On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in projective four-space provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces.Comment: Originally written in 1999. To appear in Math.

Some finiteness results for Calabi-Yau threefolds

We investigate the moduli theory of Calabi--Yau threefolds, and using Griffiths' work on the period map, we derive some finiteness results. In particular, we confirm a prediction of Morrison's Cone Conjecture.Comment: 15 pages LaTex, uses amstex, amscd. New title, paper completely rewritten, results same as in previous version

Cohomological Donaldson-Thomas theory

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces, will also be introduced. The review closes with some recent appearances, and extensions, of the cohomological DT story in the theory of knot invariants, of cluster algebras, and elsewhere.Comment: 33 pages, some references adde

Enhanced gauge symmetry and braid group actions

Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi--Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible fashion. Extensions of this result are given for nontrivial $B$-fields on K3 surfaces, explaining physical restrictions on the $B$-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local $A_{2n}$-configuration in a threefold having global $\Z/2$ monodromy.Comment: 16 pages, 5 figure

Constructing projective varieties in weighted flag varieties

We compute the Hilbert series of general weighted flag varieties and discuss a computer-aided method to determine their defining equations. We apply our results to weighted flag varieties coming from the Lie groups of type G_2 and GL(6), to construct some families of polarised projective varieties in codimensions 8 and 6, respectively.Comment: 17 pages, 1 figur

Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex

We compute the motivic Donaldson-Thomas theory of the resolved conifold, in all chambers of the space of stability conditions of the corresponding quiver. The answer is a product formula whose terms depend on the position of the stability vector, generalizing known results for the corresponding numerical invariants. Our formulae imply in particular a motivic form of the DT/PT correspondence for the resolved conifold. The answer for the motivic PT series is in full agreement with the prediction of the refined topological vertex formalism.Comment: 26 page

Focus as a grammatical notion: A case study in autism

Proper identification of the focus of an utterance is essential for discourse to proceed adequately. But how does the hearer identify the focus intended by the speaker? It is well-known that the focal constituent carries prosodic prominence, usually pitch accent. The question at the heart of this paper is how the hearer associates such accents with the notion focus. Is there a deductive step involved or is this an automated, grammatical process. I investigate the issue from a psycholinguistic perspective. In particular, I carried out a case study with an autistic speaker. I argue that given the general communicative breakdown associated with autism, the fact that this speaker uses focus adequately shows that focus is more than a domaingeneral communicative device. It must be a notion encoded in the grammar. If correct, such psycholinguistic evidence helps solidify the foundations of theoretical linguistic notions such as focus

The Syntax Of Information Structure And The PF Interface

Focus movement to a left-peripheral position has been posited for both Hungarian and Italian. In this paper I argue against a unified cartographic treatment of focus movement, which analyses both as instances of movement to [Spec, Focus0]. I raise some theoretical issues for cartography, such as the proliferation of focus heads and the difficulties with accounting for optionality. Empirically, I show that a set of properties distinguish Hungarian and Italian left-peripheral focus movement suggesting a different syntactic analysis for the two constructions. Following Hamlaoui & Szendrői’s (2015) proposal for the syntax-prosody mapping of clauses, I show that Hungarian focus movement is prosodically motivated in that it is movement targeting the position that main stress is assigned to in the prosody. I show how the same proposal extends to right-peripheral and string-medial focus in Italian and heavy NP shift in English. I discuss the typological predictions of the that it follows from the proposal that left-peripheral focus movement is always accompanied by verb movement, while right-peripheral focus movement will target a position lower than the surface position of the finite verb. Finally, I propose that Italian left-peripheral focus movement is motivated by contrastivity. This accounts for the different characteristics of the two constructions: (i) that Hungarian, but not Italian, focus movement is accompanied by the movement of the finite verb; (ii) that Hungarian left-peripheral focus is prosodically unmarked, while Italian left-peripheral focus comes with marked prosody; and (iii) that Hungarian focus movement is pragmatically unmarked, in the sense that it can answer a wh-question, while Italian focus movement is explicitly contrastive (or perhaps even mirative or corrective)