Abstract

Let Fe denote the Hirzebruch surface P(OP1⊕OP1(e)), and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves MFe,H(r,c1,c2) is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one.We then study sharp Bogomolov inequalities Δ≥δH(c1/r) for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function δH(c1/r) can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and δH(c1/r) is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute δH(c1/r) exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of MFe,H(v).

Highlights

  • Let Fe denote the Hirzebruch surface P(OP1 ⊕OP1 (e)), and let H be any ample divisor

  • To determine when the stack PH m (v) contains Hm-semistable sheaves, we study the Hm-Harder-Narasimhan filtration of the general sheaf

  • First we show the total slopes of the terms in the Hm-Harder-Narasimhan filtration lie in a narrow strip centered on the slope ν

Read more

Summary

Preliminaries

We recall basic facts concerning moduli spaces of sheaves, prioritary sheaves and Hirzebruch surfaces that will be used in the rest of the paper. Given a torsion-free sheaf V on a surface X and an ample divisor H, the total slope ν, the H-slope μH and the discriminant ∆ are defined as follows ν(V) = ch1(V) , ch[0] (V ). These quantities depend only on the Chern character of V and not on the particular sheaf. The set of ample divisors H for which a sheaf is H-Gieseker (semi)stable in general is neither open nor closed. When KX + L is anti-effective, every semistable sheaf for any polarization H is L-prioritary. (If ν(v) · F < −1 and r(v) ≥ 2, the cohomology of V can be determined by Serre duality.)

Prioritary sheaves and stability on Hirzebruch surfaces
Existence of prioritary sheaves
The generic Harder-Narasimhan filtration
Exceptional bundles and necessary conditions for stability
Sufficient conditions for stability on a del Pezzo Hirzebruch surface
Stability intervals and the stability of exceptional bundles
Sharp Bogomolov inequalities
10. Harder-Narasimhan filtrations from Kronecker modules
11. Reduction to F0 and F1
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call