# deformationsFace -- Compute the deformations associated to a face.

## Synopsis

• Usage:
deformationsFace(F,C)
deformationsFace(F,C,I)
• Inputs:
• F, a face,
• C, ,
• I, an ideal, reduced monomial
• Outputs:

## 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.