References for Algebraic Geometry, MSC 14 - XX

The following are excellent sources for the indicated areas in Algebraic Geometry.

Foundations, MSC 14A

1. Robin Hartshorne, Algebraic Geometry, Springer-Verlag Graduate Texts in Mathematics 52, 1977.

   An excellent introduction and basic reference text to the subject; discusses varieties primarily as background for the theory of schemes, which is developed in detail and used throughout the bulk of the book. Does not strive for the utmost generality, generally assuming schemes are noetherian and emphasizing algebraically closed fields (of arbitrary characteristic). Discusses sheaf cohomology, formal schemes, Serre duality and other topics in broad generality. Includes chapters on curves, surfaces, intersection theory, transcendental methods and the Weil conjectures.

2. David Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer, New York.

   A highly readable introduction to the subject. Originally a much-photocopied set of course notes (bound in red), the style is informal but extremely clear. Discusses varieties and introduces schemes; discusses flat end étale maps and their usefulness.

3. Alexandre Grothendieck, Eléments de Géometrie Algébrique, Publications Mathématiques de l'I.H.E.S., 1960.

   In French. The four volumes of this text are the definitive reference for any question in basic algebraic geometry. Developing the theory of schemes in the utmost generality, this book is difficult to read but extremely thorough. Available on the web.

4. Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, 2002.

   Liu's book, which spends 300 pages on schemes before delving into the geometry (and later arithmetic) of arithmetic surfaces, is the second most exhaustive reference on scheme theory in this list (the first being EGA). While covering essentially the same ground as Hartshorne in the theory of schemes (with the major exception that Serre duality is approached via Grothendieck duality, which is left unproven), Liu's book is less terse and includes many more number theoretically interesting examples (such as a detailed treatment of the Frobenius morphism) and, as such, does not emphasize algebraically closed fields.

5. Igor Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, Second Revised and Expanded Edition. Springer-Verlag.

   This is the first of two volumes on basic algebraic geometry. This volume deals with quasi-projective varieties, local notions and properties, divisors and differential forms, and basic intersection theory. The style is very readable and most results are proved. This volume and the second supplement Hartshorne's book well.

6. Igor Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Second Revised and Expanded Edition. Springer-Verlag.

   This volume deals with scheme theory, varieties as schemes, varieties and schemes over the complex numbers, and complex manifolds.

Curves, MSC 14H

Elliptic Curves, MSC 14H52

1. James Milne, Elliptic Curves, online course notes. Available at his website.
2. Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
3. Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
4. Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

Abelian Varieties and Schemes, MSC 14K

1. David Mumford, Abelian Varieties, Oxford University Press, London, 1970.

   This book is the canonical reference on the subject. It is written in the language of modern algebraic geometry, and provides a thorough grounding in the theory of abelian varieties. It covers the basic analytic theory of abelian varieties over , computing cohomology groups and proving various theorems. It then addresses the algebraic theory of abelian varieties, using only the theory of varieties, working towards proving the same results. In the third chapter, it applies the theory of schemes, developing some of the theory of group schemes (not necessarily commutative) but focusing on abelian varieties (rather than abelian schemes). Finally, the last chapter addresses issues combining the three previous chapters. As usual, Mumford's writing is clear and precise.