deform -- Compute the deformations associated to a Stanley-Reisner complex.

Synopsis

Usage:

deform(C)
Inputs:
- C, an embedded complex,
Outputs:
- a list,

Description

Compute a list with the homogeneous (i.e., degree(FirstOrderDeformation) zero) deformations associated to Stanley-Reisner complex C.

The result of the computation is stored in the key C.deform and this data can be stored to and loaded from a file by saveDeformations and loadDeformations.

i1 : R=QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing

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

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

o2 : Ideal of R

i3 : C1=idealToComplex I

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

o3 : complex of dim 1 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1

i4 : deform C1

                2             2             2             2             2
      x   x    x    x   x    x    x   x    x    x   x    x    x   x    x   
       0   0    0    1   1    1    2   2    2    3   3    3    4   4    4  
o4 = {--, --, ----, --, --, ----, --, --, ----, --, --, ----, --, --, ----,
      x   x   x x   x   x   x x   x   x   x x   x   x   x x   x   x   x x  
       3   2   2 3   4   3   3 4   4   0   0 4   1   0   0 1   2   1   1 2 
     ------------------------------------------------------------------------
     x   x   x   x   x   x   x   x   x   x
      1   4   0   2   1   3   2   4   0   3
     --, --, --, --, --, --, --, --, --, --}
     x   x   x   x   x   x   x   x   x   x
      0   0   1   1   2   2   3   3   4   4

o4 : 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.

Ways to use deform :

deform(Complex)

For the programmer

The object deform is a method function.