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| In [[algebraic geometry]], a '''toric variety''' or '''torus embedding''' is an [[algebraic variety]] containing an [[algebraic torus]] as an open [[dense subset]], such that the [[group action|action]] of the torus on itself extends to the whole variety. Some authors also require it to be [[normal variety|normal]]. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.
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| ==Toric Varieties from Tori==
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| The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus ''T'', the group of characters Hom(''T'','''C'''<sup>x</sup>) forms a lattice. Given a collection of points ''A'', a subset of this lattice, each point determines a map to '''C''' and thus the collection determines a map to '''C'''<sup>|A|</sup>. By taking the Zariski closure of the image of such a map, one obtains an affine variety. If the collection of lattice points ''A'' generates the character lattice, this variety is a torus embedding. In similar fashion one may produce a parametrized projective toric variety, by taking the projective closure of the above map, viewing it as a map into an affine patch of projective space.
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| Given a projective toric variety, observe that we may probe its geometry by one-parameter subgroups. Each one parameter subgroup, determined by a point in the lattice, dual to the character lattice, is a punctured curve inside the projective toric variety. Since the variety is compact, this punctured curve has a unique limit points. Thus, by partitioning the one-parameter subgroup lattice by the limit points of punctured curves, we obtain a lattice fan, a collection of polyhedral rational cones. The highest dimensional cones correspond precisely to the torus fixed points, the limits of these punctured curves.
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| ==The Toric Variety of a Fan==
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| 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.
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| For each cone σ its affine toric variety ''U''<sub>σ</sub> is the spectrum of the [[semigroup algebra]] of the [[dual cone]].
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| A '''fan''' is a collection of cones closed under taking intersections and faces. | |
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| The toric variety of a fan is given by taking the affine toric varieties of its cones and gluing them together by identifying ''U''<sub>σ</sub> with an open subvariety of ''U''<sub>τ</sub> whenever σ is a face of τ. Conversely, every fan of strongly convex rational cones has an associated toric variety. | |
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| The fan associated with a toric variety condenses some important data about the variety. For example, a variety is [[smooth scheme|smooth]] if every cone in its fan can be generated by a subset of a [[basis (universal algebra)|basis]] for the free abelian group ''N''.
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| ==Morphisms of Toric Varieties==
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| Suppose that Δ<sub>1</sub> and Δ<sub>2</sub> are fans in lattices ''N''<sub>1</sub> and ''N''<sub>2</sub>. If ''f'' is a linear map from ''N''<sub>1</sub> to ''N''<sub>2</sub> such that the image of every cone of Δ<sub>1</sub> is contained in a cone of Δ<sub>2</sub>, then ''f'' induces a morphism ''f''<sub>*</sub> between the corresponding toric varieties. This map ''f''<sub>*</sub> is proper if and only if the map ''f'' maps |Δ<sub>1</sub>| onto |Δ<sub>2</sub>|, where |Δ| is the underlying space of a fan Δ given by the union of its cones.
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| ==Resolution of Singularities==
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| 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.
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| ==The Toric Variety of a Convex Polytope==
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| 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''.
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| 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'''<sup>2</sup> may be represented by three complex coordinates satisfying
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| :<math>|z_1|^2+|z_2|^2+|z_3|^2 = 1 , \,\!</math>
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| 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:
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| :<math>(z_1,z_2,z_3)\approx e^{i\phi} (z_1,z_2,z_3) . \,\!</math>
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| The approach of toric geometry is to write
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| :<math>(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2) . \,\!</math>
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| The coordinates <math>x,y,z</math> are non-negative, and they parameterize a triangle because
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| :<math>x+y+z=1 ; \,\! </math>
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| that is,
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| :<math>\quad z=1-x-y . \,\!</math>
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| The triangle is the '''toric base''' of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of <math>z_1,z_2</math>; the phase of <math>z_3</math> can be chosen real and positive by the <math>U(1)</math> symmetry.
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| However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at <math>x=0</math> or <math>y=0</math> or <math>z=0</math> because the phase of <math>z_1,z_2,z_3</math> becomes inconsequential, respectively.
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| 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).
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| ==References==
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| *{{Citation | last1=Cox | first1=David | author1-link=David Cox (mathematician)| title=Topics in algebraic geometry and geometric modeling | url=http://www3.amherst.edu/~dacox/ | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Contemp. Math. | mr=2039974 | year=2003 | volume=334 | chapter=What is a toric variety? | pages=203–223}}
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| *{{citation| first=David A. |last=Cox
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| |first2=John B. |last2=Little
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| |first3=Hal |last3=Schenck
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| |title=Toric varieties |url=http://www.cs.amherst.edu/~dac/toric.html}}
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| *{{Citation | last1=Danilov | first1=V. I. | title=The geometry of toric varieties | mr=495499 | year=1978 | journal=Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=33 | issue=2 | pages=85–134 |doi= 10.1070/RM1978v033n02ABEH002305}}
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| * {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Introduction to toric varieties | publisher=[[Princeton University Press]] | isbn=978-0-691-00049-7 | year=1993}}
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| *{{Citation | last1=Kempf | first1=G. | last2=Knudsen | first2=Finn Faye | last3=Mumford | first3=David | author3-link=David Mumford | last4=Saint-Donat | first4=B. | title=Toroidal embeddings. I | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi= 10.1007/BFb0070318 | mr=0335518 | year=1973 | volume=339}}
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| *{{Citation | last1=Miller | first1=Ezra | title=What is ... a toric variety? | url=http://www.ams.org/notices/200805/tx080500586p.pdf | mr=2404030 | year=2008 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=55 | issue=5 | pages=586–587}}
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| *{{Citation | last1=Oda | first1=Tadao | title=Convex bodies and algebraic geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-17600-8 | mr=922894 | year=1988 | volume=15}}
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| ==External links==
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| * [http://www3.amherst.edu/~dacox/ Home page] of D. A. Cox, with several lectures on toric varieties
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| [[Category:Algebraic geometry]]
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Their distinctive coloration, lively behavior and intriguing songs make the Cordon-Bleu finch one with the more popular finch varieties as people make cautious raise finches for pets. A mid-sized finch, measuring accomplishment five inches from beak to tail, they are quite handsome ducks.
Sometimes, the requirements change mainly because the property owners want more aesthetics, though with just a little bit of protection. They work on property division by installing a fence filled with hedges about. In making a decision, you need to tell the workers where end up being like this situation.
Clinton used the metaphor of the space between the invention belonging to the club as well as the shield to describe the present situation typically the war against terrorism. He was quoted saying "this gap needs to closed". Metaphors can give intangible concepts more impact with a crowd.
A strong personal brand is built on rumors. The story of Clinton meeting President Kennedy when on a youth leadership camp applied to great effect. Simply was it mentioned in the introduction but that famous photo of Clinton shaking JFK's hand was also used within marketing fibres. Other brand building shots included a captivating moment with Hilary, an effort of him playing the saxophone, a jogging photo, one with Chelsea as well as featuring Clinton lined plan 3 past Presidents. They both helped to define Clinton the man.
Live Cattle, like corn, is a powerful market for newbies. The margins are low and glucose market is usually medium. A news report margin of $1200 controls a 40,000 pound futures contract of approximately $30,000 in value. One full cent/pound of price movement comes to $400.
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