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### 8 Toric divisors

#### 8.1 Toric divisors: Examples

##### 8.1-1 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+P;
<A divisor of a toric variety with coordinates ( 1, 1, 0, 0 )>
gap> IsBasepointFree(D);
true
gap> IsAmple(D);
true
gap> CoordinateRingOfTorus(H7,"x");
Q[x1,x1_,x2,x2_]/( x1*x1_-1, x2*x2_-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(RingMorphismOfDivisor(D));
Q[x_1,x_2,x_3,x_4]
(weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, -7 ), ( 0, 1 ) ])
^
|
[ x_1*x_2, x_1^8*x_3, x_2*x_4, x_1^7*x_3*x_4, x_1^6*x_3*x_4^2,
x_1^5*x_3*x_4^3, x_1^4*x_3*x_4^4, x_1^3*x_3*x_4^5, x_1^2*x_3*x_4^6,
x_1*x_3*x_4^7, x_3*x_4^8 ]
|
|
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> MonomsOfCoxRingOfDegree(D);
[ x_1*x_2, x_1^8*x_3, x_2*x_4, x_1^7*x_3*x_4, x_1^6*x_3*x_4^2,
x_1^5*x_3*x_4^3, x_1^4*x_3*x_4^4, x_1^3*x_3*x_4^5, x_1^2*x_3*x_4^6,
x_1*x_3*x_4^7, x_3*x_4^8 ]
gap> D2:=D-2*P;
<A divisor of a toric variety with coordinates ( 1, -1, 0, 0 )>
gap> IsBasepointFree(D2);
false
gap> IsAmple(D2);
false


#### 8.2 The GAP category

##### 8.2-1 IsToricDivisor
 ‣ IsToricDivisor( M ) ( filter )

Returns: true or false

The GAP category of torus invariant Weil divisors.

##### 8.2-2 twitter
 ‣ twitter( arg ) ( attribute )

#### 8.3 Properties

##### 8.3-1 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.

##### 8.3-2 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.

##### 8.3-3 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.

##### 8.3-4 IsBasepointFree
 ‣ IsBasepointFree( divi ) ( property )

Returns: true or false

Checks if the divisor divi is basepoint free.

##### 8.3-5 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.

##### 8.3-6 IsVeryAmple
 ‣ IsVeryAmple( divi ) ( property )

Returns: true or false

Checks if the divisor divi is very ample.

##### 8.3-7 IsNumericallyEffective
 ‣ IsNumericallyEffective( divi ) ( property )

Returns: true or false

Checks if the divisor divi is nef.

#### 8.4 Attributes

##### 8.4-1 CartierData
 ‣ CartierData( divi ) ( attribute )

Returns: a list

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

##### 8.4-2 CharacterOfPrincipalDivisor
 ‣ CharacterOfPrincipalDivisor( divi ) ( attribute )

Returns: a homalg module element

Returns the character corresponding to the principal divisor divi.

##### 8.4-3 ClassOfDivisor
 ‣ ClassOfDivisor( divi ) ( attribute )

Returns: a homalg module element

Returns the class group element corresponding to the divisor divi.

##### 8.4-4 PolytopeOfDivisor
 ‣ PolytopeOfDivisor( divi ) ( attribute )

Returns: a polytope

Returns the polytope corresponding to the divisor divi.

##### 8.4-5 BasisOfGlobalSections
 ‣ BasisOfGlobalSections( divi ) ( attribute )

Returns: a list

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

##### 8.4-6 IntegerForWhichIsSureVeryAmple
 ‣ IntegerForWhichIsSureVeryAmple( divi ) ( attribute )

Returns: an integer

Returns an integer n such that n \cdot \text{divi} is very ample.

##### 8.4-7 AmbientToricVariety
 ‣ AmbientToricVariety( divi ) ( attribute )

Returns: a variety

Returns the toric variety which contains the prime divisors of the divisor divi.

##### 8.4-8 UnderlyingGroupElement
 ‣ UnderlyingGroupElement( divi ) ( attribute )

Returns: a homalg module element

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

##### 8.4-9 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.

##### 8.4-10 DegreeOfDivisor
 ‣ DegreeOfDivisor( divi ) ( attribute )

Returns: an integer

Returns the degree of the divisor divi. This is not to be confused with the (divisor) class of divi!

##### 8.4-11 VarietyOfDivisorpolytope
 ‣ VarietyOfDivisorpolytope( divi ) ( attribute )

Returns: a variety

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

##### 8.4-12 MonomsOfCoxRingOfDegree
 ‣ MonomsOfCoxRingOfDegree( divi ) ( attribute )

Returns: a list

Returns the monoms in the Cox ring of degree equal to the (divisor) class of the divisor divi.

##### 8.4-13 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.

##### 8.4-14 RingMorphismOfDivisor
 ‣ RingMorphismOfDivisor( divi ) ( attribute )

Returns: a ring map

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.

#### 8.5 Methods

##### 8.5-1 VeryAmpleMultiple
 ‣ VeryAmpleMultiple( divi ) ( operation )

Returns: a divisor

Returns a very ample multiple of the ample divisor divi. The method will fail if divisor is not ample.

##### 8.5-2 CharactersForClosedEmbedding
 ‣ CharactersForClosedEmbedding( divi ) ( operation )

Returns: a list

Returns characters for closed embedding defined via the ample divisor divi. The method fails if the divisor divi is not ample.

##### 8.5-3 \+
 ‣ \+( divi1, divi2 ) ( operation )

Returns: a divisor

Returns the sum of the divisors divi1 and divi2.

##### 8.5-4 \-
 ‣ \-( divi1, divi2 ) ( operation )

Returns: a divisor

Returns the divisor divi1 minus divi2.

##### 8.5-5 \*
 ‣ \*( k, divi ) ( operation )

Returns: a divisor

Returns k times the divisor divi.

##### 8.5-6 MonomsOfCoxRingOfDegree
 ‣ MonomsOfCoxRingOfDegree( vari, elem ) ( operation )

Returns: a list

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

##### 8.5-7 DivisorOfGivenClass
 ‣ DivisorOfGivenClass( vari, elem ) ( operation )

Returns: a divisor

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.

##### 8.5-8 AddDivisorToItsAmbientVariety
 ‣ AddDivisorToItsAmbientVariety( divi ) ( operation )

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

##### 8.5-9 Polytope
 ‣ Polytope( divi ) ( operation )

Returns: a polytope

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

##### 8.5-10 CoxRingOfTargetOfDivisorMorphism
 ‣ CoxRingOfTargetOfDivisorMorphism( divi, string ) ( operation )

Returns: a ring

Given a toric divisor divi, it induces a toric morphism. The target of this morphism is a toric variety. This method returns the Cox ring of this target. The variables are named according to string.

#### 8.6 Constructors

##### 8.6-1 DivisorOfCharacter
 ‣ DivisorOfCharacter( elem, vari ) ( operation )

Returns: a divisor

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

##### 8.6-2 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.

##### 8.6-3 CreateDivisor
 ‣ CreateDivisor( elem, vari ) ( operation )

Returns: a divisor

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

##### 8.6-4 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.

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