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Toric sheaves on Hirzebruch orbifolds. (English) Zbl 1446.14003

The paper under review gives a description of a class of toric vector bundles over the toric Deligne-Mumford stacks in terms of the stacky fans and the maps between them. This generalizes a similar construction for the toric vector bundles over toric varieties.
The paper also finds a uniform description of the stacky fans of the weighted projective spaces.
Combining these, the paper works out the stacky fans of the “Hirzebruch orbifold” \(\mathcal H^r_{a,b}:=\mathbb P(\mathcal O\oplus \mathcal O(r))\to \mathbb P(a,b)\).
The paper also finds combinatorial description of the Euler characteristics of the moduli spaces of ranks 1 and 2 stable torsion free sheaves on \(\mathcal H^r_{a,b}\) by identifying the fixed points of the \(\mathbb C^{*2}\)-action on these moduli spaces. This is achieved by adapting Klyachko’s method to this setting.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14D23 Stacks and moduli problems
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

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