Toric Sheaves on Hirzebruch Orbifolds

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 orbifolds H ab r obtained by projectivizing O ⊕ O(r) over the weighted projective line P(a, b). Next, we give a combinatorial description of toric sheaves on H ab r 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 µ-stable torsion free sheaves of rank 1 and 2 on H ab r . As an example, we obtain explicit formulas for generating functions of Euler characteristics of locally free sheaves of rank 2 on P(1, 2) × P 1 .