Chapter 7 Local properties of plane algebraic curves - RISC

F 0 and F 1 determine a 1–dimensional linear subsystem of S 2 . The defining equation of
a general curve in this subsystem is of the form
or
λ 0 F 0 + λ 1 F 1 = 0,
(λ 0 − λ 1 )x 2 + λ 0 y 2 − λ 0 z 2 + λ 1 yz = 0.
We get a linear condition by requiring that the point (1 : 2 : 1) should be a point on
any curve. This forces λ 1 = −4λ 0 . The following curve (λ 0 = 1, λ 1 = −4) satisfies the
condition:
5x 2 + y 2 − z 2 − 4yz = 0.
It is very common for linear conditions to arise from requirements such as “all
curves in the subsystem should contain the point P as a point of multiplicity at least
r, r ≥ 1.” This means that all the partial derivatives of order < r of the defining
equation have to vanish at P. There are exactly r(r + 1)/2 such derivations, and they
are homegeneous linear polynomials in the coefficients a ijk . Thus, such a requirement
induces r(r + 1)/2 linear conditions on the curves of the subsystem.
Definition 7.3.2. A point P of multiplicity ≥ r on all the curves in a linear system is
called a base point of multiplicity r of the system. In particular, the linear subsystem of
S n which has the different points P 1 , . . .,P m as base points of multiplicity 1, is denoted
by S n (P 1 , . . .,P m ).
Definition 7.3.3. A divisor is a formal expression of the type
m∑
r i P i ,
i=1
where r i ∈ Z, and the P i are different points in P 2 (K). If all integers r i are non-negative
we say that the divisor is effective or positive.
We define the linear system of curves of degree d generated by the effective divisor
D = r 1 P 1 + · · · + r m P m as the set of all curves C of degree d such that mult Pi (C) ≥ r i ,
for i = 1, . . ., m, and we denote it by H(d, D).
Example 7.3.2. We compute the linear system of quintics generated by the effective
divisor D = 3P 1 +2 P 2 +P 3 , where P 1 = (0 : 0 : 1), P 2 = (0 : 1 : 1), and P 3 = (1 : 1 : 1).
For this purpose, we consider the generic form of degree 5:
H(x, y, z) = a 0 z 5 + a 1 yz 4 + a 2 y 2 z 3 + a 3 y 3 z 2 + a 4 y 4 z + a 5 y 5 + a 6 xz 4 + a 7 xyz 3
+a 8 xy 2 z 2 + a 9 xy 3 z + a 10 xy 4 + a 11 x 2 z 3 + a 12 x 2 yz 2 + a 13 x 2 y 2 z
+a 14 x 2 y 3 + a 15 x 3 z 2 + a 16 x 3 yz + a 17 x 3 y 2 + a 18 x 4 z + a 19 x 4 y + a 20 x 5
The linear conditions that we have to impose are:
∂ 2 H
∂x 2−i−j ∂y i ∂z j (P 1) = 0, i + j ≤ 2,
∂H
∂x (P 2) = ∂H
∂y (P 2) = ∂H
∂z (P 2) = 0, H(P 3 ) = 0
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