Characteristic polyhedra of singularities without completion -- Part II

Hironaka's characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition $(\mathrm{Pol})$ holds, or a mild condition $(*)$ on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.Comment: 46 pages; minor change

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Koll\'ar-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than $4$ are K-semistable.Comment: 21 pages. Thorough rewrite following referee's suggestions. To be published in Manuscripta Mathematic

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.Comment: 26 pages; minor changes have been adde

Ultra-thin rigid endoscope: Two-photon imaging through a graded-index multi-mode fiber

Rigid endoscopes like graded-index (GRIN) lenses are known tools in biological imaging, but it is conceptually difficult to miniaturize them. In this letter, we demonstrate an ultra-thin rigid endoscope with a diameter of only 125 microns. In addition, we identify a domain where two-photon endoscopic imaging with fs-pulse excitation is possible. We validate the ultra-thin rigid endoscope consisting of a few cm of graded-index multi-mode fiber by using it to acquire optically sectioned two-photon fluorescence endoscopic images of three-dimensional samples.Comment: 17 pages, 15 figures, submitted to Opt. Expres

Manipulation of ultracold atomic mixtures using microwave techniques

We used microwave radiation to evaporatively cool a mixture of of 133Cs and 87Rb atoms in a magnetic trap. A mixture composed of an equal number (around 10^4) of Rb and Cs atoms in their doubly polarized states at ultracold temperatures was prepared. We also used microwaves to selectively evaporate atoms in different Zeeman states.Comment: 9 pages, 6 figure

Weights in arithmetic geometry

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g., to Hasse principles and the computation of motivic cohomology, and discuss some open questions.Comment: 27 page