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More bisections by hyperplane arrangements

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 Added by Roman Karasev
 Publication date 2018
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and research's language is English




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A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the non-negative side of the arrangement ${xin R^d : p_{H}(x)ge 0}$ is the same as the measure on the non-positive side ${xin R^d : p_{H}(x)le 0}$. In 2017 Barba, Pilz & Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $dge lceil j/k rceil$. The conjecture was confirmed in the case when $dge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevic, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grunbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+ell$ measures on $R^{2^a+ell}$, where $1leq ellleq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.



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Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $mathcal{A}_I$, the regular nilpotent Hessenberg variety $mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $mathcal{A}_I$ is isomorphic to $H^*(mbox{Hess}(N,I))$ and $H^*(mbox{Hess}(S,I))^W$, the invariants in $H^*(mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borels celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)to H^*(mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $sigma$-Grobner bases. Moreover, we prove that the Teraos conjecture over finite fields implies the conjecture over the rationals.
We introduce the package textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.
180 - Takuro Abe 2020
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.
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