An Undergraduate Primer in Algebraic Geometry

This book consists of two parts. The first is devoted to an introduction to basic concepts in algebraic geometry: affine and projective varieties, some of their main attributes and examples. The second part is devoted to the theory of curves: local properties, affine and projective plane curves, resolution of singularities, linear equivalence of divisors and linear series, RiemannRoch and RiemannHurwitz Theorems.

The approach in this book is purely algebraic. The main tool is commutative algebra, from which the needed results are recalled, in most cases with proofs. The prerequisites consist of the knowledge of basics in affine and projective geometry, basic algebraic concepts regarding rings, modules, fields, linear algebra, basic notions in the theory of categories, and some elementary pointset topology.

This book can be used as a textbook for an undergraduate course in algebraic geometry. The users of the book are not necessarily intended to become algebraic geometers but may be interested students or researchers who want to have a first smattering in the topic.

The book contains several exercises, in which there are more examples and parts of the theory that are not fully developed in the text. Of some exercises, there are solutions at the end of each chapter.

Provides a self contained introduction to Algebraic Geometry for undergraduate students Contains many exercises, some of them with solution Useful for non-experts who want to learn the basics of Algebraic Geometry

Auteur
Ciro Ciliberto has been a full professor of higher geometry at the University of Roma "Tor Vergata". Recently, he retired and kept the honorary position of "Docens Turris Virgatae" at the same university. His research fields are algebraic geometry and history of mathematics. He published more than 200 research papers, most of them in top scientific journals, and various books. He spent several research stays abroad, he has been invited to give talks to several international conferences and he is the editor of various scientific journals. Among his hobbies, there are painting and writing.

Résumé
"The text is fairly self-contained ... . The clear exposition and numerous exercises also make the text suitable for self-study ... . The text is thorough and clearly written, and the numerous exercises do indeed give the reader an algebraic primer to algebraic geometry." (Kelly Jabbusch, Mathematical Reviews, June, 2022)

"New concepts in algebra that one needs, in order to understand the content of the book, are introduced ... . I find this feature as an asset of the present book. ... the present book is a valuable introduction to algebraic geometry which is nicely written ... and a pleasure to read. It can be recommended as a first book for undergraduate courses devoted to algebraic geometry, and also as a supplementary source for standard courses devoted to algebraic geometry." (Piotr Pokora, zbMATH 1471.14001, 2021)

Contenu
1 Affine and projective algebraic sets.- 2 Basic notions of elimination theory and applications.- 3 Zariski closed subsets and ideals in the polynomials ring.- 4 Some topological properties.- 5 Regular and rational functions.- 6 Morphisms.- 7 Rational maps.- 8 Product of varieties.- 9 More on elimination theory.- 10 Finite morphisms.- 11 Dimension.- 12 The Cayley form.- 13 Grassmannians.- 14 Smooth and singular points.- 15 Power series.- 16 A ne plane curves .- 17 Projective plane curves.- 18 Resolution of singularities of curves.- 19 Divisors, linear equivalence, linear series.- 20 The Riemann-Roch Theorem.