Verdier specialization via weak factorization

Let X in V be a closed embedding, with V - X nonsingular. We define a constructible function on X, agreeing with Verdier's specialization of the constant function 1 when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich et al. The main property of the specialization function is a compatibility with the specialization of the Chern class of the complement V-X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier's result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart in a motivic group. The specialization function and the corresponding Chern class and motivic aspect all have natural `monodromy' decompositions, for for any X in V as above. The definition also yields an expression for Kai Behrend's constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.Comment: Minor revision. To appear in Arkiv f\"or Matemati

Derived categories of cubic fourfolds

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.Comment: 18 page

Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non Pfaffian hypersurface but the dual Calabi-Yau becomes non commutative.Comment: 34 pages; exposition of Hochschild homology expanded; references added; introduction re-written; some imrecisions, typos and the orbit diagram in the last section correcte

Mesoscopic Analysis of Structure and Strength of Dislocation Junctions in FCC Metals

We develop a finite element based dislocation dynamics model to simulate the structure and strength of dislocation junctions in FCC crystals. The model is based on anisotropic elasticity theory supplemented by the explicit inclusion of the separation of perfect dislocations into partial dislocations bounding a stacking fault. We demonstrate that the model reproduces in precise detail the structure of the Lomer-Cottrell lock already obtained from atomistic simulations. In light of this success, we also examine the strength of junctions culminating in a stress-strength diagram which is the locus of points in stress space corresponding to dissolution of the junction.Comment: 9 Pages + 4 Figure

On Darboux-Treibich-Verdier potentials

It is shown that the four-parameter family of elliptic functions $$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$$ introduced by Darboux and rediscovered a hundred years later by Treibich and Verdier, is the most general meromorphic family containing infinitely many finite-gap potentials.Comment: 8 page

Power in the Multinational Corporation in Industry Equilibrium

Recent theories of the multinational corporation introduce the property rights model of the firm and examine whether to integrate our outsource firm activities locally or to a foreign country. This paper focus instead on the internal organization of the multinational corporation by examining the power allocation between headquarters and subsidiaries. We provide a framework to analyse the interaction between the decision to serve the local market by exporting or FDI, market acces and the optimal mode of organization of the multinational corporation. We find that subsidiary managers are given most autonomy in their decision how to run the firm at intermediate levels of local competition. We then provide comparative statics for changes in fixed FDI entry costs and trade costs, information technology, the number of local competitors, and in the size of the local market