How to find the chart around $(0,0)$ of $\{(z,w)\in\mathbb C\times \mathbb C: w^3= z^2(z^2-1)\}$? For $f(z,w)= w^3-z^2(z^2-1)$, in this case both $\frac{\partial f(z,w)}{\partial z}$ and $\frac{\partial f(z,w)}{\partial w}$ at $(0,0)$ vanishes. So the projection map does not become a chart—any hint to proceed with this?
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Note that zero set of a smooth affine plane curves form a Riemann surface. In this case at $(0,0) $ this not smooth – Infinity Mar 02 '23 at 05:57