Semicubical parabola ( $y^{2} = x^{3}$ )

A semicubical parabola is a cusped cubic curve affinely equivalent to

y^{2} = x^{3}

.

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- Coordinate ring and singularity of the semicubical parabola Semicubical parabola

Coordinate ring and singularity of the semicubical parabola

The homomorphism

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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Semicubical parabola by

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A semicubical parabola is a specific type of cubic curve that is defined mathematically and has interesting properties in both geometry and calculus. The general form of the semicubical parabola can be expressed with the equation: \[ y^2 = kx^3 \] where \( k \) is a non-zero constant. In this equation, the curve is defined in a Cartesian coordinate system, and it is symmetric about the y-axis.

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