# ideal(NormalToricVariety) -- make the irrelevant ideal

## Synopsis

• Function: ideal
• Usage:
ideal X
monomialIdeal X
• Inputs:
• X, ,
• Outputs:
• an ideal, that is homogeneous 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 more information, see Subsection 5.3 in Cox-Little-Schenck's Toric Varieties.

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 ] 0 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 ] 0 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 ] 0 3 i8 : assert (monomialIdeal C == 1) i9 : monomialIdeal affineSpace 3 o9 = monomialIdeal 1 o9 : MonomialIdeal of QQ[x ..x ] 0 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 )} 2 1 0 6 5 4 3 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 ] 0 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 irrelevant ideal is a monomial ideal, the command monomialIdeal also produces the irrelevant ideal.

 i20 : code (monomialIdeal, NormalToricVariety) o20 = -- code for method: monomialIdeal(NormalToricVariety) /usr/share/Macaulay2/NormalToricVarieties/Sheaves.m2:33:56-34:25: --source code: monomialIdeal NormalToricVariety := MonomialIdeal => X -> monomialIdeal ideal X