This post is about Hilbert Polynomials and how they can be computed using free resolutions, and is based on a talk I gave on 7/4. There is a pdf version: Free Resolutions and Hilbert Polynomials. For more detail and historical background, see [Eis05, Ch. 1] (this was also used as the basis for this talk); for general facts about commutative algebra that are used, see [AM69, Eis95].
The Segre Embedding. Let be the map defined by sending the ordered pair to in lexicographic order, where . Note that is well-defined and injective. It is called the Segre embedding. Show that the image of is a subvariety of .
Proof. Let have coordinates , and let where . Then, . We claim . But any has coordinates , and clearly any vanishes on , and so . Conversely, note that any point satisfies , and if , say, then setting gives maps to .