| ‣ IsToricDivisor( M ) | ( category ) | 
Returns: true or false
The GAP category of torus invariant Weil divisors.
| ‣ IsCartier( divi ) | ( property ) | 
Returns: true or false
Checks if the torus invariant Weil divisor divi is Cartier i.e. if it is locally principal.
| ‣ 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.
| ‣ 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.
| ‣ IsBasepointFree( divi ) | ( property ) | 
Returns: true or false
Checks if the divisor divi is basepoint free. What else?
| ‣ IsAmple( divi ) | ( property ) | 
Returns: true or false
Checks if the divisor divi is ample, i.e. if it is colored red, yellow and green.
| ‣ IsVeryAmple( divi ) | ( property ) | 
Returns: true or false
Checks if the divisor divi is very ample.
| ‣ CartierData( divi ) | ( attribute ) | 
Returns: a list
Returns the Cartier data of the divisor divi, if it is Cartier, and fails otherwise.
| ‣ CharacterOfPrincipalDivisor( divi ) | ( attribute ) | 
Returns: an element
Returns the character corresponding to principal divisor divi.
| ‣ 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.
| ‣ ClassOfDivisor( divi ) | ( attribute ) | 
Returns: an element
Returns the class group element corresponding to the divisor divi.
| ‣ PolytopeOfDivisor( divi ) | ( attribute ) | 
Returns: a polytope
Returns the polytope corresponding to the divisor divi.
| ‣ BasisOfGlobalSections( divi ) | ( attribute ) | 
Returns: a list
Returns a basis of the global section module of the quasi-coherent sheaf of the divisor divi.
| ‣ IntegerForWhichIsSureVeryAmple( divi ) | ( attribute ) | 
Returns: an integer
Returns an integer which, to be multiplied with the ample divisor divi, someone gets a very ample divisor.
| ‣ AmbientToricVariety( divi ) | ( attribute ) | 
Returns: a variety
Returns the containing variety of the prime divisors of the divisor divi.
| ‣ UnderlyingGroupElement( divi ) | ( attribute ) | 
Returns: an element
Returns an element which represents the divisor divi in the Weil group.
| ‣ 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.
| ‣ DegreeOfDivisor( divi ) | ( attribute ) | 
Returns: an integer
Returns the degree of the divisor divi.
| ‣ MonomsOfCoxRingOfDegree( divi ) | ( attribute ) | 
Returns: a list
Returns the variety corresponding to the polytope of the divisor divi.
| ‣ 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.
| ‣ 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.
| ‣ VeryAmpleMultiple( divi ) | ( operation ) | 
Returns: a divisor
Returns a very ample multiple of the ample divisor divi. Will fail if divisor is not ample.
| ‣ CharactersForClosedEmbedding( divi ) | ( operation ) | 
Returns: a list
Returns characters for closed embedding defined via the ample divisor divi. Fails if divisor is not ample.
| ‣ 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.
| ‣ 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.
| ‣ AddDivisorToItsAmbientVariety( divi ) | ( operation ) | 
Adds the divisor divi to the Weil divisor list of its ambient variety.
| ‣ Polytope( divi ) | ( operation ) | 
Returns: a polytope
Returns the polytope of the divisor divi. Another name for PolytopeOfDivisor for compatibility and shortness.
| ‣ +( divi1, divi2 ) | ( operation ) | 
Returns: a divisor
Returns the sum of the divisors divi1 and divi2.
| ‣ -( divi1, divi2 ) | ( operation ) | 
Returns: a divisor
Returns the divisor divi1 minus divi2.
| ‣ *( k, divi ) | ( operation ) | 
Returns: a divisor
Returns k times the divisor divi.
| ‣ DivisorOfCharacter( elem, vari ) | ( operation ) | 
Returns: a divisor
Returns the divisor of the toric variety vari which corresponds to the character elem.
| ‣ 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.
| ‣ CreateDivisor( elem, vari ) | ( operation ) | 
Returns: a divisor
Returns the divisor of the toric variety vari which corresponds to the Weil group element elem.
| ‣ 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.
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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