 Coordinate ring and singularity of the semicubical parabola

ID: coordinate-ring-and-singularity-of-the-semicubical-parabola

k [x, y] \to k [t], x \mapsto t^{2}, y \mapsto t^{3}

has kernel

(y^{2} - x^{3})

k [t^{2}, t^{3}]

. The quotient is therefore an integral domain, so the curve is irreducible. In characteristic zero its gradient

(- 3 x^{2}, 2 y)

vanishes at the origin, making the origin its unique singular point of an algebraic variety.

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