Blowing-Ups Describing the Polarization Change of Moduli Schemes of Semistable Sheaves of General Rank

Abstract

Let H and H′ be two ample line bundles over a smooth projective surface X, and M(H) (resp. M(H′)) the coarse moduli scheme of H-semistable (resp. H′-semistable) sheaves of fixed type (r, c ₁, c ₂). We construct a sequence of blowing-ups which describes how M(H) differs from M(H′) when r is arbitrary and the wall of fixed type separating H and H′ is not necessarily good. Means we here utilize are elementary transforms and the notion of a sheaf with flag.

Key Words:

• Birational maps
• Moduli of semistable sheaves on algebraic surface
• Wall-crossing

2000 Mathematics Subject Classification:

• 14J60
• 14D20

ACKNOWLEDGMENT

The author was supported by Japan Society for the Promotion of Science.

Notes

Communicated by L. Ein.