a branch of mathematics that studies algebraic manifolds. Algebraic manifolds are sets of points in n-dimensional space, the coordinates of which (xux x2,,... , JC, ) are solutions of a system of equations
F1 (x1 x2,. . . , xn) = 0
...................................
Fm (x1,x2,. . ., xn) = 0
where F1,. . . , Fm are polynomials in the unknowns x1, x2, . . . , xn. Every algebraic manifold has a definite dimension, which is the number of independent parameters defining a point on the manifold. Algebraic manifolds of dimension 1 are called algebraic curves, and those of dimension 2 are called algebraic surfaces. Examples of algebraic curves are provided by conic sections.
Two algebraic manifolds are said to be birationally equivalent if the coordinates of every point of one manifold can be expressed by rational functions in terms of the coordinates of a point of the other manifold, and conversely. In algebraic geometry, algebraic manifolds are usually studied to within birational equivalence, and therefore one of the basic problems of algebraic geometry is the construction of birational invariants for algebraic manifolds. The most important birational invariants are constructed with the tools of mathematical analysis (the so-called transcendental methods), in particular by means of multiple integrals over an algebraic manifold. Besides transcendental methods, the geometrical methods of projective geometry and topological methods are frequently used. Topology is invoked because certain important birational invariants—for example, the genus of a curve (see below)—of algebraic manifolds are of a topological nature. The relationship between algebraic geometry and topology becomes particularly important in view of the theorem of the Japanese mathematician Hironaka, according to which every algebraic manifold is birationally equivalent to a manifold without singular points.
The most extensively developed area of algebraic geometry is the theory of algebraic curves. The fundamental birational invariant of an algebraic curve is its genus. If an algebraic curve is plane—that is, if it is defined in Cartesian coordinates by the equation F(x, y) = 0—the genus of the curve is g = (m − 1) (m − 2)/2 − d, where m is the order of the curve and d is the number of double points. The genus of a curve is always a nonnegative integer. Curves of genus 0 are birationally equivalent to straight lines—that is, they can be represented parametrically by rational expressions. Curves of genus 1 can be parametrized by elliptic functions and are therefore called elliptic curves. Curves of genus greater than 1 can be parametrized by automorphic functions. Each curve of genus g greater than 1 is uniquely defined (to within birational equivalence) by 3g − 3 complex parameters, which themselves run through some algebraic manifold.
In the multidimensional case, the most widely studied class of algebraic manifolds is that of Abelian manifolds. These are closed submanifolds of a projective space which at the same time are groups such that multiplication is specified by rational expressions. Multiplication on such a manifold is automatically commutative. An algebraic curve is an Abelian manifold if and only if it is of genus 1—that is, if it is an elliptic curve.
The theory of algebraic curves and the theory of Abelian manifolds are closely related. Every algebraic curve of genus greater than 0 is canonically embedded in some Abelian manifold called the Jacobian manifold of the given curve. The Jacobian manifold is an important invariant of the curve and almost completely defines the curve itself.
Historically, algebraic geometry arose from the study of curves and surfaces of lower order. Third-order curves were classified by I. Newton in 1704. In the 19th century algebraic geometry gradually advanced from the study of special classes of curves and surfaces to the formulation of general problems pertaining to all manifolds. A general algebraic geometry was formulated around the turn of the 20th century by the German mathematician M. Noether, by the Italians F. Enriques and F. Severi, and others. Algebraic geometry flourished in the 20th century with the work of the French mathematician A. Weil, the American S. Lefschetz, and others. Major contributions have been made by the Soviet mathematicians N. G. Chebotarev, I. G. Petrovskii, and I. R. Shafarevich.
Algebraic geometry is one of the most rapidly developing areas of mathematics. Its methods are exerting an enormous influence on such related fields as the theory of functions of many complex variables, number theory, and also on more remote fields such as partial differential equations, algebraic topology, and group theory.
B. B. VENKOV