Remarks on the Bondal quiver

TL;DR

This paper investigates a specific subcategory of the Bondal quiver, revealing its Serre functor as a spherical twist, classifying spherical objects, and demonstrating the absence of certain stability conditions, with implications for categorical resolutions.

Contribution

It identifies the Serre functor as a spherical twist, classifies spherical objects, and shows the non-existence of Serre-invariant stability conditions in this subcategory.

Findings

Serre functor coincides with a spherical twist

Classification of spherical objects

Non-existence of Serre-invariant stability conditions

Abstract

We study an admissible subcategory of the Bondal quiver which conjecturally does not admit any Bridgeland stability conditions. Specifically, we prove that its Serre functor coincides with the spherical twist associated with a $3$-spherical object. As a consequence, we obtain a classification of the spherical objects, deduce the non-existence of Serre-invariant stability conditions, and construct a natural spherical functor from its structure as a categorical resolution of the nodal cubic curve.