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Normal Analytic Compactifications of \(\mathbb{C}^{2}\)

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Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

This is a survey of some results on the structure and classification of normal analytic compactifications of \(\mathbb{C}^{2}\). Mirroring the existing literature, we especially emphasize the compactifications for which the curve at infinity is irreducible.

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Notes

  1. 1.

    Recall that C has one place at infinity iff C meets the line at infinity at only one point Q and C is unibranch at Q.

  2. 2.

    \(\bar{X}\) is a “minimal normal compactification” (in the sense of Morrow), or in modern terminology, a minimal SNC-compactification of \(X:= \mathbb{C}^{2}\) iff (i) \(\bar{X}\) is non-singular, (ii) each \(\Gamma _{i}\) is non-singular, (iii) C ∞ has at most normal-crossing singularities, and (iv) for all \(\Gamma _{i}\) with self-intersection − 1, contracting \(\Gamma _{i}\) destroys some of the preceding properties.

  3. 3.

    Recall that an isolated singular point P on a surface Y is sandwiched if there exists a birational map Y → Y ′ such that the image of P is non-singular. Sandwiched singularities are rational [15, Proposition 1.2].

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Author information

Authors and Affiliations

  1. Weizmann Institute of Sciences, Rehovot, Israel

    Pinaki Mondal

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  1. Pinaki Mondal
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Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

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Mondal, P. (2014). Normal Analytic Compactifications of \(\mathbb{C}^{2}\) . In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_10

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