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

Transforming Popular Consciousness through the Sacralisation of the Western School:: The Meiji Schoolhouse and Tennō Worship

Im Zentrum der ideenpolitischen Bemühungen, die die japanischen Eliten der Meiji-Ära (1868– 1912) unternahmen, um das Land in modernem Gewand, doch auf traditionellen Grundlagen zur Nation zu formen, standen der Tennō-Kult und die allgemeine staatliche Pflichtschule. Zwischen der Verabschiedung der Verfassung des Kaiserlichen Japan im Jahr 1889 und dem Erlass des Kaiserlichen Erziehungs-Edikts im Jahr 1890 bestand insofern ein enger politisch-ideeller Zusammenhang. Der Artikel verfolgt die konflikthaften Auseinandersetzungen, die im letzten Drittel des 19. Jahrhunderts zwischen den Verteidigern altjapanischer Bildungstradition und den Anhängern westlicher Aufklärung, zwischen konfuzianischen Intellektuellen und westlich orientierten Demokraten, ausgetragen wurden. Er entwickelt vor diesem Hintergrund die weder selbstverständliche noch einlinige Genese der normativen Grundlagen des modernen Japan. Die Autoren beschreiben die wechselnden Inszenierungen, die in diesem Zusammenhang dem Tennō zuteil wurden: von seiner anfänglichen Rolle als öffentlich sichtbarem Symbol der Einheit der Nation (auf landesweiten Rundreisen oder Militärparaden) über den völligen Rückzug hinter die Palastmauern (in Konsequenz seiner Sakralisierung durch den erstarkenden Staats-Shintō) hin zu seiner indirekten Rückkehr in den öffentlichen Raum (in Form eines hochstilisierten und kultisch verehrten Konterfeis). Und sie schildern – unter Einbezug auch autobiographischer Reminiszenzen – sowohl die aus buddhistischer Tradition übernommene sakrale Überhöhung der modernen Schule wie die zeremoniellen Präsentationsformen des kultisch verehrten Tennō-Bildes und die von ihm ausgehenden sozialintegrativen und loyalitätsstiftenden Wirkungen

Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: The minimal dimension of a canonical measure)

In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler. He shows that the minimal dimension of the components of a canonical measure is equal to the dimension of the abelian part of the subvariety. We can apply this result to make a further contribution to the geometric Bogomolov conjecture.Comment: 30 page

Improving Teaching Methods, Student Learning Outcomes, and Curricula through Cross-Phase Teaching in Primary and Junior High School

In order to obtain a certain hint of forming a good cooperation between primary and junior high schools through Cross-Phase Teaching (CPT), it is necessary to verify the effect of CPT. We analyzed our CPT, surveyed the teachers and students by questionnaire, and interviewed the teachers, so that we might verify the effect of the CPT. While verifying, we not only recognized the change of the teachers' mind but also found the concrete way of improving their teaching