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Resolving nodes of a plane curve
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Manage episode 185577042 series 1521141
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Let X be the Zero of f(z,w) and p a node. Then the quadratic terms of the Taylor series expansion of f at p factors into distinct linear factors which lift to f= gh. Around p, the zeroes of X looks like zeroes X_g union with X_h. Delete p from all three and glue them together. The result is a Riemann surface (if f is irreducible) since at least near p, X-p is a Riemann surface, being a nonsingular curve; X_g and X_h are Riemann surfaces by implicit function theorem (by changing coordinate we get g= x + .. so dg/dx(p) is not Zero and hence locally g looks like graph of a function of y) In case of projective plane curve cut out by irred f, the result is a compact Riemann surface.
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