Around a node, f looks like it's quadratic part

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Let X be zeroes of a f(z,w). Then p is a node of X if it is a singular point (I.e. Grad f vanish at p) but hessian f is not singular at p. In other words, the Taylor expansion of f at p has no constant term or linear term, and it's quadratic term factors into distinct homogeneous linear factors l_1, l_2. Around p, zeroes of f looks like union of zeroes of l_1 and l_2. By hensel's lemma, as power series f factor as gh. Where g = l_1 + ... and h = l_2 + ...

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