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

Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

We present a finite difference implementation of a three-dimensional higher-order ice sheet model. In comparison to a conventional centred difference discretisation it enhances both numerical stability and convergence. In order to achieve these benefits the discretisation of the governing force balance equation makes extensive use of information on staggered grid points. Using the same iterative solver, a centred difference discretisation that operates exclusively on the regular grid serves as a reference. The reprise of the ISMIP-HOM experiments indicates that both discretisations are capable of reproducing the higher-order model inter-comparison results. This setup allows a direct comparison of the two numerical implementations also with respect to their convergence behaviour. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the accuracy for the resultant velocity field can be chosen higher in the novel finite difference implementation. Changing the discretisation also prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. <br><br> The improved behaviour makes the new discretisation more reliable for extensive application to real ice geometries. Higher accuracy and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented

Simply connected projective manifolds in characteristic p>0p>0 have no nontrivial stratified bundles

We show that simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles. This gives a positive answer to a conjecture by D. Gieseker. The proof uses Hrushovski's theorem on periodic points.Comment: 16 pages. Revised version contains a more general theorem on torsion points on moduli, together with an illustration in rank 2 due to M. Raynaud. Reference added. Last version has some typos corrected. Appears in Invent.math

On the full, strongly exceptional collections on toric varieties with Picard number three

We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism

A variant of the Mukai pairing via deformation quantization

We give a new method to prove a formula computing a variant of Caldararu's Mukai pairing \cite{Cal1}. Our method is based on some important results in the area of deformation quantization. In particular, part of the work of Kashiwara and Schapira in \cite{KS} as well as an algebraic index theorem of Bressler, Nest and Tsygan in \cite{BNT},\cite{BNT1} and \cite{BNT2} are used. It is hoped that our method is useful for generalization to settings involving certain singular varieties.Comment: 8 pages. Comments and suggestions welcom