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Applied Mathematics Division (Kawakita) Lab
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A lab that studies the birational geometry of algebraic varieties using the minimal model program.
AboutAbout the lab
Birational geometry of algebraic varieties, through the minimal model program.
The Kawakita Lab studies the birational geometry of algebraic varieties, within the field of algebraic geometry.
The minimal model program is a method that picks out a representative variety from each birational equivalence class by comparing canonical divisors.
The lab works on the classification of threefold divisorial contractions, and on research guided by minimal log discrepancies, an invariant of singularities.
Building on the research so far, a book has also been written.
Key terms
- Birational geometry
- Geometry that studies algebraic varieties by regarding them as the same when they can be transformed into each other by maps that give a one-to-one correspondence at almost all points
- Minimal model program
- A procedure that extracts a representative variety from each birational equivalence class through a sequence of step-by-step operations
- Singularity
- A point where a shape is not smooth, such as where it has a sharp point or crosses itself
ResearchResearch themes
The lab studies the birational geometry of algebraic varieties using the minimal model program. The minimal model program extracts a representative variety from each birational equivalence class by comparing canonical divisors.
*Illustrative imageClassification of threefold divisorial contractions
To understand three-dimensional birational geometry explicitly, the lab systematically studied threefold divisorial contractions. In the course of this work, it also proved Reid's general elephant conjecture.
*Illustrative imageTermination of flips and minimal log discrepancies
From this standpoint, the lab studies the singularities that appear in the course of the minimal model program, using minimal log discrepancies as a guide.
*Illustrative imageInversion of adjunction
When a new pair on a divisor is introduced from a pair consisting of a variety and a divisor, inversion of adjunction is the problem of comparing the singularities of the two pairs. The lab proved that the two pairs are equivalent with respect to log canonicity.
*Illustrative imageBoundedness of singularities
The lab analyzed Artinian rings given by hyperplane sections. It also proved Shokurov's conjecture that the Gorenstein index of a genuinely three-dimensional canonical singularity is at most 6.
MembersFaculty
- Associate ProfessorMasayuki Kawakita
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Du Val singularities” Aizenr / CC BY-SA 4.0 (cropped)
- Classification of threefold divisorial contractions: “Sarti's Octic with 72” Salix alba / CC BY-SA 4.0 (cropped)
- Termination of flips and minimal log discrepancies: “Surface of degree 7” Jean Constant / CC BY-SA 4.0 (cropped)
- Inversion of adjunction: “DoubleCone” Lars H. Rohwedder (User:RokerHRO) / Public domain (cropped)
- Boundedness of singularities: “Whitney umbrella” Shmurak / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Lab (Fukuhiro Ueda)): “Eisenstein 14” Jan Homann / Public domain (cropped)
- Related lab (Hikita Lab): “Young tableau for partitions of number 10 stacked” Unmismoobjetivo / CC BY 4.0 (cropped)
- Related lab (Applied Mathematics Lab (Kazuhisa Makino)): “B functies 2var versie02” Bob.v.R / CC BY-SA 4.0 (cropped)
- Related lab (Isono Lab): “Bernoullishift” Andreas Mandl / CC BY-SA 3.0 (cropped)
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