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isToricDegeneration -- Does the matching field give rise to a toric degeneration

Synopsis

Description

A matching field is said to give rise to a toric degeneration (of the corresponding variety: Grassmannian or partial flag variety) if the matching field ideal is equal to the initial ideal of the Pluecker ideal with respect the weight order that induces the matching field. For further details on each of these ideals see the functions matchingFieldIdeal and plueckerIdeal.

i1 : L = diagonalMatchingField(2, 4)

o1 = Grassmannian Matching Field for Gr(2, 4)

o1 : GrMatchingField
 
i2 : I = plueckerIdeal L

o2 = ideal(p   p    - p   p    + p   p   )
            2,3 1,4    1,3 2,4    1,2 3,4

o2 : Ideal of QQ[p   ..p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   1,4   2,4   3,4
 
i3 : J = matchingFieldIdeal L

o3 = ideal(p   p    - p   p   )
            2,3 1,4    1,3 2,4

o3 : Ideal of QQ[p   ..p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   1,4   2,4   3,4
 
i4 : J == ideal leadTerm(1, I)

o4 = true
 
i5 : isToricDegeneration L

o5 = true

In the above example, the last two tests are the same.

If the matching field provided is not defined in terms of a weight matrix then one is automatically computed for it. If the matching field is not coherent then this will produce an error.

See also

Ways to use isToricDegeneration :

For the programmer

The object isToricDegeneration is a method function.

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