Noncommutative del Pezzo surfaces and Calabi-Yau algebras

The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r 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=F_1(z), [y,z]_t=F_2(x), [z,x]_t=F_3(y). This gives a family of Calabi-Yau algebras A(F) parametrized by a complex number t and a triple F=(F_1,F_2,F_3), 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.

💡 Research Summary

The paper investigates two intertwined problems: (i) the semi‑universal Poisson deformation of the natural Poisson structure on a three‑dimensional hypersurface with an isolated quasi‑homogeneous elliptic singularity of type E₆, E₇ or E₈, and (ii) a deformation‑quantization of the coordinate ring of the corresponding del Pezzo surface.
First, the authors consider the affine hypersurface defined by a weighted homogeneous polynomial W(x,y,z) that yields an elliptic singularity of type Eᵣ (r = 6, 7, 8). The Jacobian of W induces a Poisson bracket {·,·} on ℂ