Singularities are important in
differential geometryand
algebraic geometry.
Roughly speaking they are points at which a geometric object fails to
be smooth.
I'll give a precise definition and then a discussion with some examples.
Suppose that we have a subset
X of Cn
that is defined by the
simultaneous vanishing+ of m polynomials
f1(x1,...,xn),...,
fm(x1,...,xn).
Note that in one favourite model of the universe (as a Calabi-Yau manifold)
space locally has this form.
Now take a point x on X.
Form the Jacobian matrix J which is the
mxn matrix with i,j entry dfi / dxj
and evaluate at the point x, to obtain J(x).
Then the point x is a singular point iff J(x) has rank
strictly less than n - the dimension* of X.
A point which is not singular is smooth.
OK that's the definition, what does it mean? Consider the special
case of a hypersurface, that is we just have one defining equation
f=f1. In this case the Jacobian is just
a row (df/dx1 ... df/dxn) and X
has dimension n-1. Thus for a singularity at x
we are asking that the Jacobian matrix
should vanish identically at x, that is all the partial derivatives
of f have to vanish at that point.
For example, consider a quadratic cone with defining polynomial
x2 + y2 + z2. This has a sharp
looking point at the vertex of the cone at the origin but looks smooth
everywhere else. This intuition fits with the definition because
the Jacobian (2x 2y 2z) clearly vanishes at the vertex and nowhere else.
What we are seeing here illustrates a general principle, at most points
of X the Jacobian matrix will have the correct rank
n - the dimension of X and be smooth, and only at
the smaller dimensional set defined by the vanishing of the appropriate
minors of the Jacobian can it be singular.
Here's another example, consider the plane algebraic curve defined by
y2-x3, (the cuspidal cubic curve). If you
sketch this curve then you'll see it is symmetric about the x-axis.
In the first quadrant as it moves away from the origin y
is growing much faster than x. Thus the curve has a pronounced
sharp point at the origin, which is indeed a singularity. At all other
points of the curve you can draw a well-defined tangent line but at the origin
this tangent line is not well-defined, there is a two-dimensional space
in which we could draw tangents. In general, a singularity occurs where
the tangent space is bigger than the dimension of the space under
consideration.
+ Strictly speaking when I talk about X being defined by
the simultaneous vanishing of f1,...,fm what
I meam is that when I consider the ideal in the polynomial ring
C[x1,...,xn] consisting of
all polynomials which vanish on X then this ideal is generated
by f1,...,fm.
* The dimension of such a set X given by the simultaneous vanishing
of polynomials is defined as the maximum
length of a chain of similarly defined subsets.