Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998)in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution ofA. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ

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

Mori Dream Spaces as fine moduli of quiver representations

We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw--Smith \cite{CrawSmith} beyond the toric case. Any collection of effective line bundles $\mathscr{L}=(\mathscr{O}_X, L_1,..., L_r)$ on a Mori Dream Space $X$ defines a bound quiver of sections and a map from $X$ to a toric quiver variety $|\mathscr{L}|$ called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on $\mathscr{L}$, the image realises $X$ as the fine moduli space of $\vartheta$-stable representations of the bound quiver. As an application, we show how to reconstruct del Pezzo surfaces from a full, strongly exceptional collection of line bundles.Comment: 25 pages, 2 figures; v2 section 3 simplified, typos corrected; v3 final versio

An examination of the relationship between firm size and export activity in the New Zealand lumber industry : a thesis presented in partial fulfilment of the requirements for the degree of Master of Philosophy in Development Studies, Massey University

This thesis explores the relationship between export activity and firm size, with specific reference to New Zealands lumber industry. The main purpose of this investigation is to identify firm characteristics which link firm size to export behavior. This task basically involves exploring the literature to identify possible linking variables and.conducting tests to determine whether or not these characteristics actually link export activity and firm size in New Zealand's lumber industry. · Two major areas of research were drawn upon to produce the hypotheses of this study: determinants of export activity and firm size-related variables in the lumber industry. The former body of literature is well-defined and very extensive. A great deal of empirical research has been done on firm-level export behavior (though unfortunately very little theoretical study has been done to link export behavior back to microeconomics. The second area of research is not very well­ defined. Inferences on the relationship between various characteristics and firm size are drawn from the literature on lumber production in New Zealand. These inferences are supplemented by scattered pieces of research on the linkage between firm size and firm characteristics, as well as by sensible guesses as to how certain characteristics are associated with firm size. Using these two areas of research, hypotheses were drawn as to how firm size and export activity are linked. Based upon these two areas of study then, nine characteristics were identified as possible links between firm size and export activity: proximity to a city, product quality, production cost, legal structure, foreign ownership, managerial experience and education, marketing skill, export­ related information, and managerial attitudes and ambition. It was decided to test these hypotheses by conducting a survey of New Zealand's lumber industry. This particular industry was selected because it was felt that a greater understanding of the export dynamics of this sector would assist policymakers in stimulating New Zealand's economy. In all, 26 lumber mills (out of 40 that were contacted) agreed to participate in the survey. On the whole, it was found that some characteristics do link firm size to export activity. Specifically, legal structure, managerial experience and education, and managerial attitudes and ambition were found to be significantly related to both export activity and firm size. These results suggest that firm size can be indirectly linked to export activity. However, researchers should be aware that the nature of this link could possibly vary with industry, place and time. Hence, using firm size as a predictor of export activity should be avoided until more research is conducted