Recent Developments in Commutative Algebra: "F-pure threshold as a measure of singularities"
Presenter
April 16, 2024
Keywords:
- Commutative rings
- modules
- ideals
- mixed characteristic
- Frobenius powers
- test ideals
- tight closure
- perfectoid methods
- singularities
- birational algebraic geometry
- multiplier ideals
- symbolic powers
- syzygies
- free resolutions
- homological methods
- derived categories
- polynomials
- monomial ideals
- toric varieties
- Schubert varieties
- combinatorial commutative algebra
- equivariant ideals
- maximal Cohen-Macaulay modules
- applications of representation theory
- twisted commutative algebras
- D-modules
- local cohomology
- computational commutative algebra
- graded rings and projective varieties
MSC:
- 05Exx - Algebraic combinatorics
- 11Sxx - Algebraic number theory: local fields
- 11Txx - Finite fields and commutative rings (number-theoretic aspects)
- 13-XX - Commutative algebra
- 14-XX - Algebraic geometry
- 16Exx - Homological methods in associative algebras {For commutative rings
- see \newline 13Dxx
- for general categories
- see 18Gxx}
- 18Gxx - Homological algebra in category theory
- derived categories and functors [See also 13Dxx
- 16Exx
- 20Jxx
- 55Nxx
- 55Uxx
- 57Txx]
- 19Axx - Grothendieck groups and $K_0$K_0 [See also 13D15
- 18F30]
- 19Lxx - Topological $K$K-theory [See also 55N15
- 55R50
- 55S25]
- 20Jxx - Connections of group theory with homological algebra and category theory
Abstract
Takagi and Watanabe introduced F-pure threshold as a “Frobenius analogue” of log canonical threshold from birational geometry, showed that F-pure threshold detects whether a local ring is singular, and explored its connections with other invariants. This local study of F-pure threshold, was subsequently further expanded by other authors. I will report on a joint work with Alessandro de Stefani and Luís Núñez-Betancourt, which instead takes the theory in a new direction by studying the geometric properties of F-pure threshold as a singularity measure.