Bridgeland Stability on Blow Ups and Counterexamples

TL;DR

This paper presents counterexamples to a conjecture on Bridgeland stability for threefolds and explores stability conditions on blow-ups, revealing new insights and limitations in the theory.

Contribution

It provides new counterexamples to the conjectural Bridgeland stability construction on threefolds and analyzes stability conditions on blow-ups under certain conjectural assumptions.

Findings

Counterexamples for specific threefolds containing divisors contracting to points

Existence of stability conditions on blow-ups where skyscraper sheaves are semistable

Implications for the conjectural construction of Bridgeland stability on threefolds

Abstract

We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macr\`i, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstra{\ss} elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable.