# Easy to state high-dimensional consequences of Bezout theorem

A classical consequence of Bezout's theorem for plane curves is 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?...)?

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, the course is very close to some bits of Harris book "algebraic geometry first course", and covers some bits of it)

"The roots of $f(x)$ correspond to the points at which the zero set of the polynomial $y- f(x)$ and the zero set of the polynomial $y$ intersect." -- Stephanie Fitchett "Bezout's theorem: a taste of algebraic geometry"
The curves $f(x)$ and $y=0$ are replaced by arbitrary curves in projective space.
• I see. I thought you might consider a polynomial of degree $n$ as a higher dimensional object (vector space of). Feb 11, 2013 at 16:54