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

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.