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Noncommutative del Pezzo surfaces and Calabi-Yau algebras. (English) Zbl 1204.14004
Let $A={\Bbb C}[x_1,x_2,x_3]$ be the polynomial ${\Bbb C}$-algebra in 3 variables, $t$ a non-zero complex number and choose a polynomial $\Phi_k\in {\Bbb C}[x_k]$ for each $1\leq k\leq 3$. Then the noncommutative ${\Bbb C}$-algebras ${\cal U}^t(\Phi)$ generated by $x_1,x_2,x_3$ with the relations: $x_1x_2-tx_2x_1=\Phi_3(x_3)$, $x_2x_3-tx_2x_1=\Phi_1(x_1)$, $x_3x_1-tx_1x_3=\Phi_2(x_2)$ 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 $E_r$, $6\leq r\leq 8$ considering noncommutative algebras of the form ${\cal U}^t(\Phi)/\langle\langle\Psi\rangle\rangle$, where $\langle\langle\Psi\rangle\rangle$ is the ideal generated by a central element $\Psi$, which generates the center of ${\cal U}^t(\Phi)$ if $\Phi$ 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 ${\Bbb C}^3$ with an isolated quasi-homogeneous elliptic singularity of type $E_r$.

14B07Deformations of singularities (local theory)
14H52Elliptic curves
14J32Calabi-Yau manifolds
13C14Cohen-Macaulay modules
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