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  Example  gap> H7 := Fan( [[0,1],[1,0],[0,-1],[-1,7]],[[1,2],[2,3],[3,4],[4,1]] );  gap> H7 := ToricVariety( H7 );  gap> P := TorusInvariantPrimeDivisors( H7 ); [ ,   ,   ,   ] gap> D := P[3]+P[4];  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);  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);  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));  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];  gap> IsBasepointFree(D2); false gap> IsAmple(D2); false