ALGEBRAIC GEOMETRY

the study of geometric objects by means of algebra. A Babylonian record of about 1700 BC describes a problem about a rectangle (a geometric object) and involves unknown numbers; it is, thus, an example of algebraic geometry. The Greek mathematicians of classical times had the problem: to construct the edge of a cube that shall have twice the volume of a given cube. Calling the edge of the given cube a and the edge of the sought cube x, we would write x3 = 2a3. Hippocrates of Chios (c. 430 BC) reformulated the problem as one of finding x and y such that a/x = x/y = y/2a. Menaechmus (c. 350 BC) considered the locus in a rectangular coordinate system of the points (x,y) such that x2 = ay and similarly of the points (x,y) such that xy = 2a2, recognized the first as a parabola and the second as a hyperbola, and thus found x,y from the intersection of these curves. The 17th-century French mathematicians Ren Descartes and Pierre de Fermat studied the conic sections and knew that the points (x,y) of their geometric loci satisfied algebraic equations of degree 2 in x and y. Later, Isaac Newton studied polynomial equations of degree 3 and classified the corresponding loci, cubics, into 72 kinds. Thus, Descartes, Fermat, and Newton can be credited with initiating the study of plane algebraic curves, i.e., loci given by polynomials of arbitrary degree, a basic topic in algebraic geometry. The scope and generality of modern algebraic geometry are increased by adoption of certain ideas from projective geometry. For example, two (distinct) straight lines should meet in just one point. This is correct except in the case of parallel lines. This exception can be technically circumvented by introducing so-called points at infinity and dealing with the images of the lines in the projective plane. Plane geometric curves are represented in the projective plane by homogeneous polynomial equations (a polynomial is homogeneous if all its terms have the same degree). In the projective plane any two (distinct) straight lines, parallel or not, meet in precisely one point. Another difficulty, that of counting the intersections of tangent curves, is overcome by assigning a point of tangency the multiplicity 2. In these developments some typical phenomena may be noted: in studying the geometry, the understanding of what geometry is changes; and new algebraic methods are devised. A curve in the projective plane over the complex numbers cannot be perceived in the way a real curve in the affine plane is, but geometric terminology, governed by logic, remains valid. In the plane a point is given by two coordinates (x,y), and in ordinary space by three (x,y,z). One may call, for any positive integer n, a sequence (x1, . . . , xn) of numbers a point of (affine) n-space. In n-space an algebraic variety is defined as the set of points satisfying a system of polynomial equations: F1(x1, . . . , xn) = 0, f2(x1, . . . , xn) = 0, . . . . These objects and their generalizations are the main objects of study in algebraic geometry. study of geometric objects by means of algebra. In algebraic geometry the properties of a geometrical structure are described by means of algebraic expressions. The subject arose in the algebraic study of loci (collections of all points the location of which is determined by stated conditions) in projective space. A projective space of n dimensions is denoted by Pn and is a set of elements, called points, endowed with certain allowable coordinate systems. In an allowable coordinate system x each point A of Pn is determined by a set of n + 1 numbers (x0, . . . , xn) that are not all zero. Two such sets, (x0, . . . , xn) and (x0, . . . , xn), determine the same point if, and only if, there is a number r such that xi = rxi (i = 0, . . . , n). Any set (x0, . . . , xn) corresponding to A are the homogeneous coordinates of A in the coordinate system x. If y is any other coordinate system, the relation between the x and y coordinates (x0, . . . , xn), (y0, . . . , yn) of A is given by equations that are linear (see 172), in which coefficients aij, bij are numbers independent of A, and r and s are factors of proportionality.