abstractVariety(NormalToricVariety,AbstractVariety) -- make the corresponding abstract variety

Synopsis

Function: abstractVariety
Usage:

abstractVariety (X, B)
Inputs:
- X, a normal toric variety, that is complete and simplicial
- B, an abstract variety, the base variety, when omitted the default base variety of dimension 0 is used
Optional inputs:
- ReturnType => ..., default value AbstractVariety, unused
- DefaultIntegral => ..., default value true, unused
Outputs:
- an abstract variety, determined by the dimension of X and its rational Chow ring

Description

This method converts a NormalToricVariety into an AbstractVariety, as defined in the Schubert2 package.

Since many routines from the Schubert2 package have been overloaded so that they apply directly to normal toric varieties, this method is primarily of interest to developers.

Projective space can be constructed as an AbstractVariety in a few equivalent, but not identical, ways.

i1 : tPP2 = toricProjectiveSpace 2;

i2 : aPP2 = abstractVariety tPP2

o2 = aPP2

o2 : an abstract variety of dimension 2

i3 : assert (dim tPP2 === dim aPP2)

i4 : intersectionRing aPP2

              QQ[][t ..t ]
                    0   2
o4 = ------------------------------
     (t t t , - t  + t , - t  + t )
       0 1 2     0    1     0    2

o4 : QuotientRing

i5 : intersectionRing tPP2

              QQ[][t ..t ]
                    0   2
o5 = ------------------------------
     (t t t , - t  + t , - t  + t )
       0 1 2     0    1     0    2

o5 : QuotientRing

i6 : intersectionRing abstractVariety (tPP2, base())

              QQ[][t ..t ]
                    0   2
o6 = ------------------------------
     (t t t , - t  + t , - t  + t )
       0 1 2     0    1     0    2

o6 : QuotientRing

i7 : intersectionRing abstractVariety (tPP2, base(a))

              QQ[a][t ..t ]
                     0   2
o7 = ------------------------------
     (t t t , - t  + t , - t  + t )
       0 1 2     0    1     0    2

o7 : QuotientRing

i8 : PP2 = toricProjectiveSpace 2

o8 = PP2

o8 : NormalToricVariety

i9 : intersectionRing PP2

              QQ[][t ..t ]
                    0   2
o9 = ------------------------------
     (t t t , - t  + t , - t  + t )
       0 1 2     0    1     0    2

o9 : QuotientRing

i10 : minimalPresentation intersectionRing PP2

      QQ[t ]
          2
o10 = ------
         3
        t
         2

o10 : QuotientRing

i11 : minimalPresentation intersectionRing tPP2

      QQ[t ]
          2
o11 = ------
         3
        t
         2

o11 : QuotientRing

Ways to use this method: