Fullness of exceptional collections via stability conditions -- A case study: the quadric threefold

TL;DR

This paper explores how Bridgeland stability conditions can determine the fullness of exceptional collections in derived categories, focusing on the quadric threefold and suggesting broader applications in algebraic geometry.

Contribution

It demonstrates a novel approach using stability conditions to analyze the fullness of exceptional collections and residual categories in Fano varieties, exemplified by the quadric threefold.

Findings

Established a method to assess fullness using stability conditions

Analyzed the residual category of the quadric threefold

Indicated potential for broader applications in classical results

Abstract

A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. Proving the fullness of such a collection is generally a nontrvial problem, usually solved on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its "residual" category, i.e., its right orthogonal, is the derived category of a variety. We show how one can use the existence of Bridgeland stability condition these residual categories (when they exist) to address these problems. We examine a simple case in detail: the quadric threefold $Q_3$ in $\mathbb{P}^{4}$. We also give an indication how a variety of other classical results could be justified or re-discovered via this technique., e.g., the commutativity of the Kuznetsov component of the Fano threefold $Y_4$.