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Register a new userFree hyperplane arrangements over arbitrary fields
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In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $mathbb{Q}$.
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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.
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.
In this article, we study the $k$-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the $k$-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements.
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Teraos celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Teraos conjecture is true in this class. As an application, we can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.
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.
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