Toric Sheaves on Hirzebruch Orbifolds

TL;DR

This paper develops a combinatorial framework for describing toric sheaves on Hirzebruch orbifolds, providing explicit formulas for generating functions of certain sheaf moduli on these stacks.

Contribution

It introduces a stacky fan description of vector bundles over toric Deligne-Mumford stacks and characterizes toric sheaves on Hirzebruch orbifolds, including fixed point loci of moduli schemes.

Findings

Explicit combinatorial description of toric sheaves on Hirzebruch orbifolds.

Formulas for generating functions of Euler characteristics of rank 2 sheaves.

Analysis of fixed point loci of moduli schemes of stable sheaves.

Abstract

We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $\mu$-stable torsion free sheaves of rank $1$ and $2$ on $\mathcal{H}_{r}^{ab}$. As an example, we obtain explicit formulas for generating functions of Euler characteristics of locally free sheaves of rank 2 on $\mathbb{P}(1,2) \times \mathbb{P}^1$.