Interactions between toric varieties and convex polytopes

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K03562
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Nara Medical University
• Principal Investigator: Ryo Kawaguchi 奈良県立医科大学, 医学部, 講師 (10573694)
• Project Period (FY): 2020-04-01 – 2023-03-31
• Keywords: 代数幾何学 / 偏極多様体 / 断面幾何種数 / トーリック多様体 / 凸多面体 / Wierstarss半群

Outline of Final Research Achievements

In this study, we worked on various problems in algebraic geometry and algebraic combinatorics by using the close relationship between toric varieties and convex polytopes. We published a paper on the relation between the sectional genus and the volume of a polytope, which led to exchanges and initiated joint research with polyhedral researchers. While we also obtain the numerical criterion for determining the cyclicness of Weierstrass semigroups whose degree is even number less than or equal to 10.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

図形（代数多様体）を方程式の解集合として捉える代数幾何学において, トーリック多様体は凸多面体論と深いつながりを持った特殊な多様体群であり, 重要な不変量の多くを対応する多面体の形や体積, 格子点の数といった情報から読み取ることができる. 両分野の概念の間にできるだけ多くの接点を見つけることが研究の発展に欠かせない要素であるが, 本研究では代数幾何学における偏極多様体の断面種数やWeierstrass半群といった概念を凸多面体の言葉で解釈する方法を見つけた.