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On the minimal compactification of a polynomial in two variables

Angelo Vistoli
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Abstract

Consider a primitive polynomial $f$ in two variables, thought of as a map from the affine plane to the affine line. We study the minimimal compactification of $f$; from our result one deduces in particular that if one of the fibers of $f$ has only one fiber at infinity, then all the fibers of $f$ have a simultaneous resolution of singularities at infinity. From this one gets a very simple proof of the Suzuki-Abhyankar-Moh theorem on the embeddings of the affine line in the plane.

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Angelo Vistoli. 1999-08-02. On the minimal compactification of a polynomial in two variables. https://arxiv.org/abs/math/9904035

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