How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface?
Does there exist a complex version of the implicit function theorem?
2026-03-27 19:31:42.1774639902
Alebgraic curve and Riemann surfaces
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Take $(x:y:z)$ as the homogeneous co-ordinates on the projective plane. Then the subset of the curve where $x\ne0$ being open in $C$ is again smooth. But this subset is embeddable as closed set in $\mathbf{C}^2$: dehomogenize the equation taking $y/x, z/x$ as affine co-ordinates. This gives a co-ordinate patch. Similarly do for the other co-ordinates. Verification that transition between the patches are analytic follows from the fact these transition functions are rational functions with denominators not vanishing on the intersections.