Some computations on the characteristic variety of a line arrangement

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 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_3$$\end{document}-deleted arrangement.


Introduction
An abelian local system L ρ 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 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][4][5][6][8][9][10][11][12]15,[17][18][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 computer methods, up to n = 7.
The arrangement R(2n) belongs to the class of fibered arrangements: the projection through its "center" gives a fibration of the complement M(A) over B = C \ {n − 1 points} with fiber F = C \ {n points}.
We obtain the above result by using algebraic complexes which compute the parallel transport and the monodromy of the first homology group of the fiber, which we think are interesting in themselves. For example, we deduce a restriction on essential components of the characteristic variety (Theorem 2.9) and a description of V (A) ( = V 1 (A)) as the set of points such that the transpose of the monodromy operators have a common eigenvector (Theorem 2.10).
For the arrangement R(2n), we use the algebraic complexes constructed in Sect. 2 to make explicit computations. If ω n is an n-th primitive root of 1, we find that assigning to the edges of the polygon the parameter (ω n ) k and to the diagonals the parameter (ω n ) k(n−2) , for k = 1, . . . , n − 1, lowers the rank of the boundary operator in the algebraic complex computing H 0 (B; H 1 (F; L |F )), therefore the corresponding point P n,k belongs to V (A). In case k and n are coprime, we deduce that P n,k is an isolated point in the characteristic variety. Our argument works for n 5 and uses the description (Theorem 2.10) of the characteristic variety and the remark that triple points give simple eigenvalues for the corresponding monodromy operators. We find that the shape of the common eigenvector imposes equalities among the eigenvalues of the monodromy operators that define a 0-dimensional locus.
When n and k are not coprime, our argument can also be used. In this case the shape of the common eigenvector imposes less equalities (so, a higher dimensional locus in general). For n = 8, we find (Theorem 3.7) a translated 1-dimensional component which contains the two points P 8,2 and P 8,6 (in contrast with [26, pp. 45-46]; see Remark 3.6). One can easily see that such translated component cannot be obtained by using the method of [6, Section 5.5]; therefore our method seems indeed novel. This last construction generalizes to the case n = 4m: we find an explicit 1-dimensional translated component containing P n,m and P n,3m (Theorem 3.12). When n = 4, this gives another description of the 1-dimensional translated component found in [24]. The examples in Theorems 3.7 and 3.12 are the first examples of translated 1-dimensional components to be found after [24], not obtained by the method in [6] (see Remark 3.9).
We also remark that our approach can be generalized (with some modifications) to not necessarily fibered arrangements (we will return to this in future work).

Algebraic complexes
In this section we produce an algebraic complex computing local homology for a line arrangement. This complex seems particularly convenient in case of a fibered arrangement, in which case it is smaller than the one in [12,22].
Let A be an affine line arrangement; we assume that A is defined over R because formulas are easier to write, but all we say can be generalized to general complex arrangements. Let M(A) be its complement in C 2 = {(x, y) : x, y ∈ C}; let π : C 2 → C be the projection π(x, y) = x onto the x-axis, and π = π |M(A) : M(A) → C its restriction. Let S ⊂ C 2 be the set of singular points of the arrangement, π(S) ⊂ C its projection, and B = C \ π(S). Then is a fiber bundle with fibers F x . . = (π ) −1 (x), x ∈ B. We divide the arrangement A = { 1 , . . . , n , 1 , . . . , m } into horizontal and vertical lines, meaning that j = π −1 (ξ ) for some ξ ∈ π(S), while π( i ) = C. Notice that if for all ξ ∈ π(S) the vertical line π −1 (ξ ) is in A, then M(A) = (π ) −1 (B), so the whole complement M(A) is a fiber bundle over B.
We fix a fiber F 0 = (π ) −1 (x 0 ), x 0 ∈ B, x 0 ∈ R; take a basepoint P 0 ≡ (x 0 , y 0 ) ∈ F 0 with Im(y 0 ) 0, and elementary well-ordered generators α 1 , . . . , α n of π 1 (F 0 ), where α i is constructed by using a path connecting P 0 to a small circle around i ∩ F 0 . The indices of the lines are taken according to the growing intersections with the real axis Re(C x 0 ).
We can make the same construction for any fiber F x , x ∈ B, x ∈ R, by using the basepoint P We are interested in the local homology of M(A), where we consider the abelian local system defined by taking C as an H 1 (M(A); Z)-module: the action is given by taking standard generators (small circles around the hyperplanes) into multiplication by non-zero numbers. So, such local systems correspond to (n + m)-tuples of parameters in (C * ) n+m .
We call s 1 , . . . , s n and t 1 , . . . , t m the parameters corresponding to the lines 1 , . . . , n and 1 , . . . , m , respectively. Denote by L = L s,t the corresponding local system. We also set L s = L | F 0 .
In this paper we will assume that M(A) fibers over B; it corresponds, as said before, to the case where the singular set S is contained in the union of vertical lines. Lemma 2. 1 We have where the second factor on the right vanishes if L s is non-trivial (i.e., some s i = 1).
Proof By standard constructions (see e.g. [13,Section 4.17]), This is (1) when we consider the homology groups H * (F 0 ; L s ) ∼ = H * (F 0 ; L s,t ) as π 1 (B, x 0 )-modules and the well-known correspondence between local systems and representations of the fundamental group on one fixed fiber (see [23]). The 0-th local homology group is well known to vanish for non-trivial coefficients.
So we have to compute the term H 0 (B; H 1 (F 0 ; L s )), where H 1 (F 0 ; L s ) is seen as a π 1 (B)-module (actually, an H 1 (B; Z)-module); the precise form of the action will be included in Theorem 2.8.

Lemma 2.2
If L s is non-trivial, then dim(H 1 (F 0 ; L s )) = n − 1. As generators we can take the classes of if no s i equals 1.

Proof
The fiber F 0 deformation retracts onto a wedge of n 1-spheres, given by n i=1 α i . Therefore by standard methods to compute the homology we are done.
To compute the monodromy on H 1 , we need to understand the parallel transport where γ (x, x ) is a path in B connecting x with x and L x = L |F x . We compute the parallel transport for any points x, x ∈ R \ π(S), where R is the real axis of the first coordinate (recall that we are assuming π(S) ⊂ R). We denote by α i 's. Formulas look better if we take as generators Of course, this requires each s i = 1, but since we are interested in essential components of the characteristic variety (i.e., not contained in any coordinate tori s i = 1), this is not a serious restriction.
, connecting x and x , and such that γ ∩ R = {x, x } and Im(γ (t))(x − x ) 0 for all t ∈ [0, 1] (i.e., γ leaves the real axis on the left while traveling from x to x ). We denote by τ (x, x ) the corresponding transport isomorphism (3).
Let σ = σ (x, x ) be the permutation of the indices 1, . . . , n which is obtained as follows: the i-th line in the x-ordering (considering the growing intersections with Here Proof The proof is a standard computation and it is obtained by the following steps. First, remark that any generator in (2) is obtained from Fox calculus from the com- where ϕ : is the valuation homomorphism taking α i to s i .
Next, we remark that while the path γ turns around the projection of some singularity of a half-circle, the corresponding points in the vertical line make a half-twist, giving rise to a "local permutation" which takes a sequence of consecutive numbers to the opposite sequence (i.e., k, . . . , k + h goes to k + h, . . . , k).
By induction on the number of points in π(S) separating x from x , we easily see that a generator α (x) i is taken by γ to i+1 ]) and we conclude.
The local monodromy around one point p ∈ π(S) is obtained by taking a point x ∈ R very close to p and transporting the fiber of π starting from x around a circle centered at p. If x ∈ R is a point symmetric to x with respect to p, the local monodromy μ x is the automorphism of The "local permutation" σ (x, x ) is determined by a partition 1 , 2 , . . . , h of the set {1, . . . , n}, where each is composed of consecutive numbers a, a + 1, . . . , a + r which are the indices (in the x-ordering) of the lines which intersect in a singular point P on the vertical line C p (so the multiplicity of P as a singular point of A equals # + 1).

Theorem 2.4 (Local monodromy) With the previous notations, the local monodromy around p is given by
where ( Proof The proof directly follows from (4) and As an immediate consequence of (6) we have Corollary 2.5 Let p ∈ π(S) and let P 1 , . . . , P h be the singular points of A in the vertical line C p , having multiplicities m 1 , . . . , m h , respectively (m 1 + · · · + m h = n + h). Then the local monodromy μ x has characteristic polynomial and it is diagonalizable if all eigenvalues of the form s are different from 1. Here we use s to denote the parameter corresponding to the horizontal line .
Now we consider the global monodromy μ : For each point p ∈ π(S) we denote by x p , x p ∈ R two points close to p and lying on opposite sides of p in R; assume that x p is the point lying in the segment [ p, The easiest way to compute monodromy is to take generators ; such generator circles around all the projections of the singular points p ∈ [p, x 0 ] counterclockwise. By using the parallel transport (4) we get Theorem 2.6 (Global monodromy I) The global monodromy μ is determined by Here [a, b] stands for the segment connecting the two points a and b in the real axis (of C x p and C x 0 ).
It is convenient to take elementary generators of π 1 (B, x 0 ) associated with p: the path γ p is composed with the sequence Clearly such paths give a well-ordered set of elementary generators of π 1 (B, x 0 ), so the global monodromy is determined by their images.

Theorem 2.7 (Global monodromy II)
The global monodromy μ is determined by the maps where τ (x 0 , x p ) is the parallel transport in (4) and μ x p is the local monodromy in (6).
It is also possible to write explicit formulas for (9): formulas (4) and (6) make possible to compute the images of the generators α i, i+1 ; we do not report such formulas here because we do not need their explicit form now. Notice also that (9) and (8) are related by where the p i are ordered according to increasing distance from x 0 . We now come back to the computation of H 0 (B; H 1 (F 0 ; L s )) in (1). The space B deformation retracts onto a wedge of 1-spheres, actually onto the union p∈π(S) γ p , where the unique 0-cell is x 0 . By standard methods which compute local system homology we obtain defined as where we set here t p for the parameter associated with the vertical line C p ∈ A. . Therefore either t p = 1 or t p P i ∈ s = 1 for some singular point P i ∈ C p of order at least 3 (see (7)). We obtain the following corollary.

Theorem 2.9
The essential components of the characteristic variety of A are contained in where we write ρ for the parameter associated with the line (in our previous notations it is of the shape s i for horizontal lines and t j for vertical lines).
As another immediate consequence we get the following description of the characteristic variety of A. Proof Indeed, a linear relation among the rows of the boundary matrix B is given by

Some computations
We consider here the arrangement R(2n) which is projectively given by taking the lines spanned by the edges of a regular n-polygon, together with all its n diagonals (see Figs. 1 and 3 for the case n = 6 and 7, respectively). Notice that there is an n-tuple point P which is the intersection of the diagonals. For odd n, on each diagonal we have, besides P, one double point at the intersection with the middle point of an edge, and (n − 1)/2 triple points. For even n, n = 2k, we have k diagonals passing through two opposite vertices of the n-polygon, each of them containing k triple points; and other k diagonals passing through the middle points of two opposite sides of the npolygon, each of them containing two double points and k − 1 triple points. Of course, the projection from P makes the complement M(A) a fiber bundle. We take one of the diagonals to infinity, so the projection π : C 2 → C describes M(A) as a fiber bundle over C \ {n − 1 points} with fiber C \ {n points} (see Figs. 2 and 4).
be the vertical lines, ordered according to p 1 > · · · > p n−1 . In the notations of the previous section, let x 0 ∈ R be a basepoint in the x-coordinate: we choose x 0 p 1 . Let also 1 , . . . , n be the horizontal lines, ordered according to increasing intersections with the real axis of C 0 . Fix also points x i = p i + , • σ i = (1)(2, 3)(4, 5) · · · (n − 1, n) if i is even.
(ii) For even n the local permutation σ i = σ (x i , x i ) is given by Let ω n be a primitive n-th root of the unity. We use results from the previous section to find explicit points in the characteristic variety.
Proof The case k = 0 is trivial since P n,0 = (1, . . . , 1) (2n − 1 entries). So assume k > 0. Let us consider the row vector where we set for brevity ω = ω k n . We prove that v n is an eigenvector for the transpose operator t ∂ 1 ([γ p ]) in (11) for all γ p when the parameters take the above values. This shows that such parameters lower the rank of ∂ 1 in (10), so the corresponding point belongs to the characteristic variety. Notice that by Lemma 3.1 the local monodromy around p i coincides with the local monodromy M 1 around p 1 for odd i or with the local monodromy M 2 around p 2 for even i. Therefore by the expression (9) for the monodromy it is sufficient to prove that v n is an eigenvector for t M 1 and t M 2 with parameters s i = ω, with respect to the eigenvalue ω n−2 , and that both τ (x 1 , x 2 ) −1 and τ (x 2 , x 3 ) −1 take v n into an eigenvector of t M 2 , t M 1 respectively.
The first assertion is easily proven by looking at formulas (6). With the given parameters, there are as many zero columns in t i M i − Id as triple points. So, for odd n there are (n − 1)/2 non-zero columns, at even position for t 1 M 1 − Id, at odd position for t 2 M 2 −Id; for even n, there are n/2 non-zero columns at odd position for t 1 M 1 −Id and (n − 2)/2 non-zero columns at even position for t 2 M 2 − Id. The non-zero j-th column has entries ω n−2 − 1 in position ( j, j), preceded (if it is not the first entry of the column) by ω n−2 (1 − ω), and followed (if it is not the last entry of the column) The form of T = τ (x 1 , x 2 ) −1 and T = τ (x 2 , x 3 ) −1 (with the given parameters) is derived from formula (4) and from Lemma 3.1. In case n is odd one obtains: • for odd j, T j, j = − ω −1 and T i, j = 0 for i = j; • for even j, T j−1, j = ω −1 , T j, j = T j+1, j = 1, and T i, j = 0 for all other i's.
The shape of T is completely analogous; it is enough to exchange the roles of even and odd j. For n even, the shape is the same exchanging T and T . Then it is easy to check directly that v n · T = v n · T = − ω −1 v n and we conclude.  s σ n (0) , . . . , s σ n (n−1) , t σ 2 n (0) , . . . , t σ 2 n (n−1) ) which also belongs to V . Notice that the points P n,k are fixed by this action.
With this labeling the double points are given by the intersection of the lines with indices (i, (2i) ) and the triple points are given by the intersections of the lines with indices (i, j, (i + j) ) (taking indices mod n).
The next theorem needs a lemma. where we take all possible pairs (i, j) with i = j, and the indices are mod n. Then J is 0-dimensional and defines the n 2 points P n,h,k = s i = (ω n ) hi+k , t i = (s 0 s i ) −1 , i = 0, . . . , n − 1 , h, k = 0, . . . , n − 1.
Proof (i) At first we prove the relations By multiplying the two relations (n > 3) Then one easily deduces for recurrence the relations s a 1 · · · s a k = s b 1 · · · s b k each time a j = b j (mod n). In particular, we have s n 0 = s n i for all i = 1, . . . , n−1. In conclusion, we show Now notice that any point P very close to P n,k either does not belong to the characteristic variety or it corresponds to a common eigenvector v n still having non-vanishing components, so defining the same conditions. Therefore in a neighborhood of P n,k the characteristic variety is contained in the zero locus of (14). By Lemma 3.4 the ideal I is zero-dimensional (for n 5), so we are done.

Remark 3.6
In [26, pp. 45-46] the case n = 8 is considered and the authors claim that the six points P 8,k for k = 1, . . . , 7, k = 4, are isolated points in the characteristic variety. The argument used in Theorem 3.5 works for (n, k) = 1; nevertheless, it can be refined in the following way. For n = 8 and k = 2, 6, the eigenvector v 8 in (12) has a unique zero component in the fourth coordinate: (where ω is a primitive 8-th root of the unity), and those are eigenvectors for both local monodromies t M 1 and t M 2 . As in the proof of Theorem 3.5 we look at the explicit form of the local monodromy around the first two lines, which gives the two blocks in the boundary ∂ 1 : The non-zero positions of v 8 give the equalities from the first block, and t 2 s 1 s 2 = t 2 s 3 s 4 = t 2 s 5 s 6 = t 2 s 6 s 7 = 1 (16) from the second block, which must hold in a neighborhood of P 8,2 .
Notice that the first vertical line corresponds in the projective picture to a line 0 (notations of Remark 3.3) passing through the center and cutting two opposite edges of the octagon in the middle point. The second vertical line corresponds to a diagonal 1 passing through two opposite vertices of the octagon. Among all triple points contained in these lines, the unique triple point which we are not considering in the equations (15) and (16) corresponds (in the projective picture) to that formed by 0 and the two edges of the octagon which are "parallel" to 0 (and parallel themselves).
By symmetry (or changing the coordinate system) we obtain analogous equations for all diagonals. That is, in a neighborhood of P 8,2 (and of P 8,6 ), the characteristic variety is contained in the zero locus of the ideal I which is obtained from the ideal I in (14) by deleting the four equations which correspond to the triple points determined by the four pairs of opposite edges of the octagon.
By making explicit computations we obtain that the ideal I is 1-dimensional, and we finally get for even i; where x is a parameter in C * . Such component contains the two points P 8,2 and P 8,6 (which are therefore not isolated).
Proof Once we have the explicit form (17) it is just a direct computation to see that it defines a component which is contained in the characteristic variety: we used both the expression of the matrix boundary ∂ 1 of Theorem 2.8 and also the description of the boundary in [12] (of course both of them give that (17) is contained in the characteristic variety).
It is also possible to give a proof which parallels the one in Theorem 3.2. In particular, we find an explicit common eigenvector for the transposed monodromy operators (Theorem 2.10), which is given (in terms of the parameter x) by It is immediate to see that P 8,2 and P 8,6 are contained in (17) (with parameters x = ω 2 , x = ω 6 , respectively); in fact, notice that vectors v 8,2 and v 8,6 above are obtained as v 8,2 = − ω 2 w 8 (ω 2 ) and v 8,6 = − ω 6 w 8 (ω 6 ).
As we have already remarked, in a neighborhood of these points the characteristic variety is 1-dimensional; therefore the component (17) is isolated.

Remark 3.8
Case n = 4, which corresponds to the B 3 -deleted arrangement considered in [24], can also be included in our method. If we consider any of the points P n,k , k = 1, 2, 3, and we repeat the same construction as in Theorem 3.5, we obtain the same ideal I in (14), which is in this case 1-dimensional. In fact, the zero locus of I has four irreducible 1-dimensional components: one of these has the same shape as (17), being parametrized as and it is the translated component appearing in [24]. It contains the two points P 4,1 and P 4,3 . The other three components of the zero locus of I do not belong to the characteristic variety. (P 4,2 belongs to a different component of the zero locus of I , and that component is not fully contained in the characteristic variety.)

Remark 3.9
In [6, Proposition 5.6] the authors explain a method to find translated, 1-dimensional components in terms of pointed multinets, which are a special case of multinets. The B 3 -deleted arrangement is an example of an arrangement whose 1-dimensional translated component is obtained with this method. One more example is given in [20,Chapter 5], where the arrangement Q(4) is given by ten projective lines with the following (affine) equation: If we give all the ten lines weight 1 and add the infinity line ∞ with weight 2, we obtain an arrangement of eleven lines which supports a pointed multinet, with special line ∞ . So Q(4) verifies the criterion in [6].
In case of the arrangement R(16) of Theorem 3.7, such method would apply if one could find a bigger arrangement A with 17 lines, containing R(16), which supports a pointed multinet. It is just an exercise to prove the following It follows that our example in Theorem 3.7 cannot be obtained by the method in [6].

Remark 3.11
Some other cases where (n, k) = 1 are known. For example, in case n = 9, k = 3, the point P 9,3 is contained in a 2-dimensional component coming from a multinet (see [11]).
The argument outlined in Remark 3.6 can be extended to any case n = 4m to produce a 1-dimensional translated component, generalizing (19), (17).
Proof We outline the proof of the theorem, which parallels that of Theorem 3.7. As in Theorem 3.2, by using the explicit form of the boundary one verifies that the points in (20) lower the rank of the boundary matrix, so the component is contained in V (A). We find a common eigenvector of the transposed monodromy operators, which generalizes (18), w n (x) = [1, x − 1, − x, 0, 1, x − 1, − x, 0, . . . , 1, x − 1, − x, 0, 1 (21) (the triple [1, x − 1, − x] repeats m times, separated by a 0). Now consider the point P n,m , which is contained in (20) (with x = ω m ); a similar argument holds by using P n,3m instead. As in the proof of Theorem 3.7, for this point the vector v n in (12) is given by v n = − ω m w n (ω m ), where ω is a primitive n-th root of the unity. Therefore v n has 4m − 1 entries which vanish exactly in the positions multiple of 4. The argument of Theorem 3.5 (or of Theorem 3.7) produces an ideal I which is obtained from I by deleting, for each even i among 0, . . . , 4m − 1, the m − 1 equations t i s (i/2)+2 j s (i/2)−2 j = 1, for j = 1, . . . , m − 1 (indices are as in Remark 3.3 and are taken mod n). One sees that the zero locus Z (I ) is of dimension 1 and (20) is contained in Z (I ) as an irreducible component which contains P n,m and P n,3m .
Then we conclude that such component is not contained in a higher dimensional one by the same argument as used in Theorem 3.5 (and Theorem 3.7): points of the characteristic variety which are close to P n,m impose the same conditions, so they are contained in Z (I ).

Remark 3.13
Analogous computations can be done for other points P n,k where the number and the positions of the zero entries of the common eigenvector tell us what ideal to take. For example, we made some computations finding further isolated points in V (A) for n = 12. Details and further applications will be provided later.