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

Related Topics:
Mathematics - Algebraic variety - Algebraic curve - Plane curve - Double point - Cubic curve

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:y2 = x2(x + 1)

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exhibits at (0, 0), cannot simply be parametrized near the origin.

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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

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:F(x,y) = y2 − x2(x + 1)

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so that the curve has equation

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:F(x,y) = 0,

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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.

Related Topics:
Partial derivative - Implicit function theorem

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In general for a hypersurface

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

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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

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:Xi − Pi

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

Related Topics:
Origin of coordinates - Zariski tangent space

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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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See also:

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• singularity theory