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

Abstract

Let $k$ be an algebraically closed field of characteristic zero and let $A$ be a finitely generated $k$-algebra. The Nori–Hilbert scheme of $A$, ${\mathrm{Hilb}}_A^n$, parameterizes left ideals of codimension $n$ in $A$. It is well known that ${\mathrm{Hilb}}_A^n$ is smooth when $A$ is formally smooth.

In this paper we will study ${\mathrm{Hilb}}_A^n$ 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 $n=1$, while for the latter we show that a component of ${\mathrm{Hilb}}_A^n$ 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.