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NormalToricVarieties :: abstractVariety(NormalToricVariety,AbstractVariety)

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

Synopsis

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

o4 : QuotientRing
i5 : intersectionRing tPP2

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

o7 : QuotientRing
i8 : PP2 = toricProjectiveSpace 2

o8 = normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))

o8 : NormalToricVariety
i9 : intersectionRing PP2

            QQ[][t , t , t ]
                  0   1   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