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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 Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approachtotheirsubjectmatter.Someofthemostsuccessfulandestablishedbooks intheserieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolution of teaching curricula, into very polished texts. 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More information about this series at http://www.springer.com/series/223 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 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 bothtastyanduseful.TheclassicaltextbookwaswrittenbyOrlik 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 introduction, 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 hyperplane 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 characteristic 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 surfaces, 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 polynomial 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 arrangement,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 arrangements 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 construct explicitbases forsome 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 arrangements.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.