1209.3922 (Amin Gholampour et al.)

Toric sheaves on weighted projective planes    [PDF]

Amin Gholampour, Yunfeng Jiang, Martijn Kool

We give an explicit description of toric sheaves on the weighted projective plane $\PP(a,b,c)$ viewed as a toric Deligne-Mumford stack. The stack $\PP(a,b,c)$ has an action of the torus $\T=\C^{*2}$ and by a toric sheaf we mean a coherent sheaf together with a lift of the torus action. The integers $(a,b,c)$ are not necessarily chosen coprime or mutually coprime allowing for gerbe and root stack structures. This is a first step in an attempt to give an explicit description of toric sheaves on any smooth toric Deligne-Mumford stack. Specializing the description to torsion free sheaves gives a combinatorial picture with various new features, which do not appear in the case of smooth toric varieties. As an application, we describe the fixed point locus of the moduli scheme of stable rank 1 and 2 torsion free sheaves on $\PP(a,b,c)$ with fixed $K$-group class. Summing over all $K$-group classes, we obtain formulae for generating functions of the topological Euler characteristics of these moduli schemes.

View original: http://arxiv.org/abs/1209.3922