Basic Algebraic Geometry 1: Varieties in Projective Space

by Igor R. Shafarevich, Miles Reid (translator)

Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction  to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles.

Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.

Genres: Mathematics, Science, Textbooks

Kindle Edition

First published August 8, 1994

About the author

Igor Rostislavovich Shafarevich is a Russian mathematician who has contributed to algebraic number theory and algebraic geometry. He has written books and articles that criticize socialism and was an important dissident during the Soviet regime.

Reviews

[George · Rating 4 out of 5]

This is a great book to learn from in a second course on varieties, or perhaps, after skimming a 'lighter' alternative. It is quite heavy and often a bit hard to follow. However, it is very complete, appeals to intuition, and often prefers taking the geometrical side over the algebraic, making it certainly easier to get a 'big picture', at the expense of making the proofs more complicated. Certainly I'm not the smartest person, but I often felt that I would not have been able to produce most of the proofs of the theorems, and after a thorough reading I was still left slightly uneasy. This is not a jab at the translators, but at times it is quite obvious that some details are lost in translation and the grammar to make it work seems off, which further exacerbated the issue.

I admit that I had some trouble with the often-changing non-standard notation presented in the textbook. For example, I had a hard time following his examples of the Grassmannian and Veronese varieties since the notation was so clunky and unexplained. The exercises tend to be either easy or hard, but at least you can count on there being plenty of them.

This was my experience with the first two chapters - I haven't read the last two, and it was recommended to me to not do so by a well-known geometer if one expects to go further in their studies of AG. It is better to learn from a more general source such as Hartshorne in that case, or other, more modern books on the subject.