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algebraic geometry

Encyclopdia Britannica

algebraic geometry, study of the geometric properties of solutions to polynomial [1]

equations, including solutions in dimensions beyond three. (Solutions in two and three

dimensions are first covered in plane and solid analyticgeometry[2], respectively.)

Algebraic geometry[3] emerged from analyticgeometry[4] after 1850 when topology

[5], complex analysis, and algebra

[6]were used to study algebraic curves. An algebraic

curve [7]Cis the graph of an equationf(x,y) = 0, with points at infinity added, where

f(x,y) is a polynomial, in two complex variables, that cannot be factored. Curves are

classified by a nonnegative integerknown as their genus, gthat can be calculated

from their polynomial.

The equationf(x,y) = 0 determinesy as a function ofx at all but a finite number of points

ofC. Sincex takes values in the complex numbers [8], which are two-dimensional over

the real numbers [9], the curve Cis two-dimensional over the real numbers near most of

its points. Clooks like a hollow sphere[10] with g hollow handles attached and finitely

many points pinched togethera sphere has genus 0, a torus has genus 1, and so forth.

The Riemann-Roch theorem uses integrals along paths on Cto characterize g

analytically.

A birational transformation matches up the points on two curves via maps given in both

directions by rational functions [11] of the coordinates. Birational transformations

preserve intrinsic properties of curves, such as their genus, but provide leeway for

geometers to simplify and classify curves by eliminating singularities (problematic

points).

An algebraic curve generalizes to a variety, which is the solution set ofr polynomial

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equations in n complex variables. In general, the difference nr is the dimension of the

varietyi.e., the number of independent complex parameters near most points. For

example, curves have (complex) dimension one and surfaces have (complex) dimension

two. The French mathematician Alexandre Grothendieck [12] revolutionized algebraic

geometry in the 1950s by generalizing varieties to schemes and extending the

Riemann-Roch theorem.

Arithmetic geometry combines algebraic geometry and number theory [13] to study

integer solutions of polynomial equations. It lies at the heart of the British mathematician

Andrew Wiles [14]s 1995 proof of Fermats last theorem [15].

Robert Alan BixHarry Joseph DSouza

Additional Reading

Phillip A. Griffiths,Introduction to Algebraic Curves, trans. from Chinese (1989),

develops the topological and analytical properties of complex curves. Vol. 1 of Igor R.

Shafarevich,Basic Algebraic Geometry, 2nd rev. and expanded ed., 2 vol. (1994;

originally published in Russian, 1988), elegantly demonstrates the power of modern

approaches to higher-dimensional algebraic geometry. Robin Hartshorne,Algebraic

Geometry (1977, reprinted with corrections, 1997), is the best prcis of Grothendiecks

work in the foundations of algebraic geometry. Simon Singh, Fermats Enigma (also

published as Fermats Last Theorem, 1997), gives a historical introduction to Fermatslast theorem and its proof.

http://www.britannica.com/EBchecked/topic/468893/polynomial1.

http://www.britannica.com/EBchecked/topic/22548/analytic-geometry2.

http://www.britannica.com/EBchecked/topic/229851/geometry3.

http://www.britannica.com/EBchecked/topic/22548/analytic-geometry4.

http://www.britannica.com/EBchecked/topic/599686/topology5.

http://www.britannica.com/EBchecked/topic/14885/algebra6.

http://www.britannica.com/EBchecked/topic/1053865/algebraic-curve7.

http://www.britannica.com/EBchecked/topic/129992/complex-number8.

http://www.britannica.com/EBchecked/topic/492990/real-number9.

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http://www.britannica.com/EBchecked/topic/559619/sphere10.

http://www.britannica.com/EBchecked/topic/492008/rational-function11.

http://www.britannica.com/EBchecked/topic/246803/Alexandre-Grothendieck12.

http://www.britannica.com/EBchecked/topic/422325/number-theory13.

http://www.britannica.com/EBchecked/topic/643734/Andrew-John-Wiles14.

http://www.britannica.com/EBchecked/topic/204685/Fermats-last-theorem15.

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