent published (and unpublished) work from both authors.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 proo…
reviewed by: Jorge Neves
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 7438189
zbMATH Keywords: graded algebras
zbMATH Keywords: symmetric algebra
zbMATH Keywords: Rees algebra
zbMATH Keywords: Cremona transformations
zbMATH Keywords: Jacobian ideal
zbMATH Keywords: elimination
zbMATH Keywords: Gauss map
zbMATH Keywords: Aluffi algebras
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110640694
OpenAlex ID: W4226409417
description: scientific article; zbMATH DE number 7438189