Example: Plane Curve
For example, suppose given a plane curve C defined by a polynomial equation
- F(X,Y) = 0
and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get a 'linearised' equation reading
- L(X,Y) = 0
in which all terms XaYb have been discarded if a + b > 1.
We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.)
It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2.
Read more about this topic: Zariski Tangent Space
Famous quotes containing the words plane and/or curve:
“As for the dispute about solitude and society, any comparison is impertinent. It is an idling down on the plane at the base of a mountain, instead of climbing steadily to its top.”
—Henry David Thoreau (18171862)
“The years-heired feature that can
In curve and voice and eye
Despise the human span
Of durancethat is I;
The eternal thing in man,
That heeds no call to die.”
—Thomas Hardy (18401928)