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

TL;DR

This paper investigates the smoothness properties of the Nori-Hilbert scheme for 2-Calabi Yau algebras, revealing it is generally not smooth except in specific cases involving simple representations.

Contribution

It demonstrates that the Nori-Hilbert scheme is only smooth for certain 2-Calabi Yau algebras, extending known results from formally smooth algebras to this class.

Findings

Smooth only for n=1 in surface group algebras

Smooth components contain only simple representations in preprojective algebras

Generalization to arbitrary 2-Calabi Yau algebras under certain conditions

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 $A$, parameterizes left ideals of codimension $n$ in $A,$ and it is well known to be smooth when $A$ is formally smooth. In this paper we will study the Nori - Hilbert scheme for $2-$Calabi Yau algebras. The main examples of these are surface group algebras and preprojective algebras. For the former we show that the Nori-Hilbert scheme is smooth for $n=1$ only, while for the latter we show that the smooth components that contain simple representations are precisely those that only contain simple representation. Under certain conditions we can generalize this last statement to arbitrary $2-$Calabi Yau algebras.