Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces
AffiliationDepartment of Mathematics, University of Arizona
KeywordsElliptic divisibility sequences
Geography of surfaces
MetadataShow full item record
PublisherSpringer Science and Business Media LLC
CitationUlmer, D., & Urzúa, G. (2022). Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces. Selecta Mathematica, New Series, 28(2).
JournalSelecta Mathematica, New Series
Rights© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021.
Collection InformationThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at firstname.lastname@example.org.
AbstractWe consider elliptic surfaces E over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or p> 3 and E is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and K2 unbounded.
Note12 month embargo; published: 30 December 2021
VersionFinal accepted manuscript
Showing items related by title, author, creator and subject.
The development of a mass spectrometry-based technique that uses low energy ion-surface collisions to characterize surfacesWysocki, Vicki H.; Angelico, Vincent James (The University of Arizona., 2002)Low energy (tens of eV) ion-surface collisions carried out in a tandem mass spectrometer are investigated as a tool to characterize self-assembled monolayer (SAM) films. The target films are prepared by spontaneous chemisorption of thiol-based (HS-R) compounds onto Au (111) substrates. Most of the films used as targets contain alkane or fluoro-alkane backbones, some with unique groups in the terminal position (e.g., -CD₃, -OH, -OC(O)CF₃). Pyrazine is the most frequently used probe ion, however in certain cases other small organic molecules are also used. Common interactions between the impinging ion and the target film that vary as a function of film characteristics include, but are not limited to, reactive scattering, neutralization and T → V conversion. Pyrazine ion readily reacts when colliding with hydrocarbon films at 20-eV, forming product ions that incorporate a hydrogen atom or a methyl group. Several examples of the utility of these processes to characterize film properties are presented. For hydrocarbon films, ion-surface reactions of pyrazine ion resulting in addition of a hydrogen atom or a methyl group are shown to vary with the quality, chemical composition and orientation of the target film. Experiments with isotopically labeled films show that the ion beam interacts predominantly with the end groups of the film, however interactions with underlying groups increase as the film or substrate quality decreases. The orientation difference of odd and even chain length n-alkanethiols produces a measurably different degree of hydrogen addition with the higher free energy odd chain length orientation being more reactive. The composition of mixed component films (H, D or H, F) is tracked by measuring the abundance of unique reaction products, energy transfer (translational to vibrational conversion) and charge exchange properties. When mixed films containing deuterium labeled and unlabeled n-alkanethiols are subjected to collisions of 20-eV pyrazine ion, the D-addition ion abundance increases linearly with the surface concentration of D-containing alkane chains. When mixed films containing different ratios of H and F components are the target, several processes track with the changing population of surface species. As the target films become more fluorocarbon in nature H-addition decreases, total ion current reaching the detector increases, and dissociation increases. Several properties of electron transfer from the film to the ion are examined. When the probe ion and collision energy remain consant, charge exchange is shown to be primarily governed by the work function of the film and the thickness of the adsorbed layer. Fluorocarbon films, which have a higher work function than hydrocarbon films, consistently show less charge exchange. When comparing hydrocarbon films of varying chain lengths (ranging from 15 to 18 carbons), a increase of ∼1% in total ion current measured at the detector is observed for each additional methylene in the chain.
Near Surface Composition and Reactivity of Indium Tin Oxide: An Evaluation Towards Surface Chemical Concepts and Relevance in Titanyl Phthalocyanine Photovoltaic DevicesArmstrong, Neal R.; Brumbach, Michael T.; Armstrong, Neal R.; Wysocki, Vicki H.; Pemberton, Jeanne E.; Zheng, Zhiping; Lichtenberger, Dennis L. (The University of Arizona., 2007)Photovoltaics manufactured using organic materials as a substitute for inorganic materials may provide for cheaper production of solar cells if their efficiencies can be made comparable to existing technologies. Photovoltaic devices are comprised of layered structures where the electrical, chemical, and physical properties at the multiple interfaces play a significant role in the operation of the completed device. This thesis attempts to establish a relationship between interfacial properties and overall device performance by investigation of both the organic/organic heterojunction interface, as well as the interface between the inorganic substrate and the first organic layer with useful insights towards enhancing the efficiency of organic solar cells.It has been proposed that residual chemical species may act as barriers to charge transfer at the interface between the transparent conductor (TCO) and the first organic layer, possibly causing a large contact resistance and leading to reduced device performance. Previous work has investigated the surface of the TCO but no baseline characterization of carbon-free surfaces has previously been given. In this work clean surfaces are investigated to develop a fundamental understanding of the intrinsic spectra such that further analyses of contaminated surfaces can be presented systematically and reproducibly to develop a chemical model of the TCO surface.The energy level offset at the organic/organic heterojunction has been proposed to relate to the maximum potential achievable for a solar cell under illumination, however, few experimental observations have been made where both the interface characterization and device performance are presented. Photovoltaic properties are examined in this work with titanyl phthalocyanine used as a novel donor material for enhancement of spectral absorption and optimization of the open-circuit potential. Characterization of the interface between TiOPc and C60 coupled with characterization of the interface between copper phthalocyanine and C60 shows that the higher ionization potential of TiOPc does correlate to greater open circuit potentials.Examination of photovoltaic behavior using equivalent circuit modeling relates the importance of series resistance and recombination to the homogeneity of the solar cell structure.