Brauer groups and quotient stacks

A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.Comment: American J. Math, to appear. (Latex2e, 17pp

There is no degree map for 0-cycles on Artin stacks

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (i) the degree of an ordinary point is non-zero, and (ii) degrees are compatible with closed immersions.Comment: 3 page

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where G = GL(n,\C), and where $K$ is one of the symmetric subgroups O(n,\C) or Sp(n,\C). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.Comment: Minor revisions and corrections suggested by referees. Final version, to appear in Transformation Group

An extremal effective survey about extremal effective cycles in moduli spaces of curves

We survey recent developments and open problems about extremal effective divisors and higher codimension cycles in moduli spaces of curves.Comment: Submitted to the Proceedings of the Abel Symposium 2017. Comments are welcom

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.Comment: 23 pages; this is an essentially extended version of the previous preprint: the construction of an equivariant cycle (co)homology and the spectral sequence (generalizing the long exact localization sequence) are adde

Tilt Texture Domains on a Membrane and Chirality induced Budding

We study the equilibrium conformations of a lipid domain on a planar fluid membrane where the domain is decorated by a vector field representing the tilt of the stiff fatty acid chains of the lipid molecules, while the surrounding membrane is fluid and structureless. The inclusion of chirality in the bulk of the domain induces a novel budding of the membrane, which preempts the budding induced by a decrease in interfacial tension.Comment: 5 pages, 3 figure

A FRAP model to investigate reaction-diffusion of proteins within a bounded domain: a theoretical approach

Temporally and spatially resolved measurements of protein transport inside cells provide important clues to the functional architecture and dynamics of biological systems. Fluorescence Recovery After Photobleaching (FRAP) technique has been used over the past three decades to measure the mobility of macromolecules and protein transport and interaction with immobile structures inside the cell nucleus. A theoretical model is presented that aims to describe protein transport inside the nucleus, a process which is influenced by the presence of a boundary (i.e. membrane). A set of reaction-diffusion equations is employed to model both the diffusion of proteins and their interaction with immobile binding sites. The proposed model has been designed to be applied to biological samples with a Confocal Laser Scanning Microscope (CLSM) equipped with the feature to bleach regions characterised by a scanning beam that has a radially Gaussian distributed profile. The proposed model leads to FRAP curves that depend on the on- and off-rates. Semi-analytical expressions are used to define the boundaries of on- (off-) rate parameter space in simplified cases when molecules move within a bounded domain. The theoretical model can be used in conjunction to experimental data acquired by CLSM to investigate the biophysical properties of proteins in living cells.Comment: 25 pages. Abstracts Proceedings, The American Society for Cell Biology, 46th Annual Meeting, December 9-13, 2006, San Dieg