反射的凸多面体を中心とした格子凸多面体の分類理論及び正規性に関する探究

• Project/Area Number: 19J00312
• Research Category: Grant-in-Aid for JSPS Fellows
• Allocation Type: Single-year Grants
• Section: 国内
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: 土谷 昭善 東京大学, 大学院数理科学研究科, 特別研究員(PD)
• Project Period (FY): 2019-04-25 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 格子凸多面体 / 正規凸多面体 / トーリック多様体 / 反射的凸多面体 / Fano多様体 / Castelnuovo多様体 / トーリックイデアル / 正規性 / 完全縮約グラフ / Nef分割 / 理想グラフ

Outline of Research at the Start

本研究は，格子凸多面体の研究における現在の流行である，格子凸多面体の分類問題への取り組み，また格子凸多面体の正規性に関する2つの予想，「小田予想」と「Stanley予想 」を解決することを目的としている．特に，可換代数や代数幾何と繋がりが深い反射的凸多面体に注目する，具体的な研究計画としては，(1)中心的対称な反射的凸多面体の分類アルゴリズムの開発，(2)正規反射的 凸多面体に付随するδ多項式がunimodalであることの証明，そして(3)非特異反射的凸多面体が正規であることの証明，に取り組み，上記問題の解決を目指す.

Outline of Annual Research Achievements

本研究は格子凸多面体の分類問題と正規性に関する問題を，特に反射的凸多面体に関して調べるものである．本年度は特に反射的凸多面体の面の構造について調べた．これまで知られている結果として，任意の格子凸多面体はある反射的凸多面体の面になっている．この結果に注目して以下の問題を考えた．
問題：性質Aを持つ任意の格子凸多面体は，性質Aを持つある反射的凸多面体の面となるか．
この問題を特に，性質Aとして正規性や非特異性について考えた結果，正しいことが証明できた．この結果の応用として，まず正規ではあるが，正則単模三角形を持たない反射的凸多面体の存在を示すことができる．つまり，特殊な格子凸多面体の例から，同じ特殊な性質を持つ反射的凸多面体を構成することが可能となった．またもう一つの応用として，本研究の最終目標である小田予想「非特異格子凸多面体は正規である」に対して，次のようなことがわかった．
定理「小田予想は非特異反射的凸多面体に対して証明すれば十分である．」
非特異凸多面体は各次元に無限個存在するが，非特異反射的凸多面体は各次元に有限個しかないため，調べるべき対象が極めて少なくなった．さらにこの定理から，代数幾何学的には，小田予想は「トーリックFano多様体は常に射影的正規である」を示せれば十分であることがわかる．この結果をベルリン自由大学のChristian Haase氏とともに共著論文として現在執筆中である．

Research Progress Status

令和3年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和3年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] Rigid Gorenstein toric Fano varieties arising from directed graphs (2022)
  • Author(s): Kara Selvi、Portakal Irem、Tsuchiya Akiyoshi
  • Journal Title: Collectanea Mathematica Volume: － Issue: 2 Pages: 333-351
  • DOI: 10.1007/s13348-022-00350-z
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Edge rings with q-linear resolutions (2022)
  • Author(s): Mori Kenta、Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Journal of Algebra Volume: 593 Pages: 550-567
  • DOI: 10.1016/j.jalgebra.2021.11.018
  • Peer Reviewed / Open Access
• [Journal Article] Castelnuovo polytopes (2021)
  • Author(s): Akiyoshi Tsuchiya
  • Journal Title: Michigan Mathematical Journal Volume: －
  • Peer Reviewed
• [Journal Article] Symmetric edge polytopes and matching generating polynomials (2021)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Combinatorial Theory Volume: 1 Issue: 0
  • DOI: 10.5070/c61055371
  • Peer Reviewed / Open Access
• [Journal Article] Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs (2021)
  • Author(s): Tran Tan Nhat、Tsuchiya Akiyoshi
  • Journal Title: Comptes Rendus. Mathematique Volume: 359 Issue: 6 Pages: 665-674
  • DOI: 10.5802/crmath.210
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Regularity and a-invariant of Cameron?Walker graphs (2021)
  • Author(s): Hibi Takayuki、Kimura Kyouko、Matsuda Kazunori、Tsuchiya Akiyoshi
  • Journal Title: Journal of Algebra Volume: 584 Pages: 215-242
  • DOI: 10.1016/j.jalgebra.2021.05.007
  • Peer Reviewed
• [Journal Article] The h*-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their γ-Positivity (2021)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Discrete & Computational Geometry Volume: 66 Issue: 2 Pages: 701-722
  • DOI: 10.1007/s00454-020-00236-6
  • Peer Reviewed / Open Access
• [Journal Article] Gorenstein polytopes with trinomial h*-polynomials (2021)
  • Author(s): Akihiro Higashitani, Benjamin Nill and Akiyoshi Tsuchiya
  • Journal Title: Beitrage zur Algebra und Geometrie Volume: ー Issue: 3 Pages: 667-685
  • DOI: 10.1007/s13366-020-00513-8
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Rigid Gorenstein toric Fano varieties arising from directed graphs (2021)
  • Author(s): Selvi Kara、Irem Portakal、Akiyoshi Tsuchiya
  • Journal Title: arXiv:2103.0640 Volume: －
  • Open Access / Int'l Joint Research
• [Journal Article] PQ-type adjacency polytopes of join graphs (2021)
  • Author(s): Hidefumi Ohsugi、Akiyoshi Tsuchiya
  • Journal Title: arXiv:2103.15045 Volume: －
  • Open Access
• [Journal Article] Classification of lattice polytopes with small volumes (2020)
  • Author(s): Hibi Takayuki、Tsuchiya Akiyoshi
  • Journal Title: Journal of Combinatorics Volume: 11 Issue: 3 Pages: 495-509
  • DOI: 10.4310/joc.2020.v11.n3.a4
  • Peer Reviewed
• [Journal Article] Levelness of Order Polytopes (2020)
  • Author(s): Haase Christian、Kohl Florian、Tsuchiya Akiyoshi
  • Journal Title: SIAM Journal on Discrete Mathematics Volume: 34 Issue: 2 Pages: 1261-1280
  • DOI: 10.1137/19m1292345
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Edge rings of bipartite graphs with linear resolutions (2020)
  • Author(s): Tsuchiya Akiyoshi
  • Journal Title: Journal of Algebra and Its Applications Volume: － Issue: 09 Pages: 2150163-2150163
  • DOI: 10.1142/s0219498821501632
  • Peer Reviewed
• [Journal Article] Nef‐partitions arising from unimodular configurations (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Mathematische Nachrichten Volume: 293 Issue: 9 Pages: 1791-1800
  • DOI: 10.1002/mana.201900347
  • Peer Reviewed / Open Access
• [Journal Article] Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Selecta Mathematica Volume: 26 Issue: 4 Pages: 59-59
  • DOI: 10.1007/s00029-020-00588-0
  • Peer Reviewed / Open Access
• [Journal Article] Integer decomposition property for Cayley sums of order and stable set polytopes (2020)
  • Author(s): Takayuki Hibi, Hidefumi Ohsugi, Akiyoshi Tsuchiya
  • Journal Title: Michigan Mathematical Journal Volume: 掲載確定 Issue: 4 Pages: 765-778
  • DOI: 10.1307/mmj/1585792887
  • Peer Reviewed
• [Journal Article] Enriched order polytopes and enriched Hibi rings (2020)
  • Author(s): Hidefumi Ohsugi, Akiyoshi Tsuchiya
  • Journal Title: European Journal of Mathematics Volume: published online Issue: 1 Pages: 48-68
  • DOI: 10.1007/s40879-020-00403-2
  • Peer Reviewed / Open Access
• [Journal Article] Symmetric edge polytopes and matching generating polynomials (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: arXiv:2008.08621 Volume: －
  • Open Access
• [Journal Article] Edge rings with q-linear resolutions (2020)
  • Author(s): Kenta Mori、Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: arXiv:2010.02854 Volume: －
  • Open Access
• [Journal Article] Castelnuovo polytopes (2020)
  • Author(s): Akiyoshi Tsuchiya
  • Journal Title: arXiv:2010.13617 Volume: －
  • Open Access
• [Journal Article] Enriched chain polytopes (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Israel Journal of Mathematics Volume: － Issue: 1 Pages: 485-500
  • DOI: 10.1007/s11856-020-2012-1
  • Peer Reviewed
• [Journal Article] Edge rings of bipartite graphs with linear resolutions (2020)
  • Author(s): Tsuchiya Akiyoshi
  • Journal Title: Journal of Algebra and its Applications Volume: ー
  • Peer Reviewed
• [Journal Article] Nef-partitions arising from unimodular configurations (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Mathematische Nachrichten Volume: ー
  • Peer Reviewed
• [Journal Article] The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity (2020)
  • Author(s): Ohsugi Hidefumi、Tsuchiya Akiyoshi
  • Journal Title: Discrete and Computational Geometry Volume: ー
  • Peer Reviewed
• [Journal Article] Two enriched poset polytopes (2020)
  • Author(s): Okada Soichi、Tsuchiya Akiyoshi
  • Journal Title: arXiv:2003.12271 Volume: ー
  • Open Access
• [Journal Article] Odd Cycles and Hilbert Functions of Their Toric Rings (2019)
  • Author(s): Hibi Takayuki、Tsuchiya Akiyoshi
  • Journal Title: Mathematics Volume: 8 Issue: 1 Pages: 22-22
  • DOI: 10.3390/math8010022
  • Peer Reviewed / Open Access
• [Journal Article] Gorenstein simplices with a given δ-polynomial (2019)
  • Author(s): Hibi Takayuki、Tsuchiya Akiyoshi、Yoshida Koutarou
  • Journal Title: Discrete Mathematics Volume: 342 Issue: 12 Pages: 111619-111619
  • DOI: 10.1016/j.disc.2019.111619
  • Peer Reviewed
• [Journal Article] Perfectly contractile graphs and quadratic toric rings (2019)
  • Author(s): Ohsugi Hidefumi、Shibata Kazuki、Tsuchiya Akiyoshi
  • Journal Title: arXiv:1911.12345 Volume: ー
  • Open Access
• [Presentation] Ehrhart theory on adjacency polytopes (2022)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: Mini-Symposium on Lattice Polytopes
  • Int'l Joint Research / Invited
• [Presentation] Castelnuovo凸多面体 (2021)
  • Author(s): 土谷昭善
  • Organizer: オンライン可換環論セミナー
• [Presentation] Castelnuovo凸多面体 (2021)
  • Author(s): 土谷昭善
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] グラフに付随するGorensteinトーリックFano多様体の剛性 (2021)
  • Author(s): 土谷昭善
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Castelnuovo toric varieties (2021)
  • Author(s): 土谷昭善
  • Organizer: 特異点論セミナー
  • Invited
• [Presentation] Rigid Gorenstein toric Fano varieties arising from finite graphs (2021)
  • Author(s): 土谷昭善
  • Organizer: 第42回可換環論シンポジウム
• [Presentation] Castelnuovo polytopes (2021)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: (Polytop)ics: Recent advances on polytopes
  • Int'l Joint Research
• [Presentation] Canonical triangulations of enriched order polytopes (2021)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: AlCoVE: an Algebraic Combinatorics Virtual Expedition
  • Int'l Joint Research / Invited
• [Presentation] Ehrhart theory of locally anti-blocking lattice polytopes (2021)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: 12th polymake conference
  • Int'l Joint Research / Invited
• [Presentation] 対称辺凸多面体とマッチング生成多項式 (2021)
  • Author(s): 土谷昭善
  • Organizer: 日本数学会年会
• [Presentation] Perfectly contractile graphs and quadratic toric rings (2020)
  • Author(s): 土谷昭善
  • Organizer: オンライン研究集会 組合せ論と可換環論
  • Invited
• [Presentation] 対称辺凸多面体の体積公式と蔵元モデルとの関係 (2020)
  • Author(s): 土谷昭善
  • Organizer: 2020年度応用数学合同研究集会
• [Presentation] Nef-partitions arising from unimodular configurations (2020)
  • Author(s): 土谷 昭善
  • Organizer: 特異点論セミナー
  • Invited
• [Presentation] Construction of nef-partitions via the technique of Groebner bases (2020)
  • Author(s): 土谷 昭善
  • Organizer: 東京可換環論セミナー
  • Invited
• [Presentation] Two enriched poset polytopes (2019)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: Oberseminar DIGO: Diskrete Mathematik, Geometrie und Optimierung (Goethe-Universitat Frankfurt)
  • Int'l Joint Research / Invited
• [Presentation] Two enriched poset polytopes (2019)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: Toric Topology 2019 in Okayama
  • Int'l Joint Research / Invited
• [Presentation] Locally anti-blocking lattice polytopes and their $h^*$-polynomials (2019)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: Commutative Algebra and Lattice Polytopes
  • Int'l Joint Research / Invited
• [Presentation] Gamma-positivity in Ehrhart theory (2019)
  • Author(s): 土谷 昭善
  • Organizer: Discrete Geometric Structure Seminar
  • Invited
• [Presentation] 反射的凸多面体とグレブナー基底 (2019)
  • Author(s): 土谷 昭善
  • Organizer: 組合せ論と可換代数オータムセミナー
  • Invited
• [Presentation] Ehrhart theory and interior polynomials (2019)
  • Author(s): Akiyoshi Tsuchiya
  • Organizer: Recent advances in matroids and Tutte polynomials (Hokkaido University)
  • Int'l Joint Research
• [Presentation] Enriched Hibi rings (2019)
  • Author(s): 土谷 昭善
  • Organizer: 東京可換環論セミナー
  • Invited
• [Presentation] Two enriched poset polytopes (2019)
  • Author(s): 土谷 昭善
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Enriched Hibi ring (2019)
  • Author(s): 土谷 昭善
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] 格子凸多面体に付随する$h^*$多項式と二部グラフに付随する内部多項式 (2019)
  • Author(s): 土谷 昭善
  • Organizer: Japanese Conference on Combinatorics and its Applications (JCCA2019), ミニシンポジウム（グラフ理論）
  • Invited
• [Presentation] Two enriched poset polytopes (2019)
  • Author(s): 土谷 昭善
  • Organizer: COMAセミナー
  • Invited
• [Presentation] Cayley sums of lattice polytopes and Oda's question (2019)
  • Author(s): 土谷 昭善
  • Organizer: 特異点論セミナー
  • Invited