Noncommutative del Pezzo surfaces and Calabi-Yau algebras
Abstract
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- t.vu): [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.