The Nori-Hilbert scheme is not smooth for 2-Calabi Yau algebras

Let k be an algebraically closed field of characteristic zero and let A be a finitely generated k-algebra. The Nori-Hilbert scheme of A, Hilbn parameterizes left ideals of codimension n in A. It is well known that A is smooth when A is formally smooth. In this paper we will study Hilbn A for 2-Calabi-Yau algebras. Impor-tant examples include the group algebra of the fundamental group of a In this paper we will study Hilbn compact orientable surface of genus g, and preprojective algebras. For the former, we show that the Nori-Hilbert scheme is smooth only for n = 1, while for the latter we show that a component of Hilbn containing a simple representation is smooth if and only if it only contains simple representations. Under certain conditions, we generalize this last statement to arbitrary 2-Calabi-Yau algebras.