GAP (ToricVarieties) - Chapter 3: Toric varieties

Checks if the toric variety vari has an orbifold, which is, in the toric case, equivalent to the simpliciality of the fan.

3.3 Toric varieties: Attributes

3.3-1 AffineOpenCovering

Returns a torus invariant affine open covering of the variety vari. The affine open cover is computed out of the cones of the fan.

3.3-2 CoxRing

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.

3.3-3 ListOfVariablesOfCoxRing

3.3-4 ClassGroup

3.3-5 PicardGroup

3.3-6 TorusInvariantDivisorGroup

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.

3.3-7 MapFromCharacterToPrincipalDivisor

Returns a map which maps an element of the character group into the torus invariant Weil group of the variety vari. This has to viewn as an help method to compute divisor classes.

3.3-8 Dimension

3.3-9 DimensionOfTorusfactor

3.3-10 CoordinateRingOfTorus

Returns the coordinate ring of the torus of the variety vari. This method is not implemented, you need to call it with a second argument, which is a list of strings for the variables of the ring.

3.3-11 IsProductOf

If the variety vari is a product of 2 or more varieties, the list contain those varieties. If it is not a product or at least not generated as a product, the list only contains the variety itself.

3.3-12 CharacterLattice

The method returns the character lattice of the variety vari, computed as the containing grid of the underlying convex object, if it exists.

3.3-13 TorusInvariantPrimeDivisors

The method returns a list of the torus invariant prime divisors of the variety vari.

3.3-14 IrrelevantIdeal

3.3-15 MorphismFromCoxVariety

The method returns the quotient morphism from the variety of the Cox ring to the variety vari.

3.3-16 CoxVariety

3.3-17 FanOfVariety

3.3-18 CartierTorusInvariantDivisorGroup

Returns the the group of Cartier divisors of the variety vari as a subgroup of the divisor group.

3.3-19 NameOfVariety

Returns the name of the variety vari if it has one and it is known or can be computed.

3.3-20 twitter

This is a dummy to get immediate methods triggered at some times. It never has a value.

3.4 Toric varieties: Methods

3.4-1 UnderlyingSheaf

3.4-2 CoordinateRingOfTorus

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.

3.4-3 \*

3.4-4 CharacterToRationalFunction

Computes the rational function corresponding to the character grid element elem or to the list of integers elem. To compute rational functions you first need to compute to coordinate ring of the torus of the variety vari.

3.4-5 CoxRing

Computes the Cox ring of the variety vari. vars needs to be a string containing one variable, which will be numbered by the method.

3.4-6 WeilDivisorsOfVariety

3.4-7 Fan

3.5 Toric varieties: Constructors

3.5-1 ToricVariety

3.6 Toric varieties: Examples

3.6-1 The Hirzebruch surface of index 5

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> IsAffine( H5 );
false
gap> IsOrbifold( H5 );
true
gap> IsProjective( H5 );
true
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> CoordinateRingOfTorus(H5,"x");;
Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-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> DivisorOfCharacter([1,2],H5);
<A principal divisor of a toric variety with coordinates [ 9, 2, 1, -2 ]>
gap> BasisOfGlobalSections(last);
[ |[ x1_*x2_^2 ]| ]

3 Toric varieties

3.1 Toric variety: Category and Representations

3.1-1 IsToricVariety

3.2 Toric varieties: Properties

3.2-1 IsNormalVariety

3.2-2 IsAffine

3.2-3 IsProjective

3.2-4 IsComplete

3.2-5 IsSmooth

3.2-6 HasTorusfactor

3.2-7 HasNoTorusfactor

3.2-8 IsOrbifold