Chapter 7 Local properties of plane algebraic curves - RISC

Solving them one gets that the linear system is defined by:
H(x, y, z) = a 3 y 3 z 2 − 2a 3 y 4 z + a 3 y 5 + (−a 9 − a 10 ) xy 2 z 2 + a 9 xy 3 z + a 10 xy 4 +
(−a 13 − a 14 − a 15 − a 16 − a 17 − a 18 − a 19 − a 20 ) 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
4
3
y2
1
–4 –3 –2 –1 0 1 2 3 4
x
–1
–2
–3
–4
4
y
2
–4 –2 0 2 x 4
–2
–4
Figure 7.4: real part of C 1⋆,z (left),
real part of C 2⋆,z (right)
Hence, the dimension of the system is 10. Finally, we take two particular curves in
the system, C 1 and C 2 , defined by the polynomials:
H 1 (x, y, z) = 3 y 3 z 2 − 6 y 4 z + 3 y 5 − xy 3 z + xy 4 − 5 x 2 y z 2 + 2 x 2 y 2 z + x 3 y 2
+x 4 z + y x 4 ,
H 2 (x, y, z) = y 3 z 2 − 2 y 4 z + y 5 − 8 3 z2 xy 2 + 3 xy 3 z − 1 3 xy4 − 8 x 2 y z 2 + 2 x 2 y 2 z
+y 3 x 2 + 2 y x 3 z + x 3 y 2 − x 4 z + 2 y x 4 + x 5 ,
respectively. In the figure the real part of the affine curves C 1⋆,z and C 2⋆,z are plotted,
respectively.
Theorem 7.3.1. If two curves F 1 , F 2 ∈ S n have n 2 points in common, and exactly
mn of these lie on an irreducible curve of degree m, then the other n(n − m) common
points lie on a curve of degree n − m.
Proof: Let G be an irreducible curve of degree m containing exactly mn of the n 2
common points of F 1 and F 2 . We choose an additional point P on G and determine a
curve F in the system
λ 1 F 1 + λ 2 F 2 = 0
containing P. So F = λ 1 F 1 +λ 2 F 2 , for some λ 1 , λ 2 ∈ K. G and F have at least mn+1
points in common, so by Theorem 7.2.2 (Bézout’s Theorem) they must have a common
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