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.