Prehistoric times (about 400 BC-1630). d)，
Exploration stage (exploration, 1630- 1795),
The golden age of projective geometry (1795- 1850),
The era of riemann sum's double rational geometry (1850- 1866),
Development and chaos period (1866- 1920),
The period when new structures and new ideas appeared (1920- 1950),
The last stage, the most glorious period in the history of algebraic geometry, is the era of layers and schemas (1950-).
The object of algebraic geometry is the algebraic curve in Euclidean plane, that is, the trajectory defined by polynomial P(x, y)=0, such as the simplest plane algebraic curve-straight line and circle. Since ancient Greece, people have begun to study conic curves and some simple cubic and quartic algebraic curves. From the foregoing, it can be seen that the study of the common zero set of algebraic equations is inseparable from the coordinate representation, so the real study has to start from the coordinate representation of geometric figures created by Descartes and Fermat, but this has already happened in the17th century. Analytic geometry has quite complete results for algebraic curves and surfaces. Starting from Newton, cubic algebraic curves are classified and 72 classes are obtained.
Since then, the classification problem has become an important problem in algebraic geometry, and these problems have become the driving force of a lot of research work. On the other hand, it is precisely because the classification of cubic or quartic algebraic curves is too complicated that it promotes the transition from analytic geometry to algebraic geometry, that is, classification and general theoretical research on a rougher level.
/kloc-the basic problem of AG (representing algebraic geometry, the same below) in the 0/8th century is the intersection of algebraic curves or algebraic surfaces, which is equivalent to the elimination problem in algebraic equations. The basic achievement of this period is Bezo Theorem (Bezhu Theorem):
Let x and y be two different curves in p 2, the degrees are d and e respectively, let x # y = {p _ 1, p _ 2, ... p _ s} be their intersections, and the number of intersections of each point is recorded as I(X, y; P_j), then
∑I(X，Y； P_j)=de .
With the rise of projective geometry in19th century, the method of projective geometry began to be used to study algebraic curves, in which infinite points and imaginary points were introduced. Algebraic curves were expressed by homogeneous polynomials and projective coordinates P (X_0, X_ 1, X_2)=0, and complex coordinates were allowed. 1834, German mathematician Prukl reached a conclusion about plane curve. This formula relates the algebraic characteristics and geometric characteristics of plane algebraic curves, such as degree and inflection point. In particular, it is proved that the ordinary cubic algebraic curve (that is, elliptic curve) has nine inflection points. In 1839, he also found that the quartic curve has 28 double tangents, of which at most 8 are real numbers.
The above is an overview of algebraic geometry in the first three stages.