Flops and Clusters in the Homological Minimal Model Program

Suppose that $f\colon X\to\mathrm{Spec}\, R$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of $f$ are at most one dimensional, so by [VdB1d] there is a noncommutative ring $\Lambda$ derived equivalent to $X$. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of $\Lambda$ is finite dimensional, improving a result of [DW2]. We further show that the mutation functor of [S6][IW4] is functorially isomorphic to the inverse of the Bridgeland--Chen flop functor in the case when the factor of $\Lambda$ is finite dimensional. These results then allow us to jump between all the minimal models of $\mathrm{Spec}\, R$ in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw--Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying $R$-module generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander--McKay correspondence to dimension three.Comment: 58 pages. Last update had an old version of Section 4, no other changes. Final version, to appear Invent. Mat

Reconstruction algebras of type D (II)

This is the third in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin quivers. This paper is the companion to [W12] and deals with dihedral groups G = DDn,q which have rank two special CM modules. We show that such reconstruction algebras are described by combining a preprojective algebra of type D~D~ with some reconstruction algebra of type A

The supply chain in air capability acquisition by the New Zealand Defence Force : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in Defence and Security Studies at Massey University, Manawatū, New Zealand

Over the last decade the New Zealand Government has acquired and introduced into operational service, two important platforms for air power capability, namely the new NH90, and SH-2G(I) Seasprite helicopters. The NH90 purchased new, and the Seasprite purchased second hand, are at different stages in their capability life cycles. The introductions of these aircraft have challenged support and sustainment within the Royal New Zealand Air Force (RNZAF) supply chain, which has been hampered by organisational factors such as the lack of capability and sustainment corporate knowledge, resource constraints, culture, and insufficient priority being given to Integrated Logistic Support (ILS) model In-Service. Equally aircraft specific issues such as their product maturity, and relationships also challenge the supply chain. The most significant level of aircraft acquisition is still yet to come as the Government progresses towards the replacement of the RNZAF surveillance and mobility capability. Therefore it is vital to understand the effect on support and sustainment from recent acquisitions

Reconstruction Algebras of Type D (I)

This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G=D_{n,q} for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=G-Hilb) to the same level of difficulty as the toric case.Comment: 31 pages, final versio

Faithful actions from hyperplane arrangements

We show that if X is a smooth quasiprojective 3–fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of X. The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (or mutation functor). This allows us to relate compositions of the flop functor (or mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of 3–fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of a result of Brav and Thomas (Math. Ann. 351 (2011) 1005–1017) for Kleinian singularities

Gopakumar-Vafa invariants do not determine flops

Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new, and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analytic classification of 3-fold flops via contraction algebras.Comment: 10 pages, final versio

Bordering seafarers at sea and onshore

This study uses a historically informed lens of coloniality, bordering, and intersectionality to analyze maritime bordering discourses and practices that target seafarers recruited from the Global South who embody the border in their everyday lives. In seeking to explain the current context exemplified by the sacking of P&O Ferry workers and the recruitment of “foreign agency” crews in March 2022, the study foregrounds 19th- and 20th-century maritime bordering legislation on ships and onshore, focusing on public-/private-bordering partnerships between governments, shipping companies, and unions. Archival research on British Indian seafarers employed by P&O a century ago and analysis of contemporary media and political discourses relating to “foreign agency crews” are drawn on to consider the implications of earlier bordering discourses and practices for 21st-century British citizenship and belonging. Attending to imperial bordering regulations that created the racialized and class-defined labor category of lascars explains the “common sense” designations of seafarers recruited in the Global South and their families as potential “illegal migrants,” and in doing so, it constitutes the long history of the public/private partnerships that constitute the UK's “hostile environment” immigration policies