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deformationsFace -- Compute the deformations associated to a face.

Synopsis

Description

Compute the homogeneous (i.e., degree(FirstOrderDeformation) zero) deformations associated to a face F of the complex C.

The additional parameter I should be the Stanley-Reisner ideal of C and can be given to avoid computation of the Stanley-Reisner ideal if it is already known. Usually this is not necessary: Once I is computed it is stored in C.ideal, so deformationsFace(F,C,I) is equivalent to deformationsFace(F,C). Note also that all methods producing a complex from an ideal (like idealToComplex) store the ideal in C.ideal.

The deformations and C are stored in F.deform = {C, deformations}. Note that usually C is not ofComplex F.

i1 : R=QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing
 
i2 : I=ideal(x_0*x_1*x_2,x_3*x_4)

o2 = ideal (x x x , x x )
             0 1 2   3 4

o2 : Ideal of R
 
i3 : C1=idealToComplex I

o3 = 2: x x x  x x x  x x x  x x x  x x x  x x x  
         0 1 3  0 2 3  1 2 3  0 1 4  0 2 4  1 2 4

o3 : complex of dim 2 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 9, 6, 0, 0}, Euler = 1
 
i4 : F=C1.fc_0_0

o4 = x
      0

o4 : face with 1 vertex
 
i5 : deformationsFace(F,C1)

                        2     2
      x   x   x   x    x     x
       0   0   0   0    0     0
o5 = {--, --, --, --, ----, ----}
      x   x   x   x   x x   x x
       4   3   2   1   3 4   1 2

o5 : List
 
i6 : F=C1.fc_0_1

o6 = x
      1

o6 : face with 1 vertex
 
i7 : deformationsFace(F,C1)

                        2     2
      x   x   x   x    x     x
       1   1   1   1    1     1
o7 = {--, --, --, --, ----, ----}
      x   x   x   x   x x   x x
       4   3   2   0   3 4   0 2

o7 : List
 
i8 : F=C1.fc_1_0

o8 = x x
      0 1

o8 : face with 2 vertices
 
i9 : deformationsFace(F,C1)

      x x
       0 1
o9 = {----}
      x x
       3 4

o9 : List
 
i10 : F=C1.fc_2_0

o10 = x x x
       0 1 3

o10 : face with 3 vertices
 
i11 : deformationsFace(F,C1)

o11 = {}

o11 : List
 

i12 : R=QQ[x_0..x_4]

o12 = R

o12 : PolynomialRing
 
i13 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)

o13 = ideal (x x , x x , x x , x x , x x )
              0 1   1 2   2 3   3 4   0 4

o13 : Ideal of R
 
i14 : C1=idealToComplex I

o14 = 1: x x  x x  x x  x x  x x  
          0 2  0 3  1 3  1 4  2 4

o14 : complex of dim 1 embedded in dim 4 (printing facets)
      equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1
 
i15 : F=C1.fc_0_1

o15 = x
       1

o15 : face with 1 vertex
 
i16 : deformationsFace(F,C1)

                 2
       x   x    x
        1   1    1
o16 = {--, --, ----}
       x   x   x x
        4   3   3 4

o16 : List
 
i17 : F=C1.fc_1_1

o17 = x x
       0 3

o17 : face with 2 vertices
 
i18 : deformationsFace(F,C1)

o18 = {}

o18 : List

Caveat

To homogenize the denominators of deformations (which are supported inside the link) we use globalSections to deal with the toric case. Speed of this should be improved. For ordinary projective space globalSections works much faster.

See also

Ways to use deformationsFace :

For the programmer

The object deformationsFace is a method function.