Math 8205
Spring 1998

Hint for Part d. of Exercise 1

Let X be the closure of U in Aⁿ, and let Y be the closure of (U) in A^n-1. It may help to replace Y by a distinguished open subset Y_g for some g A(Y). One would also want to replace X by ^-1(Y_g) X = X_g . (This notation is not literally correct, of course, but the correct interpretation is fairly clear.) By doing this, i.e. replacing X and Y with X_g Aⁿ⁺¹ and Y_g Aⁿ respectively, we are still working with a linear projection. {Recall from Fall Quarter that if g is a polynomial in n variables, then the distinguished open subset U_g Aⁿ⁺¹ is isomorphically to a closed subvariety of Aⁿ⁺¹ ... }

This "new" projection has the additional property that X is mapped onto Y.

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