Hyperplane Arrangements: An Introduction

208 Pages·2017·2.95 MB·English

by  Alexandru Dimca (auth.)

Universitext
Alexandru Dimca
Hyperplane
Arrangements
An Introduction
Universitext
Universitext
Series editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Carles Casacuberta
Universitat de Barcelona, Barcelona, Spain
Angus MacIntyre
Queen Mary University of London, London, UK
Kenneth Ribet
University of California, Berkeley, CA, USA
Claude Sabbah
École polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France
Endre Süli
University of Oxford, Oxford, UK
Wojbor A. Woyczyński
Case Western Reserve University, Cleveland, OH, USA
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Alexandru Dimca
Hyperplane Arrangements
An Introduction
123
Alexandru Dimca
UniversitéCôte d’Azur
Nice
France
ISSN 0172-5939 ISSN 2191-6675 (electronic)
Universitext
ISBN978-3-319-56220-9 ISBN978-3-319-56221-6 (eBook)
DOI 10.1007/978-3-319-56221-6
LibraryofCongressControlNumber:2017935563
MathematicsSubjectClassification(2010): 32S22,32S55,32S35,14F35,14F40,14F45,52C35
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Preface
Hyperplane arrangement theory is a very active area of research, combining ideas
from combinatorics, algebraic topology and algebraic geometry in a blend that is
both tastyanduseful.Theclassical textbookwaswrittenbyOrlik andTerao[180]
about 25 years ago, when the subject was already vast and mature. A continuation
by the same authors was produced 10 years later, to discuss important progress
involving local systems and twisted cohomology, see [183].
On the combinatorial side, the book by Stanley [213] is an excellent introduc-
tion, see also the monograph [29] by Björner, Las Vergnas, Sturmfels, White,
Ziegler, as well as Cartier’s report [40]. The recent and detailed monograph by
De Concini and Procesi [58] adds new important directions to the subject, by
looking at polytopes and polyhedra, matroids and root systems, splines and num-
bers of integral points in polytopes.
The results continued toaccumulate overrecent years, anda numberof leading
expertsintheareajoinedeffortstoproducearathercompletesurveyofthecurrent
situation, but their book is still a project, see [50] for a preliminary version.
Our aim in writing these notes was more modest: We intended to write an
introduction to hyperplane arrangement theory which is both accessible and
motivating. On the accessibility side, we have recalled in the first few chapters
many of the basic results from the book by Orlik and Terao [180], choosing to
avoidtheproofsthatarelongandtechnical.Asforthemotivatingside,ourchoice
oftopicsinthelatter chapterswasinspiredbythecurrentfrontiers ofresearchand
includesbothnewresultsandopenproblems.Asaresult,someimportanttopicsare
not treated at all, e.g., the Lie algebras attached to arrangements, which have a
whole chapter devoted to them in [50]. Other subjects, such as free arrangements,
aretreatedfrom a verypersonalviewpoint. One ofourmainconcernswas tokeep
the book at a reasonable size, so that the resulting text is more an invitation to
explore a beautiful area of mathematics and/or to embark on a related research
project,ratherthanareferencemonograph.Theexercisesattheendofeachchapter
are a good test of the understanding of the material and should make the book (or
parts of it) readily usable in a graduate course.
v
vi Preface
We describe the contents of this book in more detail. Chapter 1 is a brief
introduction,inwhichwepointouttheinterplaybetweencombinatorics,topology,
geometry and arithmetic in the realm of line arrangements in the plane, where the
reader’s intuition can be strongly supported by drawings. Each of the themes
introducedhereisdevelopedfullyinalaterchapter.Weincludeadiscussionofthe
Sylvester–Gallai property for real line arrangements, both the classical projective
version in Theorem 1.5 and an affine version in Theorem 1.4. The proof of both
results is inspired by Hirzebruch’s approach in [135].
InChap.2,wecollectthebasicdefinitionsandresultsinvolvingtheintersection
lattice LðAÞ of a hyperplane arrangement A; we explain the key induction tech-
nique based on triples of hyperplane arrangements (A,A0,A00) and apply it to
deduce the main properties of the characteristic polynomial v(A;t) and of the
Poincarépolynomialp(A;t).ThecharacteristicpolynomialentersintoZaslavsky’s
Theorem 2.8, expressing the number of regions (resp. bounded regions) of the
complementofarealarrangementAintermsofv(A,(cid:2)1).Thesenumbersarealso
related to the number of singularities of the polynomial Q on the complement
MðAÞ, for any essential affine arrangement A:QðxÞ¼0, see Theorem 2.9. We
also introduce the supersolvable arrangements and state the factorization property
of their Poincaré polynomials in Theorem 2.4. In the last section, we define the
graphic arrangements and state the fact that the chromatic polynomial of a simple
graph C coincides with the characteristic polynomial of the associated hyperplane
arrangementAC,seeTheorem2.10.Finally,wediscussthereflectionarrangements
and introduce the main arrangements in this class, namely the monomial arrange-
mentsAðr;r;nÞinExample2.24andthefullmonomialarrangementsAðr;1;nÞin
Example 2.23.
The purely combinatorial definition of the Orlik–Solomon algebra A(cid:3)ðAÞ of a
hyperplane arrangement A is given in Chap. 3, and the fundamental result stating
that this algebra is isomorphic to the cohomology algebra of the complex hyper-
plane arrangement complement MðAÞ is proved in Theorem 3.5. To do this, we
assume a technical result on the behavior of the Orlik–Solomon algebras with
respecttotriples,seeTheorem3.1.Inthischapter,wealsomentionatensorproduct
decomposition of the Orlik–Solomon algebra of a supersolvable arrangement, see
Theorem3.3,aswellasanalternativedefinitionoftheOrlik–Solomonalgebraofa
projective hyperplane arrangement, see Theorem 3.4.
InChap.4,wediscusstheminimalityofthecomplementMðAÞanditsrelation
to the degree of the gradient mapping of the defining equation for A, see
Theorem 4.4. In Remark 4.2, we collect some basic results on the topology of the
union of the hyperplanes in A, namely on the hypersurface NðAÞ¼fx2Cn :
QðxÞ¼0g. Then, we mention two beautiful results of June Huh, the first on the
log-concavity of the coefficients of the Poincaré polynomial pðA;tÞ, see
Theorem 4.6, and the second on the relation between the degree of the gradient
mappingofaprojectivehypersurfaceV andthemultiplicitiesofitssingularities,see
Theorem4.7.WhenV isalinearrangement,wegiveanelementarynewproofofa
more precise version of the latter result in Theorem 4.8. In this chapter, we also
discuss the fundamental group of the complement MðAÞ and the arrangements
Preface vii
whose complements are Kðp;1Þ-spaces. We state Deligne’s result which says that
real simplicial hyperplane arrangements give rise to such Kðp;1Þ-spaces, see
Theorem 4.12, as well as Bessis’ result which says that the same holds for the
complex reflection arrangements, see Theorem 4.15. We introduce the fiber type
arrangements, show their relation to the Kðp;1Þ-spaces in Theorem 4.15, and then
statethesurprisingfactthatacentralarrangementissupersolvableifandonlyifitis
fibertype,seeTheorem5.3.Thisisanotherdeepconnectionbetweencombinatorics
and topology in this subject. Some very interesting groups occur as fundamental
group of complements MðAÞ, for instance the Stallings group and the Bestvina–
Brady groups are discussed in Remark 4.11.
In Chap. 5, we start our discussion of the Milnor fiber F associated to a central
hyperplanearrangementA,themonodromyactiononthecohomology H(cid:3)ðFÞand
the relation between monodromy eigenspaces and the twisted cohomology of the
complement MðA0Þ, where A0 is the projective arrangement associated to A, see
Proposition 5.4. Then, we state a very general vanishing result on the twisted
cohomologyofthecomplementMðA0Þwithcoefficientsinarankonelocalsystem,
a result due to D. Cohen, P. Orlik and the author, see Theorem 5.3, whose con-
sequences are used several times in the sequel. A largely unexplored subject,
namelyanaturalcandidatefortheMilnorfiberofanaffinehyperplanearrangement,
is discussed in Remark 5.3, where the Bestvina–Brady groups occur again
naturally.
Characteristic varieties (resp. resonance varieties) are jumping loci for some
cohomologygroupswhicharetopologically(resp.algebraically)defined,andareat
the center of many research papers published in the last decade in relation to both
hyperplane arrangements and, more generally, the fundamental groups of smooth
quasi-projective algebraic varieties. We discuss the relation between the charac-
teristic varieties and the homology of finite abelian covers, in particular cyclic
covers of prime order in Proposition 6.3 and congruence covers in Theorem 6.2.
ThisbringsustothepolynomialperiodicitypropertiesofthefirstBettinumbersof
suchcovers,seeTheorem6.3,andtothesmoothsurfacesobtainedascoveringsof
P2 ramified over a line arrangement, in particular to the Hirzebruch covering sur-
faces, a.k.a. Hirzebruch–Kummer surfaces, see Theorem 6.4. The main results in
this chapter are the Tangent Cone Theorem 6.1 and its relation with the multinet
structuresintroducedbyM.FalkandS.Yuzvinsky,seeTheorem6.6.Thisbringsin
unexpected and beautiful relations with the theory of pencils of projective plane
curves studied in classical algebraic geometry, see Theorem 6.5. After a brief
discussion of the translated components of the characteristic varieties, we treat in
great detail the deleted B -line arrangement. This was the first example of a
3
hyperplane arrangement having a translated component of a characteristic variety,
and it was discovered by Suciu in [217].
InChap.7,wefirstconsidermoregeneralsmoothquasi-projectivevarietiesand
proveaversionoftheTangentConeTheoreminthissetting,usingthelogarithmic
connections, and following the main ideas from [112], [204], see Theorem 7.3.
Then,wediscussthemixedHodgestructureonthecohomologyofthehyperplane
viii Preface
complement MðAÞ and of the Milnor fiber F. We define the corresponding
spectrum and state the key results of N. Budur and M. Saito which say that this
spectrum is determined by the intersection lattice and which give an explicit for-
mulainthecaseofalinearrangementinP2,seeTheorems7.6and7.8.InRemark
7.4, we relate the topology of the Milnor fiber F associated to a projective line
arrangement to the topology of two natural compactifications of F, one with iso-
lated singularities and the other a smooth surface. Next, we discuss an arithmetic
property of algebraic varieties Y defined over the rationals Q, namely the poly-
nomial count property, see Definition 7.5. Following Katz [131], this property is
related to Hodge theoretic properties of Y, i.e., to the property of Y being coho-
mologically Tate. This property holds when Y ¼MðAÞ, the complement of a
central or projective arrangement A. When Y is the Milnor fiber F of such an
arrangement,then thispropertyisrelated tothetriviality ofthemonodromy action
onallthecohomologygroupsofF,seePropositions7.6and7.8,Theorem7.10and
Example 7.10. A discussion of Hodge–Deligne polynomials of line arrangements
with only nodes and triple points completes this chapter.
Chapter 8 starts with a discussion offree projective hypersurfaces, stressing the
relationwiththeJacobiansyzygiesofthedefiningequationf ¼0.Aftergivingthe
general definitions, we give a proof of Kyoji Saito’s Criterion in this setting, see
Theorem 8.1, valid for any projective hypersurface and not only for hyperplane
arrangements. The factorization property for pðA;tÞ, when A is a free arrange-
ment,isgiveninTheorem8.3,whilethefactthatanysupersolvablearrangementis
free occurs in Theorem 8.4. The freeness of reflection arrangements is stated in
Theorem 8.2. For the monomial line arrangement, the full monomial line
arrangement and the Hessian arrangement, bases of the Jacobian syzygy modules
are given inExample 8.6. The locally free arrangements A occur in Theorem 8.5,
and the Chern classes of their logarithmic 1-form vector bundles X1ðlogAÞ are
determinedinTheorem8.6.Tamehyperplanearrangementsareshowntoplayakey
role in the Logarithmic Comparison Theorem 8.7 due to J. Wiens and
S. Yuzvinsky.
Then,wemovetothecaseofcurves(andinparticular,linearrangements)inP2
and state several characterizations of such free curves in Proposition 8.2 and
Theorem8.8.Westatearecentresultofours,seeTheorem8.9,relatingthefreeness
of a line arrangement to the multiplicities of its intersection points. This result (or
the freeness of supersolvable arrangements) is used to construct free line arrange-
ments with arbitrary exponents in Theorem 8.10.
Inthethirdsection,wediscussaspectralsequenceapproachtothecomputation
of the Alexander polynomial of a plane curve, see Theorems 8.12 and 8.15. The
initial term of this spectral sequence is given by the cohomology of the Koszul
complex associated to the partial derivatives fx;fy;fz, and this is why the Jacobian
syzygies play a major role in this approach. First, we consider the case when the
curve C :f ¼0 and describe an algorithm to compute the second page of this
spectral sequence. This algorithm is very fast, as we can use the software Singular
todetermineabasisoftheJacobiansyzygymodule.Thegeneralcaseisconsidered
Preface ix
next, and here, we reduce everything to some huge systems of linear equations,
see Eq. (8.42). In this case, the necessary computer time increases.
In the final section, we show that this approach is effective, by looking at the
monomial line arrangement and the Hessian arrangement: In each case, we con-
struct explicitbases for some eigenspaces ofthemonodromy actiononH1ðFÞ,see
Theorems 8.17 and 8.19. Some examples are included to show that our new
approach can be successfully applied even beyond the class of free line arrange-
ments.Tostatetheresultsinaconvenientway,weintroducethepoleorderspectra,
in analogy to the spectrum, see Eq. (8.44). For some of these results, the compu-
tations based on effective algorithms developed jointly with Sticlaru in [99–101]
and using the computer algebra systems Singular [55] and CoCoA [45] play an
important role.
Thesenoteshavegrownoutofseveralsources.First,thereweremylectureson
hyperplane arrangements in Nice, followed in particular by my former Ph.D. stu-
dents N. Abdallah, P. Bailet, T.A.T. Dinh and H. Zuber. Then, my lectures in
ASSMS GCU Lahore, followed by S. Ahmad, I. Ahmed, S. Nazir, K. Shabbir,
H. Shaker and R. Zahid. And finally, the series of lectures which I gave in the
USTC summer school on “Hyperplanes arrangements” in Hefei, attended by
Y.Liu,Z.Wang,K.T.WongandS.Yun.Ithankallofthemfortheirinterestinthe
subject, their questions and their comments, which shaped the presentation into its
present form.
Hyperplane arrangements were not my “first love.” I became interested in this
beautifulsubjectwhenIwasalreadya(relatively)maturemathematician.Inspiteof
that,thespecialistsinthisfieldhavetreatedmeextremelywell:Theytookthetime
toexplaintomemanythings,andabovealltheyallowedmetoparticipateintheir
veryfriendlymeetingsallovertheworld.Forallthis,Ithankthemall.Discussions
with G. Lehrer, S. Papadima, M. Saito, A. Suciu and M. Yoshinaga had a direct
impact on these notes, and I am very grateful to them.