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 A, parameterizes left ideals of codimension n in A: It is well known that Hilb _A ⁿ is smooth when A is formally smooth. In this paper we will study Hilb_A ⁿ for 2-Calabi-Yau algebras. Important examples include the group algebra of the fundamental group of a compact orientable surface of genus g, and preprojective algebras. For the former, we show that the Nori-Hilbert scheme is smooth only for n D 1, while for the latterwe showthat a component of Hilb _A ⁿcontaining 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.