JournalsjncgVol. 10, No. 2pp. 745–774

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

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

In this paper we will study HilbAn{\mathrm{Hilb}^n _A} for 2–Calabi–Yau algebras. Important examples include the group algebra of the fundamental group of a compact orientable surface of genus gg, and preprojective algebras. For the former, we show that the Nori–Hilbert scheme is smooth only for n=1n=1, while for the latter we show that a component of HilbAn{\mathrm{Hilb}^n _A} 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