GAP (toric) - Chapter 3: Affine toric varieties

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### 3. Affine toric varieties

This chapter concerns toric commands which deal with the coordinate rings of affine toric varieties U_sigma.

#### 3.1 Ideals defining affine toric varieties

##### 3.1-1 IdealAffineToricVariety

`> IdealAffineToricVariety`( L ) | ( function )

Input: L is a list generating a cone (as in `DualSemigroupGenerators`).
Output: the GAP ideal defining the toric variety associated to the cone generated by the vectors in L.

This computation is not very efficient and should not be used for ideals with many generators. For example, if you take L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]; then `IdealAffineToricVariety(L);` can exhaust GAP's memory allocation.

    gap> J:=IdealAffineToricVariety([[1,0],[3,4]]);
    [ two-sided ideal in PolynomialRing(..., [ x_1, x_2 ]), (3 generators) ]
    gap> GeneratorsOfIdeal(J);
    [ -x_2^2+x_1, -x_2^3+x_1^2, -x_2^4+x_1^3 ]

##### 3.1-2 EmbeddingAffineToricVariety

`> EmbeddingAffineToricVariety`( L ) | ( function )

Input: L is a list generating a cone (as in `DualSemigroupGenerators`).
Output: the toroidal embedding of X=Spec(`IdealAffineToricVariety(L)`) (given as a list of multinomials).

    gap> phi:=EmbeddingAffineToricVariety([[1,0],[3,4]]);
    [ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ]
    gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
    gap> phi:=EmbeddingAffineToricVariety(L);
    [ x_3, x_2, x_1/x_5, x_1/x_6 ]

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