# Stability Conditions for 3-fold Flops

## TL;DR

This paper characterizes the space of Bridgeland stability conditions for certain 3-fold flopping contractions, revealing that the Stringy K"ahler Moduli Space (SKMS) is always a sphere with a specific number of points removed, depending on the curve length.

## Contribution

It provides the first description of the SKMS for all smooth irreducible 3-fold flops, linking stability conditions to hyperplane arrangements and covering spaces.

## Key findings

- Connected components of stability conditions relate to hyperplane arrangements.
- The SKMS is a sphere with 3, 4, 6, 8, 12, or 14 points removed.
- The arrangements are not Coxeter in general.

## Abstract

Let $f\colon X\to\mathrm{Spec}\, R$ be a 3-fold flopping contraction, where $X$ has at worst Gorenstein terminal singularities and $R$ is complete local. We describe the space of Bridgeland stability conditions on the null subcategory $\mathscr{C}$ of the bounded derived category of $X$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory $\mathscr{D}$, which consists of complexes supported on the exceptional locus. We show that a connected component of stability conditions on $\mathscr{C}$ is the universal cover of the complexified complement of the real hyperplane arrangement associated to $X$ via the Homological MMP, and more generally that a connected component of normalised stability conditions on $\mathscr{D}$ is a regular covering space of the infinite hyperplane arrangement constructed in Iyama-Wemyss [IW9]. Neither arrangement is Coxeter in general. As a consequence, we give the first description of the Stringy K\"ahler Moduli Space (SKMS) for all smooth irreducible 3-fold flops. The answer is surprising: we prove that the SKMS is always a sphere, minus either 3, 4, 6, 8, 12 or 14 points, depending on the length of the curve.