Chapter 7 Local properties of plane algebraic curves - RISC

factor D(z) intersection point multipl. of intersection
x 4 (2z − 1) 2 P 1 = (0 : 2 : 1) 4
y z + 1/3 P 2 = (−3 : 0 : 1) 1
3x + y z − 1 P 3 = (1 : −3 : 1) 1
x + 2y z + 1 P 4 = (−2 : 1 : 1) 1
x + y z + 1 P 5 = (−1 : 1 : 1) 1
x − y z − 1 P 6 = (1 : 1 : 1) 1
Table 7.2.
4
y
2
–4 –2 0 2 x 4
–2
–4
Figure 7.3: Joint picture of the real parts of C ⋆,z and D ⋆,z
Furthermore, since (0 : 0 : 1) is not on the cubics nor on any line connecting their
intersection points, the multiplicity of intersection is 4 for P 1 , and 1 for the other
points. It is also interesting to observe that P 1 is a double point on each cubic (compare
Theorem 7.2.3. (5)).
Some authors introduce the notion of multiplicity of intersection axiomatically (see,
for instance, [Ful69], Sect. 3.3). The following theorem shows that these axioms are
satified for our definition.
Theorem 7.2.3. Let C and D be two projective plane curves, without common components,
defined by the polynomials F and G, respectively, and let P ∈ P 2 (K). Then
the following holds:
(1) mult P (C, D) ∈ N.
(2) mult P (C, D) = 0 if and only if P ∉ C ∩ D. Furthermore, mult P (C, D) depends
only on those components of C and D containing P.
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