# ideal(NormalToricVariety) -- make the irrelevant ideal

## Synopsis

• Usage:
ideal X
• Function: ideal
• Inputs:
• Outputs:
• an ideal, in the total coordinate ring of X

## Description

The irrelevant ideal is a reduced monomial ideal in the total coordinate ring that encodes the combinatorics of the fan. For each maximal cone in the fan, it has a minimal generator, namely the product of the variables not indexed by elements of the list corresponding to the maximal cone.

For projective space, the irrelevant ideal is generated by the variables.

 `i1 : PP4 = toricProjectiveSpace 4;` ```i2 : B = ideal PP4 o2 = ideal (x , x , x , x , x ) 4 3 2 1 0 o2 : Ideal of QQ[x , x , x , x , x ] 0 1 2 3 4``` `i3 : assert (isMonomialIdeal B and B == radical B)` ```i4 : monomialIdeal PP4 o4 = monomialIdeal (x , x , x , x , x ) 0 1 2 3 4 o4 : MonomialIdeal of QQ[x , x , x , x , x ] 0 1 2 3 4``` `i5 : assert (B == monomialIdeal PP4)`

For an affine toric variety, the irrelevant ideal is the unit ideal.

 `i6 : C = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}}, {{0,1,2,3}});` ```i7 : ideal C o7 = ideal 1 o7 : Ideal of QQ[x , x , x , x ] 0 1 2 3``` `i8 : assert (monomialIdeal C == 1)` ```i9 : monomialIdeal affineSpace 3 o9 = monomialIdeal 1 o9 : MonomialIdeal of QQ[x , x , x ] 0 1 2``` `i10 : assert (ideal affineSpace 3 == 1)`

The irrelevant ideal for a product of toric varieties is intersection of the irrelevant ideal of the factors.

 `i11 : X = toricProjectiveSpace (2) ** toricProjectiveSpace (3);` `i12 : S = ring X;` ```i13 : B = ideal X o13 = ideal (x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , 2 6 2 5 2 4 2 3 1 6 1 5 1 4 1 3 0 6 0 5 ----------------------------------------------------------------------- x x , x x ) 0 4 0 3 o13 : Ideal of S``` ```i14 : primaryDecomposition B o14 = {ideal (x , x , x ), ideal (x , x , x , x )} 0 1 2 3 4 5 6 o14 : List``` ```i15 : dual monomialIdeal B o15 = monomialIdeal (x x x , x x x x ) 0 1 2 3 4 5 6 o15 : MonomialIdeal of S```

For a complete simplicial toric variety, the irrelevant ideal is the Alexander dual of the Stanley-Reisner ideal of the fan.

 `i16 : Y = smoothFanoToricVariety (2,3);` ```i17 : dual monomialIdeal Y o17 = monomialIdeal (x x , x x , x x , x x , x x ) 0 2 0 3 1 3 1 4 2 4 o17 : MonomialIdeal of QQ[x , x , x , x , x ] 0 1 2 3 4``` ```i18 : sort apply (max Y, s -> select (# rays Y, i -> not member (i,s))) o18 = {{0, 1, 2}, {0, 1, 4}, {0, 3, 4}, {1, 2, 3}, {2, 3, 4}} o18 : List``` ```i19 : primaryDecomposition dual monomialIdeal Y o19 = {monomialIdeal (x , x , x ), monomialIdeal (x , x , x ), monomialIdeal 0 1 2 0 1 4 ----------------------------------------------------------------------- (x , x , x ), monomialIdeal (x , x , x ), monomialIdeal (x , x , x )} 0 3 4 1 2 3 2 3 4 o19 : List```

Since the irrelevent ideal is a monomial ideal, the command monomialIdeal also produces the irrelevant ideal.

 ```i20 : code (monomialIdeal, NormalToricVariety) o20 = -- code for method: monomialIdeal(NormalToricVariety) /home2/dan/src/M2/M2/Macaulay2/packages/NormalToricVarieties.m2:1160:56 monomialIdeal NormalToricVariety := MonomialIdeal => X -> monomialIdeal ----------------------------------------------------------------------- -1160:79: --source code: ideal X```