2016
DOI: 10.4171/jncg/232
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Noncommutative versions of some classical birational transformations

Abstract: In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct an analogue of the Cremona transform for non-commutative projective planes.

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Cited by 6 publications
(14 citation statements)
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“…As noted in [21], the theory of Ore sets and Ore localization works fine for Zindexed algebras (recall that the construction of the derived category by inverting quasi-isomorphisms is such a general case of localization). As usual, a D-module is uniform if any two non-zero submodules have non-zero intersection, or equivalently, any non-zero submodule is essential.…”
Section: Z-indexed Algebras and Torsion Theorymentioning
confidence: 99%
“…As noted in [21], the theory of Ore sets and Ore localization works fine for Zindexed algebras (recall that the construction of the derived category by inverting quasi-isomorphisms is such a general case of localization). As usual, a D-module is uniform if any two non-zero submodules have non-zero intersection, or equivalently, any non-zero submodule is essential.…”
Section: Z-indexed Algebras and Torsion Theorymentioning
confidence: 99%
“…The author wishes to thank Michel Van den Bergh for providing many interesting ideas and for reading through the results multiple times. The author is also very grateful to Susan J. Sierra for inviting him to the University of Edinburgh and for posing the question about the nature of the inverses to the birational transformations considered in [10]. This question directly lead to the current paper.…”
Section: Acknowledgementsmentioning
confidence: 90%
“…4. Summary of the results in [10] In [10] we construct noncommutative versions of the birational transformation P 1ˆP1 P 2 and the Cremona Transform P 2 P 2 using the following recipe:…”
Section: Z-algebrasmentioning
confidence: 99%
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