By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal's_theorem .
I am curious if there are some other statements that you find pretty that can be formulated (almost) as elementarily as Pascal's theorem and proven using higher dimensional Bezout's theorem? For example, is there some statement that involves quadrics, planes and lines (cubics?…)?
Motivation. I ask this question since I want to finish to teach my (introductory) course in algebraic geometry by higher-dimensional Bezout theorem (using Hilbert polynomials, ect), and I would be extremely happy to give some pretty application :). To give you an idea of the level of the course, it is based on some bits of Harris book "Algebraic geometry first course",
Disclaimer. I don't doubt the usefulness of Bezout theorem and am sorry if the original question sounded like I doubt it. On the contrary I based the elementary course in algebraic geometry that I teach on this theorem. Namely, the course starts with Bezout for plane curves (using resultants), intorduces projective spaces and varieties, goes through Hilbert basis theorem and Hylbert polynomials (last section of Atiyah-Macdonald) and then as an applications we get a proof of a simplest version of Bezout's theorem in high dimension.
Also, It would be difficult for me to explain what I mean by pretty in math (for myself)
but still I feel that the using of this word is justified, because we, mathematicians use this word… Sometimes we disagree on what is pretty, but personally I find pretty huge amount of facts in algebraic geometry. In other words I will be happy to see any application that can be stated in the language on the level of my course.
In the comment I put the link to the question on stackexchange
Best Answer
Here is an excerpt from my class notes, inspired by the discussion in Joe Harris' book on desingularizing curves as an application of Bezout in space.
The degree of a curve is the number of intersections with a general hyperplane. The projection of a spanning curve still spans. The degree of a rational map ƒ:D--->E of curves, is the common number of preimages of most points of the image curve E, and in characteristic zero, a map of geometric degree one is birational.
Strong Bezout Theorem in space: If E is an irreducible curve of degree d spanning P^n, and p is a singular point of E, then a general hyperplane H of Pn passing through p, meets E away from p in at most d-2 points.
Corollary: If an irreducible curve E of degree d spanning P^n is projected into P^(n-1) from a singular point of E, then the product of the degree of the projected curve by the degree of the projection map on E, is ≤ d-2.
Corollary: An irreducible curve in P^n of degree < n cannot span P^n. Proof: Otherwise can project down eventually to a line spanning P^m, with m ≥ 2, contradiction.
Corollary: The projection map from a point of itself, of an irreducible curve of degree < 2n from P^n to P^(n-1), is birational.
Theorem. If D is any irreducible plane curve of degree ≥ 3, then D is isomorphic to a curve in P^n of degree d < 2n, and any such curve is birational to a non singular curve. Proof: The Veronese map :D-->P^N by homogeneous polynomials of degree d-2, where N = [d(d-1)/2] - 1, is an embedding of D as a curve spanning P^N and having degree d(d-2). But 2N = > d(d-2), precisely when d > 2, which holds by hypothesis. If the re - embedded curve D has a singular point, we project from it and lower the degree by at least 2, and the dimension of the ambient space by exactly one. We continue as long as the curve has singularities. Since the projected curve always spans, the degree can never drop below the dimension of the space, so the projection map always has degree one on the curve, i.e. the projected curve is always birational to the original curve. As long as there are singularities, the degree goes down at least twice as fast as the dimension, so the projected curve always has degree less than twice the dimension of the ambient space. Eventually then we either get to a point where the projected curve has no more singularities, or we get to a singular curve of degree 3 spanning P^2, and one more projection is birational to P^1. QED.