52 research outputs found
Quiver flag varieties and multigraded linear series
This paper introduces a class of smooth projective varieties that generalise
and share many properties with partial flag varieties of type A. The quiver
flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique
source) and a dimension vector r is a fine moduli space of stable
representations of Q. Quiver flag varieties are Mori Dream Spaces, they are
obtained via a tower of Grassmann bundles, and their bounded derived category
of coherent sheaves is generated by a tilting bundle. We define the multigraded
linear series of a weakly exceptional sequence of locally free sheaves E =
(O_X,E_1,...,E_\rho) on a projective scheme X to be the quiver flag variety |E|
= M_\vartheta(Q,r) of a pair (Q,r) encoded by E. When each E_i is globally
generated, we obtain a morphism \phi_|E| : X -> |E| realising each E_i as the
pullback of a tautological bundle. As an application we introduce the
multigraded Plucker embedding of a quiver flag varietyComment: 23 pages. Final version includes minor changes, to appear in Duke
Math Journa
The Special McKay correspondence as an equivalence of derived categories
We give a new moduli construction of the minimal resolution of the
singularity of type 1/r(1,a) by introducing the Special McKay quiver. To
demonstrate that our construction trumps that of the G-Hilbert scheme, we show
that the induced tautological line bundles freely generate the bounded derived
category of coherent sheaves on X by establishing a suitable derived
equivalence. This gives a moduli construction of the Special McKay
correspondence for abelian subgroups of GL(2).Comment: 17 pages. Final version, to appear in Quart. J. Mat
Cohomology of wheels on toric varieties
We describe explicitly the cohomology of the total complex of certain
diagrams of invertible sheaves on normal toric varieties. These diagrams,
called wheels, arise in the study of toric singularities associated to dimer
models. Our main tool describes the generators in a family of syzygy modules
associated to the wheel in terms of walks in a family of graphs.Comment: 17 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1111.6018; final version 21 page
Projective toric varieties as fine moduli spaces of quiver representations
This paper proves that every projective toric variety is the fine moduli
space for stable representations of an appropriate bound quiver. To accomplish
this, we study the quiver with relations corresponding to the
finite-dimensional algebra where
is a list of line bundles on a
projective toric variety . The quiver defines a smooth projective toric
variety, called the multilinear series , and a map . We provide necessary and sufficient conditions for the induced
map to be a closed embedding. As a consequence, we obtain a new geometric
quotient construction of projective toric varieties. Under slightly stronger
hypotheses on , the closed embedding identifies with the fine
moduli space of stable representations for the bound quiver .Comment: revised version: improved exposition, corrected typos and other minor
change
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