Shigefumi Mori

Shigefumi Mori is a Japanese mathematician whose work established key parts of the minimal model program for three-dimensional algebraic varieties. He received the Fields Medal in 1990.

Connect

Algebraic geometry: The study of geometric objects defined by polynomial equations, using tools from algebra and geometry. Mori’s research developed within this broad field.
Minimal model program: A framework for simplifying projective varieties through birational transformations and classifying their outcomes. Mori proved foundational three-dimensional results that made the program work beyond surfaces.
Yujiro Kawamata: A Japanese algebraic geometer known for major contributions to the minimal model program. Kawamata developed influential results alongside Mori in higher-dimensional birational geometry.
Extremal ray: A boundary ray of a convex cone that cannot be expressed as a sum of distinct rays in that cone. Mori’s cone theorem identifies these rays as directions for simplifying a variety.
Enriques–Kodaira classification: The classification of compact complex surfaces into broad families using their geometric invariants. The success of surface classification motivated efforts to extend birational methods to higher dimensions.
Algebraic variety: A geometric object described locally by polynomial equations. Varieties are the spaces whose birational geometry Mori helped classify.
Mori theory: The study of extremal rays and contractions in the birational geometry of algebraic varieties. This body of techniques bears Mori’s name and drives his central contributions.
Miles Reid: A British mathematician known for work on algebraic geometry, especially three-dimensional varieties and singularities. Reid’s work on threefolds intersects closely with the geometry Mori transformed.
Cone theorem: A theorem describing the cone of curves of a projective variety in relation to its canonical divisor. Mori’s theorem reveals the contraction directions central to his program.
Castelnuovo contraction theorem: A theorem characterizing when a curve on a smooth algebraic surface can be contracted to a smooth point. Surface contractions supplied a precedent for Mori’s higher-dimensional contraction theory.