normalToricVariety(Ring) -- get the associated normal toric variety

Synopsis

Function: normalToricVariety
Usage:

normalToricVariety S
Inputs:
- S, a ring,
Optional inputs:
- CoefficientRing => a ring, default value QQ, not used
- MinimalGenerators => a Boolean value, default value false, not used
- Variable => a symbol, default value x, not used
- WeilToClass => a matrix, default value null, not used
Outputs:
- a normal toric variety,

If a polynomial ring is constructed as the total coordinate ring of normal toric variety, then this method returns the associated variety.

i1 : PP3 = toricProjectiveSpace 3;

i2 : S = ring PP3

o2 = S

o2 : PolynomialRing

i3 : gens S

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

o3 : List

i4 : degrees S

o4 = {{1}, {1}, {1}, {1}}

o4 : List

i5 : normalToricVariety S

o5 = PP3

o5 : NormalToricVariety

i6 : assert (PP3 === normalToricVariety S)

i7 : variety S

o7 = PP3

o7 : NormalToricVariety

i8 : assert (PP3 === variety S)

If the polynomial ring is not constructed from a variety, then this method produces an error: "no variety associated with ring".

i9 : S = QQ[x_0..x_2];

i10 : gens S

o10 = {x , x , x }
        0   1   2

o10 : List

i11 : degrees S

o11 = {{1}, {1}, {1}}

o11 : List

i12 : assert (try (normalToricVariety S; false) else true)

i13 : assert (try (variety S; false) else true)

This methods does not determine if a ring could be realized as the total coordinate ring of a normal toric variety.