GAP (ToricVarieties) - Chapter 3: Toric varieties

Goto Chapter: Top 1 2 3 4 5 6 7 8 Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

3 Toric varieties
3.1 Toric variety: Category and Representations
3.1-1 IsToricVariety
3.2 Toric varieties: Properties
3.2-1 IsNormalVariety
3.2-2 IsAffine
3.2-3 IsProjective
3.2-4 IsComplete
3.2-5 IsSmooth
3.2-6 HasTorusfactor
3.2-7 HasNoTorusfactor
3.2-8 IsOrbifold
3.3 Toric varieties: Attributes
3.3-1 AffineOpenCovering
3.3-2 CoxRing
3.3-3 ListOfVariablesOfCoxRing
3.3-4 ClassGroup
3.3-5 PicardGroup
3.3-6 TorusInvariantDivisorGroup
3.3-7 MapFromCharacterToPrincipalDivisor
3.3-8 Dimension
3.3-9 DimensionOfTorusfactor
3.3-10 CoordinateRingOfTorus
3.3-11 IsProductOf
3.3-12 CharacterLattice
3.3-13 TorusInvariantPrimeDivisors
3.3-14 IrrelevantIdeal
3.3-15 MorphismFromCoxVariety
3.3-16 CoxVariety
3.3-17 FanOfVariety
3.3-18 CartierTorusInvariantDivisorGroup
3.3-19 NameOfVariety
3.3-20 twitter
3.4 Toric varieties: Methods
3.4-1 UnderlyingSheaf
3.4-2 CoordinateRingOfTorus
3.4-3 \*
3.4-4 CharacterToRationalFunction
3.4-5 CoxRing
3.4-6 WeilDivisorsOfVariety
3.4-7 Fan
3.5 Toric varieties: Constructors
3.5-1 ToricVariety
3.6 Toric varieties: Examples
3.6-1 The Hirzebruch surface of index 5

### 3 Toric varieties

#### 3.1 Toric variety: Category and Representations

##### 3.1-1 IsToricVariety

`‣ IsToricVariety`( M ) | ( category )

Returns: `true` or `false`

The GAP category of a toric variety.

#### 3.2 Toric varieties: Properties

##### 3.2-1 IsNormalVariety

`‣ IsNormalVariety`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari is a normal variety.

##### 3.2-2 IsAffine

`‣ IsAffine`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari is an affine variety.

##### 3.2-3 IsProjective

`‣ IsProjective`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari is a projective variety.

##### 3.2-4 IsComplete

`‣ IsComplete`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari is a complete variety.

##### 3.2-5 IsSmooth

`‣ IsSmooth`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari is a smooth variety.

##### 3.2-6 HasTorusfactor

`‣ HasTorusfactor`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari has a torus factor.

##### 3.2-7 HasNoTorusfactor

`‣ HasNoTorusfactor`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari has no torus factor.

##### 3.2-8 IsOrbifold

`‣ IsOrbifold`( vari ) | ( property )

Returns: `true` or `false`

Checks if the toric variety vari has an orbifold, which is, in the toric case, equivalent to the simpliciality of the fan.

#### 3.3 Toric varieties: Attributes

##### 3.3-1 AffineOpenCovering

`‣ AffineOpenCovering`( vari ) | ( attribute )

Returns: a list

Returns a torus invariant affine open covering of the variety vari. The affine open cover is computed out of the cones of the fan.

##### 3.3-2 CoxRing

`‣ CoxRing`( vari ) | ( attribute )

Returns: a ring

Returns the Cox ring of the variety vari. The actual method requires a string with a name for the variables. A method for computing the Cox ring without a variable given is not implemented. You will get an error.

##### 3.3-3 ListOfVariablesOfCoxRing

`‣ ListOfVariablesOfCoxRing`( vari ) | ( attribute )

Returns: a list

Returns a list of the variables of the cox ring of the variety vari.

##### 3.3-4 ClassGroup

`‣ ClassGroup`( vari ) | ( attribute )

Returns: a module

Returns the class group of the variety vari as factor of a free module.

##### 3.3-5 PicardGroup

`‣ PicardGroup`( vari ) | ( attribute )

Returns: a module

Returns the Picard group of the variety vari as factor of a free module.

##### 3.3-6 TorusInvariantDivisorGroup

`‣ TorusInvariantDivisorGroup`( vari ) | ( attribute )

Returns: a module

Returns the subgroup of the Weil divisor group of the variety vari generated by the torus invariant prime divisors. This is always a finitely generated free module over the integers.

##### 3.3-7 MapFromCharacterToPrincipalDivisor

`‣ MapFromCharacterToPrincipalDivisor`( vari ) | ( attribute )

Returns: a morphism

Returns a map which maps an element of the character group into the torus invariant Weil group of the variety vari. This has to viewn as an help method to compute divisor classes.

##### 3.3-8 Dimension

`‣ Dimension`( vari ) | ( attribute )

Returns: an integer

Returns the dimension of the variety vari.

##### 3.3-9 DimensionOfTorusfactor

`‣ DimensionOfTorusfactor`( vari ) | ( attribute )

Returns: an integer

Returns the dimension of the torus factor of the variety vari.

##### 3.3-10 CoordinateRingOfTorus

`‣ CoordinateRingOfTorus`( vari ) | ( attribute )

Returns: a ring

Returns the coordinate ring of the torus of the variety vari. This method is not implemented, you need to call it with a second argument, which is a list of strings for the variables of the ring.

##### 3.3-11 IsProductOf

`‣ IsProductOf`( vari ) | ( attribute )

Returns: a list

If the variety vari is a product of 2 or more varieties, the list contain those varieties. If it is not a product or at least not generated as a product, the list only contains the variety itself.

##### 3.3-12 CharacterLattice

`‣ CharacterLattice`( vari ) | ( attribute )

Returns: a module

The method returns the character lattice of the variety vari, computed as the containing grid of the underlying convex object, if it exists.

##### 3.3-13 TorusInvariantPrimeDivisors

`‣ TorusInvariantPrimeDivisors`( vari ) | ( attribute )

Returns: a list

The method returns a list of the torus invariant prime divisors of the variety vari.

##### 3.3-14 IrrelevantIdeal

`‣ IrrelevantIdeal`( vari ) | ( attribute )

Returns: an ideal

Returns the irrelevant ideal of the cox ring of the variety vari.

##### 3.3-15 MorphismFromCoxVariety

`‣ MorphismFromCoxVariety`( vari ) | ( attribute )

Returns: a morphism

The method returns the quotient morphism from the variety of the Cox ring to the variety vari.

##### 3.3-16 CoxVariety

`‣ CoxVariety`( vari ) | ( attribute )

Returns: a variety

The method returns the Cox variety of the variety vari.

##### 3.3-17 FanOfVariety

`‣ FanOfVariety`( vari ) | ( attribute )

Returns: a fan

Returns the fan of the variety vari. This is set by default.

##### 3.3-18 CartierTorusInvariantDivisorGroup

`‣ CartierTorusInvariantDivisorGroup`( vari ) | ( attribute )

Returns: a module

Returns the the group of Cartier divisors of the variety vari as a subgroup of the divisor group.

##### 3.3-19 NameOfVariety

`‣ NameOfVariety`( vari ) | ( attribute )

Returns: a string

Returns the name of the variety vari if it has one and it is known or can be computed.

##### 3.3-20 twitter

`‣ twitter`( vari ) | ( attribute )

Returns: a ring

This is a dummy to get immediate methods triggered at some times. It never has a value.

#### 3.4 Toric varieties: Methods

##### 3.4-1 UnderlyingSheaf

`‣ UnderlyingSheaf`( vari ) | ( operation )

Returns: a sheaf

The method returns the underlying sheaf of the variety vari.

##### 3.4-2 CoordinateRingOfTorus

`‣ CoordinateRingOfTorus`( vari, vars ) | ( operation )

Returns: a ring

Computes the coordinate ring of the torus of the variety vari with the variables vars. The argument vars need to be a list of strings with length dimension or two times dimension.

##### 3.4-3 \*

`‣ \*`( vari1, vari2 ) | ( operation )

Returns: a variety

Computes the categorial product of the varieties vari1 and vari2.

##### 3.4-4 CharacterToRationalFunction

`‣ CharacterToRationalFunction`( elem, vari ) | ( operation )

Returns: a homalg element

Computes the rational function corresponding to the character grid element elem or to the list of integers elem. To compute rational functions you first need to compute to coordinate ring of the torus of the variety vari.

##### 3.4-5 CoxRing

`‣ CoxRing`( vari, vars ) | ( operation )

Returns: a ring

Computes the Cox ring of the variety vari. vars needs to be a string containing one variable, which will be numbered by the method.

##### 3.4-6 WeilDivisorsOfVariety

`‣ WeilDivisorsOfVariety`( vari ) | ( operation )

Returns: a list

Returns a list of the currently defined Divisors of the toric variety.

##### 3.4-7 Fan

`‣ Fan`( vari ) | ( operation )

Returns: a fan

Returns the fan of the variety vari. This is a rename for FanOfVariety.

#### 3.5 Toric varieties: Constructors

##### 3.5-1 ToricVariety

`‣ ToricVariety`( conv ) | ( operation )

Returns: a ring

Creates a toric variety out of the convex object conv.

#### 3.6 Toric varieties: Examples

##### 3.6-1 The Hirzebruch surface of index 5

    gap> H5 := Fan( [[-1,5],[0,1],[1,0],[0,-1]],[[1,2],[2,3],[3,4],[4,1]] );
    <A fan in |R^2>
    gap> H5 := ToricVariety( H5 );
    <A toric variety of dimension 2>
    gap> IsComplete( H5 );
    true
    gap> IsAffine( H5 );
    false
    gap> IsOrbifold( H5 );
    true
    gap> IsProjective( H5 );
    true
    gap> TorusInvariantPrimeDivisors(H5);
    [ <A prime divisor of a toric variety with coordinates [ 1, 0, 0, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 1, 0, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 0, 1, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 0, 0, 1 ]> ]
    gap> P := TorusInvariantPrimeDivisors(H5);
    [ <A prime divisor of a toric variety with coordinates [ 1, 0, 0, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 1, 0, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 0, 1, 0 ]>,
      <A prime divisor of a toric variety with coordinates [ 0, 0, 0, 1 ]> ]
    gap> A := P[ 1 ] - P[ 2 ] + 4*P[ 3 ];
    <A divisor of a toric variety with coordinates [ 1, -1, 4, 0 ]>
    gap> A;
    <A divisor of a toric variety with coordinates [ 1, -1, 4, 0 ]>
    gap> IsAmple(A);
    false
    gap> CoordinateRingOfTorus(H5,"x");;
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> D:=CreateDivisor([0,0,0,0],H5);
    <A divisor of a toric variety with coordinates 0>
    gap> BasisOfGlobalSections(D);
    [ |[ 1 ]| ]
    gap> D:=Sum(P);
    <A divisor of a toric variety with coordinates [ 1, 1, 1, 1 ]>
    gap> BasisOfGlobalSections(D);
    [ |[ x1_ ]|, |[ x1_*x2 ]|, |[ 1 ]|, |[ x2 ]|,
      |[ x1 ]|, |[ x1*x2 ]|, |[ x1^2*x2 ]|,
      |[ x1^3*x2 ]|, |[ x1^4*x2 ]|, |[ x1^5*x2 ]|,
      |[ x1^6*x2 ]| ]
    gap> DivisorOfCharacter([1,2],H5);
    <A principal divisor of a toric variety with coordinates [ 9, 2, 1, -2 ]>
    gap> BasisOfGlobalSections(last);
    [ |[ x1_*x2_^2 ]| ]

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 Ind

generated by GAPDoc2HTML