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On special values for pencils of plane curve singularities. (English) Zbl 1079.14035
Let \((F_t: t\in \mathbb{P}^1)\) be a pencil of plane curves singularities , where \(F_t=f-tg\), with \(f,g \) two coprime power series without constant term in \(\mathbb{C}\{x,y\}\) and \(F_{\infty}=g\). Let \(\mu_{0}^t =\mu_{0}(F_t )\) be the Milnor number of the fiber. The author’s main result is a formula for the jumps \(\mu_{0}^t-\mu_{0}^{\text{min}}\) , which involves the meromorphic fraction \(f/g\) considered on the branches of the Jacobian curve \(J(F)=0\). As an application, the author presents a new proof of the description (due to Le Dũng Tráng [Funct. Anal. Appl. 8, 127–131 (1974); translation from Funkts. Anal. Prilozh. 8, No.2, 45–49 (1974; Zbl 0351.32007)]) of the special values of the pencil by means of the discriminant curve.
14H20 Singularities of curves, local rings
32S10 Invariants of analytic local rings
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