next | previous | forward | backward | up | index | toc

# der(CentralArrangement,List) -- compute the module of logarithmic derivations

## Synopsis

• Function: der
• Usage:
der(A, m)
der(A)
• Inputs:
• A, , a central arrangement of hyperplanes
• m, a list, an optional list of multiplicities, one for each hyperplane
• Optional inputs:
• Strategy => , default value null, that specifies the algorithm. If an arrangement has (squarefree) defining polynomial $Q$, then the logarithmic derivations are those derivations $D$ for which $D(Q)$ is in the ideal $(Q)$. The Popescu strategy assumes that the arrangement is simple and implements this definition. By contrast, the default strategy treats all arrangements as multiarrangements.
• Outputs:
• , whose image is the module of logarithmic derivations corresponding to the (multi)arrangement ${\mathcal A}$; see below.

## Description

The module of logarithmic derivations of an arrangement defined over a ring $S$ is, by definition, the submodule of $S$-derivations $D$ 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.

In this package, we grade derivations so that a constant coefficient derivation (i.e. a derivation $D$ which takes linear forms to constants) has degree 0. In the literature, this is often called polynomial degree.

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 the ideal $(f_i^{m_i})$ for each linear form $f_i$. See Günter M. Ziegler, Multiarrangements of hyperplanes and their freeness, in Singularities (Iowa City, IA, 1986), 345-359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989.

The $j$th column of the output matrix expresses the $j$th 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 0 0 | {1} | -1 -x+y 0 | {1} | -1 -x+z -xy+xz+yz-z2 | {1} | -1 y 0 | {1} | -1 z yz-z2 | {1} | -1 -x+y+z xz-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 ${\mathcal A}$ is treated as if it were essential: that is, the intersection of all the hyperplanes is the origin. So, the rank of the matrix produced by der equals the rank of the arrangement. For instance, although the $A_3$ arrangement is not essential, der will produce a rank 3 matrix.

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

A hyperplane arrangement ${\mathcal A}$ is free if the module of derivations is a free $S$-module. Not all arrangements are free.

 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 : der A o7 = {1} | -1 0 0 0 | {1} | -1 z -z -x-y-z | {1} | -1 -y -x-z 0 | {1} | -1 0 -z -y | 4 4 o7 : Matrix R <-- R i8 : betti res prune image der A 0 1 o8 = total: 4 1 1: 1 . 2: 3 1 o8 : BettiTally

The Popescu strategy produces a different presentation of the module of logarithm derivations. For instance, in the following example, the first three rows of column 0 means that $x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y} + z\frac{\partial}{\partial z}$ is a logarithmic derivation of $\mathcal A$, and the last row of column 0 means that applying this derivation to $xyz(x+y+z)$ produces $4xyz(x+y+z)$.

 i9 : der(A, Strategy => Popescu) o9 = {0} | x 0 0 | {0} | y -yz yz | {0} | z yz xz+z2 | {1} | 4 y-z x+3z | 4 3 o9 : Matrix R <-- R

If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:

 i10 : R = QQ[x,y] o10 = R o10 : PolynomialRing i11 : prune image der arrangement {x,y,x-y,y-x,y,2*x} -- rank 2 => free 2 o11 = R o11 : R-module, free, degrees {2:3} i12 : prune image der(arrangement {x,y,x-y}, {2,2,2}) -- same 2 o12 = R o12 : R-module, free, degrees {2:3}