Relations between some invariants of algebraic varieties in positive characteristic

We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfacesComment: 13 page

Parallel patterns and trends in functional structures in extinct island mammals

Endemic mammalian species on islands are generally known to have followed a different evolutionary pathway than their mainland relatives. General patterns, such as body size trends, have been described regularly. However, most island mammal species are unique and each of them is adapted to a specific local niche as part of an equally specific ecological assemblage. Therefore, comparing island species across taxa, islands and time is inherently dangerous without understanding the adaptational value of the studied feature in the compared taxa and without taking the ecological setting of the taxa into account. In this contribution, general and recurring patterns are described per taxon. Some features, like body mass change and sturdy limbs, are relatively general, whereas most features, like bone fusions and change of orbital axis, occur only in a very few taxa. Some features are even contradictory, such as brain size and degree of hypsodonty, with each taxon having its own particular design. In conclusion, general patterns are more often than not just trends and need to be applied with caution

Effective divisors on projectivized Hodge bundles and modular Forms

We construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a ℙ1‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k‐Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf

Concept of a laser-plasma based electron source for sub-10 fs electron diffraction

We propose a new concept of an electron source for ultrafast electron diffraction with sub-10~fs temporal resolution. Electrons are generated in a laser-plasma accelerator, able to deliver femtosecond electron bunches at 5 MeV energy with kHz repetition rate. The possibility of producing this electron source is demonstrated using Particle-In-Cell simulations. We then use particle tracking simulations to show that this electron beam can be transported and manipulated in a realistic beamline, in order to reach parameters suitable for electron diffraction. The beamline consists of realistic static magnetic optics and introduces no temporal jitter. We demonstrate numerically that electron bunches with 5~fs duration and containing 1.5~fC per bunch can be produced, with a transverse coherence length exceeding 2~nm, as required for electron diffraction