Algebraic Geometry - Computer Graphics and Geometric Modeling - page 794

10.16.11. Theorem. Let V be an irreducible projective variety in P n (k) for an alge-
braically closed field k and assume that f Πk[X 1 ,X 2 ,...,X n+1 ] is a homogeneous poly-
nomial that does not vanish on V . If V f is the subvariety of V defined by the equation
f = 0, then dim V f = dim V - 1.

Proof. We follow the argument in [Shaf94]. By hypothesis, V f π V , so that dim V f <
dim V by Theorem 10.16.10. Set V 1 = V f and f 0 = f. Now pick a point in each irre-
ducible component of V 1 and let f 1 be a homogeneous polynomial that does not vanish
on any of these points. Let V 2 be the subvariety of one of the irreducible components
of V 1 defined by the condition that f 1 = 0. Continue on in this way to get a sequence
of irreducible varieties V i and homogeneous polynomials f i so that

VV V 2 …… … ...

1

and dim V i+1 < dim V i . Since the dimension of the varieties is decreasing there is a d,
d £ dim V , so that V d π f and V d+1 = f. By definition the polynomials f 0 , f 1 ,..., f d have
no common zeros and so we can define a map

d

()

j : VP Æ k

by

[ () = [

() () ()

]

j x f x , f x ,..., f d x .

0 1

By Theorem 10.13.37, j : V Æj( V ) is a finite map and by Theorem 10.13.33 j( V ) is a
subvariety of P d (k). It follows that dim V = dim j( V ) = d. Note also that j( V ) = P d (k)
by Theorem 10.16.10. Clearly we must have dim V i = dim V i+1 + 1. In particular, dim
V f = dim V - 1 and the theorem is proved.

Returning to the proof of Theorem 10.16.9, we first observe

(4) ¤ (5): This is easy because of the connection between ideals and varieties.

Therefore, to finish the proof of Theorem 10.16.9 we have the choice of proving that
(3) is equivalent to (4) or (5).

(3) ¤ (4): Theorem 10.16.10 clearly implies that there cannot be a strictly decreas-
ing sequence of irreducible varieties of longer length than d. Repeated application of
Theorem 10.16.11 shows that there is at least one of that length in the projective case.
In the affine case this follows from an affine version of Theorem 10.16.11. To finish
the proof we must show that every maximal such sequence has length d. For that it
suffices to show that if W 1 and W 2 are irreducible varieties with W 2 properly contained
in W 1 and if

dim W W < dim - 1

2 1

then there is an irreducible subvariety W ¢ of W 1 with the property that

WWW …¢ … and dim W ¢ = dim W - 1 .

1 2 1