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The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1108–1131 | Cite as

On Plane Cremona Transformations of Fixed Degree

  • Cinzia BisiEmail author
  • Alberto Calabri
  • Massimiliano Mella
Article

Abstract

We study the quasi-projective variety \(\operatorname{Bir}_{d}\) of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety \(\operatorname{Bir}_{d}^{\circ}\) where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that \(\operatorname{Bir}_{d}\) is connected for each d and \(\operatorname{Bir}_{d}^{\circ}\) is connected when d<7.

Keywords

Plane Cremona transformations Homaloidal nets De Jonquières transformations 

Mathematics Subject Classification

14E07 

Notes

Acknowledgements

It is a pleasure to thank Ciro Ciliberto for the reference in Enriques–Chisini. Furthermore, we thank the anonymous referee for a thorough reading, suggestions, and remarks that improved the paper, in particular for pointing out a mistake in Definition 17.

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Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  • Cinzia Bisi
    • 1
    Email author
  • Alberto Calabri
    • 1
  • Massimiliano Mella
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità di FerraraFerraraItalia

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