Some computations on the characteristic variety of a line arrangement
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Some computations on the characteristic variety of a line arrangement Oscar Papini1 · Mario Salvetti1 Dedicated to the memory of S¸ tefan Papadima Received: 1 April 2019 / Revised: 28 November 2019 / Accepted: 17 July 2020 / Published online: 17 August 2020 © The Author(s) 2020
Abstract We find monodromy formulas for line arrangements that are fibered with respect to the projection from one point. We use them to find 0-dimensional translated components in the first characteristic variety of the arrangement R(2n) determined by a regular n-polygon and its diagonals. We also find new 1-dimensional translated components which generalize the well-known case of the B3 -deleted arrangement. Keywords Hyperplane arrangements · Characteristic varieties · Local cohomology Mathematics Subject Classification 32S22 · 55N25 · 32S50
1 Introduction An abelian local system Lρ on the complement M(A) of a hyperplane arrangement A in C N is defined by choosing one non-zero complex number ρ for each hyperplane ∈ A. For a generic choice of parameters ρ , it is well known that homology concentrates in the top dimension N (see e.g. [14]). The i-th characteristic variety of A is the subvariety Vi (A) = (ρ )∈A ∈ (C∗ )r : dim(Hi (M(A); Lρ )) > 0
Partially supported by: progetto PRA “Geometria e Topologia delle Varietà” (2018), Università di Pisa; INdAM; MIUR.
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Mario Salvetti [email protected] Oscar Papini [email protected]
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Department of Mathematics, University of Pisa, 56126 Pisa, Italy
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Some computations on the characteristic variety of the line…
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where r = # A. Broad literature on this subject is known, also in connection with the theory of resonance varieties and the study of the cohomology of the associated Milnor fiber (see e.g. [1,3–6,8–12,15,17–19,21,24,25,27,28]). Probably the main problem is understanding if the characteristic varieties are combinatorially determined (see e.g. [16]). This is known to be true for their “homogeneous part”, which corresponds to the resonance variety by the tangent cone theorem (see e.g. [7] for general references). Meanwhile the translated components of the characteristic variety are still not well understood; in particular, the geometric description which is known for the translated components of dimension at least 1 does not work in the same way for the 0-dimensional translated components (see also [2]). For these reasons we think that it can be useful to produce more examples (interesting in themselves) such that the characteristic variety has some translated 0-dimensional essential component. In this paper we outline a different approach, based on an (apparently new) elementary description of the characteristic variety. We consider here the case N = 2. We find that the arrangement R(2n) determined by a regular n-polygon and its diagonals produces the above mentioned phenomenon for n 5, namely we find φ(n) translated 0-dimensional components in its characteristic variety (here φ is the Euler function). This fact was experimentally observed in [20] by using c
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