GAP (ToricVarieties) - Chapter 8: Toric divisors

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8 Toric divisors
8.1 Toric divisors: Category and Representations
8.1-1 IsToricDivisor
8.2 Toric divisors: Properties
8.2-1 IsCartier
8.2-2 IsPrincipal
8.2-3 IsPrimedivisor
8.2-4 IsBasepointFree
8.2-5 IsAmple
8.2-6 IsVeryAmple
8.3 Toric divisors: Attributes
8.3-1 CartierData
8.3-2 CharacterOfPrincipalDivisor
8.3-3 ToricVarietyOfDivisor
8.3-4 ClassOfDivisor
8.3-5 PolytopeOfDivisor
8.3-6 BasisOfGlobalSections
8.3-7 IntegerForWhichIsSureVeryAmple
8.3-8 AmbientToricVariety
8.3-9 UnderlyingGroupElement
8.3-10 UnderlyingToricVariety
8.3-11 DegreeOfDivisor
8.3-12 MonomsOfCoxRingOfDegree
8.3-13 CoxRingOfTargetOfDivisorMorphism
8.3-14 RingMorphismOfDivisor
8.4 Toric divisors: Methods
8.4-1 VeryAmpleMultiple
8.4-2 CharactersForClosedEmbedding
8.4-3 MonomsOfCoxRingOfDegree
8.4-4 DivisorOfGivenClass
8.4-5 AddDivisorToItsAmbientVariety
8.4-6 Polytope
8.4-7 +
8.4-8 -
8.4-9 *
8.5 Toric divisors: Constructors
8.5-1 DivisorOfCharacter
8.5-2 DivisorOfCharacter
8.5-3 CreateDivisor
8.5-4 CreateDivisor
8.6 Toric divisors: Examples
8.6-1 Divisors on a toric variety

### 8 Toric divisors

#### 8.1 Toric divisors: Category and Representations

##### 8.1-1 IsToricDivisor

`‣ IsToricDivisor`( M ) | ( category )

Returns: `true` or `false`

The GAP category of torus invariant Weil divisors.

#### 8.2 Toric divisors: Properties

##### 8.2-1 IsCartier

`‣ IsCartier`( divi ) | ( property )

Returns: `true` or `false`

Checks if the torus invariant Weil divisor divi is Cartier i.e. if it is locally principal.

##### 8.2-2 IsPrincipal

`‣ IsPrincipal`( divi ) | ( property )

Returns: `true` or `false`

Checks if the torus invariant Weil divisor divi is principal which in the toric invariant case means that it is the divisor of a character.

##### 8.2-3 IsPrimedivisor

`‣ IsPrimedivisor`( divi ) | ( property )

Returns: `true` or `false`

Checks if the Weil divisor divi represents a prime divisor, i.e. if it is a standard generator of the divisor group.

##### 8.2-4 IsBasepointFree

`‣ IsBasepointFree`( divi ) | ( property )

Returns: `true` or `false`

Checks if the divisor divi is basepoint free. What else?

##### 8.2-5 IsAmple

`‣ IsAmple`( divi ) | ( property )

Returns: `true` or `false`

Checks if the divisor divi is ample, i.e. if it is colored red, yellow and green.

##### 8.2-6 IsVeryAmple

`‣ IsVeryAmple`( divi ) | ( property )

Returns: `true` or `false`

Checks if the divisor divi is very ample.

#### 8.3 Toric divisors: Attributes

##### 8.3-1 CartierData

`‣ CartierData`( divi ) | ( attribute )

Returns: a list

Returns the Cartier data of the divisor divi, if it is Cartier, and fails otherwise.

##### 8.3-2 CharacterOfPrincipalDivisor

`‣ CharacterOfPrincipalDivisor`( divi ) | ( attribute )

Returns: an element

Returns the character corresponding to principal divisor divi.

##### 8.3-3 ToricVarietyOfDivisor

`‣ ToricVarietyOfDivisor`( divi ) | ( attribute )

Returns: a variety

Returns the closure of the torus orbit corresponding to the prime divisor divi. Not implemented for other divisors. Maybe we should add the support here. Is this even a toric variety? Exercise left to the reader.

##### 8.3-4 ClassOfDivisor

`‣ ClassOfDivisor`( divi ) | ( attribute )

Returns: an element

Returns the class group element corresponding to the divisor divi.

##### 8.3-5 PolytopeOfDivisor

`‣ PolytopeOfDivisor`( divi ) | ( attribute )

Returns: a polytope

Returns the polytope corresponding to the divisor divi.

##### 8.3-6 BasisOfGlobalSections

`‣ BasisOfGlobalSections`( divi ) | ( attribute )

Returns: a list

Returns a basis of the global section module of the quasi-coherent sheaf of the divisor divi.

##### 8.3-7 IntegerForWhichIsSureVeryAmple

`‣ IntegerForWhichIsSureVeryAmple`( divi ) | ( attribute )

Returns: an integer

Returns an integer which, to be multiplied with the ample divisor divi, someone gets a very ample divisor.

##### 8.3-8 AmbientToricVariety

`‣ AmbientToricVariety`( divi ) | ( attribute )

Returns: a variety

Returns the containing variety of the prime divisors of the divisor divi.

##### 8.3-9 UnderlyingGroupElement

`‣ UnderlyingGroupElement`( divi ) | ( attribute )

Returns: an element

Returns an element which represents the divisor divi in the Weil group.

##### 8.3-10 UnderlyingToricVariety

`‣ UnderlyingToricVariety`( divi ) | ( attribute )

Returns: a variety

Returns the closure of the torus orbit corresponding to the prime divisor divi. Not implemented for other divisors. Maybe we should add the support here. Is this even a toric variety? Exercise left to the reader.

##### 8.3-11 DegreeOfDivisor

`‣ DegreeOfDivisor`( divi ) | ( attribute )

Returns: an integer

Returns the degree of the divisor divi.

##### 8.3-12 MonomsOfCoxRingOfDegree

`‣ MonomsOfCoxRingOfDegree`( divi ) | ( attribute )

Returns: a list

Returns the variety corresponding to the polytope of the divisor divi.

##### 8.3-13 CoxRingOfTargetOfDivisorMorphism

`‣ CoxRingOfTargetOfDivisorMorphism`( divi ) | ( attribute )

Returns: a ring

A basepoint free divisor divi defines a map from its ambient variety in a projective space. This method returns the cox ring of such a projective space.

##### 8.3-14 RingMorphismOfDivisor

`‣ RingMorphismOfDivisor`( divi ) | ( attribute )

Returns: a ring

A basepoint free divisor divi defines a map from its ambient variety in a projective space. This method returns the morphism between the cox ring of this projective space to the cox ring of the ambient variety of divi.

#### 8.4 Toric divisors: Methods

##### 8.4-1 VeryAmpleMultiple

`‣ VeryAmpleMultiple`( divi ) | ( operation )

Returns: a divisor

Returns a very ample multiple of the ample divisor divi. Will fail if divisor is not ample.

##### 8.4-2 CharactersForClosedEmbedding

`‣ CharactersForClosedEmbedding`( divi ) | ( operation )

Returns: a list

Returns characters for closed embedding defined via the ample divisor divi. Fails if divisor is not ample.

##### 8.4-3 MonomsOfCoxRingOfDegree

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

Returns: a list

Returns the monoms of the Cox ring of the variety vari with degree to the class group element elem. The variable elem can also be a list.

##### 8.4-4 DivisorOfGivenClass

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

Returns: a list

Computes a divisor of the variety divi which is member of the divisor class presented by elem. The variable elem can be a homalg element or a list presenting an element.

##### 8.4-5 AddDivisorToItsAmbientVariety

`‣ AddDivisorToItsAmbientVariety`( divi ) | ( operation )

Adds the divisor divi to the Weil divisor list of its ambient variety.

##### 8.4-6 Polytope

`‣ Polytope`( divi ) | ( operation )

Returns: a polytope

Returns the polytope of the divisor divi. Another name for PolytopeOfDivisor for compatibility and shortness.

##### 8.4-7 +

`‣ +`( divi1, divi2 ) | ( operation )

Returns: a divisor

Returns the sum of the divisors divi1 and divi2.

##### 8.4-8 -

`‣ -`( divi1, divi2 ) | ( operation )

Returns: a divisor

Returns the divisor divi1 minus divi2.

##### 8.4-9 *

`‣ *`( k, divi ) | ( operation )

Returns: a divisor

Returns k times the divisor divi.

#### 8.5 Toric divisors: Constructors

##### 8.5-1 DivisorOfCharacter

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

Returns: a divisor

Returns the divisor of the toric variety vari which corresponds to the character elem.

##### 8.5-2 DivisorOfCharacter

`‣ DivisorOfCharacter`( lis, vari ) | ( operation )

Returns: a divisor

Returns the divisor of the toric variety vari which corresponds to the character which is created by the list lis.

##### 8.5-3 CreateDivisor

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

Returns: a divisor

Returns the divisor of the toric variety vari which corresponds to the Weil group element elem.

##### 8.5-4 CreateDivisor

`‣ CreateDivisor`( lis, vari ) | ( operation )

Returns: a divisor

Returns the divisor of the toric variety vari which corresponds to the Weil group element which is created by the list lis.

#### 8.6 Toric divisors: Examples

##### 8.6-1 Divisors on a toric variety

    gap> H7 := Fan( [[0,1],[1,0],[0,-1],[-1,7]],[[1,2],[2,3],[3,4],[4,1]] );
    <A fan in |R^2>
    gap> H7 := ToricVariety( H7 );
    <A toric variety of dimension 2>
    gap> P := TorusInvariantPrimeDivisors( H7 );
    [ <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> D := P[3]+P[4];
    <A divisor of a toric variety with coordinates [ 0, 0, 1, 1 ]>
    gap> IsBasepointFree(D);
    true
    gap> IsAmple(D);
    true
    gap> CoordinateRingOfTorus(H7,"x");
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> Polytope(D);
    <A polytope in |R^2>
    gap> CharactersForClosedEmbedding(D);
    [ |[ 1 ]|, |[ x2 ]|, |[ x1 ]|, |[ x1*x2 ]|, |[ x1^2*x2 ]|,
      |[ x1^3*x2 ]|, |[ x1^4*x2 ]|, |[ x1^5*x2 ]|,
      |[ x1^6*x2 ]|, |[ x1^7*x2 ]|, |[ x1^8*x2 ]| ]
    gap> CoxRingOfTargetOfDivisorMorphism(D);
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> RingMorphismOfDivisor(D);
    <A "homomorphism" of rings>
    gap> Display(last);
    Q[x_1,x_2,x_3,x_4]
    (weights: [ [ 0, 0, 1, -7 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ])
      ^
      |
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6,
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3,
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
      |
      |
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> ByASmallerPresentation(ClassGroup(H7));
    <A free left module of rank 2 on free generators>
    gap> Display(RingMorphismOfDivisor(D));
    Q[x_1,x_2,x_3,x_4]
    (weights: [ [ 1, -7 ], [ 0, 1 ], [ 1, 0 ], [ 0, 1 ] ])
      ^
      |
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6,
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3,
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
      |
      |
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> MonomsOfCoxRingOfDegree(D);
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6,
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3,
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
    gap> D2:=D-2*P[2];
    <A divisor of a toric variety with coordinates [ 0, -2, 1, 1 ]>
    gap> IsBasepointFree(D2);
    false
    gap> IsAmple(D2);
    false

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