Toric variety

In algebraic geometry, a toric variety or torus embedding is a normal variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety.

The toric variety of a fan

Suppose that N is a finite-rank 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 Δ₁ and Δ₂ are fans in lattices N₁ and N₂. If f is a linear map from N₁ to N₂ such that the image of every cone of Δ₁ is contained in a cone of Δ₂, then f induces a morphism f_* between the corresponding toric varieties. This map f_* is proper if and only if the map f maps |Δ₁| onto |Δ₂|, 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 CP² may be represented by three complex coordinates satisfying

$|z_1|^2+|z_2|^2+|z_3|^2 = 1 , \,\!$

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:

$(z_1,z_2,z_3)\approx e^{i\phi} (z_1,z_2,z_3) . \,\!$

The approach of toric geometry is to write

$(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2) . \,\!$

The coordinates x,y,z are non-negative, and they parameterize a triangle because

$x+y+z=1 ; \,\!$

that is,

$\quad z=1-x-y . \,\!$

The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of z₁,z₂; the phase of z₃ can be chosen real and positive by the U(1) symmetry.

However, the two-torus 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 z₁,z₂,z₃ 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 0042-1316, MR495499 
• Fulton, William (1993), Introduction to toric varieties, Princeton University Press, ISBN 978-0-691-00049-7 
• Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal embeddings. I, Lecture Notes in Mathematics, 339, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318, MR0335518 
• Miller, Ezra (2008), "What is ... a toric variety?", Notices of the American Mathematical Society 55 (5): 586–587, ISSN 0002-9920, 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: Springer-Verlag, ISBN 978-3-540-17600-8, MR922894 

External links

• Home page of D. A. Cox, with several lectures on toric varieties