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50 2. Plane Algebraic Curves

Exercises
d
1. Show that the dimension of H is the binomial coefficient n+d − 1.
n
n
2. If P 1 , P 2 ,and P 3 are three points in P 2 not on a line, then show that the set of conics
through P 1 , P 2 ,and P 3 form a two-dimensional subspace of the five-dimensional space
2
H of conics in P 2 . Show that the set S of conics through 1 : 0 : 0, 0 : 1 : 0, and 0 : 0 : 1
2
can be parametrized by the coefficiens a : b : c of the equations axy + bwy + cwx = 0.

Describe the subfamilies of S consisting of conics which also go through w : x : y and

which also go through two distinct points w : x : y and w : x : y .

§3. Elements of Intersection Theory for Plane Curves
The following result gives some indication of the number of intersection points be-
tween two plane curves.
(3.1) Proposition. Let C f and C g be two plane algebraic curves of degrees m and
n, respectively, defined over k. If for some extension K of k the set C f (K) ∩ C g (K)
has strictly more than mn points, then C f and C g have an entire curve in common.
Proof. Suppose C f (K) ∩ C g (K) contains mn + 1 points. Join these points by lines
and with a projective tranformation move the two curves so that the point (0, 0, 1) is
not on any of these lines. Decompose the polynomials f and g with respect to the
variable y:
m
f (w, x, y) = a 0 y + a 1 y m−1 + ··· + a m ,
m
g(w, x, y) = b 0 y + b 1 y n−1 +· · · + b n ,
where a i (w, x) and b j (w, x) are homogeneous of degrees i and j, respectively. By
(4.3) of the Appendix the resultant R( f, g)(w, x) is homogeneous of degree mn.
Moreover, R( f, g)(w, x) = 0 for w, x ∈ K if and only if there exists y in K such
that w : x : y ∈ C f (C) ∩ C g (K). Since there exists mn + 1 points (w i , x i , y i ) in
C f (K) ∩ C g (K) for i = 0,... , mn, it follows that the polynomial

(x i w − w i x)
0≤i≤mn
of degree mn + 1 divides the polynomial R( f, g)(w, x) of degree mn. From this
we deduce that R( f, g) = 0, and, hence, f and g have a common factor h. Then
C h ⊂ C f ∩ C g , and this proves the theorem.
The above theorem is a corollary of Bezout’s theorem which says that over an
algebraically closed field the intersection of a plane curve of degree m with a plane
curve of degree n will have exactly mn points in common when the intersection mul-
tiplicity is assigned to each intersection point, or the two curves will have a common
subcurve. This is a more difficult result and requires both an analysis of singular
points and a good definition of intersection multiplicity.
d
In (2.5) we described the space H (k) of hypersurfaces of degree d over k in
n
P n . For two plane curves C f and C f of degree d over k we can speak of the line