Nonlinear Equations - UFRJ

[SEC. 2.4: GROUP ACTION AND NORMALIZATION 21
3. Let E the hyperplane y n = 0. The canonical projection π :
k n → E maps the zero-set of (f) onto E.
4. Furthermore, (y 1 , . . . , y n−1 ) has exactly d preimages by π in the
zero-set of f if and only if
.
Discr yn f(y 1 , . . . , y n−1 , y n ) ≠ 0
Again, when f is irreducible, for L in a Zariski-open set, the
polynomial in item 5 is not uniformly zero. Hence, we may say that
for f irreducible, a generic line intersects the zero-set of f in exactly
d points.
Proof of Proposition 2.14. The coefficient of y d n in (f ◦L)(y) is a polynomial
in the coefficients of L. We will show that this polynomial is
not uniformly zero. Then, for generic L, it suffices to multiply f by
a non-zero constant to recover the situation of Proposition 2.11. The
other items of this Proposition follow immediately.
Let f = F 0 + · · · + F d where each F i is homogeneous of degree d.
The field k is algebraically closed, hence infinite, so there are
α 1 , · · · , α d−1 so that F d (α 1 , · · · , α d−1 , 1) ≠ 0. Then there is L ∈ G
that takes e n into c[α 1 , · · · , α n−1 , 1] for c ≠ 0.
Then up to a non-zero multiplicative constant,
f ◦ L = x d n + (low order terms in x n )
We may extend the construction above to quotient by arbitrary
ideals. Let J be an ideal in k[x 1 , . . . , x n ]. Then the quotient A =
k[x 1 , . . . , x n ]/J is finitely generated. (For instance, by the cosets
x i + J).
We say that an ideal p of a ring R is prime if and only if, for all
f, g ∈ R with fg ∈ p, f ∈ p or g ∈ p.
Given an ideal J, let Z(J) = {x ∈ k n : f(x) = 0∀f ∈ J} denote
its zero-set.
Lemma 2.15 (Noether’s normalization). Let k be an algebraically
closed field, and let A ≠ {0} be a finitely generated k-algebra. Then: