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Quadro-Cubic Cremona Transformations and Feigin-Odesskii-Sklyanin Algebras with 5 Generators

  • V. RubtsovEmail author
Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 273)

Abstract

We study different algebraic and geometric properties of Heisenberg (H-) invariant Poisson polynomial algebras with 5 generators. These algebras are unimodular, and the elliptic Feigin-Odesskii-Sklyanin Poisson algebras \(q_{n,k}(Y)\) constitute the main important example. We discuss all the quadratic H-invariant Poisson tensors on \({\mathbb C}^5\). For the Sklyanin algebras \(q_{5,1}(Y)\) and \(q_{5;2}(Y)\), we explicitly write the Poisson morphisms on the moduli space of the vector bundles on the normal elliptic curve Y in \(\mathbb P^4\), studied by Polishchuk and Odesskii-Feigin as the quadro-cubic Cremona transformation on \(\mathbb P^4\).

Keywords

Cremona transformations Feigin-Odesskii-Sklyanin algebras Sklyanin elliptic algebras 
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Notes

Acknowledgements

During this work the author benefited from many useful discussions and suggestions of many colleagues. He is thankful to Brent Pym with whom he discussed the holomorphic Poisson geometry, to Sasha Polishchuk for his help with Algebraic Geometry and to Alexander Odesskii who taught him the Elliptic Algebras. Igor Reider had clarified some question and helped to understand the relation with his own unpublished results.

Some parts of this paper are based on previously published results of the author in a collaboration with Giovanni Ortenzi and Serge Tagne Pelap as well as with A. Odesskii. He is grateful to them for their collaboration.

A special thank to Rubik Pogossyan who had verified with Mathematica Package the author’s hint statement about the form and exact value of the determinant of the Jacobian for cubic “Inverse” Cremona map.

He had benefited from a lot of numerous conversations with Boris Feigin and Alexei Gorodentsev on various related subjects.

Finally, this work would have never been written without two author’s talks on the Moscow HSE Bogomolov Laboratory Seminar in June 2016 and on the 1st International Conference of Mathematical Physics at Kezenoy-Am in November 2016. He greatly acknowledges the invitation of Misha Verbitsky to Moscow and of V. Buchstaber and A. Mikhailov to the Chechen Republic. His special thanks to Dima Grinev and Sotiris Konstantinou-Rizos for their hospitality, excellent organisation of the Conference at Kezenoy-Am and inspiration. He also thanks Sotiris Konstantinou-Rizos for his great patience and valuable help while the text was being prepared to submission.

His work was partly supported by the Russian Foundation for Basic Research (Projects 18-01-00461 and 16-51-53034-GFEN). Part of this work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project 1.12873.2018/12.1.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Maths DepartmentUniversity of AngersAngersFrance
  2. 2.Theory Division, Math. Physics Lab, ITEPMoscowRussia

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