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V. A. Ginzburg University of Chicago Chicago, United States ginzburg@math.uchicago.edu

Noncommutative del Pezzo surfaces
The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type Er , r = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type Er provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra C [x, y , z ] to a noncommutative algebra with generators x, y , z and the following 3 relations (where [u, v ]t = uv - tv u): [x, y ]t = F1 (z ), [y , z ]t = F2 (x), [z , x]t = F3 (y )

This gives a family of Calabi-Yau algebras A(F ) parametrized by a complex number t and a triple F = (F1 , F2 , F3 ), of polynomials in one variable of specifically chosen degrees. Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form A(F )/ g where g stands for the ideal of A(F ) generated by a central element g , which generates the center of the algebra A(F ) if F is generic enough.

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