Chapter 7 Local properties of plane algebraic curves - RISC

For analyzing a singular point P on a curve C we need to know its multiplicity but
also the multiplicities of the tangents. If all the r tangents at the r-fold point P are
different, then this singularity is of well-behaved type. For instance, when we trace the
curve through P we can simply follow the tangent and then approximate back onto
the curve. This is not possible any more when some of the tangents are the same.
Definition 7.1.4. A singular point P of multiplicity r on an affine plane curve C is
called ordinary iff the r tangents to C at P are distinct, and non-ordinary otherwise.
non-ordinary. An ordinary double point is a node.
Example 7.1.2. We consider the following curves in A 2 (C). Pictures of such curves
can be helpful, but we have to keep in mind that we can only plot the real components
of such curves, i.e. the components in A 2 (R). Interesting phenomena, such as singular
points etc., might not be visible in the real plane.
A : a(x, y) = y−x 2 = 0 B : b(x, y) = y 2 −x 3 +x = 0 C : c(x, y) = y 2 −x 3
D : d(x, y) = y 2 −x 3 −x 2 = 0 E : e(x, y) = (x 2 +y 2 ) 2 +3x 2 y−y 3 F : f(x, y) = (x 2 +y 2 ) 3 −4x 2 y 2
The curves A, B are non-singular. The only singular point on C, D, E and F is the
origin (0, 0).
The linear forms in the equations for A and B define the tangents to these curves
at the origin.
The forms of lowest degree in C, D, E and F are y 2 , y 2 −x 2 = (y −x)(y +x), 3x 2 y −
y 3 = y( √ 3x − y)( √ 3x + y) and −4x 2 y 2 . The factors of these forms of lowest degree
determine the tangents to these curves at the origin.
Intersecting F by the line x = y we get the point P = ( √ 1 1
2
, √2 ).
∂f
∂x (P) = (6x5 + 12x 3 y 2 + 6xy 4 − 8xy 2 )(P) = √ 2,
∂f
∂y (P) = √ 2.
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