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Noncommutative del Pezzo surfaces and Calabi-Yau algebras. (English) Zbl 1204.14004
Let A=[x 1 ,x 2 ,x 3 ] be the polynomial -algebra in 3 variables, t a non-zero complex number and choose a polynomial Φ k [x k ] for each 1k3. Then the noncommutative -algebras 𝒰 t (Φ) generated by x 1 ,x 2 ,x 3 with the relations: x 1 x 2 -tx 2 x 1 =Φ 3 (x 3 ), x 2 x 3 -tx 2 x 1 =Φ 1 (x 1 ), x 3 x 1 -tx 1 x 3 =Φ 2 (x 2 ) are noncommutative deformations of A and form a family of Calabi-Yau algebras. Here it constructs a deformation-quantization of the coordinate ring of a del Pezzo surface of type E r , 6r8 considering noncommutative algebras of the form 𝒰 t (Φ)/Ψ, where Ψ is the ideal generated by a central element Ψ, which generates the center of 𝒰 t (Φ) if Φ is generic enough. Also it shows that the family of del Pezzo surfaces of type E r provides a semiuniversal Poisson deformation of the Poisson structure inherited by hypersurfaces in 3 with an isolated quasi-homogeneous elliptic singularity of type E r .
14B07Deformations of singularities (local theory)
14H52Elliptic curves
14J32Calabi-Yau manifolds
13C14Cohen-Macaulay modules