Research on singularities of algebraic varieties

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03428
• 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: The University of Tokyo
• Principal Investigator: Ishii Shihoko 東京大学, 大学院数理科学研究科, 特任教授 (60202933)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: singularities / minimal log discrepancy / log canonical threshold / arc space

Outline of Final Research Achievements

I researched invariants of a pair consisting of a smooth variety and a coherent ideal. The achievements are the following:
1. If the ideal is ``general" on a 3-dimensional variety, then the invariant mld is computed by two weighted blowups.
2. We associate a pair in positive characteristic to a pair in characteristic 0, that inherits the properties of the pair. Here, the pair in characteristic 0 consists of a fractional ideal instead of an ideal. The next goal is to study fractional ideals in characteristic 0.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

対の不変数は無限個の対象を使って定義されることが多い．例えば mld は全てのprime divisor 上のlog discrepancies の infimum で定義される．これを有限個の対象を調べるだけで不変数が計算できる，ということを示しているのが上記 １　の結果である．
標数０の多様体については，色々な良い性質が成り立ち，対の不変数についても計算がしやすいが，正標数の多様体については多くの困難がある．上記２の結果は，困難な正標数の研究をより易しい標数０の研究に帰着させる第一歩である．