Paul Mücksch

Preprints

(Click on the title to reveal more information.)

• Projective dimension of weakly chordal graphic arrangements

  Abstract:
  A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

  With Takuro Abe, Lukas Kühne and Leonie Mühlherr
  Preprint - arXiv:2307.06021 (2023), 20 pages. [arXiv]


• Flag-accurate arrangements

  Abstract:
  In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper.
  More specifically, let A be a free arrangement of rank ℓ. Suppose that for every 1 ≤ d ≤ ℓ, the first d exponents of A -- when listed in increasing order -- are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of A of dimension d. Following [MR21], we call such an arrangement A with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call A flag-accurate.
  We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate.
  The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.

  With Gerhard Röhrle and Tan Nhat Tran
  Preprint - arXiv:2302.00343 (2023), 53 pages. [arXiv]


• Erratum to: Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complex

  Abstract:
  This is an Erratum to ED's paper ``Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complex'', Int. Math. Res. Not., 2008, 39.
  There, a special decomposition of the Salvetti complex of a real hyperplane arrangement is constructed. This construction is based on a choice of an arbitrary linear extension of the arrangement's tope poset. While working on [Mücksch2022], PM noticed that, for the above-mentioned construction to work, the chosen linear extension cannot be arbitrary. The results in Section 4 remain valid by work of Lofano and Paolini, who proved that an appropriate total ordering of the chambers exists for every real hyperplane arrangement. The corresponding claim for non-realizable oriented matroids remains open.

  With Emanuele Delucchi
  Preprint (2022). [local]

Papers (published or accepted)

(Click on the title to reveal more information.)

• Modular flats of oriented matroids and poset quasi-fibrations

  Abstract:
  We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration -- a notion derived from Quillen's fundamental Theorem B from algebraic K-theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a K(pi,1)-space -- a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups -- analogous to the realizable case.
  Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements.
  We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.

  Trans. Amer. Math. Soc. Ser. B 11 (2024), 306-328 [arXiv, journal]


• On Formality and Combinatorial Formality for hyperplane arrangements

  Abstract:
  A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal.
  The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality.
  Our second main theorem shows that formality is hereditary, i.e.~is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e. asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

  With Tilman Möller and Gerhard Röhrle
  Discrete & Computational Geometry (2023). [arXiv, journal, poster]


• On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements

  Abstract:
  We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a Künneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky’s freeness criterion. Furthermore, our approach affords a new formulation of Terao’s freeness conjecture and a more general problem.

  Mathematische Annalen (2022) [arXiv, journal, SN-SharedIt]


• Accurate arrangements

  Abstract:
  Let A be a Coxeter arrangement of rank ℓ. In 1987 Orlik, Solomon and Terao conjectured that for every 1 ≤ d ≤ ℓ, the first d exponents of A - when listed in increasing order - are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of A of dimension d.
  This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free.
  We call a general free arrangement with this natural property involving their free restrictions accurate. In this paper we initialize their systematic study.
  Our principal result shows that MAT-free arrangements, a notion recently introduced by Cuntz and Mücksch, are accurate.
  This theorem in turn directly implies this special property for all ideal subarrangements of Weyl arrangements. In particular, this gives a new, simpler and uniform proof of the aforementioned conjecture of Orlik, Solomon and Terao for Weyl arrangements which is free of any case-by-case considerations.
  Another application of a slightly more general formulation of our main theorem shows that extended Catalan arrangements, extended Shi arrangements, and ideal-Shi arrangements share this property as well.
  We also study arrangements that satisfy a slightly weaker condition, called almost accurate arrangements, where we simply disregard the ordering of the exponents involved. This property in turn is implied by many well established concepts of freeness such as supersolvability and divisional freeness.

  With Gerhard Röhrle
  Advances in Mathematics, 383 (2021) [arXiv, journal]


• MAT-free reflection arrangements

  Abstract:
  We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization of the Multiple Addition Theorem by Abe and Terao. We give classifications of the irreducible complex reflection arrangements which are MAT-free respectively MAT2-free. Furthermore, we ask some questions concerning relations to other classes of free arrangements.

  With Michael Cuntz
  Electronic Journal of Combinatorics, 27(1) (2020), #P1.28. [arXiv, journal]


• Supersolvable simplicial arrangements

  Abstract:
  Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation.

  With Michael Cuntz
  Advances in Applied Mathematics, 107 (2019), 32 – 73. [arXiv, journal]


• Recursively free reflection arrangements

  Abstract:
  Let A(W) be the reflection arrangement of the finite complex reflection group W. By Terao's famous theorem, the arrangement A(W) is free. In this paper we classify all reflection arrangements which belong to the smaller class of recursively free arrangements. Moreover for the case that W admits an irreducible factor isomorphic to G₃₁ we obtain a new (computer free) proof for the non-inductive freeness of A(W). Since our classification implies the non-recursive freeness of the reflection arrangement A(G₃₁), we can prove a conjecture by Abe about the new class of divisionally free arrangements which he recently introduced.

  Journal of Algebra, 474 (2017), 24 – 48. [arXiv, journal]

Other publications

• On Yuzvinsky’s lattice sheaf cohomology for hyperplane arrangements Oberwolfach Report No. 5/2021, 10.14760/OWR-2021-5