 Toric variety

In algebraic geometry, a toric variety or torus embedding is a normal variety containing an algebraic torus as a dense subset, such that the action of the torus on itself extends to the whole variety.
Contents
The toric variety of a fan
Suppose that N is a finiterank free abelian group. A strongly convex rational polyhedral cone in N is a convex cone (of the real vector space of N) with apex at the origin, generated by a finite number of vectors of N, that contains no line through the origin. These will be called "cones" for short.
For each cone σ its affine toric variety U_{σ} is the spectrum of the semigroup algebra of the dual cone.
A fan is a collection of cones closed under taking intersections and faces.
The toric variety of a fan is given by taking the affine toric varieties of its cones and glueing them together by identifying U_{σ} with an open subvariety of U_{τ} whenever σ is a face of τ. Conversely, every fan of strongly convex rational cones has an associated toric variety.
The fan associated with a toric variety condenses some important data about the variety. For example, a variety is smooth if every cone in its fan can be generated by a subset of a basis for the free abelian group N.
Morphisms of toric varieties
Suppose that Δ_{1} and Δ_{2} are fans in lattices N_{1} and N_{2}. If f is a linear map from N_{1} to N_{2} such that the image of every cone of Δ_{1} is contained in a cone of Δ_{2}, then f induces a morphism f_{*} between the corresponding toric varieties. This map f_{*} is proper if and only if the map f maps Δ_{1} onto Δ_{2}, where Δ is the underlying space of a fan Δ given by the union of its cones.
Resolution of singularities
A toric variety is nonsingular if its cones of maximal dimension are generated by a basis of the lattice. This implies that every toric variety has a resolution of singularities given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.
The toric variety of a convex polytope
The fan of a rational convex polytope in N consists of the cones over its proper faces. The toric variety of the polytope is the toric variety of its fan. A variation of this construction is to take a rational polytope in the dual of N and take the toric variety of its polar set in N.
The toric variety has a map to the polytope in the dual of N whose fibers are topological tori. For example, the complex projective plane CP^{2} may be represented by three complex coordinates satisfying
where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following U(1) action:
The approach of toric geometry is to write
The coordinates x,y,z are nonnegative, and they parameterize a triangle because
that is,
The triangle is the toric base of the complex projective plane. The generic fiber is a twotorus parameterized by the phases of z_{1},z_{2}; the phase of z_{3} can be chosen real and positive by the U(1) symmetry.
However, the twotorus degenerates into three different circles on the boundary of the triangle i.e. at x = 0 or y = 0 or z = 0 because the phase of z_{1},z_{2},z_{3} becomes inconsequential, respectively.
The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case).
References
 Cox, David (2003), "What is a toric variety?", Topics in algebraic geometry and geometric modeling, Contemp. Math., 334, Providence, R.I.: Amer. Math. Soc., pp. 203–223, MR2039974, http://www3.amherst.edu/~dacox/
 Cox, David A.; Little, John B.; Schenck, Hal, Toric varieties, http://www.cs.amherst.edu/~dac/toric.html
 Danilov, V. I. (1978), "The geometry of toric varieties", Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 33 (2): 85–134, doi:10.1070/RM1978v033n02ABEH002305, ISSN 00421316, MR495499
 Fulton, William (1993), Introduction to toric varieties, Princeton University Press, ISBN 9780691000497
 Kempf, G.; Knudsen, Finn Faye; Mumford, David; SaintDonat, B. (1973), Toroidal embeddings. I, Lecture Notes in Mathematics, 339, Berlin, New York: SpringerVerlag, doi:10.1007/BFb0070318, MR0335518
 Miller, Ezra (2008), "What is ... a toric variety?", Notices of the American Mathematical Society 55 (5): 586–587, ISSN 00029920, MR2404030, http://www.ams.org/notices/200805/tx080500586p.pdf
 Oda, Tadao (1988), Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Berlin, New York: SpringerVerlag, ISBN 9783540176008, MR922894
External links
 Home page of D. A. Cox, with several lectures on toric varieties
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