The Cone Over a Projective Variety. Let be a nonempty algebraic set, and let be the map which sends the point with affine coordinates to the point with homogeneous coordinates . We define the affine cone over to be
- Show that is an algebraic set in , whose ideal is equal to , considered as an ordinary ideal in .
- is irreducible if and only if is.
Sometimes we consider the projective closure of in . This is called the projective cone over .
Proof of a. since consists of all scalar multiples of points in , i.e., all points in that are zeros of the generators of .
Proof of b. is irreducible if and only is prime if and only if is prime (by a) if and only if is irreducible.
Proof of c. by Problem 2.6.