Projectivity of the moduli of higher rank PT-stable pairs on a threefold

TL;DR

The article proves projectivity for the moduli space of higher-rank PT-stable objects on a smooth projective threefold by constructing determinantal line bundles on the PT moduli stack and analyzing their positivity. It first constructs a globally generated line bundle on the moduli stack that separates points, leveraging Mehta–Ramanathan-type restriction to curves and Beentjes–Ricolfi embeddings. In the coprime case, it then derives an explicit ample line bundle on the coarse moduli space by combining three determinantal line bundles $oldsymbol{L}_2$, $oldsymbol{L}_1$, and $oldsymbol{L}_0$, with the Todd class used to ensure descent to the good moduli space. The approach bypasses direct GIT, using stack-theoretic descent and restriction techniques to establish semiampleness and ampleness, ultimately proving projectivity of the moduli space under coprime invariants. This extends the PT framework to higher rank and provides concrete tools for compactifying and studying higher-rank PT-pair moduli spaces.

Abstract

We construct a globally generated line bundle on the moduli stack of higher-rank PT-semistable objects over a smooth projective threefold and analyze the extent to which it separates points. Furthermore, 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.

Theorems & Definitions (58)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Proposition 2.3: lo-PT1, lo-PT2
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof