Rationality of Euler-Chow series and finite generation of Cox rings

In this paper we work with a series whose coefficients are the Euler characteristic of Chow varieties of a given projective variety. For varieties where the Cox ring is defined, it is easy to see that in this case the ring associated to the series is the Cox ring. If this ring is noetherian then the series is rational. It is an open question whether the converse holds. In this paper we give an example showing the converse fails. However we conjecture that it holds when the variety is rationally connected. As an evidence of this conjecture, It is proved that the series is not rational, and in a sense defined, not algebraic, in the case of the blowup of the projective plane at nine or more points in general position. Furthermore, we also treat some other examples of varieties with infinitely generated Cox ring, studied by Mukai and Hassett-Tschinkel. These are the first examples known where the series is not rational. We also compute the series for Del Pezzo surfaces.Comment: 26 pages. In this last version we correct many typos and add a cite of a work of Artebani and Laface in Theorem 1.6 which was brought to our attention. More typo correction

Fluorous membrane-based separations and reactions

Porous alumina membranes were rendered compatible with fluorous liquids by surface modification with a carboxylic acid terminated perfluoropolyether (Krytox 157FSH). FTIR and contact angle measurements demonstrate the success of the modification.Fluorous liquids are readily imbibed by modified alumina membranes, resulting in fluorous supported liquid membranes. Fluorine-containing organic solutes are selectively transported through the fluorous supported liquid membranes. Selectivity is defined as the permeability of a fluorous tagged solute over an analogous organic compound. The membrane modification conditions (reagents, concentrations, reaction time) were optimized to maximize the selectivity. The membrane pore size affects the solute permeabilities and selectivities.Two series of homologous esters of perfluoroalkanoic acids with different organic moieties were studied. Permeability increased for both series as the perfluoroalkyl chain was lengthened. This shows that the difference in permeabilities is dominated by partitioning rather than diffusion. We further measured the partition coefficients of the homologs. The free energy of transfer of a ¨CCF2¨C group (ethanol to perfluorinated solvents) is -1.1 kJ/mol. The experimental values of the partition coefficients are well correlated with the ¡®mobile order and disorder¡¯ theory. This provides an easy way to estimate partition coefficients in any biphasic system, even for solvents that are fluorous mixtures. Diffusion coefficients were determined from permeabilities and partition coefficients based on the solution-diffusion model of permeability. Measured values are satisfactorily related to the Stokes-Einstein equation, especially for higher homologs. This investigation enables the prediction of transport properties of the fluorous supported alumina membranes.Krytox 157FSH, which is virtually insoluble in any but fluorous solvents, was deposited on the fluorous-modified alumina membranes. FTIR shows the presence of the H-bond-based carboxylic acid dimers for these adsorbed Krytox 157FSH molecules. The carboxylate groups of Krytox 157FSH extract cations from aqueous solutions

Purity of the stratification by Newton polygons and Frobenius-periodic vector bundles

This thesis includes two parts. In the first part, we show a purity theorem for stratifications by Newton polygons coming from crystalline cohomology, which says that the family of Newton polygons over a noetherian scheme have a common break point if this is true outside a subscheme of codimension bigger than 1. The proof is similar to the proof of [dJO99, Theorem 4.1]. In the second part, we prove that for every ordinary genus-2 curve X over a finite field k of characteristic 2 with automorphism group Z/2Z × S_3, there exist SL(2,k[[s]])-representations of π_1(X) such that the image of π_1(X^-) is infinite. This result produces a family of examples similar to Laszlo's counterexample [Las01] to a question regarding the finiteness of the geometric monodromy of representations of the fundamental group [dJ01]

An improvement of de Jong-Oort’s purity theorem

Consider an F-crystal over a noetherian scheme S. De Jong-Oort’s purity theorem states that the associated Newton polygons over all points of S are constant if this is true outside a subset of codimension bigger than 1. In this paper we show an improvement of the theorem, which says that the Newton polygons over all points of S have a common break point if this is true outside a subset of codimension bigger than 1