‣ IsAffineToricVariety( M ) | ( category ) |
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.
Affine toric varieties have no additional properties. Remember that affine toric varieties are toric varieties, so every property of a toric variety is a property of an affine toric variety.
‣ CoordinateRing( vari ) | ( attribute ) |
Returns: a ring
Returns the coordinate ring of the affine toric variety vari. The computation is mainly done in ToricIdeals package.
‣ 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 ring of the affine toric variety vari.
‣ CoordinateRing( vari, indet ) | ( operation ) |
Returns: a variety
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.
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> IsOrbifold(C3); true gap> IsSmooth(C3); true gap> CoordinateRingOfTorus(C3,"x"); Q[x1,x1_,x2,x2_,x3,x3_]/( x3*x3_-1, x2*x2_-1, x1*x1_-1 ) gap> CoordinateRing(C3,"x"); Q[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"
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