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The Representation Type of Determinantal Varieties

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Abstract

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves \(\mathcal {E}\) of arbitrary high rank on a general standard (resp. linear) determinantal scheme \(X\subset \mathbb {P}^{n}\) of codimension c ≥ 1, n − c ≥ 1 and defined by the maximal minors of a t × (t + c−1) homogeneous matrix \(\mathcal {A}\). The sheaves \(\mathcal {E}\) are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type provided the degrees of the entries of the matrix \(\mathcal {A}\) satisfy some weak numerical assumptions; and (2) we determine values of t, n and n − c for which a linear standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type.

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Authors and Affiliations

  1. Faculty of Technology, Art and Design, Oslo and Akershus University College, PB. 4, St. Olavs Plass, 0130, Oslo, Norway

    Jan O. Kleppe

  2. Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Rosa M. Miró-Roig

Authors
  1. Jan O. Kleppe
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  2. Rosa M. Miró-Roig
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Correspondence to Rosa M. Miró-Roig.

Additional information

Presented by Henning Krause.

Rosa M. Miró-Roig Partially supported by MTM2013-45075-P

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Kleppe, J.O., Miró-Roig, R.M. The Representation Type of Determinantal Varieties. Algebr Represent Theor 20, 1029–1059 (2017). https://doi.org/10.1007/s10468-017-9673-4

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  • DOI: https://doi.org/10.1007/s10468-017-9673-4

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