Derived categories of Burniat surfaces and exceptional collections

Abstract

We construct an exceptional collection ${\rm Y}$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal{A}$ of ${\rm Y}$ is an "almost phantom" category: it has trivial Hochschild homology, and $K_0(\mathcal A)=\bZ_2^6$.

15 pages, 1 figure; further remarks expanded

References in corpus (6)

• Hochschild homology and semiorthogonal decompositions
• Geometric Phantom Categories
• Determinantal Barlow surfaces and phantom categories
• On the derived category of the classical Godeaux surface
• Exceptional collections of line bundles on the Beauville surface
• Compactifications of spaces of Landau-Ginzburg models

Cited by in corpus (22)

• Semiorthogonal decompositions in algebraic geometry
• Geometric Phantom Categories
• Exceptional collections on isotropic Grassmannians
• Determinantal Barlow surfaces and phantom categories
• On the derived category of the classical Godeaux surface
• Exceptional collections of line bundles on the Beauville surface
• Height of exceptional collections and Hochschild cohomology of quasiphantom categories
• Exceptional collections, and the Neron-Severi lattice for surfaces
• On the Jordan-Hölder property for geometric derived categories
• Nonexistence of semiorthogonal decompositions and sections of the canonical bundle
• Cycles, derived categories, and rationality
• Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
• Exceptional collections on Dolgachev surfaces associated with degenerations
• Descent conditions for generation in derived categories
• Quasiphantom categories on a family of surfaces isogenous to a higher product
• G-theory of root stacks and equivariant K-theory
• Exceptional collection of objects on some fake projective planes
• Note on Applications of Toric Systems on Surfaces
• Examples of Mori dream surfaces of general type with $p_g=0$
• Non-commutative Toric Varieties
• Weil transfer of noncommutative motives
• Torsion birational motives of surfaces and unramified cohomology