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# 14: Algebraic geometry

## Introduction

Algebraic geometry combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" -- ostensibly a statement in number theory -- was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves.

## History

See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)

## Applications and related fields

• geometry (in particular for conics and curves),
• algebra (since algebraic geometry is commutative ring theory...)
• number theory (especially for Diophantine analysis).

## Subfields

• 14A: Foundations
• 14C: Cycles and subschemes
• 14D: Families, fibrations
• 14E: Birational geometry [Mappings and correspondences]
• 14H: Curves
• 14J: Surfaces and higher-dimensional varieties. For analytic theory, see 32JXX
• 14K: Abelian varieties and schemes
• 14L: Algebraic groups [Group schemes]. For linear algebraic groups, see 20GXX. For Lie algebras, See 17B45
• 14M: Special varieties
• 14N: Projective and enumerative geometry [Classical methods and problems]. See also 51: Geometry.
• 14P: Real algebraic and real analytic geometry
• 14R: Affine geometry [new in 2000]

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Textbooks: Hartshorn; ...

For a more elementary introduction see Reid, Miles: "Undergraduate algebraic geometry", London Mathematical Society Student Texts, 12. Cambridge University Press, Cambridge-New York, 1988. 129 pp. ISBN 0-521-35559-1; 0-521-35662-8 MR90a:14001

Some survey articles:

• Reid, Miles: "Young person's guide to canonical singularities", Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345--414; Proc. Sympos. Pure Math., 46, Part 1; Amer. Math. Soc., Providence, RI, 1987. MR89b:14016
• Deligne, Pierre: "À quoi servent les motifs?" (French; "What is the use of motives?") Motives (Seattle, WA, 1991), 143--161; Proc. Sympos. Pure Math., 55, Part 1; Amer. Math. Soc., Providence, RI, 1994. MR95c:14013
• Procesi, Claudio. "A primer of invariant theory", Brandeis Lecture Notes, 1. Brandeis University, Waltham, Mass., 1982. 218 pp. MR86d:14045

## Software and tables

• Notice of software for computing zeta functions of (projective) curves.
• A Maple package that computes intersection numbers on algebraic varieties, etc.
• Singular Package: Singular is a computer algebra system for singularity theory and algebraic geometry developed by G.-M. Greuel, G. Pfister, H. Schönemann, and others, at the Department of Mathematics of the University of Kaiserslautern. Singular can compute with ideals and modules generated by polynomials or polynomial vectors over polynomial or power series rings or, more generally, over the localization of a polynomial ring with respect to any ordering on the set of monomials which is compatible with the semigroup structure. Singular is available via anonymous ftp. Precompiled versions of Singular are available for Sun Sparc 2(SunOS 4.1), Sun Sparc 10 (SunOS 5.3), HP9000/300, HP9000/700, Linux, Silicon Graphics, IBM RS/6000, Next (m68k based), MSDOS, and Macintosh.

Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies.