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

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

See also

Ways to use this method:

The source of this document is in NormalToricVarieties/ChowDocumentation.m2:132:0.