Creating a normal toric variety reorders the rays?

[Kumar Sannidhya Shukla]

Dear Macaulay2 users,

If I create a normal toric variety from a ray list and a cone list, and then look at the maximal cones in the fan of the toric variety, the numbering assigned to the rays appears changed. For example, consider the following snippet

i2 : rayList = {{1, 0}, {1, 1}, {-1, 0}, {-1, -1}}

o2 = {{1, 0}, {1, 1}, {-1, 0}, {-1, -1}}

o2 : List

i3 : coneList = {{0, 1}, {2, 3}}

o3 = {{0, 1}, {2, 3}}

o3 : List

i4 : X = normalToricVariety (rayList, coneList)

o4 = X

o4 : NormalToricVariety

i5 : maxCones fan X

o5 = {{1, 3}, {0, 2}}

o5 : List

Shouldn't the output in o5 be the same as the coneList defined in i3? What causes this change? Is there criterion to determine how the constructor normalToricVariety() will change the order of the rays? Is this related to this recent post?

Best regards,
Kumar.

[Greg]

Dear Kumar,

The Polyhedra package, which provides the method fan, changes the order of the rays.  If you stick with the methods from the NormalToricVarieties package, then you will avoid this discrepancy.  In particular, the cones in the normal toric variety X are labeled as in coneList.

All the best,

Greg.

i1 : needsPackage "NormalToricVarieties";

i2 : rayList = {{1,0}, {1,1}, {-1,0}, {-1,-1}}

o2 = {{1, 0}, {1, 1}, {-1, 0}, {-1, -1}}

o2 : List

i3 : coneList = {{0,1},{2,3}}

o3 = {{0, 1}, {2, 3}}

o3 : List

i4 : X = normalToricVariety(rayList, coneList)

o4 = X

o4 : NormalToricVariety

i5 : rays X

o5 = {{1, 0}, {1, 1}, {-1, 0}, {-1, -1}}

o5 : List

i6 : max X

o6 = {{0, 1}, {2, 3}}

o6 : List

i7 : maxCones fan X

o7 = {{1, 3}, {0, 2}}

o7 : List

i8 : rays fan X

o8 = | -1 1 -1 1 |

     | 0  0 -1 1 |

              2       4

o8 : Matrix ZZ  <-- ZZ

[Kumar Sannidhya Shukla]

Thanks Greg! That clears it up.

I am trying to create the Stanley-Reisner ring associated to the toric variety by calling stanleyReisnerRing fan X. The problem is this reorders the variables in the polynomial ring, for example, I would expect the monomial x_0x_1 to not be in the SR ideal since {0, 1} forms a cone but this monomial does appear in the quotient in the SR ring constructed using the stanleyReisnerRing method. Is there  a way to create the SR ring so that the name of the variables correspond to the rays?

(I suppose one way would be to construct the fan directly and not from the toric variety)

Best regards,

Kumar.

[Greg]

Dear Kumar,

For a normal toric variety X, the corresponding Stanley–Reisner ideal is dual monomialIdeal X or ideal dual monomialIdeal X.  In particular, the Alexander dual of the irrelevant ideal equals the Stanley–Reisner ideal.  Hence, the corresponding Stanley–Reisner ring is (ring X)/dual monomialIdeal X; depending on the applications, one may want to change the grading group on the polynomial ring associated to X.

All the best,

Greg.