Toric variety

A toric variety is a special algebraic variety and thus an object from algebraic geometry , a branch of mathematics . The study of toric varieties is also known as toric geometry .

Definitions

Algebraic torus

An algebraic torus over is an algebraic group that is isomorphic to an algebraic group of form . ${\displaystyle T}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{\ast }\times \dotsb \times \mathbb {C} ^{\ast }}$

Toric varieties as toric embeddings

A toric variety is an irreducible algebraic variety that contains an algebraic torus as a Zariski open subset such that the group connection of the torus can be continued into an algebraic group operation of the torus on the whole variety. Here algebraically means that the group operation is given by a morphism of algebraic varieties. ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle T\times X\to X}$

Some authors also require a toric variety to be normal . An algebraic variety is called normal if the local ring is a normal ring at every point of the variety .

Relation to the convex geometry

Polyhedral lattice cones

Let be a lattice, that is, a free Abelian group of finite rank . A pointed convex rational polyhedral cone is a pointed convex cone in vector space that is generated from a finite number of vectors . In the following we will briefly speak of a cone. ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N_{\mathbb {R} }=N\otimes _{\mathbb {Z} }\mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle N}$

Each -cone may be a dual cones are assigned. To do this, consider the dual vector space for the dual lattice and define it . ${\displaystyle N}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma ^{\vee }}$ ${\displaystyle M=Hom(N,\mathbb {Z} )}$ ${\displaystyle M_{\mathbb {R} }=M\otimes _{\mathbb {Z} }\mathbb {R} }$ ${\displaystyle \sigma ^{\vee }=\{u\in M|\langle u,v\rangle \geq 0\forall v\in \sigma \}}$

subjects

A fan to a lattice is a finite set of -cones, in which for every cone all its sides are contained and in which every two cones whose intersection is one side of both cones. Thus the concept of a fan is analogous to the concept of a geometrical simplicial complex in algebraic topology . ${\displaystyle N}$ ${\displaystyle N}$

Toric varieties of lattice cones and fans

A cone is first assigned its dual cone . For this one considers the commutative semigroup . It turns out ( Gordan's Lemma ) that this semigroup is finitely generated and the monoidalgebra is therefore a finitely generated commutative algebra. The maximum spectrum of this algebra then has the structure of a normal toric affine variety and all normal affine toric varieties are obtained in this way. ${\displaystyle N}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma ^{\vee }}$ ${\displaystyle S_{\sigma }=\sigma ^{\vee }\cap M}$ ${\displaystyle \mathbb {C} [S_{\sigma }]}$ ${\displaystyle \mathbb {C} }$

Affine toric varieties coming from the cones of a fan can be glued together to form an abstract toric variety . In this way all normal toric varieties are obtained.

See also

• Tropical geometry : a branch of algebraic geometry that is also strongly linked to discrete mathematics .

Individual evidence

1. ^ Oda: Lectures on Torus Embeddings and Applications. 1978, 1.1 Algebraic tori.
2. ↑ Cox: Toric varieties. 2011, Theorem 3.1.1.
3. ↑ Fulton: Introduction to Toric Varieties. 1993, definition in 1.1.
4. ↑ Cox: Toric varieties. 2011, definition 3.1.2.
5. ↑ Cox: Toric varieties. 2011, Proposition 1.2.17.
6. ↑ Cox: Toric varieties. 2011, Theorem 1.2.18., Theorem 1.3.5.
7. ↑ Cox: Toric varieties. 2011, Theorem 3.1.5.
8. ↑ Cox: Toric varieties. 2011, Corollary 3.1.8.

literature

• David A. Cox , John B. Little , Henry K. Schenck: Toric varieties. American Mathematical Society, Providence 2011, ISBN 978-0-8218-4819-7 .
• Ludger Kaup: Lectures on Toric varieties. Konstanzer Schriften in Mathematik und Informatik, No. 130, version from spring 2002, ISSN  1430-3558 , ( PDF; 2.2 MB ).
• Günter Ewald : Combinatorial convexity and algebraic geometry. Springer, New York 1996, ISBN 0-387-94755-8 .
• William Fulton : Introduction to toric varieties. Princeton University Press, Princeton, NJ. 1993, ISBN 0-691-03332-3 .
• Tadao Oda : Convex bodies and algebraic geometry: an introduction to the theory of toric varieties. Springer, Berlin, 1988, ISBN 3-540-17600-4 .
• Jean-Luc Brylinski : Eventails et variétés toriques. in Séminaire sur les singularités des surfaces Springer, Berlin 1980, ISBN 3-540-09746-5 .
• VI Danilov : The geometry of toric varieties. Russian Math. Surveys 33: 2, 1978, pp. 97–154 ( PDF; 2.9 MB ).
• Tadao Oda: Lectures on Torus Embeddings and Applications. Springer, Berlin 1978, ISBN 3-540-08852-0 .
• George R. Kempf , Finn Faye Knudsen, David B. Mumford , B. Saint-Donat: Toroidal Embeddings I. Springer, Berlin 1973, ISBN 978-3-540-06432-9 .

Web links

• Page by David A. Cox with various scripts on toric varieties
• Lecture notes on toric varieties by Mircea Mustaţă from the University of Michigan