Relative stable pairs and a non-Calabi-Yau wall crossing

TL;DR

This paper extends the definition of Bryan-Steinberg invariants to non-Calabi-Yau threefolds and conjectures a relation with Pandharipande-Thomas invariants, supported by a specific contraction case analysis.

Contribution

It introduces Bryan-Steinberg invariants for a broader class of threefolds and proposes a conjectural relation to PT invariants, expanding the scope of enumerative geometry.

Findings

Conjectured a relation between Bryan-Steinberg and PT invariants for certain threefolds.

Verified the conjecture for a specific rational curve contraction using advanced techniques.

Reduced complex cases to Calabi-Yau situations via degeneration and localization methods.

Abstract

Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau, Bryan-Steinberg defined enumerative invariants associated to such maps called $f$-relative stable (or Bryan-Steinberg) invariants. When $X$ has Gorenstein singularities and $f$ has relative dimension one, they compared these invariants to the Donaldson-Thomas, or equivalently the Pandharipande-Thomas invariants of $Y$. We define Bryan-Steinberg invariants for maps $f$ as above without assuming that $Y$ is Calabi-Yau. For $X$ with Gorenstein and rational singularities, $f$ of relative dimension one, and for insertions from $X$ and arbitrary descendant levels, we conjecture a relation between the generating functions of Bryan-Steinberg and Pandharipande-Thomas invariants of $Y$. We check the conjecture for the contraction $f: Y\to X$ of a rational curve $C$ with normal bundle $N_{C/Y}\cong \mathcal{O}_C(-1)^{\oplus 2}$ using degeneration and localization techniques to reduce to a Calabi-Yau situation, which we then treat using Joyce's motivic Hall algebra.