Localizations at hyperplane arrangements: combinatorics and D-modules

Let R:=k[x₁,...,x_n] be the ring of polynomials in n variables where k is a field of characteristic zero and let A_n:=R< d₁,...,d_n > be the associated Weyl algebra. Let f be the defining polynomial of an arrangement of hyperplanes. In this work we want to study the localization module R_f. We already know that R_f is isomorphic to the quotient ring A_n/Ann_{A_n} (1/f).

Our motivation is the natural question: When is Ann_{A_n}(1/f) generated by operators of order one?

Let Ann⁽¹⁾_{A_n} (1/f) be the ideal contained in Ann_{A_n} (1/f) defined by operators of order one and let M˜^{log f} be the module A_n/Ann⁽¹⁾_{A_n} (1/f).
The answer to our question is YES if and only if the characteristic cycles of R_f and M˜^{log f} coincide.

Computing the characteristic cycle of the localization module R_f

The characteristic cycle of the localization module R_f can be completely described using the combinatorial approach given by J. Àlvarez Montaner, R. García López and S. Zarzuela in Local cohomology modules, subspace arrangements and monomial ideals , Advances in Mathematics , 174,(2003), 35-53 .

The general version of Corollary 1.3 in this paper, namely the computation of the characteristic cycle of local cohomology modules supported on arrangements of linear varieties, has been implemented by Julian Pfeifle in the software package polymake (old version 2.0.2). The script interfaces with the computer algebra system Singular for intermediate computations.

Remark: What we really compute is the characteristic cycle of the local cohomology module H¹_(f)(R) so we have to add the conormal space T_X^* X in order to get the characteristic cycle of R_f.

Source Codes:

Makefile , commalg : To start the computation and tools to interface with Singular.

ChainComplex , commalg , SimplicialComplex_as_FaceMap , Smith_normal_form : Polymake files to deal with simplicial cohomology.

primarydecomp2poset : To compute the poset associated to the arrangement.

poset2simplicialcomplex : To compute the simplicial complex associated to the poset and its homology.

Examples:

Orlik-Terao's arrangements:

f=xyzt(x+y)(x+z)(x+t)(y+z)(y+t)(z+t)(x+y+z)(x+y+t)(x+z+t)(y+z+t)(x+y+z+t): polymake result

f=xyztuv(x+y)(x+z)(x+t)(y+z)(y+t)(z+t)(x+y+z)(x+y+t)(x+z+t)(y+z+t)(x+y+z+t)(u+v): polymake result

Arrangements in A⁵_k:

f=xyztu(x+y)(z+t)(x+y+z+t+u): polymake result

Arrangements in A⁶_k:

f=xyztuv(x+y)(x+y+z+t+u+v): polymake result

f=xyztuv(x+y)(z+t+u)(x+y+z+t+u+v): polymake result

Arrangements in A⁷_k:

f=xyztuvw(x+y)(x+y+z+t+u+v+w): polymake result

Remark: Polymake produces a huge file with lots of information on the poset associated to the arrangement. We have to look for the multiplicities of the characteristic cycle and these may be found at the section FACE_LABELS. The minimal associated primes have multiplicity 1.

Computing the characteristic cycle of the module M˜^{log
f}

Computing a presentation of M˜^{log f} can be done in almost any Computer Algebra system since we have to use Groebner basis over R. We used Macaulay 2 since we also have to compute its characteristic ideal. To do so we need the package D-modules for Macaulay 2.

For each component of the characteristic variety we have to compute its generic multiplicity. It boils down to a computation in a commutative polynomial ring in 2n variables. We have done it with Maple.

Source Codes:

charideal.m2 : To compute the characteristic ideal of M˜^{log f} .

multiplicity.mws : To compute the generic multiplicity of the conormal space to the origin. To compute the generic multiplicity of any other component we only have to modify slightly this script.

Examples:

Orlik-Terao's arrangements:

f=xyzt(x+y)(x+z)(x+t)(y+z)(y+t)(z+t)(x+y+z)(x+y+t)(x+z+t)(y+z+t)(x+y+z+t): m2 result , maple result

f=xyztuv(x+y)(x+z)(x+t)(y+z)(y+t)(z+t)(x+y+z)(x+y+t)(x+z+t)(y+z+t)(x+y+z+t)(u+v): m2 result , maple result

Arrangements in A⁵_k:

f=xyztu(x+y)(z+t)(x+y+z+t+u): m2 result , maple result

Arrangements in A⁶_k:

f=xyztuv(x+y)(x+y+z+t+u+v): m2 result , maple result

f=xyztuv(x+y)(z+t+u)(x+y+z+t+u+v): m2 result , maple result

Arrangements in A⁷_k:

f=xyztuvw(x+y)(x+y+z+t+u+v+w): m2 result , maple result

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Last modified: Wednesday, September 21 2005