gap> F := Fan( [[1,0,0],[0,1,0],[0,0,1]], [[1,2,3]] ); <A fan in |R^3> gap> C3 := ToricVariety( F ); <A toric variety of dimension 3> gap> IsAffine( C3 ); true gap> Dimension( C3 ); 3
More conveniently, we can build affine toric varieties from a cone:
gap> IsAffine( ProjectiveSpace( 1 ) ); false gap> C:=Cone( [[1,0,0],[0,1,0],[0,0,1]] ); <A cone in |R^3> gap> C3:=ToricVariety(C); <An affine normal toric variety of dimension 3> gap> Dimension(C3); 3 gap> IsSimplicial( C3 ); true gap> IsOrbifold(C3); true gap> IsSmooth(C3); true gap> IsProjective( C3 ); false gap> DimensionOfTorusfactor( C3 ); 0 gap> CoordinateRingOfTorus(C3,"x"); Q[x1,x1_,x2,x2_,x3,x3_]/( x1*x1_-1, x2*x2_-1, x3*x3_-1 ) gap> CoordinateRing(C3,"x"); Q[x_1,x_2,x_3] gap> ListOfVariablesOfCoordinateRing( C3 ); [ "x_1", "x_2", "x_3" ] gap> MorphismFromCoordinateRingToCoordinateRingOfTorus( C3 ); <A monomorphism of rings> gap> C3; <An affine normal smooth toric variety of dimension 3> gap> StructureDescription( C3 ); "|A^3" gap> ConeOfVariety( C3 ); <A smooth pointed simplicial cone in |R^3 with 3 ray generators> gap> Cone( C3 ); <A smooth pointed simplicial cone in |R^3 with 3 ray generators> gap> IrrelevantIdeal( C3 ); <A graded principal torsion-free (left) ideal given by a cyclic generator> gap> CartierTorusInvariantDivisorGroup( C3 ); <A free left submodule given by 3 generators>
gap> v:=Cone( [[1,0,0],[0,1,0]] ); <A cone in |R^3> gap> v:=ToricVariety(v); <An affine normal toric variety of dimension 3> gap> DimensionOfTorusfactor( v ); 1 gap> CartierTorusInvariantDivisorGroup( v ); <A free left submodule given by 3 generators> gap> ConeOfVariety( v ); <A pointed cone in |R^3 of dimension 2 with 2 ray generators> gap> Cone( v ); <A pointed cone in |R^3 of dimension 2 with 2 ray generators>
gap> v2:=Cone( [[1,1],[-1,1]] ); <A cone in |R^2> gap> v2:=ToricVariety(v2); <An affine normal toric variety of dimension 2> gap> IsSmooth( v2 ); false gap> Display( v2 ); An affine normal non smooth toric variety of dimension 2. gap> ConeOfVariety( v * v2 ); <A pointed cone in |R^5>
‣ IsAffineToricVariety ( M ) | ( filter ) |
Returns: true or false
The GAP category of an affine toric variety. All affine toric varieties are toric varieties, so everything applicable to toric varieties is applicable to affine toric varieties.
‣ CoordinateRing ( vari ) | ( attribute ) |
Returns: a ring
Returns the coordinate ring of the affine toric variety vari.
‣ ListOfVariablesOfCoordinateRing ( vari ) | ( attribute ) |
Returns: a list
Returns a list containing the variables of the CoordinateRing of the variety vari.
‣ MorphismFromCoordinateRingToCoordinateRingOfTorus ( vari ) | ( attribute ) |
Returns: a morphism
Returns the morphism between the coordinate ring of the variety vari and the coordinate ring of its torus. This defines the embedding of the torus in the variety.
‣ ConeOfVariety ( vari ) | ( attribute ) |
Returns: a cone
Returns the cone of the affine toric variety vari.
‣ CoordinateRing ( vari, indet ) | ( operation ) |
Returns: a ring
Computes the coordinate ring of the affine toric variety vari with indeterminates indet.
‣ Cone ( vari ) | ( operation ) |
Returns: a cone
Returns the cone of the variety vari. Another name for ConeOfVariety for compatibility and shortness.
The constructors are the same as for toric varieties. Calling them with a cone will result in an affine variety.
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