Detecting projectivity in sheaves associated to representations of infinitesimal groups

Let G be an infinitesimal group scheme of finite height r and V(G) the scheme which represents 1-parameter subgroups of G. We consider sheaves over the projectivization P(G) of V(G) constructed from a G-module M. We show that if P(G) is regular then the sheaf H^[1](M) is zero if and only if M is projective. In general, H^[1] defines a functor from the stable module category and we prove that its kernel is a thick triangulated subcategory. Finally, we give examples of G such that P(G) is regular and indicate, in characteristic 2, the connection to the BGG correspondence. Along the way we will provide new proofs of some known results and correct some errors in the literature.Comment: 26 pages, grant acknowledgement, journal reference, and DOI added to previous versio

Director Field Configurations around a Spherical Particle in a Nematic Liquid Crystal

We study the director field around a spherical particle immersed in a uniformly aligned nematic liquid crystal and assume that the molecules prefer a homeotropic orientation at the surface of the particle. Three structures are possible: a dipole, a Saturn-ring, and a surface-ring configuration, which we investigate by numerically minimizing the Frank free energy supplemented by a magnetic-field and a surface term. In the dipole configuration, which is the absolutely stable structure for micron-size particles and sufficiently strong surface anchoring, a twist transition is found and analyzed. We show that a transition from the dipole to the Saturn ring configuration is induced by either decreasing the particle size or by applying a magnetic field. The effect of metastability and the occurence of hysteresis in connection with a magnetic field are discussed. The surface-ring configuration appears when the surface-anchoring strength W is reduced. It is also favored by a large saddle-splay constant K_24. A comparison with recent experiments by Poulin et al. gives a lower bound for W, i.e., W > 0.6 erg/cm^2 for the interface of water and pentylcyanobiphenyl (5CB) in the presence of the surfactant sodium dodecyl sulfate.Comment: 11 pages, Revtex, 16 eps figures, submitted to Eur. Phys. J.