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. 2.9b.
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.
Projective Closure of an Affine Variety. If is an affine variety, we identify with an open set by the homeomorphism . Then we can speak of , the closure of in , which is called the projective closure of .
- Show that is the ideal generated by , using the notation of the proof of (2.2).
- Let be the twisted cubic of (Ex. 1.2). Its projective closure is called the twisted cubic curve in . Find generators for and , and use this example to show that if generate , then do not necessarily generate .
Proof of a. Suppose . Then vanishes on , so . Since , we have that . Similarly, if , then it is the sum of polynomials of the form , each one of which is in after homogenization, and so .
Proof of b. Recall . Now , and so , but this is not equal to .