![]() ![]() This is because of several results, like Lefschetz's principle by which doing (algebraic) geometry over an algebraically closed field of characteristic $0$ is essentially equivalent to doing it over the complex numbers furthermore, Chow's theorem guarantees that all projective complex manifolds are actually algebraic, meaning that differential geometry deals with the same objects as algebraic geometry in that case, i.e. So the study of algebraic geometry in the applied and computational sense is fundamental for the rest of geometry.įrom a pure mathematics perspective, the case of projective complex algebraic geometry is of central importance. So when any software plots a transcendental surface (or manifold), it is actually displaying a polynomial approximation (an algebraic variety). by truncating the series), which is actually what calculators and computers do when computing trigonometric functions for example. all curves intersect at least at a point, giving the beautiful Bézout's theorem.įrom a purely practical point of view, one has to realize that all other analytic non-polynomial functions can be approximated by polynomials (e.g. Besides, working within projective varieties, enlarging our ambient space with the points at infinity, also helps since then we are dealing with topologically compact objects and pathological results disappear, e.g. Working over the complex numbers is actually more natural, as it is the algebraic closure of the reals and so it simplifies a lot the study tying together the whole subject, thanks to elementary things like the fundamental theorem of algebra and the Hilbert Nullstellensatz. Thus the basic plane curves over the real numbers can be studied by the algebraic properties of polynomials. Equations of second order turned out to comprise all the classical conic sections in fact the conics classification in the affine, Euclidean and projective cases (over the real and complex numbers) is the first actual algebraic geometry problem that every student is introduced to: the classification of all possible canonical forms of polynomials of degree 2 (either under affine transformations or isometries in variables $(x,y)$, or projective transformations in homogeneous variables $$). linear polynomials, are the straight lines, planes, linear subspaces and hyperplanes. The most basic equations one could imagine to start studying were polynomials on the coordinates of your plane or space, or in a number field in general, as they are the most basic constructions from the elementary arithmetic operations. ![]() The main motivation started with Pierre de Fermat and René Descartes who realized that to study geometry one could work with algebraic equations instead of drawings and pictures (which is now fundamental to work with higher dimensional objects, since intuition fails there). ![]() Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct or subtle way. Where you can really start at a slow pace (following his undergraduate textbook) to get up to the surface classification theorem.
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