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

  • Raf Bocklandt

    University of Amsterdam, Netherlands
  • Federica Galluzzi

    Università di Torino, Italy
  • Francesco Vaccarino

    Politecnico di Torino, Italy


Let be an algebraically closed field of characteristic zero and let be a finitely generated -algebra. The Nori–Hilbert scheme of , , parameterizes left ideals of codimension in . It is well known that is smooth when is formally smooth.

In this paper we will study for 2–Calabi–Yau algebras. Important examples include the group algebra of the fundamental group of a compact orientable surface of genus , and preprojective algebras. For the former, we show that the Nori–Hilbert scheme is smooth only for , while for the latter we show that a component of 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.

Cite this article

Raf Bocklandt, Federica Galluzzi, Francesco Vaccarino, The Nori–Hilbert scheme is not smooth for 2-Calabi–Yau algebras. J. Noncommut. Geom. 10 (2016), no. 2, pp. 745–774

DOI 10.4171/JNCG/247