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

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 $Y_4$.

arXiv comment20 pages

References in corpus (8)

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• A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
• A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold
• Geometric Phantom Categories
• Stability conditions in families
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