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§3. Elements of Intersection Theory for Plane Curves 51

d
determined by these two points in H (k) and describe it with projective coordinates
2

a : a ∈ P 1 (k) as the set of curves of the form aC f + a C f = C af +a f . This

one-dimensional family is called a pencil of curves of degree d over k. Classically
two-dimensional families are called nets and three-dimensional families are called
webs.
For example, two distinct lines L and L have a point P in common and the pencil

aL + a L where a : a ∈ P 1 (k) is the pencil of all lines through P and defined over
1
k. Every line in the dual projective space P = H is a pencil, and it is determined

2 2
by the unique point in P 2 contained in all members of the pencil.

As for intersection properties of conics, we consider two conics C and C with
three intersection points on the line L. By (3.1) the conics have the line L in common

and C = L ∪ M, C = L ∪ M , where M and M are lines. The pencil aC + a C
becomes L ∪ (aM + a M ) which is effectively a pencil of lines.

(3.2) Proposition. Let C and C be two conics with exactly four distinct points,
P 1 , P 2 , P 3 , and P 4 in common, all defined over an infinite field k. Then any other

conic C through P 1 , P 2 , P 3 , and P 4 is the form aC + a C .
Proof. Observe that no three of P 1 , P 2 , P 3 ,and P 4 lie on a line by the previous

argument. Since k is infinite, there is a fifty point P on C distinct from the P 1
which can be taken to the intersection point if C is the union of two lines. Choose

a : a such that P is on the conic aC +a C . Thus C and aC +a C have five points

in common. By (3.1) the equation of C , even if it is reducible, divides the equation

of aC + a C so that C = aC + a C . This proves the proposition.

This previous proposition has a version for cubics which is basic for the proof
that the group law on the nonsingular cubic satisfies the associative law.

(3.3) Theorem. Let D and D be two cubic curves intersecting at exactly nine points

in P 2 (k) all defined over an infinite field k. If D is a plane cubic curve through eight

of the intersection points, then it goes through the ninth and has the form D =

aD + a D .
Proof. First, observe that no four of the nine intersection points lie on a line, for

otherwise the line would be a common component of D and D by (3.1). No seven
of the nine intersection points lie on a conic, for otherwise a component of the conic
would be common to both D and D by (3.1). In either case the existence of such

a common component would contradict the fact that there are exactly nine points of

D ∩ D .

If D is not of the form aD+a D , then aD+a D +a D is a two-dimensional

family of cubics, and for any pair of distinct points, P and P in the projective plane,

we can find a : a : a in P 2 (k) such that aD + a D + a D goes through these
points. Now we will refine the statements in the previous paragraph.
Suppose that P 1 , P 2 ,and P 3 are intersection points which lie on a line L. Choose
P on L, and choose P off L and off the conic C through P 4 , P 5 , P 6 , P 7 ,and

P 8 . Then the cubic aD + a D + a D going through P , P , and the eight points

P 1 ,... , P 8 has L and C as components by (3.1). This contradicts the choice of P
off of L and C. Hence no three intersection points lie on a line.