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2 results

  • A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

  • David Favero, Daniel Kaplan, Tyler L. Kelly
  • Journal:
    Forum of Mathematics, Sigma / Volume 8 / 2020
    Published online by Cambridge University Press:
    16 November 2020, e56
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  • We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

  • Cluster complexes via semi-invariants

  • Part of
  • Kiyoshi Igusa, Kent Orr, Gordana Todorov, Jerzy Weyman
  • Journal:
    Compositio Mathematica / Volume 145 / Issue 4 / July 2009
    Published online by Cambridge University Press:
    01 July 2009, pp. 1001-1034
    Print publication:
    July 2009
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  • We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.