With the introduction of Cartesian coordinates by R. Descartes and P. de Fermat in the seventeenth century, it was soon realized that equations in two variables generally define curves in the plane. Descartes had already made the distinction between curves that can be described by polynomial equations and those that cannot. He called the former geometrical and the latter mechanical curves. Nowadays, these are called algebraic and transcendental curves, respectively.
Algebraic geometry is the area of mathematics that studies the properties of sets (or loci) defined as the set of common zeros of a collection of polynomial equations on the coordinates of the points of some Cartesian coordinate system. Such sets are called algebraic sets or algebraic varieties. If they are one-dimensional, they are called algebraic curves, and if two-dimensional, algebraic surfaces. For example, a subset of the plane defined as the solution set of an equation f(x,y)=0, where f(x,y) is a polynomial in two variables, is an algebraic curve. The total degree of the polynomial is called the degree of the curve. For example, lines are algebraic curves defined by linear equations, so they have a degree of one. The equation x2+y2=r2 defines a circle of radius r, and thus the circle is an algebraic curve of degree two. More generally, conics are algebraic curves of degree two, cubics are algebraic curves of degree three, and so on.
Some of the earlier questions in Algebraic Geometry had to do with counting points on curves with certain properties; for instance, determining how many points two curves can have in common. Bézout's theorem states that two plane algebraic curves of degrees m and n intersect in at most mn points unless they have a common component; that is, unless there exists an algebraic curve which is a subset of both curves. Other questions had to do with how many inflexion points a curve can have or for how many points on a curve the tangent line goes through a specific point. Several of these questions are answered by the Plücker formulas, obtained by J. Plücker in 1834.
Some of these counting questions led to realizations that there were some "missing points," since the problems would have a certain number of solutions in general, but in special cases some of these solutions would disappear. This motivated the introduction of the complex projective plane. The complex projective plane is defined in the following way. First one considers a Cartesian plane in which the (x,y) coordinates are allowed to take complex numbers as values. Second, one introduces "points at infinity." These are ideal points, one for each direction, and are introduced in such a way that parallel lines meet at these points at infinity. With the introduction of these extra points, it turns out that the "missing points" can be accounted for. The points in the complex projective plane can be described by projective coordinates (x:y:z), which are not all zero, by requiring that (x:y:z) and (ux:uy:uz) represent the same point for any nonzero number u. The points (x:y:1) then correspond to the point (x,y) in Cartesian coordinates, and the points of the form (x:y:0) give the points at infinity.
Parallel to the study of algebraic curves was the study of algebraic functions, due to N. Abel, K. Jacobi, and, most importantly, B. Riemann with his theory of Riemann surfaces. This work provided an intrinsic theory of algebraic curves and later was used to study their geometric properties.
The intensive study of algebraic surfaces came much later, toward the end of the nineteenth century, first with M. Noether and then with the Italian school of Algebraic Geometry, some of whose most prominent members were C. Segre, F. Enriques, G. Castelnuovo, and F. Severi. They developed the theory of algebraic surfaces and higher dimensional varieties. One of their major achievements was Enriques's classification of algebraic surfaces. The members of the Italian school were criticized, however, for being imprecise, in that their intuition was not backed up by rigorous mathematical proof.
In the twentieth century there was a push for developing solid foundations for Algebraic Geometry and for fully justifying the results of the Italian school. This work was carried out initially by B. L. van der Waerden, A. Weil, O. Zariski, and others, building upon the Abstract Algebra developed by E. Artin and E. Noether. Meanwhile, there were also new results, such as H. Hironaka's resolution of singularities, a process for smoothing out algebraic varieties. A major event occurred in the fifties when A. Grothendieck unified Number Theory and Algebraic Geometry with his theory of schemes. He also showed how to use methods from Topology, such as cohomology groups, in an algebraic setting. These novel techniques solved many problems and, at the same time, provided a general unified foundation for the field.
In the late twentieth century, some of the main developments were the work of S. Mori and others on the classification of three-dimensional algebraic varieties and the introduction of methods derived from String Theory, an area of Physics.
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