Classification of Q-Fano 3-folds

2023 Fiscal Year Final Research Report

• Project/Area Number: 16K05090
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Gakushuin University (2018-2023); The University of Tokyo (2016-2017)
• Principal Investigator: Takagi Hiromichi 学習院大学, 理学部, 教授 (30322150)
• Project Period (FY): 2016-04-01 – 2024-03-31
• Keywords: Fano 3-fold / Jordan algebra / triple system

Outline of Final Research Achievements

A 3-dimensional projective variety with at most terminal singularities and with ample anti-canonical divisor is called a Fano 3-fold, and one such that the anti-canonical divisor generates the group of numerical equivalence classes of Q-Cartier divisors is called a prime Fano 3-fold. During my research period, I have been actively researching the classification of Fano 3-folds with codimension 4 in weighted projective spaces. It is expected that such Fano 3-folds are divided into 143 classes, and each class have two types of prime Fano 3-folds: one related to P2×P2 and one related to P1×P1×P1. The result obtained within the research period was to construct examples of such prime Fano 3-folds related to P2×P2 for 141 classes systematically.

Free Research Field

代数多様体の分類理論

Academic Significance and Societal Importance of the Research Achievements

Fano 3-foldの分類は、Gino Fanoが非特異な素Fano 3-foldの分類に成功して以来、実りある研究の歴史を持つ。特に、森理論の登場により、Fano 3-foldは、3次元射影多様体のモデルの一つの重要なクラスをなすことが明らかになり、新たな意義を獲得して現在に至る。Fano 3-foldは、有限のクラスに分かれるものの、その数は膨大であることが知られており、その分類の全体像はいまだ明らかになっていない。その中で、当該研究の成果は、余次元４の素Fano 3-foldの例の組織的な構成であり、分類の全体像の一端に迫るものとして、意義あるものと言える。