Building on work of Segre and Kollár on cubic hypersurfaces, we construct over imperfect fields of characteristic $p\geq 3$ particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose rational points are Zariski dense are not necessarily unirational. A likewise behavior holds for certain cubic surfaces in characteristic $p=2$.

We give a geometric interpretation of sheaf cohomology for higher degrees $n\geq 1$ in terms of torsors on the member of degree $d=n-1$ in hypercoverings of type $r=n-2$, endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.

Synchrotron-based X-ray techniques are used increasingly to characterize actinide element speciation in heterogeneous media related to nuclear waste disposal safety. Especially techniques offering added temporal, spatial and energy resolved information are advancing our understanding of f-element physics and chemistry in general and of actinide element waste disposal in particular. Examples of investigations of uranium containing systems using both highly (energy) resolved X-ray emission spectroscopy (HRXES) techniques and spatially resolved techniques with focused X-ray beams are presented in this paper: polarization dependent partial fluorescence yield X-ray absorption near edge structure (PD-PFY-XANES) spectroscopic studies of a single Cs2UO2Cl4 crystal, which experimentally reveal a splitting of the σ, π, and δ components of the 6d valence states [1], and characterization of UO2/Mo thin films prepared on different substrates using a combination of techniques (2D and 3D micro- and nano-X-ray fluorescence, XANES and both holographic and ptychographic tomography).

We give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus 1 curves.

Some smooth Calabi–Yau threefolds in characteristic two and three that do not lift to characteristic zero are constructed. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles coincides with the Waldhausen K-theory of perfect complexes. Examples show that, on non-separated schemes, usually many coherent sheaves are not quotients of vector bundles.

Working in characteristic two, we classify nonsmooth Enriques surfaces with normal crossing singularities. Using Kato’s theory of logarithmic structures, we show that such surfaces are smoothable and lift to characteristic zero, provided they are d-semistable.