Basic Algebraic Geometry 2

1. Front Matter

   Pages I-XIV

   PDF

2. Book 2: Schemes and Varieties

   1. Front Matter

      Pages 1-1

      PDF

   2. Schemes

      • Igor R. Shafarevich

      Pages 3-47

   3. Varieties

      • Igor R. Shafarevich

      Pages 49-111

3. Book 3: Complex Algebraic Varieties and Complex Manifolds

   1. Front Matter

      Pages 113-113

      PDF

   2. The Topology of Algebraic Varieties

      • Igor R. Shafarevich

      Pages 115-148

   3. Complex Manifolds

      • Igor R. Shafarevich

      Pages 149-199

   4. Uniformisation

      • Igor R. Shafarevich

      Pages 201-228

4. Back Matter

   Pages 229-262

   PDF

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 second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the ``Shafarevich conjecture''.

The style of  Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of  Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.

• algebraic geometry
• complex algebraic varieties
• geometry
• schemes

From the book reviews:

“I find the book wonderfully put together, and I am sure the reader will learn a lot, either from systematic study or from browsing particular topics. … In each chapter, the theorems, propositions, corollaries, examples, remarks, etc., each have their own independent numbering system, running consecutively throughout the chapter. This makes it a real chore to track any internal reference in the book.” (Robin Hartshorne, SIAM Review, Vol. 56 (4), December, 2014)

“This is the English translation of the third edition of the second volume of the author’s classic standard text ‘Basic algebraic geometry’ … . a perfect first introduction to various aspects of both classic and modern algebraic geometry as a whole. … the present second volume may serve as an excellent source for students and non-specialists to get themselves prepared for the study of more advanced books on abstract algebraic geometry, complex algebraic geometry, and moduli theory likewise.” (Werner Kleinert, zbMATH, Vol. 1277, 2014)

• Algebra Section, Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

  Igor R. Shafarevich

Igor Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry.