Survey on the geometric Bogomolov conjecture

This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on non-archimedean analytic geometry.Comment: 57 pages. This is an expanded lecture note of a talk at "Non-archimedean analytic Geometry: Theory and Practice" (24--28 August, 2015). It has been submitted to the conference proceedings. Appendix adde

Algebraic rank on hyperelliptic graphs and graphs of genus 33

Let $\bar{G} = (G, \omega)$ be a vertex-weighted graph, and $\delta$ a divisor class on $G$. Let $r_{\bar{G}}(\delta)$ denote the combinatorial rank of $\delta$. Caporaso has introduced the algebraic rank $r_{\bar{G}}^{\operatorname{alg}}(\delta)$ of $\delta$, by using nodal curves with dual graph $\bar{G}$. In this paper, when $\bar{G}$ is hyperelliptic or of genus $3$, we show that $r_{\bar{G}}^{\operatorname{alg}}(\delta) \geq r_{\bar{G}}(\delta)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a non-hyperelliptic curve of genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.Comment: 16 page

Rank of divisors on hyperelliptic curves and graphs under specialization

Let $(G, \omega)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, \omega)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with reduction graph $(G, \omega)$ such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph $(G, \omega)$ in general, how the existence of such $X$ relates the Riemann--Roch formulae for $X$ and $(G, \omega)$, and also how the existence of such $X$ is related to a conjecture of Caporaso.Comment: 34 pages. The proof of Theorem 1.13 has been significantly simplifie

Solvent resistant thermoplastic aromatic poly(imidesulfone) and process for preparing same

A process for preparing a thermoplastic poly(imidesulfone) is disclosed. This resulting material has thermoplastic properties which are generally associated with polysulfones but not polyimides, and solvent resistance which is generally associated with polyimides but not polysulfones. This system is processable in the 250 to 350 C range for molding, adhesive and laminating applications. This unique thermoplastic poly(imidesulfone) is obtained by incorporating an aromatic sulfone moiety into the backbone of an aromatic linear polyimide by dissolving a quantity of a 3,3',4,4'-benzophenonetetracarboxylic dianhydride (BTDA) in a solution of 3,3'-diaminodiphenylsulfone and bis(2-methoxyethyl)ether, precipitating the reactant product in water, filtering and drying the recovered poly(amide-acid sulfone) and converting it to the poly(imidesulfone) by heating

Process for preparing solvent resistant, thermoplastic aromatic poly(imidesulfone)

A process for preparing a thermoplastic poly(midesulfone) is disclosed. This resulting material has thermoplastic properties which are generally associated with polysulfones but not polyimides, and solvent resistant which is generally associated with polyimides but not polysulfones. This system is processable in the 250 to 350 C range for molding, adhesive and laminating applications. This unique thermoplastic poly(imidesulfone) is obtained by incorporating an aromatic sulfone moiety into the backbone of an aromatic linear polyimide by dissolving a quantity of a 3,3',4,4'-benzophenonetetracarboxylic dianhydride (BTDA) in a solution of 3,3'-diaminodiphenylsulfone and bis(2-methoxyethyl)ether, precipitating the reactant product in water, filtering and drying the recovered poly(amide-acid sulfone) and converting it to the poly(imidesulfone) by heating