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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 14: Algebraic geometry

Introduction

History

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
• 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, 68W30
• 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.

Other web sites with this focus

• Preprint server(Duke)
• Here are the AMS and Goettingen resource pages for area 14.

Selected topics at this site

• Effective solutions of systems of polynomial equations (literature)
• Generalities on systems of polynomial equations (resolvents, Bezout's theorem)
• Bezout's theorem counts points (sort of).
• How many points in a variety of dimension zero? (Bernshtein's theorem and algorithms)
• Note that any variety can be described by quadratics alone.
• How many lines pass through four given lines in R^3 (two; generalize?) This is Enumerative Geometry (use the Schubert calculus: 14N10).
• Some exposure to conics (quadratics) and elliptic curves (cubics) suggests there ought to a "canonical form" for varieties in general. There isn't, but you might want to think about why.
• A brief introduction to Abelian varieties
• Classifying cubic polynomials under the narrow equivalence relation of rotational equivalence.
• Are there other rational functions which could be used to make groups? This is essentially the study of formal groups.
• Formal group examples related to Jacobian varieties
• Formal groups and elliptic curves.
• Is a stably rational variety actually rational? No.
• What are motives? And what does motivic mean?
• Can we find varieties birationally equivalent to V but with more rational points?
• Describe parametrically the family of lines simultaneously tangent to three spheres.
• Using elimination to show two curves have two points of tangency
• When is a real algebraic variety a bounded set?
• Does a set of real polynomials have a real solution? Tarski's Elimination of Quantifiers.