Alastair Craw

Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_\theta, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter \theta. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_\theta for some parameter \theta. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also uncover explicit equivalences between the derived categories of moduli M_\theta for parameters lying in adjacent GIT chambers.

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

Cellular resolutions of noncommutative toric algebras from superpotentials

Q

Quiver flag varieties and multigraded linear series

with relations

Fiber Fans and Toric Quotients

R

Cohomology of wheels on toric varieties

corresponding to the finite-dimensional algebra

Multigraded linear series and recollement

\mathop{\rm End}\nolimits( \textstyle\bigoplus\nolimits_{i=0}^{r} L_i )

Reconstructing toric quiver flag varieties from a tilting bundle

where

An introduction to motivic integration

{\cal L} := ({\scr O}_X,L_1, \ldots, L_r)

Dirichlet Branes and Mirror Symmetry

is a list of line bundles on a projective toric variety