Singular point of an algebraic variety Guide, Meaning , Facts, Information and Description

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.

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.

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