bibliography for algebraic geometry

bibliography for algebraic geometry

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.

Alexander Grothendieck

and J. Dieudonné.: 1960, Eléments de geometrie algébrique.,

Publ. Inst. des Hautes Etudes de Science

,

4

.

4.

Alexander Grothendieck.

Séminaires en Géometrie Algèbrique- 4

, Tome 1, Exposé 1
(or the Appendix to Exposée 1, by ‘N.

Bourbaki

’ for more detail and a large number of results.
AG4 is

http://modular.fas.harvard.edu/sga/sga/pdf/index.html

freely available in French;
also available here is an extensive

http://

planetmath

.org/?op=getobj&from=books&id=158

Abstract in English.

5.

Alexander Grothendieck. 1962. SÃÂ©minaires en GÃÂ©omÃÂ©trie AlgÃÂ©brique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux CohÃÂ¨rents et ThÃÂ©orÃÂ¨mes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud).,

http://modular.fas.harvard.edu/sga/sga/2/index.html

Typewritten manuscript available in French;

http://planetmath.org/?op=getobj&from=books&id=78

see also a brief summary in English

In French, with a few abstracts available in English (v. above two refs). The four volumes of this text are a very important reference for questions in basic algebraic geometry. Developing the theory of schemes in the utmost generality, this book may be difficult to read in some places but it is extremely thorough. Available

http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short

on the web.

6.

Alexander Grothendieck, 1984. “Esquisse d’un Programme”, (1984 manuscript),

finally published in “Geometric Galois Actions”

, L. Schneps, P. Lochak, eds.,

London Math. Soc. Lecture Notes

242

, Cambridge University Press, 1997, pp.5-48;
English transl., ibid., pp. 243-283. MR 99c:14034 .

7.

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.

8.

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.

9.

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.

http://www.jmilne.org/math/CourseNotes/math679.html

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

ℂ

, 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.

Title

bibliography for algebraic geometry

Canonical name

BibliographyForAlgebraicGeometry

Date of creation

2013-03-22 14:14:21

Last modified on

2013-03-22 14:14:21

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alozano (2414)

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alozano (2414)

Numerical id

18

Author

alozano (2414)

Entry type

Bibliography

Classification

msc 14-00

Related topic

WeilDivisorsOnSchemes

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