Algebraic Geometry - Computer Graphics and Geometric Modeling - page 800

10.18.7. Theorem. Let V be a variety of pure dimension d in C n or P n ( C ). Almost
all transforms of a given (n - d)-dimensional affine or projective plane, respectively,
intersect V in a common fixed number s of distinct points. If V is a hypersurface with
minimal polynomial f, then s is the degree of f.

Proof. See [Kend77].

Note. The set of linear transformations forms a manifold and the expression “almost
all transforms” in the last and next several theorems means all transforms except
possibly those on a proper lower-dimensional submanifold (a set of measure
zero). The planes for which the theorems hold are those that intersect the variety
in question transversally. The planes for which the theorems do not hold is a set
of measure zero. Another way of looking at the expression “almost all transforms” is
via Grassmann manifolds and have it mean “for all n-dimensional linear subspaces in
an open dense subset of the Grassmann variety with respect to the Zariski topology.”

Definition. If V is any variety of pure dimension d in C n or P n ( C ), then the number
s in Theorem 10.18.7 is called the degree of V and is denoted by deg V .

Note that the second part of Theorem 10.18.7 implies that the new definition
of the degree of a variety V agrees with the definition in Section 10.5 when V is a
hypersurface.
The degree is a global property. There is a local version of the degree that says
that near a point the number of points in the intersection with a linear subspace does
not change. We shall describe this also. It is the analog of the multiplicity for plane
curves.

10.18.8. Theorem. Let p be a point of a pure r-dimensional variety V in C n or P n ( C )
and let L be any (n - r)-dimensional affine or projective plane in C n or P n ( C ), respec-
tively, that intersects V properly at p . Then for almost all transforms L ¢ of L suffi-
ciently close to L , V « L ¢ consists of a common fixed number d of points in an
arbitrarily small neighborhood of p .

Proof. See [Kend77].

Definition. If p is a point of a pure r-dimensional variety V in C n or P n ( C ), then the
number d in Theorem 10.18.8 is called the multiplicity of intersection of V and L at p
and is denoted by i( V , L ; p ).

10.18.9. Theorem. Let p be a point of a pure r-dimensional variety V in C n or P n ( C ).
For almost all transforms L ¢ of any (n - r)-dimensional affine or projective plane L in
C n or P n ( C ), respectively, with both L ¢ and L containing p , the number i( V , L ¢; p ) is
defined and equal to a common fixed number d. If V is a hypersurface with minimal
polynomial f, then d is the order of f at p .

Proof. See [Kend77]. Recall that the order of an arbitrary polynomial at a
point is the smallest degree of all the monomials appearing in an expansion
about p .