Let be a purely inseparable extension of degree . Let be a smooth curve in and let be a -point of . Prove that the divisor can be defined by polynomials in .

*Proof.* Let vanish at with multiplicity one. Then, vanishes at with multiplicity .

Let be a purely inseparable extension of degree . Let be a smooth curve in and let be a -point of . Prove that the divisor can be defined by polynomials in .

*Proof.* Let vanish at with multiplicity one. Then, vanishes at with multiplicity .

Let be a finite Galois extension. The Galois group acts on coordinatewise. Let be a closed algebraic set of . Prove that the following are equivalent:

- the set can be defined by polynomials in ;
- the set is invariant under the -action.

*Proof.* Suppose vanishes on . Let , and set . If is defined by polynomials in then and so is Galois-invariant.

Conversely, if is Galois-invariant, then if vanishes on then does also. Denoting to be the elementary symmetric polynomials of for , each and they vanish on . We claim that these polynomials define . For, if every vanished at a point , then every is zero for they are the roots of the polynomial . So the define .

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 .

Let be the image of the 2-uple embedding of in . This is the *Veronese surface*. If is a closed curve (a *curve* is a variety of dimension 1), show that there exists a hypersurface such that .

*Proof.* The Veronese embedding is defined by , . If is defined by , then , and so defines a hypersurface . Thus .

*The* -Uple *Embedding*. For given , let be all the monomials of degree in the variables , where . We define a mapping by sending the point to the point obtained by substituting the in the monomials . This is called the -uple *embedding* of in . For example, if , then , and the image of the 2-uple embedding of in is a conic.

- Let be the homomorphism defined by sending to , and let be the kernel of . Then is a homogeneous prime ideal, and so is a projective variety in .
- Show that the image of is exactly . (One inclusion is easy. The other will require some calculation).
- Now show that is a homeomorphism of onto the projective variety .
- Show that the twisted cubic curve in (Ex. 2.9) is equal to the 3-uple embedding of in , for suitable choice of coordinates.

*Proof of a.* is homogeneous since it is generated by homogeneous elements in , for each is sent to an element of degree . is prime since maps onto a subring of , which is an integral domain, so is prime by the first isomorphism theorem.

*Proof of b.* , so . Conversely, . If , then it is identically zero for all , so , i.e. .

*Proof of c.* By *b* it suffices to show that the map is continuous with continuous inverse.

*Proof of d.* maps , which is the twisted cubic curve by Ex. 2.9*b*.

*Linear Varieties in .* A hypersurface defined by a linear polynomial is called a *hyperplane*.

- Show that the following two conditions are equivalent for a variety in :
- can be generated by linear polynomials.
- can be written as an intersection of hyperplanes.

In this case we say that is a

*linear variety*in . - If is a linear variety of dimension in , show that is minimally generated by linear polynomials.
- Let be linear varieties in , with . If , then . Furthermore, if , then is a linear variety of dimension .

*Proof of a.* for linear implies for hyperplanes. If , then , where we note each is a linear polynomial. Note we can drop the radical since our generators are linear.

*Proof of b.* Suppose can be generated by , i.e., $\dim S(Y) > r+1$. By Problem this implies $\dim Y > r$, a contradiction.

*Proof of c.* Consider their respective affine cones ; they have dimension respectively by Problem 2.10. Then, their intersection in has dimension by the hypothesis . Since , we finally have that by Problem 2.10. is linear by *a*.