Francesco Vaccarino: The Nori - Hilbert scheme of 2-Calabi Yau algebras is not smooth

The n-th Nori-Hilbert scheme of an associative algebra A is the scheme whose rational points parameterize the ideals of A having codimension n. When A is commutative this is the usual Hilbert scheme of n-points on X = Spec A. It is classical result that, when X is a smooth curve or surface, then this Hilbert scheme is smooth as well. If A is associative its n-th Nori Hilbert scheme it is know to be smooth when A is formally smooth and, therefore, of global dimension one i.e. a smooth non commutative curve. We study the case in which A has global dimension 2 specializing to 2-Calabi Yau algebras. We will exhibit an example in which the Nori-Hilbert scheme is irreducible and not smooth. We further show that the Hilbert to Chow morphism, which in this case has codomain the moduli space of n dimensional linear representations of A, is not birational. All results are based on a careful analysis of the local geometry of the representation scheme of A, performed by using the interplay between deformation theory and Hochschild cohomology.