This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Toric variety | |
|---|---|
| Name | Toric variety |
| Field | Algebraic geometry |
Toric variety A toric variety is an algebraic variety containing an algebraic torus as a dense open subset on which the torus acts algebraically; it occupies a central place in modern Algebraic geometry, connecting combinatorial, geometric, and arithmetic viewpoints. Toric varieties were developed through contributions from figures associated with institutions such as Moscow State University, Princeton University, and University of California, Berkeley, and they interact with topics studied at events like the International Congress of Mathematicians and projects supported by organizations such as the National Science Foundation. The theory links concrete constructions used in work by researchers at Harvard University, Massachusetts Institute of Technology, and École Normale Supérieure to computational tools employed at centers including Centre National de la Recherche Scientifique and Max Planck Institute for Mathematics.
Toric varieties generalize classical objects studied by mathematicians in the tradition of David Hilbert, Heinz Hopf, and Bernhard Riemann through a synthesis that involves combinatorial data from polyhedral geometry familiar to researchers at Courant Institute of Mathematical Sciences and Cambridge University. They provide explicit examples used in seminars at Institute for Advanced Study and are fundamental in the work of authors affiliated with Princeton University Press and collectors of lecture notes from conferences at Mathematical Sciences Research Institute. The subject is taught in graduate courses at institutions such as Yale University and University of Chicago and appears in monographs published by academic presses linked to scholars at Columbia University and Stanford University.
One standard construction of a toric variety uses a fan of strongly convex rational polyhedral cones inside a lattice; this combinatorial method was systematized by mathematicians from Moscow State University and popularized in expositions associated with professors at Oxford University and University of Michigan. Another approach builds toric varieties as Proj of a semigroup algebra coming from a rational polytope, a technique used in lectures at Imperial College London and workshops at ETH Zurich. Definitions rely on the algebraic torus, often denoted by (k^*)^n, and employ tools found in texts from Springer Science+Business Media and notes circulated by scholars at University of Tokyo and Seoul National University. Constructions connect to categorical perspectives developed in seminars at Institut des Hautes Études Scientifiques and to explicit coordinate charts used in computational platforms maintained by teams at Wolfram Research.
The correspondence between fans and normal toric varieties reflects deep links akin to correspondences studied in the setting of Grothendieck's schemes and was elaborated in papers by authors connected to Université Paris-Sud and Rutgers University. Lattice polytopes correspond to projective toric varieties in a manner parallel to classical correspondences explored by researchers from Brown University and Brownian motion-adjacent probabilists who study polytopal structures; this perspective appears in expository articles from University of Cambridge and collaborative projects at University of Oxford. The Cox ring construction unifies coordinate ring descriptions and quotients by diagonalisable groups, techniques taught in graduate seminars at University of California, Los Angeles and developed in work affiliated with École Polytechnique.
Basic examples include affine toric varieties defined by a cone, and projective toric varieties associated to lattice polytopes such as the simplex and cube; these examples are exhibited in course notes from University of Illinois Urbana-Champaign and texts used at University of Washington. Smooth projective toric varieties correspond to fans satisfying simplicity conditions; classification results echo programmatic classifications carried out at research centers including Clay Mathematics Institute and are illustrated in catalogs prepared by mathematicians at Duke University and University of Bonn. Famous instances like projective space and Hirzebruch surfaces appear in lectures at Princeton University and in seminars led by faculty at Cornell University.
Geometric properties of toric varieties, such as singularities, intersection theory, and cohomology, connect to invariants studied by mathematicians affiliated with Max Planck Institute for Mathematics and Institut Henri Poincaré. The study of equivariant cohomology and moment maps relates toric varieties to symplectic geometry topics presented at the International Congress on Mathematical Physics and to constructions used by researchers at University of California, Santa Barbara. Topological descriptions via orbit decompositions are standard in lectures at University of Texas at Austin and in collaboration projects involving scholars from University of Toronto.
Toric varieties serve as testing grounds for conjectures in birational geometry, mirror symmetry, and tropical geometry—areas researched at Kavli Institute for Theoretical Physics, Perimeter Institute, and laboratories tied to European Research Council grants. They appear in computations of Gromov–Witten invariants in work conducted by groups at Harvard University and in algorithmic implementations at companies like Google that leverage combinatorial structures. Related topics include lattice polytopes, Newton polytopes studied at Institute of Mathematics, Chinese Academy of Sciences, and connections to moduli problems discussed at symposia organized by American Mathematical Society.