singular point

Singular point of an algebraic variety

In mathematics, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curve

y² = x²(x + 1)

exhibits at (0, 0), cannot simply be parametrized near the origin. A plot of this curve is below with the singular point at the origin. Singular points commonly occur where a graph crosses over itself:

The reason for that algebraically is that both sides of the equation show powers higher than 1 of the variables x and y. In terms of differential calculus, if

F(x,y) = y² − x²(x + 1),

so that the curve has equation

F(x,y) = 0,

then the partial derivatives of F with respect to both x and y vanish at (0,0). This means that if we try to use the implicit function theorem to express y as a function of x near y = 0, we shall fail; and indeed no linear combination of x and y is a function of another essentially different one, so that this is a geometric condition not tied to any choice of coordinate axes.

In general for a hypersurface

F(x, y, z, ...) = 0

the singular points are those at which all the partial derivatives simultaneously vanish. A general algebraic variety V being defined by several polynomials, or in algebraic terms an ideal of polynomials, the condition on a point P to be a singular point of V is that none of those polynomials have a non-zero linear (degree 1) term, when written in terms of variables

X_i − P_i

that make P the origin of coordinates. See Zariski tangent space for geometric and algebraic interpretation.

Points of V that are not singular are non-singular. Apart from some technical questions that can be caused by non-zero characteristic, it is always true that most points are non-singular.

It is important to note that the geometric criterion for a point of a variety to be singular (mentioned earlier), that it is a point where the variety is not "locally flat", can be very hard to recognize for varieties over a general field. The work of Milnor and others shows that, over the complex numbers, the statement is precisely true in every reasonable interpretation. But, as Milnor points out, over the real numbers "The equation $y^3 + 2x^2u - x^4 = 0$ ... can actually be solved for $y$ as a real analytic function of $x$ " (so that the variety it defines is the graph of a real analytic function, and therefore a real analytic manifold) "but this equation also defines a variety having a singular point at the origin". Obviously the "geometric" meaning of "locally flat" over fields of finite characteristic, or ultrametric fields, is even more vexed.

Singular points of smooth mappings

As the notion of singular points is a purely local property the above definition can extended to cover the wider class of smooth mappings, (functions from M to Rⁿ where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the jets of the mapping. The k-th jet is the Taylor series of the mapping truncated at degree k and deleting the constant term.