Global F-regularity, Fano variety and the finiteness of Frobenius direct images

2021 Fiscal Year Final Research Report

• Project/Area Number: 16K05092
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Agriculture and Technology
• Principal Investigator: Hara Nobuo 東京農工大学, 工学(系)研究科(研究院), 教授 (90298167)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Keywords: 正標数 / フロベニウス直像 / 有限F表現型（FFRT） / 2次元正規次数環 / ファノ多様体 / 大域的F正則 / デルペッツォ曲面 / 代数幾何

Outline of Final Research Achievements

We studied the structure of iterated Frobenius direct images on algebraic varieties and their singularities in positive characteristic, from the viewpoint of the finite F-representation type (FFRT). Our results are as follows.
1. (joint work with Ryo Ohkawa) We studied the FFRT property of 2-dimensional normal graded rings (quasi-homogeneous singularities) in positive characteristic p using the method of algebraic stacks. We proved that a 2-dimensional normal graded ring has FFRT if it has a log terminal singularity; but it does not have FFRT otherwise, except for some exceptional cases depending on p.
2. We studied the FFRT property of the anti-canonical ring of a quintic del Pezzo surface X in positive characteristic. We constructed a self-dual indecomposable vector bundle of rank 3 that appears as a direct summand of self-dual Frobenius direct images on X. We have also shown that the anti-canonical ring has FFRT in characteristics 2 and 3.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

本研究は，正標数p，すなわち素数pについて，任意の数をp回足すと0になってしまう世界で，多項式系の零点集合として定義される図形（代数多様体）の大域的および局所的な性質を研究するものです．正標数の数学は暗号符号などへの応用もありますが，本研究はこれらの応用と直接的には関係せず，正標数特有の時として奇妙にも映る現象の中に，純粋数学的な意義と美しさを見出して，これを探求するものです．