Simis A. Graded Algebras in Algebraic Geometry 2022

Textbook in PDF format

The objective of this book is to look at certain commutative graded algebras that appear frequently in algebraic geometry. By studying classical constructions from geometry from the point of view of modern commutative algebra, this carefully-written book is a valuable source of information, offering a careful algebraic systematization and treatment of the problems at hand, and contributing to the study of the original geometric questions. In greater detail, the material covers aspects of rational maps (graph, degree, birationality, specialization, combinatorics), Cremona transformations, polar maps, Gauss maps, the geometry of Fitting ideals, tangent varieties, joins and secants, Aluffi algebras. The book includes sections of exercises to help put in practice the theoretic material instead of the mere complementary additions to the theory.
Geometric motivation
Graded algebras
Rational maps, I
Rational maps, II
The gradient ideal
Cremona transformations, I
Cremona transformations, II
Steps in elimination
The Gauss map
Tangent varieties
Embedded joins and secants
Aluffi algebras