I have a basic problem. I want to calculate the coordinate Ring of a variety, but I can not calculate the ideal related to the variety. For example I don't know how can I find the ideal for this variety : $$ \{ (t^2, t^3) ~\mid~ t \in\Bbb C\}. $$ Any help would be great thanks.
1 Answer
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Hint:
With the tools of differential geometry, you can check the origin is a cuspidal point, with the $x$-axis as a tangent.
You can eliminate $t$ to show the points of the curve satisfy the equation $y^2=x^3$.
Conversely, to show all points of the curve $y^2=x^3$, you can consider the intersections of a variable line through the origin: $y=tx$, which cuts the curve in the origin (double intersection) and a third point. Check this third point has the required parameterisation w.r.t. $t$.
