66 research outputs found
Construction of noncommutative surfaces with exceptional collections of length 4
Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler
forms on a free abelian group of rank 4 having the properties of the Euler form
of a smooth projective surface. There are two types of solutions: one
corresponding to (and noncommutative
quadrics), and an infinite family indexed by the natural numbers. For
there are commutative and noncommutative surfaces having this Euler form,
whilst for there are no commutative surfaces. In this paper we
construct sheaves of maximal orders on surfaces having these Euler forms,
giving a geometric construction for their numerical blowups.Comment: 24 pages, see also companion paper arXiv:1811.0881
Embeddings of algebras in derived categories of surfaces
By a result of Orlov there always exists an embedding of the derived category
of a finite-dimensional algebra of finite global dimension into the derived
category of a high-dimensional smooth projective variety. In this article we
give some restrictions on those algebras whose derived categories can be
embedded into the bounded derived category of a smooth projective surface. This
is then applied to obtain explicit results for hereditary algebras.Comment: 13 pages; revised versio
Examples violating Golyshev's canonical strip hypotheses
We give the first examples of smooth Fano and Calabi-Yau varieties violating
the (narrow) canonical strip hypothesis, which concerns the location of the
roots of Hilbert polynomials of polarised varieties. They are given by moduli
spaces of rank 2 bundles with fixed odd-degree determinant on curves of
sufficiently high genus, hence our Fano examples have Picard rank 1, index 2,
are rational, and have moduli. The hypotheses also fail for several other
closely related varieties.Comment: 8 pages, 1 table, 1 figur
The point variety of quantum polynomial rings
We show that the reduced point variety of a quantum polynomial algebra is the
union of specific linear subspaces in , we describe its
irreducible components and give a combinatorial description of the possible
configurations in small dimensions.Comment: 10 pages, an extended version of arxiv.org/abs/1506.0651
Derived categories of noncommutative quadrics and Hilbert squares
A noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin–Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture and provide some evidence in the case of smooth projective surfaces
Derived categories of (nested) Hilbert schemes
In this paper we provide several results regarding the structure of derived
categories of (nested) Hilbert schemes of points. We show that the criteria of
Krug-Sosna and Addington for the universal ideal sheaf functor to be fully
faithful resp. a -functor are sharp. Then we show how to embed
multiple copies of the derived category of the surface using these fully
faithful functors. We also give a semiorthogonal decomposition for the nested
Hilbert scheme of points on a surface, and finally we give an elementary proof
of a semiorthogonal decomposition due to Toda for the symmetric product of a
curve.Comment: 20 pages, added reference to result in new version of Jiang--Leung
preprin
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