Algebraic Geometry - Computer Graphics and Geometric Modeling - page 702

10.5.14. Theorem. Let X be any hypersurface in P n (k) and let p be any point in X .
We may always choose a coordinate system, so that the plane at infinity relative to
that coordinate system is neither a component of X nor does it contain p .

Proof. Over C , there are only a finite number of planar components of X . Therefore,
we can choose a hyperplane Y which is none of these and which does not contain the
point p either. By Theorem 10.3.2 we now define a coordinate system with Y its plane
at infinity.

What Theorems 10.3.6 and 10.5.14 allow us to do is that if we want to analyze a
projective variety in P n ( C ) in the neighborhood of a point we may always assume that
the point and a neighborhood of it always lie in C n .

10.6 Singularities and Tangents of Plane Curves

One thing we would like to have stand out in the course of reading this chapter is the
constant interplay between algebra and geometry. Seeing the geometry is especially
useful when the subject matter gets very abstract, which tends to be the case in alge-
braic geometry. We are starting to get to some very important concepts in algebraic
geometry that one can arrive at in different ways. By and large, insofar as it is possi-
ble, we emphasize a geometric approach to minimize the amount of algebraic back-
ground required of the reader. Nevertheless there are unavoidable technical aspects
to definitions and theorems if we want to state things precisely; therefore, let us start
with a brief overview of how our particular approach is motivated by some intimate
connections between algebra and geometry.
Let C be an affine plane curve and f(X,Y) its minimal polynomial. In the last
section we defined the degree of the curve C to be the degree of f(X,Y). The degree
of a polynomial is a well-defined standard algebraic invariant associated to a poly-
nomial. Here is a geometric definition of this invariant of the curve. Let

xx =+ l t

0

yy =+ m t

0

be the parametric equations for an arbitrary line L through a point p = (x 0 ,y 0 ) on the
curve. If f(X,Y) has degree d, then

() = ( )

gt fx + l ty , + m t

0 0

is a polynomial in t of degree e £ d (e could be less than d). The roots of the poly-
nomial g correspond to intersections of the line L with the curve C . A geometric
interpretation of the degree of a plane curve is then that it is the maximum number
of points that a line can intersect the curve.
How can one define tangent lines to the curve C at p ? Suppose that

() =

k k + 1 e

gt c t + c t + ... + c t ,

k k + 1 e