Hyperflächensingularitäten/Zweidimensional/Quasihomogen/Zusätzliche Zariskiform/Ableitung/Fakt/Beweis

Beweis

Es ist unter Verwendung von Fakt

\begin{matrix} d D & = d \frac{- δ_{2} y d x + δ_{1} x d y}{\partial_{3} (F)} \\ = - δ_{2} d \frac{y}{\partial_{3} (F)} \land d x + δ_{1} d \frac{x}{\partial_{3} (F)} \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} (- δ_{2} (\partial_{3} (F) d y - y d (\partial_{3} (F))) \land d x + δ_{1} (\partial_{3} (F) d x - x d (\partial_{3} (F))) \land d y) \\ = \frac{1}{(\partial_{3} (F))^{2}} (- δ_{2} (\partial_{3} (F) d y - y (\partial_{2} \partial_{3} (F) d y + \partial_{3} \partial_{3} (F) d z)) \land d x + δ_{1} (\partial_{3} (F) d x - x (\partial_{1} \partial_{3} (F) d x + \partial_{3} \partial_{3} (F) d z)) \land d y) \\ = \frac{1}{(\partial_{3} (F))^{2}} (δ_{2} \partial_{3} (F) - δ_{2} y \partial_{2} \partial_{3} (F) + δ_{2} y \frac{\partial_{3}^{2} (F) \partial_{2} (F)}{\partial_{3} (F)} + δ_{1} \partial_{3} (F) - δ_{1} x \partial_{1} \partial_{3} (F) + δ_{1} x \frac{\partial_{3}^{2} (F) \partial_{1} (F)}{\partial_{3} (F)}) d x \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} ((δ_{1} + δ_{2}) \partial_{3} (F) + (- δ_{2} y \partial_{2} - δ_{1} x \partial_{1}) \partial_{3} (F) + (δ_{2} y \partial_{2} (F) + δ_{1} x \partial_{1} (F)) \frac{\partial_{3}^{2} (F)}{\partial_{3} (F)}) d x \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} ((δ_{1} + δ_{2}) \partial_{3} (F) + (- δ_{2} y \partial_{2} - δ_{1} x \partial_{1}) \partial_{3} (F) - δ_{3} z \partial_{3} (F) \frac{\partial_{3}^{2} (F)}{\partial_{3} (F)}) d x \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} ((δ_{1} + δ_{2}) \partial_{3} (F) + (- δ_{2} y \partial_{2} - δ_{1} x \partial_{1}) \partial_{3} (F) - δ_{3} z \partial_{3}^{2} (F)) d x \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} ((δ_{1} + δ_{2}) \partial_{3} (F) + (- δ_{1} x \partial_{1} - δ_{2} y \partial_{2} - δ_{3} z \partial_{3}) \partial_{3} (F)) d x \land d y \\ = \frac{1}{(\partial_{3} (F))^{2}} ((δ_{1} + δ_{2}) \partial_{3} (F) - (δ - δ_{3}) \partial_{3} (F)) d x \land d y \\ = \frac{1}{\partial_{3} (F)} (δ_{1} + δ_{2} + δ_{3} - δ) d x \land d y . \end{matrix}

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