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Kaji Lab
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A lab of algebraic geometry that studies geometric figures inside projective space.
AboutAbout the lab
Exploring the properties of figures in projective space.
Kaji Lab studies algebraic geometry, and within it the projective geometry of algebraic varieties.
The lab investigates figures such as algebraic curves, vector bundles, and homogeneous spaces from the viewpoint of how they behave inside projective space.
Specifically, it studies the structure of secant varieties of projective varieties and the properties of Gauss maps.
It also works on the phenomena peculiar to the special case where the number system of the coefficients has positive characteristic.
Key terms
- Projective variety
- A figure defined using coordinates, considered inside a special space called projective space
- Gauss map
- A map that assigns to each point on a figure the direction tangent at that point
- Positive characteristic
- A property of a number system; the special case in which adding a number to itself repeatedly gives zero
ResearchResearch themes
Within algebraic geometry, the lab studies the projective geometry of algebraic varieties. It investigates what structure figures have inside projective space, using secant varieties and Gauss maps as clues.
*Illustrative imageSecant varieties of projective varieties
The lab studies the structure of secant varieties, the figures formed by the lines joining pairs of points on a projective variety.
*Illustrative imageGauss maps of projective varieties
The lab investigates the properties of Gauss maps, which assign to each point on a projective variety the direction tangent at that point.
*Illustrative imagePhenomena specific to positive characteristic
The lab studies phenomena that appear when the number system of the coefficients has positive characteristic, a special case, and that differ from those in characteristic 0.
MembersFaculty
- ProfessorHajime Kaji
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: “Modell einer Kubik mit vier Doppelpunkten -Schilling VII, 5 - 48” Oliver Zauzig / CC BY-SA 3.0 (cropped)
- Secant varieties of projective varieties: “Worlds First Diagrid Hyperboloid by Shukhov 1896” Sergei Arssenev / CC BY-SA 3.0 (cropped)
- Gauss maps of projective varieties: “Erinaceus roumanicus 2013 G5” George Chernilevsky / Public domain (cropped)
- Phenomena specific to positive characteristic: “Romanesco” Dietmar Rabich / CC BY-SA 4.0 (cropped)
- Related lab (永井研究室): “Klein bottle glass2” PaulT (Gunther Tschuch) / CC BY-SA 4.0 (cropped)
- Related lab (本間研究室): “Spinning top shadow” Armenak Margarian / CC BY-SA 4.0 (cropped)
- Related lab (池田研究室): “Mathematisches-Gipsmodell.Fläche3.Ordnung - Kopie” Museopedia / CC BY-SA 4.0 (cropped)
- Related lab (尾﨑研究室): “Minoritenkirche rose window - Vienna” MrPanyGoff / CC BY-SA 3.0 (cropped)
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