Determinantal Barlow surfaces and phantom categories

Abstract

We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

27 pages; 1 figure; Macaulay2 code for the paper available at http://www.math.uni-hamburg.de/home/boehning/research/BarlowM2/M2scripts

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• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• Geometric Phantom Categories
• On the derived category of the classical Godeaux surface
• Derived categories of Burniat surfaces and exceptional collections
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Cited by in corpus (21)

• Semiorthogonal decompositions in algebraic geometry
• Geometric Phantom Categories
• Exceptional collections on isotropic Grassmannians
• Derived categories of Burniat surfaces and exceptional collections
• 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
• Combinatorial aspects of exceptional sequences on (rational) surfaces
• Exceptional collections on Dolgachev surfaces associated with degenerations
• A Phantom on a Rational Surface
• Quasiphantom categories on a family of surfaces isogenous to a higher product
• On cyclic strong exceptional collections of line bundles on surfaces
• Deformation of exceptional collections
• Bloch's conjecture for Catanese and Barlow surfaces
• Fano visitors, Fano dimension and orbifold Fano hosts
• On the derived category of the Hilbert scheme of points on an Enriques surface
• Thick subcategories on weighted projective curves and nilpotent representations of quivers
• Non-commutative Toric Varieties