der -- Module of logarithmic derivations

Synopsis

• Usage:

  der(A) or der(A,m)

• Inputs:
  • A, a central hyperplane arrangement, a central arrangement, possibly with repeated hyperplanes
  • m, a list, optional list of multiplicities, one for each hyperplane
• Optional inputs:
  • Strategy => ..., default value null,
• Outputs:
  • a matrix, A matrix whose image is the module of logarithmic derivations corresponding to the (multi)arrangement A; see below.

Description

More generally, if the linear form f_i is given a positive integer multiplicity m_i, then the logarithmic derivations are those D with the property that D(f_i) is in ideal(f_i^(m_i)) for each linear form f_i.

The jth column of the output matrix expresses the jth generator of the derivation module in terms of its value on each linear form, in order.

i1 : R = QQ[x,y,z];

i2 : der arrangement {x,y,z,x-y,x-z,y-z}

o2 = {1} | 1 y-z   -yz+z2      |
     {1} | 1 y     0           |
     {1} | 1 x-z   -xz+z2      |
     {1} | 1 0     0           |
     {1} | 1 x     0           |
     {1} | 1 x+y-z xy-xz-yz+z2 |

             6       3
o2 : Matrix R  <--- R

This method is implemented in such a way that any derivations of degree 0 are ignored. Equivalently, the arrangement A is forced to be essential: that is, the intersection of all the hyperplanes is the origin.

i3 : prune image der typeA(3)

                 3
o3 = (QQ[x ..x ])
          1   4

o3 : QQ[x ..x ]-module, free, degrees {1..3}
         1   4

i4 : prune image der typeB(4) -- A is said to be free if der(A) is a free module

                 4
o4 = (QQ[x ..x ])
          1   4

o4 : QQ[x ..x ]-module, free, degrees {1, 3, 5, 7}
         1   4

i5 : R = QQ[x,y,z];

i6 : A = arrangement {x,y,z,x+y+z}

o6 = {x, y, z, x + y + z}

o6 : Hyperplane Arrangement 

i7 : betti res prune image der A

            0 1
o7 = total: 4 1
         1: 1 .
         2: 3 1

o7 : BettiTally

i8 : R = QQ[x,y]

o8 = R

o8 : PolynomialRing

i9 : prune image der arrangement {x,y,x-y,y-x,y,2*x}   -- rank 2 => free

      2
o9 = R

o9 : R-module, free, degrees {2:3}

i10 : prune image der(arrangement {x,y,x-y}, {2,2,2})  -- same thing

       2
o10 = R

o10 : R-module, free, degrees {2:3}