Math 8203
Fall 1997

Comments about Exercise 8

  1. One reasonably effective way to accomplish the change of variables is the following:
    t1 = x1 + a1 xn, t2 = x2 + a2 xn,..., tn-1 = x2 + an-1 xn ;
    tn = an xn ,
    where a1,..., an are elements of K, with an \neq 0. The coefficient of tnd will then be:
    Fd (a1,..., an).
    Since K is infinite, we can find a point [a1,..., an-] \in Pn-1 where Fd (a1,..., an) is nonzero. Since K is algebraically closed, we can rescale the coefficients in this change of variables to make the coefficient of tnd be equal to 1.
  2. Suppose that we have changed variables so that f(x1,..., xn) is monic with respect to xn, and let P = (a1,..., an-1) \in An-1. Then the polynomial:
    g(x) = f(a1,..., an-1,x)
    has degree = d because of the fact that f is monic with respect to xn. Therefore g(x) has d roots, if they are counted with suitable multiplicities. It follows that at least one point (and at most d points) of X are mapped to P by the linear projection \pi: An -> An-1.
  3. This is similar to part (b) of Exercise 7.
  4. This is similar to part (c) of Exercise 7. In particular, use the hint which is linked to that exercise. Note that if we're willing to use the Nullstellensatz, then this problem becomes considerably easier. Indeed, by applying the Nullstellensatz, we see that I(X) is the principal ideal (f), which is a prime ideal because the polynomial ring is factorial and f is an irreducible element. Without the Nullstellensatz, it is not hard to prove this claim directly, but we must say something, for instance along the lines of the previously mentioned hint.

REMARK. Let \varphi: X -> An-1 be induced by the projection \pi: An -> An-1. The proof of part b shows that \varphi is surjective and also finite to one. Indeed, there are at most d points of X which map to each point P \in An-1. Since the points of An-1. depend in an obvious way on n-1 independent parameters, it makes sense to say that An-1. has dimension = n-1. Since \varphi is surjective and finite-to-one, this indicates that it could also make sense to say that the hypersurface X \subset has dimension = n-1.

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