The module of logarithmic derivations of an arrangement defined over a ring
S is, by definition, the submodule of
S-derivations with the property that
D(f_i) is contained in the ideal generated by
f_i for each linear form
f_i in the arrangement.
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.
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.
i1 : prune image der typeA(3)
3
o1 = (QQ [x , x , x , x ])
1 2 3 4
o1 : QQ [x , x , x , x ]-module, free, degrees {1, 2, 3}
1 2 3 4
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i2 : prune image der typeB(4) -- A is said to be free if der(A) is a free module
4
o2 = (QQ [x , x , x , x ])
1 2 3 4
o2 : QQ [x , x , x , x ]-module, free, degrees {1, 3, 5, 7}
1 2 3 4
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not all arrangements are free:
i3 : R = QQ[x,y,z];
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i4 : A = arrangement {x,y,z,x+y+z}
o4 = A
o4 : Hyperplane Arrangement
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i5 : betti res prune image der A
0 1
o5 = total: 4 1
1: 1 .
2: 3 1
o5 : BettiTally
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If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:
i6 : R = QQ[x,y]
o6 = R
o6 : PolynomialRing
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i7 : prune image der arrangement {x,y,x-y,y-x,y,2*x} -- rank 2 => free
2
o7 = R
o7 : R-module, free, degrees {3, 3}
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i8 : prune image der(arrangement {x,y,x-y}, {2,2,2}) -- same thing
2
o8 = R
o8 : R-module, free, degrees {3, 3}
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