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.