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Register a new userReids recipe and derived categories
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Timothy Logvinenko
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We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of computing a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reids recipe, effectively providing a categorification of the latter.
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For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reids recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one isolated singularity.
Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other. In particular, in the case of algebraic surfaces, we explain the relationship between Blochs conjecture, Chow-theoretic decompositions of the diagonal, categorical representability, and the existence of phantom subcategories of the derived category.
A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the quadratic Fano correspondence (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperkahler variety as a Fano visitor.
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove that they coincide generically: Hassetts cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and we show that Kuznetsovs cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.
We construct an exceptional collection $Upsilon$ 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 $Upsilon$ is an almost phantom category: it has trivial Hochschild homology, and $K_0(mathcal A)=bZ_2^6$.
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