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.
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
Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif.,
1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)
- geometry (in particular for conics and curves),
- algebra (since algebraic geometry is commutative ring theory...)
- number theory (especially for Diophantine analysis).
- 14A: Foundations
- 14B: Local theory, see also 32SXX
- 14C: Cycles and subschemes
- 14D: Families, fibrations
- 14E: Birational geometry [Mappings and correspondences]
- 14F: (Co)homology theory, see also 13DXX
- 14G: Arithmetic problems. Diophantine geometry, see also 11DXX, 11GXX
- 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
- 14Q: Computational aspects in algebraic geometry, see also 12-04, 68Q40
- 14R: Affine geometry [new in 2000]
Browse all (old) classifications for this area at the AMS.
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
"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
- Notice of software for computing zeta
functions of (projective) curves.
- A Maple package that computes intersection numbers on algebraic varieties, etc.
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.
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Last modified 1999/05/12 by Dave Rusin. Mail: email@example.com