2021/03/28 by Barbara Bolognese, Bolognese, Barbara, Domenico Fiorenza +1
Computer Science · Mathematics · #14D22 #14F05 #14J45 #18E30 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2103.15205
openalex publication_date 2021/03/28 · openalex created_date 2021/04/13 · openalex updated_date 2026/07/28
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 Q3 in ℙ4. 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 Y4.