3.5-3 \*
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> IsSimplicial( H5 ); true gap> IsAffine( H5 ); false gap> IsOrbifold( H5 ); true gap> IsProjective( H5 ); true gap> ithBettiNumber( H5, 0 ); 1 gap> DimensionOfTorusfactor( H5 ); 0 gap> Length( AffineOpenCovering( H5 ) ); 4 gap> MorphismFromCoxVariety( H5 ); <A "homomorphism" of right objects> gap> CartierTorusInvariantDivisorGroup( H5 ); <A free left submodule given by 8 generators> 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> WeilDivisorsOfVariety( H5 );; gap> CoordinateRingOfTorus( H5 ); Q[x1,x1_,x2,x2_]/( x1*x1_-1, x2*x2_-1 ) gap> CoordinateRingOfTorus( H5,"x" ); Q[x1,x1_,x2,x2_]/( x1*x1_-1, x2*x2_-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> divi := DivisorOfCharacter( [ 1,2 ],H5 ); <A principal divisor of a toric variety with coordinates ( 9, -2, 2, 1 )> gap> BasisOfGlobalSections( divi ); [ |[ x1_*x2_^2 ]| ] gap> ZariskiCotangentSheafViaPoincareResidueMap( H5 );; gap> ZariskiCotangentSheafViaEulerSequence( H5 );; gap> EQ( H5, ProjectiveSpace( 2 ) ); false gap> H5B1 := BlowUpOnIthMinimalTorusOrbit( H5, 1 ); <A toric variety of dimension 2> #@if IsPackageMarkedForLoading( "TopcomInterface", ">= 2021.08.12" ) gap> H5_version2 := DeriveToricVarietiesFromGrading( [[0,1,1,0],[1,0,-5,1]], false ); [ <A toric variety of dimension 2> ] gap> H5_version3 := ToricVarietyFromGrading( [[0,1,1,0],[1,0,-5,1]] ); <A toric variety of dimension 2> #@fi gap> NameOfVariety( H5 ); "H_5" gap> Display( H5 ); A projective normal toric variety of dimension 2. The torus of the variety is RingWithOne( ... ). The class group is <object> and the Cox ring is RingWithOne( ... ).
Another example
gap> P2 := ProjectiveSpace( 2 ); <A projective toric variety of dimension 2> gap> IsNormalVariety( P2 ); true gap> AffineCone( P2 ); <An affine normal toric variety of dimension 3> gap> PolytopeOfVariety( P2 ); <A polytope in |R^2 with 3 vertices> gap> IsIsomorphicToProjectiveSpace( P2 ); true gap> IsIsomorphicToProjectiveSpace( H5 ); false gap> Length( MonomsOfCoxRingOfDegree( P2, [1,2,3] ) ); 28 gap> IsDirectProductOfPNs( P2 * P2 ); true gap> IsDirectProductOfPNs( P2 * H5 ); false
gap> rays := [ [1,0,0], [-1,0,0], [0,1,0], [0,-1,0], [0,0,1], [0,0,-1], > [2,1,1], [1,2,1], [1,1,2], [1,1,1] ]; [ [ 1, 0, 0 ], [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ], [ 0, 0, -1 ], [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ] gap> cones := [ [1,3,6], [1,4,6], [1,4,5], [2,3,6], [2,4,6], [2,3,5], [2,4,5], > [1,5,9], [3,5,8], [1,3,7], [1,7,9], [5,8,9], [3,7,8], > [7,9,10], [8,9,10], [7,8,10] ]; [ [ 1, 3, 6 ], [ 1, 4, 6 ], [ 1, 4, 5 ], [ 2, 3, 6 ], [ 2, 4, 6 ], [ 2, 3, 5 ], [ 2, 4, 5 ], [ 1, 5, 9 ], [ 3, 5, 8 ], [ 1, 3, 7 ], [ 1, 7, 9 ], [ 5, 8, 9 ], [ 3, 7, 8 ], [ 7, 9, 10 ], [ 8, 9, 10 ], [ 7, 8, 10 ] ] gap> F := Fan( rays, cones ); <A fan in |R^3> gap> T := ToricVariety( F ); <A toric variety of dimension 3> gap> [ IsSmooth( T ), IsComplete( T ), IsProjective( T ) ]; [ true, true, false ] gap> SRIdeal( T ); <A graded torsion-free (left) ideal given by 23 generators>
gap> rays := [ [1,0],[-1,0],[0,1],[0,-1] ]; [ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ] gap> cones := [ [1,3],[1,4],[2,3],[2,4] ]; [ [1,3],[1,4],[2,3],[2,4] ] gap> weights := [ [1,0],[1,0],[0,1],[0,1] ]; [ [1,0],[1,0],[0,1],[0,1] ] gap> weights2 := [ [1,1],[1,1],[1,2],[1,2] ]; [ [1,1],[1,1],[1,2],[1,2] ] gap> tor1 := ToricVariety( rays, cones, weights, "x1,x2,y1,y2" ); <A toric variety of dimension 2> gap> CoxRing( tor1 ); Q[x2,y2,y1,x1] (weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ]) gap> tor2:= ToricVariety( rays, cones, weights, "q" ); <A toric variety of dimension 2> gap> CoxRing( tor2 ); Q[q_2,q_4,q_3,q_1] (weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ]) gap> tor3:= ToricVariety( rays, cones, weights ); <A toric variety of dimension 2> gap> CoxRing( tor3 ); Q[x_2,x_4,x_3,x_1] (weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ]) gap> tor4:= ToricVariety( rays, cones, weights2, "x1,x2,z1,z2" ); <A toric variety of dimension 2> gap> CoxRing( tor4 ); Q[x2,z2,z1,x1] (weights: [ ( 1, 1 ), ( 1, 2 ), ( 1, 2 ), ( 1, 1 ) ])
The following example shows how to create the projective space \(\mathbb{P}^2\) from the grading of its Cox ring. Note that this functionality requires the package TopcomInterface.
gap> g := [[1,1,1]]; [ [ 1,1,1 ] ] gap> v1 := ToricVarietyFromGrading( g ); <A toric variety of dimension 2> gap> CoxRing( v1 ); Q[x_1,x_2,x_3] (weights: [ 1, 1, 1 ])
The following example shows how to create the resolved conifold(s) from the grading of its Cox ring.
gap> g2 := [[1,1,-1,-1]]; [ [ 1,1,-1,-1 ] ] gap> v2 := ToricVarietiesFromGrading( g2 ); [ <A toric variety of dimension 3>, <A toric variety of dimension 3> ] gap> CoxRing( v2[ 1 ] ); Q[x_1,x_2,x_3,x_4] (weights: [ 1, -1, -1, 1 ]) gap> Display( SRIdeal( v2[ 1 ] ) ); x_2*x_3 A (left) ideal generated by the entry of the above matrix (graded, degree of generator: -2) gap> Display( SRIdeal( v2[ 2 ] ) ); x_1*x_4 A (left) ideal generated by the entry of the above matrix (graded, degree of generator: 2)
The following code exemplifies blowups of the 3-dimensional affine space.
gap> rays := [ [1,0,0], [0,1,0], [0,0,1] ]; [ [1,0,0], [0,1,0], [0,0,1] ] gap> max_cones := [ [1,2,3] ]; [ [1,2,3] ] gap> fan := Fan( rays, max_cones ); <A fan in |R^3> gap> C3 := ToricVariety( rays, max_cones, [[0],[0],[0]], "x1,x2,x3" ); <A toric variety of dimension 3> gap> B1C3 := BlowupOfToricVariety( C3, "x1,x2,x3", "u0" ); <A toric variety of dimension 3> gap> [ IsComplete( B1C3 ), IsOrbifold( B1C3 ), IsSmooth( B1C3 ) ]; [ false, true, true ] gap> B2C3 := BlowupOfToricVariety( B1C3, "x1,u0", "u1" ); <A toric variety of dimension 3> gap> Rank( ClassGroup( B2C3 ) ); 3 gap> B3C3 := BlowupOfToricVariety( B2C3, "x1,u1", "u2" ); <A toric variety of dimension 3> gap> CoxRing( B3C3 ); Q[x3,x2,x1,u0,u1,u2] (weights: [ ( 0, 1, 0, 0 ), ( 0, 1, 0, 0 ), ( 0, 1, 1, 1 ), ( 0, -1, 1, 0 ), ( 0, 0, -1, 1 ), ( 0, 0, 0, -1 ) ])
Likewise, we can also perform blowups of the 3-dimensional projective space.
gap> rays := [ [1,0,0], [0,1,0], [0,0,1], [-1,-1,-1] ]; [ [1,0,0], [0,1,0], [0,0,1], [-1,-1,-1] ] gap> max_cones := [ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ]; [ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ] gap> fan := Fan( rays, max_cones ); <A fan in |R^3> gap> P3 := ToricVariety( rays, max_cones, [[1],[1],[1],[1]], "x1,x2,x3,x4" ); <A toric variety of dimension 3> gap> B1P3 := BlowupOfToricVariety( P3, "x1,x2,x3", "u0" ); <A toric variety of dimension 3> gap> [ IsComplete( B1P3 ), IsOrbifold( B1P3 ), IsSmooth( B1P3 ) ]; [ true, true, true ] gap> B2P3 := BlowupOfToricVariety( B1P3, "x1,u0", "u1" ); <A toric variety of dimension 3> gap> Rank( ClassGroup( B2C3 ) ); 3 gap> B3P3 := BlowupOfToricVariety( B2P3, "x1,u1", "u2" ); <A toric variety of dimension 3> gap> CoxRing( B3P3 ); Q[x4,x3,x2,x1,u0,u1,u2] (weights: [ ( 1, 0, 0, 0 ), ( 1, 1, 0, 0 ), ( 1, 1, 0, 0 ), ( 1, 1, 1, 1 ), ( 0, -1, 1, 0 ), ( 0, 0, -1, 1 ), ( 0, 0, 0, -1 ) ])
Also, we can perform blowups of a generalized Hirzebruch 3-fold.
gap> vars := "u,s,v,t,r"; "u,s,v,t,r" gap> rays := [ [0,0,-1],[1,0,0],[0,1,0],[-1,-1,-17],[0,0,1] ]; [ [0,0,-1],[1,0,0],[0,1,0],[-1,-1,-17],[0,0,1] ] gap> cones := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ]; [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ] gap> weights := [ [1,-17], [0,1], [0,1], [0,1], [1,0] ]; [ [1,-17], [0,1], [0,1], [0,1], [1,0] ] gap> H3fold := ToricVariety( rays, cones, weights, vars ); <A toric variety of dimension 3> gap> B1H3fold := BlowupOfToricVariety( H3fold, "u,s", "u1" ); <A toric variety of dimension 3> gap> CoxRing( B1H3fold ); Q[t,u,r,v,u1,s] (weights: [ ( 0, 1, 0 ), ( 1, -17, 1 ), ( 1, 0, 0 ), ( 0, 1, 0 ), ( 0, 0, -1 ), ( 0, 1, 1 ) ])
This example easily extends to an entire sequence of blowups.
gap> vars := "u,s,v,t,r,x,y,w"; "u,s,v,t,r,x,y,w" gap> rays := [ [0,0,-1,-2,-3], [1,0,0,-2,-3], [0,1,0,-2,-3], [-1,-1,-17,-2,-3], > [0,0,1,-2,-3], [0, 0, 0, 1, 0], > [0, 0, 0, 0, 1], [0, 0, 0, -2, -3] ]; [ [0,0,-1,-2,-3], [1,0,0,-2,-3], [0,1,0,-2,-3], [-1,-1,-17,-2,-3], [0,0,1,-2,-3], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 0, 0, -2, -3] ] gap> cones := [ [1,2,3,6,7], [1,2,3,6,8], [1,2,3,7,8], [1,2,4,6,7], [1,2,4,6,8], > [1,2,4,7,8], [1,3,4,6,7], [1,3,4,6,8], [1,3,4,7,8], [2,3,5,6,7], > [2,3,5,6,8], [2,3,5,7,8], [2,4,5,6,7], [2,4,5,6,8], [2,4,5,7,8], > [3,4,5,6,7], [3,4,5,6,8], [3,4,5,7,8] ]; [ [ 1, 2, 3, 6, 7 ], [ 1, 2, 3, 6, 8 ], [ 1, 2, 3, 7, 8 ], [ 1, 2, 4, 6, 7 ], [ 1, 2, 4, 6, 8 ], [ 1, 2, 4, 7, 8 ], [ 1, 3, 4, 6, 7 ], [ 1, 3, 4, 6, 8 ], [ 1, 3, 4, 7, 8 ], [ 2, 3, 5, 6, 7 ], [ 2, 3, 5, 6, 8 ], [ 2, 3, 5, 7, 8 ], [ 2, 4, 5, 6, 7 ], [ 2, 4, 5, 6, 8 ], [ 2, 4, 5, 7, 8 ], [ 3, 4, 5, 6, 7 ], [ 3, 4, 5, 6, 8 ], [ 3, 4, 5, 7, 8 ] ] gap> w := [ [1,-17,0], [0,1,0], [0,1,0], [0,1,0], [1,0,0], [0,0,2], [0,0,3], > [-2,14,1] ]; [ [1,-17,0], [0,1,0], [0,1,0], [0,1,0], [1,0,0], [0,0,2], [0,0,3], [-2,14,1] ] gap> base := ToricVariety( rays, cones, w, vars ); <A toric variety of dimension 5> gap> b1 := BlowupOfToricVariety( base, "x,y,u", "u1" ); <A toric variety of dimension 5> gap> b2 := BlowupOfToricVariety( b1, "x,y,u1", "u2" ); <A toric variety of dimension 5> gap> b3 := BlowupOfToricVariety( b2, "y,u1", "u3" ); <A toric variety of dimension 5> gap> b4 := BlowupOfToricVariety( b3, "y,u2", "u4" ); <A toric variety of dimension 5> gap> b5 := BlowupOfToricVariety( b4, "u2,u3", "u5" ); <A toric variety of dimension 5> gap> b6 := BlowupOfToricVariety( b5, "u1,u3", "u6" ); <A toric variety of dimension 5> gap> b7 := BlowupOfToricVariety( b6, "u2,u4", "u7" ); <A toric variety of dimension 5> gap> b8 := BlowupOfToricVariety( b7, "u3,u4", "u8" ); <A toric variety of dimension 5> gap> b9 := BlowupOfToricVariety( b8, "u4,u5", "u9" ); <A toric variety of dimension 5> gap> b10 := BlowupOfToricVariety( b9, "u5,u8", "u10" ); <A toric variety of dimension 5> gap> b11 := BlowupOfToricVariety( b10, "u4,u8", "u11" ); <A toric variety of dimension 5> gap> b12 := BlowupOfToricVariety( b11, "u4,u9", "u12" ); <A toric variety of dimension 5> gap> b13 := BlowupOfToricVariety( b12, "u8,u9", "u13" ); <A toric variety of dimension 5> gap> b14 := BlowupOfToricVariety( b13, "u9,u11", "u14" ); <A toric variety of dimension 5> gap> b15 := BlowupOfToricVariety( b14, "u4,v", "d" ); <A toric variety of dimension 5> gap> final_space := BlowupOfToricVariety( b15, "u3,u5", "u15" ); <A toric variety of dimension 5>
This sequence of blowups can also be performed with a single command.
gap> final_space2 := SequenceOfBlowupsOfToricVariety( base, > [ ["x,y,u","u1"], > ["x,y,u1","u2"], > ["y,u1","u3"], > ["y,u2","u4"], > ["u2,u3","u5"], > ["u1,u3","u6"], > ["u2,u4","u7"], > ["u3,u4","u8"], > ["u4,u5","u9"], > ["u5,u8","u10"], > ["u4,u8","u11"], > ["u4,u9","u12"], > ["u8,u9","u13"], > ["u9,u11","u14"], > ["u4,v","d"], > ["u3,u5","u15"] ] ); <A toric variety of dimension 5> gap> [ IsComplete( final_space2 ), IsOrbifold( final_space2 ), > IsSmooth( final_space2 ) ]; [ true, true, false ]
‣ IsToricVariety ( M ) | ( filter ) |
Returns: true or false
Checks if an object is a toric variety.
‣ IsCategoryOfToricVarieties ( object ) | ( filter ) |
Returns: true
or false
The GAP category of toric varieties.
‣ twitter ( vari ) | ( attribute ) |
Returns: a ring
This is a dummy to get immediate methods triggered at some times. It never has a value.
‣ IsNormalVariety ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is a normal variety.
‣ IsAffine ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is an affine variety.
‣ IsProjective ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is a projective variety.
‣ IsSmooth ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is smooth.
‣ IsComplete ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is complete.
‣ HasTorusfactor ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari has a torus factor.
‣ HasNoTorusfactor ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari has no torus factor.
‣ 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.
‣ IsSimplicial ( vari ) | ( property ) |
Returns: true or false
Checks if the toric variety vari is simplicial. This is a convenience method equivalent to IsOrbifold.
‣ 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.
‣ 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.
‣ ListOfVariablesOfCoxRing ( vari ) | ( attribute ) |
Returns: a list
Returns a list of the variables of the cox ring of the variety vari.
‣ ClassGroup ( vari ) | ( attribute ) |
Returns: a module
Returns the class group of the variety vari as factor of a free module.
‣ 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.
‣ 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 be viewed as a help method to compute divisor classes.
‣ MapFromWeilDivisorsToClassGroup ( vari ) | ( attribute ) |
Returns: a morphism
Returns a map which maps a Weil divisor into the class group.
‣ Dimension ( vari ) | ( attribute ) |
Returns: an integer
Returns the dimension of the variety vari.
‣ DimensionOfTorusfactor ( vari ) | ( attribute ) |
Returns: an integer
Returns the dimension of the torus factor of the variety vari.
‣ CoordinateRingOfTorus ( vari ) | ( attribute ) |
Returns: a ring
Returns the coordinate ring of the torus of the variety vari. This is by default done with the variables x1 to xn where n is the dimension of the variety. To use a different set of variables, a convenience method is provided and described in the methods section.
‣ ListOfVariablesOfCoordinateRingOfTorus ( vari ) | ( attribute ) |
Returns: a list
Returns the list of variables in the coordinate ring of the torus of the variety vari.
‣ IsProductOf ( vari ) | ( attribute ) |
Returns: a list
If the variety vari is a product of 2 or more varieties, the list contains those varieties. If it is not a product or at least not generated as a product, the list only contains the variety itself.
‣ 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.
‣ TorusInvariantPrimeDivisors ( vari ) | ( attribute ) |
Returns: a list
The method returns a list of the torus invariant prime divisors of the variety vari.
‣ IrrelevantIdeal ( vari ) | ( attribute ) |
Returns: an ideal
Returns the irrelevant ideal of the Cox ring of the variety vari.
‣ SRIdeal ( vari ) | ( attribute ) |
Returns: an ideal
Returns the Stanley-Reißner ideal of the Cox ring of the variety vari.
‣ MorphismFromCoxVariety ( vari ) | ( attribute ) |
Returns: a morphism
The method returns the quotient morphism from the variety of the Cox ring to the variety vari.
‣ CoxVariety ( vari ) | ( attribute ) |
Returns: a variety
The method returns the Cox variety of the variety vari.
‣ FanOfVariety ( vari ) | ( attribute ) |
Returns: a fan
Returns the fan of the variety vari. This is set by default.
‣ CartierTorusInvariantDivisorGroup ( vari ) | ( attribute ) |
Returns: a module
Returns the the group of Cartier divisors of the variety vari as a subgroup of the divisor group.
‣ PicardGroup ( vari ) | ( attribute ) |
Returns: a module
Returns the Picard group of the variety vari as factor of a free module.
‣ 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.
‣ ZariskiCotangentSheaf ( vari ) | ( attribute ) |
Returns: a f.p. graded S-module
This method returns a f. p. graded S-module (S being the Cox ring of the variety), such that the sheafification of this module is the Zariski cotangent sheaf of vari.
‣ CotangentSheaf ( vari ) | ( attribute ) |
Returns: a f.p. graded S-module
This method returns a f. p. graded S-module (S being the Cox ring of the variety), such that the sheafification of this module is the cotangent sheaf of vari.
‣ EulerCharacteristic ( vari ) | ( attribute ) |
Returns: a non-negative integer
This method computes the Euler characteristic of vari.
‣ UnderlyingSheaf ( vari ) | ( operation ) |
Returns: a sheaf
The method returns the underlying sheaf of the variety vari.
‣ 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.
3.5-3 \*
‣ \* ( vari1, vari2 ) | ( operation ) |
Returns: a variety
Computes the categorial product of the varieties vari1 and vari2.
‣ 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. This computation needs to know the coordinate ring of the torus of the variety vari. By default this ring is introduced with variables x1 to xn where n is the dimension of the variety. If different variables should be used, then CoordinateRingOfTorus has to be set accordingly before calling this method.
‣ CoxRing ( vari, vars ) | ( operation ) |
Returns: a ring
Computes the Cox ring of the variety vari. vars needs to be a string. We allow for two different formats. Either, it is a string which does not contain ",". Then this string will be index and the resulting strings are then used as names for the variables of the Cox ring. Alternatively, one can also use a string containing ",". In this case, a "," is considered as separator and one can provide individual names for all variables of the Cox ring.
‣ WeilDivisorsOfVariety ( vari ) | ( operation ) |
Returns: a list
Returns a list of the currently defined Divisors of the toric variety.
‣ Fan ( vari ) | ( operation ) |
Returns: a fan
Returns the fan of the variety vari. This is a rename for FanOfVariety.
‣ Factors ( vari ) | ( operation ) |
‣ BlowUpOnIthMinimalTorusOrbit ( vari, p ) | ( operation ) |
‣ ZariskiCotangentSheafViaEulerSequence ( arg ) | ( function ) |
‣ ZariskiCotangentSheafViaPoincareResidueMap ( arg ) | ( function ) |
‣ ithBettiNumber ( vari, p ) | ( operation ) |
‣ NrOfqRationalPoints ( vari, p ) | ( operation ) |
‣ ToricVariety ( vari ) | ( operation ) |
‣ ToricVariety ( vari ) | ( operation ) |
‣ ToricVariety ( conv ) | ( operation ) |
Returns: a variety
Creates a toric variety out of the convex object conv.
‣ ToricVariety ( rays, cones, degree_list ) | ( operation ) |
Returns: a variety
Creates a toric variety from a list rays of ray generators and cones cones. Beyond the functionality of the other methods, this constructor allows to assign specific gradings to the homogeneous variables of the Cox ring. With respect to the order in which the rays appear in the list rays, we assign gradings as provided by the third argument degree_list . The latter is a list of integers.
‣ ToricVariety ( rays, cones, degree_list, var_list ) | ( operation ) |
Returns: a variety
Creates a toric variety from a list rays of ray generators and cones cones. Beyond the functionality of the other methods, this constructor allows to assign specific gradings and homogeneous variable names to the ray generators of this toric variety. With respect to the order in which the rays appear in the list rays, we assign gradings and variable names as provided by the third and fourth argument. These are the list of gradings degree_list and the list of variables names var_list. The former is a list of integers and the latter a list of strings.
‣ ToricVarietiesFromGrading ( a, list, of, lists, of, integers ) | ( operation ) |
Returns: a list of toric varieties
Given a \(\mathbb{Z}^n\)-grading of a polynomial ring, this method computes all toric varieties, which are normal and have no-torus factor and whose Cox ring obeys the given \(\mathbb{Z}^n\)-grading.
‣ ToricVarietyFromGrading ( a, list, of, lists, of, integers ) | ( operation ) |
Returns: a toric variety
Given a \(\mathbb{Z}^n\)-grading of a polynomial ring, this method computes a toric variety, which is normal and has no-torus factor and whose Cox ring obeys the given \(\mathbb{Z}^n\)-grading.
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