Projectivity of Moduli Spaces in the Higher-Rank DT/PT Correspondence

TL;DR

This paper proves the projectivity of certain moduli spaces related to higher-rank Donaldson-Thomas and Pandharipande-Thomas invariants on threefolds, using explicit line bundle constructions.

Contribution

It constructs explicit ample line bundles on moduli spaces of PT-stable objects, establishing their projectivity in higher-rank DT/PT wall crossing.

Findings

Constructed globally generated line bundle on moduli stack.

Established projectivity of coarse moduli space when rank and degree are coprime.

Proved projectivity on walls between Gieseker and PT stability for certain threefolds.

Abstract

We study projectivity of moduli spaces on the DT/PT wall crossing in Bridgeland and polynomial stability on a smooth, projective threefold. First, we construct a globally generated line bundle on the moduli stack of higher-rank PT-semistable objects and analyze the extent to which it separates points. Next, when the rank and degree are coprime, we refine our construction to obtain an explicit ample line bundle on the corresponding coarse moduli space of PT-stable objects, thereby establishing its projectivity. Finally, we consider certain classes of threefolds for which PT-stability is realized as a Bridgeland stability condition and establish projectivity also on the wall separating the Gieseker and PT chambers.