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The irregularity of cyclic multiple planes after Zariski

A formula for the irregularity of a cyclic multiple plane associated to a branch curve that has arbitrary singularities and is transverse to the line at infinity is established. The irregularity is expressed as a sum of superabundances of linear systems associated to some multiplier ideals of the branch curve and the proof rests on the theory of standard cyclic coverings. Explicit computations of multiplier ideals are performed and some applications are presented.


Introduction
Let f (x, y) = 0 be an affine equation of a curve B ⊂ P 2 and H ∞ be the line at infinity. The projective surface S 0 ⊂ P 3 defined by the affine equation z n = f (x, y) is called the n-cyclic multiple plane associated to B and H ∞ . In [20], Zariski obtains his famous result for the irregularity of certain n-cyclic multiple planes.
Zariski's Theorem. Let B be an irreducible curve of degree b, transverse to the line at infinity H ∞ and with only nodes and cusps as singularities. Let S 0 ⊂ P 3 be the n-cyclic multiple plane associated to B and H ∞ , and let S be a desingularization of S 0 . The surface S is irregular if and only if n and b are both divisible by 6 and the linear system of curves of degree 5b/6 − 3 passing through the cusps of B is superabundant. In this case, where Z is the support of the set of cusps.
The aim of this paper is to present a generalization of Zariski's Theorem to a branch curve that has arbitrary singularities and is transverse to the line at infinity bringing to the fore the theory of cyclic coverings as developed in [18]. The irregularity will be expressed as a sum of superabundances of linear systems associated to some multiplier ideals of the branch curve B. We refer to [4] for the notion of multiplier ideal. To state the main result in Section 2, we recall here that if the rational c varies from a very small positive value to 1, then one can attach a collection of multiplier ideals J (cB) that starts at O P 2 , diminishes exactly when c equals a jumping number-they represent an increasing discrete sequence of rationals-and finally ends at I B .
Theorem (2.1). Let B be a plane curve of degree b and let H ∞ be a line transverse to B. Let S be a desingularization of the n-cyclic multiple plane associated to B and H ∞ . If J(B, n) is the subset of subunitary jumping numbers of B that live in where Z(ξB) is the subscheme defined by the multiplier ideal J (ξB).
In case the singularities of B are locally given by x p = y q such as equations, explicit computations of the jumping numbers and of the multiplier ideals will enable us to apply the above theorem to various examples in Section 4. An example in Remark 4.4 shows that the irregularity may jump in case the position of H ∞ with respect to B becomes special.
Generalizations of Zariski's Theorem are discussed in several papers and the proofs are based on different points of view. First, Zariski's original argument divides naturally into three parts. He describes the canonical system of S in terms of the conditions imposed by the singularities of S 0 that correspond to the cusps. Then he establishes the formula where Z denotes the support of the set of cusps. To finish, he invokes the topological result proved in [21]: If n is the power of a prime and B is irreducible, then the n-cyclic multiple plane is regular. The theorem follows from the examination of the terms that vanish in the previous sum when the degree of the cyclic multiple plane covers an unbounded sequence of powers of primes. Second, in [5], Esnault establishes a formula, similar to (1), for the irregularity of the bcyclic multiple plane S 0 , where b is the degree of the branch curve B that possesses arbitrary isolated singularities. She uses the techniques of logarithmic differential complexes, the existence of a mixed Hodge structure on the complex cohomology of the associated Milnor fibrethe complement of S 0 with respect to the plane that contains B-and Kawamata-Viehweg Vanishing Theorem. In [1], Artal-Bartolo interprets Esnault's formula for irregularity and applies it to produce two new Zariski pairs. Two plane curves B 1 , B 2 ⊂ P 2 are called a Zariski pair if they have the same degree and homeomorphic tubular neighbourhood in P 2 , but the pairs (P 2 , B 1 ) and (P 2 , B 2 ) are not homeomorphic. Zariski was the first to discover that there are two types of plane sextics with six cuspidal singularities: there are the ones where the cusps lie on a conic, and the ones where the cusps don't lie on a plane conic. In [19], Vaquié gives a formula for the irregularity of a cyclic covering of degree n of a nonsingular algebraic surface Y ramified along a reduced curve B of degree b with respect to some projective embedding and a nonsingular hyperplane section H that intersects B transversally. His formula is stated in terms of superabundances of the set of singularities of B and the proof also uses the techniques of logarithmic differential complexes. The superabundances involved are given by ideal sheaves that coincide in fact to the multiplier ideals. Vaquié's paper is one among several to introduce the notion of multiplier ideals implicitly and we refer to [4] for this issue.
Third, in [11], Libgober applies methods from knot theory to study the n-multiple plane S 0 . His results are expressed in terms of Alexander polynomials and extend Zariski's Theorem to irreducible curves B with arbitrary singularities and to lines H ∞ with arbitrary position with respect to B. Later on, in [12,13,14], he deals with the case of reducible curves B having transverse intersection with the line at infinity and the irregularity of the multiple plane is expressed using quasiadjunction ideals. The technique is based on mixed Hodge theory, and the result is a particular case in a vaster study, pursued in the above mentioned papers, where the homotopy groups of the complements of various divisors in smooth projective varieties are explored. These groups are related to the Hodge numbers of cyclic or more generally abelian coverings ramified along the considered divisors, as well as to the position of their singularities. We refer the reader to [16] for more ample details and references and to [15] for the relation between the quasiadjunction ideals and the multiplier ideals.
Our argument will follow Zariski's ideas. The multiple plane is transformed into a standard cyclic covering of the plane through a sequence of blowing-ups of P 3 . Then an analog of the formula (1) is obtained thanks to the theory of cyclic coverings: Finally Theorem 2.1 is established using Kawamata-Viehweg-Nadel Vanishing Theorem.
Remark. The above formula coincides with Vaquié's in [19] when the latter is interpreted for a plane curve B and a line H transverse to it. At the same time, Vaquié's formula in its general form might be obtained by the argument we make use of in establishing Theorem 2.1 if Vaquié's general setting were to be considered.
The paper is organized as follows. In Section 1 the theory of cyclic coverings and some facts about multiplier ideals are recalled. Next, in Section 2 it is shown how through a sequence of blowing-ups a cyclic multiple plane is transformed into a standard cyclic covering of the plane and Theorem 2.1 is proved. Explicit computations of the jumping numbers and multiplier ideals are performed in Section 3, using the theory of clusters. Finally, in Section 4 some applications are presented.
Notation and conventions. All varieties are assumed to be defined over C. Standard symbols and notation in algebraic geometry will be freely used. Moreover, if D is a divisor on the variety X, we shall often write H i (X, D) and h i (X, D) instead of H i (X, O X (D)) and h i (X, O X (D)) respectively. If L is an invertible sheaf on X, then we shall regularly denote by L a divisor such that L ≃ O X (L).
Acknowledgements. I started this paper during a one week stay at the University of Pisa in the spring of 2004. I would like to thank Rita Pardini for her hospitality and for the friendly talks we had.
The paper owes Mihnea Popa its present form. I would like to record my debt to his reading of a preliminary version in the autumn of 2005 and to his encouragements to generalize the results I obtained at that time.
Finally, I would like to thank my colleagues Laurent Evain and Jean-Philippe Monnier for the conversations they put up with throughout this period.

Preliminaries
We shall summarize, in a form convenient for further use, some properties of cyclic coverings and of multiplier ideals.

Cyclic coverings
Let X be a variety and let G be the finite abelian group of order n. If G acts faithfully on X, then the quotient Y = X/G exists and X is called an abelian covering of Y with group G. The map π : X → Y is a finite morphism, π * O X is a coherent sheaf of O Y -algebras, and If X is normal and Y is smooth, then π is flat which is equivalent to π * O X locally free. The action of G on π * O X decomposes it into the direct sum of eigensheaves associated to the characters χ ∈ G, The action of G on L χ is the multiplication by χ and To understand the ring structure of π * O X we suppose that every component D of the ramification locus is 1-codimensional. Such a component is associated to its stabilizer subgroup H ⊂ G and to a character ψ ∈ H that generates H: ψ corresponds to the induced representation of H on the cotangent space to X at D. Dualizing the inclusion H ⊂ G, such a couple (H, ψ) is equivalent to a group epimorphism f : G → Z/m f , where m f = |H|; for any χ ∈ G, the induced representation χ| H is given by ψ f (χ) • .
Here and later on, a • denotes the smallest non-negative integer in the equivalence class of a ∈ Z/m, and F the set of all group epimorphisms from G to different Z/mZ.
Let B f be the divisor whose components belong to the branch locus and are exactly those covered by components of the ramification locus associated to the group epimorphism f . The ring structure is given by the following isomorphisms (see [18]): for any χ, χ ′ ∈ G, with ε(f, χ, χ ′ ) = 0 or 1, depending on whether or not f (χ) The next proposition is formulated for cyclic groups, since it is in this case that will be used in the sequel. We refer again to [18] for the case of abelian groups. Proposition 1.1. Let π : X → Y be a cyclic covering with X normal, Y smooth and every component of the ramification locus 1-codimensional. If χ generates G, then for every k = 1, . . . , n, In particular, for k = n equation (3) becomes Proof. For the proof we need to define the sequence (ζ m,r k ) k≥0 : for m and r fixed positive integers with r ≤ m, and for k ≥ 0, put ζ m,r k = 1 if ⌊kr⌋ • m < r, and ζ m,r k = 0 otherwise. Obviously this sequence is m-periodic, ζ m,r m = ζ m,r 0 = 1 and in case m > r, ζ m,r 1 = 0. Now, from the hypotheses, χ spans the group of characters. Taking Then, summing over j from 1 to k, j represents the number of 1's among the first k terms in the sequence (ζ n,b j ) j , hence k j=1 ζ n,b j = ⌊kb/n⌋ and (3) follows. Formula (4) is obvious, since χ n = 1. Conversely, to every set of data L χ , B f , with f ∈ F, that satisfies (4), using (3), we define the line bundles L χ k and associate in a natural way the standard cyclic covering π : Spec (⊕ k L −1 χ k ) → Y , unique up to isomorphisms of cyclic coverings. The line bundles L χ k verify equation (2) and, consequently, ⊕ k L −1 χ k is endowed with a ring structure. Now, the standard covering thus obtained may not be normal; in fact it is not normal precisely above the multiple components of the branch locus (see [18] Corollary 3.1).

The normalization procedure for standard cyclic coverings
Let f : G → Z/m f be a group epimorphism, so m f = ord(Imf ), and let B f = rC + R, with C irreducible and not a component of R, and r ≥ 2. X is not normal along the pull-back of C. The normalization procedure along this multiple component of the branch locus splits into three steps showing how to end up with a new covering X ′ → X → Y , with X ′ normal along the pull-back of C (see [18]). The steps are given by the comparison between the multiplicity r and the order m f of the stabilizer subgroup.
Step 1. If B f = rC + R with r ≥ m f , then set q and r ′ by the Euclidean division r = qm f + r ′ , and construct a new set of building data by putting Step 2. If B f = rC + R with r < m f and (r, m f ) = d > 1, then the natural composition is considered in order to construct a 'less non-normal' covering. Notice that the induced multiplicity and the corresponding subgroup order become relatively prime.
Step 3. If B f = rC + R with r < m f and (r, m f ) = 1, then the composition  Here the only functions f : G → Z/n involved in (2) are given by χ → 1 and by χ → n − 1. In this way, the standard n-cyclic covering S 0 → P 2 is normal and has a singular point above P , the intersection of H 0 and H ∞ . To desingularize it, we consider the blow-up surface Bl P P 2 , with E the exceptional divisor and the induced cyclic covering S → Bl P P 2 . We have nL χ ≡ H 0 + (n − 1)H ∞ + nE and the induced covering S is not normal above E. The normalization procedure leads to S ′ → Bl P P 2 defined by S ′ is a geometrically ruled surface and the pull-back of E is a rational section with self-intersection −n, i.e. S ′ is the Hirzebruch surface F n .

Multiplier ideals
Let X be a smooth variety, D ⊂ X be an effective Q-divisor and µ : Y → X be an embedded resolution for D. Assume that the support of the Q-divisor K Y |X −µ * D is a union of irreducible smooth divisors with normal crossing intersections. Then µ * O Y (K Y |X − ⌊µ * D⌋) is an ideal sheaf J (D) on X. We will denote by Z(D) the subscheme defined by this ideal. Hence I Z(D) = J (D). Showing that J (D) is independent of the choice of the resolution, see [10], we have: The sheaf computing the multiplier ideal verifies the following vanishing result: for every Therefore, applying the Leray spectral sequence, we obtain that for every i Moreover, Kawamata-Viehweg-Nadel Vanishing Theorem. Let X be a smooth projective variety. If L is a Cartier divisor and D is an effective Q-divisor on X such that L − D is a nef and big Q-divisor, then Definition-Lemma. (see [4]) Let B ⊂ X be an effective divisor and P ∈ B be a fixed point. Then there is an increasing discrete sequence of rational numbers ξ i := ξ(B, P ), The rational numbers ξ i 's are called the jumping numbers of B at P .
with Z(ξB) the subscheme defined by the multiplier ideal of ξB.
The proof splits naturally into four parts. First, we show that there is a sequence of blowing-ups of P 3 such that S 1 , the strict transform of the multiple plane S 0 ⊂ P 3 , becomes a standard cyclic covering of the plane defined by nL ′ χ ≡ B + (βn − b)H ∞ , with β = ⌈b/n⌉ and L ′ χ = O P 2 (β). Second we choose a desingularization of B such that its total transform on µ : Y → P 2 is a divisor with normal crossing intersections, a log resolution. It induces a standard cyclic covering S 2 . We apply the normalization procedure to it and obtain a normalization S of S 0 , defined by the line bundle L χ and that has only Hirzebruch-Jung singularities. S Third, we compute the line bundles L χ k 's in terms of the pull-back µ * O P 2 (1) and the exceptional configuration on Y and get the irregularity of S as a sum of some h 1 's. Finally, the result is obtained by applying the Kawamata-Viehweg-Nadel Vanishing Theorem. The first step is given by: Let S 0 be the n-multiple plane associated to the curve B of degree b and the line H ∞ . There exists a sequence of blowing-ups S 1 → S 0 such that S 1 is the standard cyclic covering of the plane determined by Proof. Let [x 0 , . . . , x 3 ] be a homogeneous system of coordinates in P 3 . Let Λ be the plane defined by x 3 = 0, and let B, H ∞ ⊂ Λ be defined by F (x 0 , x 1 , x 2 ) = 0 and x 0 = 0, respectively. H ∞ will be called the line at infinity. The projective n-cyclic multiple plane S 0 ⊂ P 3 is defined by then the situation is slightly more complicated since now N lies on S 0 . Let Ξ be the plane spanned by H ∞ and N . In the open set x 3 = 0, S 0 is defined by Second, we see X 1 as P(O Λ ⊗ O Λ (1)) → Λ and make an elementary transform of X 1 along L ∞ . We blow up X 1 along L ∞ (the trace of the new exceptional divisor on E is denoted by L ∞ ). Then, we contract the strict transform of Ξ to L ∞ . We obtain X 2 = P(O Λ ⊗O Λ (2)) → Λ. The new exceptional divisor becomes an F 1 through H ∞ and L ∞ , and will be denoted by Ξ. On u and look at x = u  as to local coordinates on Λ. Then The complement of E in X β seen through p : X β − E → Λ, coincides with the total space of O Λ (β). The coordinate z coincides with the tautological section of p * O Λ (β). We conclude that S 0 is the standard cyclic covering determined by For the next step in the proof of Theorem 2.1 we need several preliminary results.
Proposition 2.3. Let Y be smooth and let π : X → Y be a standard cyclic covering of degree n determined by For a fixed g ∈ F, the branching divisor B g is supposed to have a multiple component, say B g = rC + R with r > 1. Let X ′ → Y be the standard cyclic covering obtained from X after the normalization procedure has been applied to the multiple component rC.
then for every k = 1, . . . , n − 1, Proof. If r ≥ m g , then r = qm g + r 1 , with 0 ≤ r 1 < r. The covering data are modified to If (r 1 , m g ) = d > 1, then the map g 2 : The covering data are modified to Finally, if the multiplicity of C, r 1 /d, is an integer greater than 1 and prime to m g /d, then the map g 3 : G The covering data are modified to Using (6), (8) and (10) (9) and (7), we get successively and finally, by the Euclidean division of r to m g , Proposition 2.4. Let X be a normal projective variety and Y be a smooth projective variety. Let π : X → Y be a cyclic covering. If ω X is a dualizing sheaf for X, then Proof. We recall the following construction from [7], III, Ex.6.10 and Ex.7.2 valid for X and Y be projective schemes and π : X → Y a finite morphism. For any quasi-coherent If F is coherent on X and G is quasi-coherent on Y , then there is a natural isomorphism π * Hom(F, π ! G) ≃ Hom(π * F, G). It yields the natural isomorphism If ω Y is the canonical sheaf for Y , then it follows that π ! ω Y is a dualizing sheaf for X. Hence Lemma 2.5. Let S 1 → Y be a normal standard cyclic covering of surfaces defined by the line bundle L. If S 1 has only rational singularities and S → S 1 denotes a desingularization of S 1 , then 1 ) and using the formulae for π * ω S and π * O S , we get By Serre duality, the required equality follows.
One more notation is in order. Let P be a singular point of B and let µ : Y → P 2 be a desingularization of B at P , with E P,1 , E P,2 , . . . be the irreducible components of the fibre µ −1 (P ) ⊂ Y . E P will denote this finite array of irreducible curves, and if c is a finite array of rational numbers c 1 , c 2 , . . ., then Proof of Theorem 2.1. For any integer n, S 0 ⊂ P 3 , the n-cyclic multiple plane associated to B and H ∞ is considered. B is assumed to be reduced and transverse to H ∞ . By Proposition 2.2 there is a convenient sequence of blowing-ups such that S 1 , the strict transform of S 0 , becomes a standard cyclic covering of the plane defined by nL ′ χ ≡ B + (βn − b)H ∞ , with β = ⌈b/n⌉ and L ′ χ ≡ βH. We choose a desingularization of B such that its total transform by µ : Y → P 2 is a divisor with normal crossing intersections. S 1 induces a standard cyclic covering S 2 defined by We apply the normalization procedure to S 2 to end up with a normalization S of S 0 that has only Hirzebruch-Jung singularities (see [18], Proposition 3.3). By Proposition 2.3, if S is defined by the line bundle L χ , then the last equality resulting from βk − ⌊k(βn − b)/n⌋ = ⌈kb/n⌉. Here, ⌊kc P /n⌋ · E P denotes α ⌊kc P,α /n⌋ E P,α . From Lemma 2.5, since H · (−L χ k ) = − ⌈kb/n⌉ < 0, it follows that In order to end the proof we have to take account in the formula above, of the vanishing of certain h 1 's and of the equality of the others with certain superabundances of linear systems on the projective plane.
, with Z(k/nB) the scheme defined by the multiplier ideal of k/nB. Indeed, by (12) and (5), it follows that justifying the claim.
Using (13) and the above claim, the irregularity is given by If k/n ∈ J(B, n), then either k/n is not a jumping number of B, or it is, but kb/n is not an integer. In the former case, if ξ is the biggest jumping number for B smaller than k/n, then, since ⌈kb/n⌉ − ξ > 0, by Kawamata-Viehweg-Nadel Vanishing Theorem. In the latter case, we apply the same argument, now using ⌈kb/n⌉ − kb/n > 0. The result follows.
Corollary 2.6. Under the hypotheses of Theorem 2.1, if furthermore B is supposed to be an irreducible plane curve, then with J ′ (B, n) the subset of J(B, n) that contains those rationals ξ for which the denominator can not be the power of a prime.
Proof. In [21] the following topological result is established: If q is the power of a prime and B is irreducible and transverse to H ∞ , then the q-cyclic multiple plane is regular. By inspecting the formula for the irregularity given in Theorem 2.1 for q-multiple planes associated to B and H ∞ , q a power of a prime such that there exists a jumping number l/q ∈ J(B, n), we obtain the corollary.

The case of specified singularities
Explicit versions of Theorem 2.1 may be formulated as soon as the multiplier ideals and the jumping numbers can be evaluated. Such an explicit version is obtained, for example, if the singularities of B are locally characterized by the equation x dp − y dq = 0, with p, q and d positive integers and p, q relatively prime. Turning to Definition 3.6 where the cluster K p,q (α, β) is introduced and if Z p,q (α, β) is the subscheme associated to it, we have: Corollary 3.1. Let B be a plane curve of degree b with each of its singular points of type, either A 1 , or given locally by the equation x dp − y dq = 0. Let H ∞ be a line transverse to B and let S be a desingularization of the n-cyclic multiple plane associated to B and H ∞ . Then The sum ranges over the couples (α, β) such that αp + βq dpq < 1 and αp + βq dpq ∈ 1 gcd(b, n) Z. In addition, the couple of positive integers ( α, β) is defined by and Z p,q ( α, β) = ∪ P Z p,q ( α, β) P , with P ∈ Sing C not of type A 1 .
In this section we mainly want to establish this corollary. We need to control the jumping numbers and the multiplier ideals associated to a curve with this type of singularities. The multiplier ideals and their jumping numbers are known in this case; see for example [3] and [4], or [9] for the case of monomial ideals in general. We like to present a different argument based on Enriques diagrams for the particular case of two unknowns, since it will provide a simple interpretation of the multiplier ideals involved, and also, could provide an algorithm for the generalization to an arbitrary singular point of a curve on a surface.

Clusters and Enriques diagrams
Let X be a surface and P ∈ X a smooth point. A point Q is called infinitely near to P if Q ∈ X ′ , µ : X ′ → X is a composition of blowing-ups and Q lies on the exceptional configuration that maps to P .

Definition.
A cluster in X, centered at a smooth point P is a finite set of weighted infinitely near points to P , K = {P w 1 1 , . . . , P wr r }, with P 1 = P . Let µ : Y → X be the composition of blowing-ups Y = Y r+1 → Y r → · · · → Y 1 = X, with Y α+1 = Bl Pα Y α . Since the points infinitely near P are partially ordered-the point Q precedes the point R if and only if R is infinitely near Q-the points of a cluster are partially ordered. In the sequel, if K is a cluster, then all points preceding a point that belongs to K are in K, possibly with weight 0.
Let K be a cluster centered at P . Each point P α corresponds to an exceptional divisor E α ⊂ Y α+1 . All its strict transforms will also be denoted by E α and the total transform of each E α will be denoted by F α . When needed, the strict transform of E α on Y β will be denoted by E Definitions. Let K be a cluster. A point P β is said to be proximate to P α if P β lies on E α , the exceptional divisor corresponding to the blowing-up at P α , or on one of its strict transforms.
A cluster K is said to satisfy the proximity relations if for every P α in K, Lemma 3.2 (see also [2], Theorem 4.2). Let K = {P w 1 1 , . . . , P wr r } be a cluster that contains a point P α for which the proximity relation is not satisfied.
for every β with P β proximate to P α , and w ′ γ = w γ otherwise, then K and K ′ define the same subscheme in K ′ is said to be obtained from K by the unloading procedure. Starting from K, iterated applications of this procedure lead to a cluster K that satisfies the proximity relations and defines the same subscheme in X. K is called the unloaded cluster. Notice that Remark. If w r < 0, then the proximity relation is not satisfied at P r since w r = 0. When the unloading procedure of Lemma 3.2 is applied to a cluster with non-negative weights, it may happen that a weight becomes negative, or more precisely, becomes −1. But it is to be noticed that the negative weight is eventually rubbed out by the next applications of the procedure, and that the unloaded cluster has only non-negative weights. Moreover, the unloaded cluster associated to a cluster with non-positive weights is the empty cluster, the one with all its weights equal to 0.

Definition.
A gridded tree is a couple (T, g), where T = T (V, A) is an oriented tree with V the set of vertices and A the set of arcs, and g is a map g : A → {slant, horizontal, vertical}.
Definition. Let T be a gridded tree. A horizontally (vertically) L-shape branch of T is an ordered chain of arcs, such that each begins where the previous ends, and such that all are horizontal (vertical), but the first.
Notice that an L-shape branch is completely determined by the subset of incident vertices of it. Moreover, an arc is an L-shape branch, regardless its value through g.
Definition. Let T be a gridded tree. A segment is a maximal chain of arcs of the same type through g, arcs that are also maximal L-shape branches.
Example 3.3. Let p < q be relatively prime positive integers. T p,q will denote the gridded tree associated to the Euclidean algorithm. If r 0 = a 1 r 1 + r 2 , . . . , r m−2 = a m−1 r m−1 + r m and r m−1 = a m r m , with r 0 = q and r 1 = p, then T p,q has d segments containing a 1 , . . . , a m−1 and respectively a m vertices each. The first segment is slanted and the others are alternatively, either horizontal or vertical, starting with a horizontal one.
Definition. An Enriques diagram is an weighted gridded tree.
Clusters and Enriques diagrams carry the same information as the lemma below asserts, and it will often be convenient to argue using diagrams.
Definition. A point of a cluster is said to be free if it is proximate to exactly one point of the cluster. A point is said to be a satellite if it is proximate to exactly two points of the cluster.
Lemma 3.4 (see [6]). There exists an unique map from the set of clusters in X centered at a smooth point P to the set of Enriques diagrams such that: 1. for every cluster K = {P w 1 1 , . . . , P wr r } the set of vertices of the image tree is V = {P 1 , . . . , P r } with the weights given by the integers w 1 , w 2 , . . . , w r ; 2. at every point ends at most one arc; 3. a point P α is satellite if and only if there is either a horizontal or a vertical arc that ends at the vertex P α ; 4. if there is an arc that begins at the vertex P α and ends at the vertex P β then P β ∈ E (β) α , and the converse is true if P β is free; 5. P β is proximate to P α if and only if there is an L-shape branch that starts at P α and ends at P β ; 6. the strict transforms E α and E β intersect on Y if and only if the Enriques diagram contains a maximal L-shape branch that has P α and P β as its extremities; 7. an arc that begins at a vertex of a free point and ends at a vertex of a satellite point is horizontal.

The minimal unloaded clusters associated to a T p,q tree
Let p < q be relatively prime positive integers. All clusters treated in this subsection will be associated to the gridded tree T p,q introduced in Example 3.3. The intent is to look for a characterization of the minimal unloaded clusters modeled on T p,q . We refer to Lemma 3.7 for the result. Depending on the context its vertices will be denoted either by P α i.e. using one subscript 1 ≤ α ≤ r = a 1 + · · · + a m , or by P k,i , i.e. using two subscripts 1 ≤ k ≤ d, 1 ≤ i ≤ a k . T 3,5 is represented below in the latter notation.
Let K be a cluster. We define the proximity matrix of K by Π = ||p αβ ||, where the elements of the diagonal equal 1 and, for every α = β, the element p αβ equals −1 if P β is proximate to P α and 0 if not. Notice that along the α column of Π, the non-zero elements not on the diagonal correspond to the points to which P α is a satellite.
The proximity matrix is the decomposition matrix of the strict transforms E α 's in terms of the total transforms F α 's. Hence if K = {P w 1 1 , . . . , P wr r }, then on Y , and c = w Π −1 , where w = (w 1 , . . . , w r ) and similarly c = (c 1 , . . . , c r ). The formula and induction on α tell us that the coefficient of E r in the decomposition of a total transform corresponding to a point lying on the kth segment in terms of strict transforms equals the remainder r k introduced in Example 3.3.
Lemma 3.5. If K = {P w 1 1 , . . . , P wr r } is an unloaded cluster centered at P , then the coefficient of E r is of the form ap + bq, with a, b non-negative integers.
Proof. We shall denote the weights on the kth segment of the Enriques diagram for the cluster K by w k,1 , w k,2 , . . . , w k,a k and the coefficient of E r by c r . We shall successively transform the cluster, each time considering the last segment that contains non-zero weights, unless this segment is the first or the second one. The transformation is the following: if the last segment with non-zero weights is the segment k + 1, and if w k,a k = i w k+1,i , then put and leave the other weights unchanged. It is easy to see that the cluster K ′ defined by the weights w ′ k,i is again unloaded and that the coefficient of E r remains unchanged. Hence the same process can be applied till eventually K is transformed into the cluster K ′ with non-zero weights only on the first and, at the most, the second segments of the Enriques diagram. K ′ is unloaded and where q = a 1 p + r 2 .
Besides, since for each couple of non-negative integers a, b there exists an unloaded cluster with ap + bq the coefficient of E r -for example the cluster with w 1,1 = a + b, w 1,i = b for i = 1, w 2,1 = b and w k,i = 0 otherwise-, the ideal sheaf µ * O Y (−(ap + bq)E r ) defines the minimal unloaded cluster having ap + bq the coefficient of E r . It is natural to ask the question whether we can decide if an unloaded cluster is minimal only by inspection of its weights, or equivalently its associated divisor. The answer is yes and is given by the lemma hereafter. It will deal with clusters satisfying the following condition: ( * ) for every ordered chain of maximal L-shape branches determined by the points P α 1 , . . . , P α l , i.e. each P α j precedes P α j+1 and the jth maximal L-shape branch starts at P α j and ends at P α j+1 , then Definition 3.6. If a and b are non negative integers, K p,q (a, b) denotes the minimal unloaded cluster associated to the T p,q tree and whose coefficient of the last strict transform equals ap + bq. Proof. We start by showing that a minimal unloaded cluster K min always satisfies ( * ). Indeed, if not, there would exist a chain of maximal L-shape branches such that p α j ≥ w α j − w α j for 1 ≤ j ≤ l, and that l 1 w α j − w α j ≥ l 1 (p α j − 1) + 2. Furthermore, since there would be at least two points such that p α = w α − w α , we may always assume w α 1 − w α 1 = p α 1 , w α j − w α j = p α j − 1 for 2 ≤ j ≤ l − 1, and w α l − w α l = p α l . Now for this chain, we could apply the inverse of the unloading procedure successively at P α 1 , P α 2 , . . . , P α l to end up with an unloaded cluster with the same coefficient for E r as K min , hence a contradiction.
To end the proof, we assume that K = K min , K and K min having the same coefficient for E r , and show that K does not satisfy ( * ). Since it is satisfied by K min , we notice that along each segment of K min , there is at most one jump of height 1. We may further assume that w 1,1 ≥ w min 1,1 + 1. To make up for the apparent increase of c m,am (= c r ) by at least r 1 = p due to the difference between w 1,1 and w min 1,1 , some of the weights along the next segments of K must be smaller than the corresponding weights of K min , but not along the first segment. Looking at the points on the second segment and at the first point of the third segment for this counterbalance problem, we notice that at most one of their weights may not diminish, otherwise ( * ) will not be satisfied somewhere along the following segments. Two possibilities can appear. First, all a 2 weights of the second segment satisfy w 2,i ≤ w min 2,i − 1 and, either there exists an i such that w 3,i ≤ w min 3,i − 1, or w 3,i = w min 3,i for all i's. In the former case ( * ) is not verified at P 1,a 1 , P 3,1 , P 3,2 , . . . , P 3,i , and in the latter the same counterbalance problem must be solved for a difference of r 3 units, starting with the fourth segment. Second, the inequalities w 2,i ≤ w min 2,i − 1 are verified for all but one point of the second segment, and w 3,1 ≤ w min 3,1 − 1. There is a counterbalance problem left for r 2 units, starting with the third segment. Eventually, the counterbalance problem is pushed on to the last segment and hence ( * ) will not be satisfied there for K.

3.3
The multiplier ideals and the jumping numbers for (x dp , y dq ) Let P be a singular point of B ⊂ X given locally by x dp + y dq = 0, with p, q and d positive integers and p ≤ q relatively prime, and let µ : Y → X be the minimal log resolution of B at P . The exceptional configuration of µ is given by the gridded tree T p,q . As before, we shall denote by E r the last strict transform.
Proof. It is sufficient to determine the coefficient of E r in µ * B. By Example 3.3 and the decomposition of the F α 's in terms of the strict transforms, we have that the coefficient of E r is a 1 r 2 1 + · · · + a m r 2 m = dpq.
Proposition 3.9. If c r is the coefficient of E r in −K Y |X + ⌊µ * ξB⌋, then the multiplier ideal J (ξB) is given by i.e. is the ideal sheaf associated to the minimal cluster that contains c r E r .
Let P α 1 , . . . , P α l be ordered points that determine a chain of maximal L-shape branches.
where the matrix −Π Π t is the intersection matrix of the strict transforms E α 's on the surface Y , for every 1 ≤ j ≤ l, and since c Π t Π = (0, . . . , 0, d), we have Putting l = 1 we observe that if the proximity relation is not satisfied at P α , then w α − w α = −1. But the unloading procedure of Lemma 3.2 at P α changes the vector w−w into the vector w − w + (Π t Π) α . It follows that the unloading procedure does not change the inequalities in (15) for the new cluster. We conclude that the unloaded cluster satisfies ( * ), hence, by Lemma 3.7, the result. Proof. Let ξ be a jumping number and K ′ the corresponding unloaded cluster and let, by Lemma 3.5, c ′ r = a ′ p + b ′ q be its coefficient for E r with a ′ , b ′ non-negative integers. By Lemma 3.8 and Proposition 3.9, we have ⌊dpqξ⌋ − (p + q − 1) = ap + bq + 1 ≤ a ′ p + b ′ q where a and b are non-negative integers, and a ′ p+b ′ q is the first integer combination of p and q with this property. So, by the definition of the jumping numbers, ξ = ((a+1)p+(b+1)q)/(dpq).
Proof of Corollary 3.1. We know that the irregularity is given by with J(B, n) the subset of jumping numbers ξ of B of the form k/n, 0 < k < n, and such that ξb is an integer. By Proposition 3.10, it is sufficient to describe the subscheme associated to the multiplier ideal J (ξB) for every ξ = (αp + βq)/(dpq) ∈ J(B, n). By Proposition 3.9, Lemma 3.8 and Lemma 3.7, the subscheme is given by the minimal unloaded cluster whose coefficient for the last strict transform is the first integer combination of p and q not smaller than (α − 1)p + (β − 1)q + 1.
Remark 3.11. Since many of the applications in the next section will be for a curve B with singularities of a given type A • , we interpret Corollary 3.1 for this situation. If P 1 , . . . , P r are the infinitely near points to P = P 1 involved in the minimal log resolution of an A • type singularity at P , we shall denote by Z α ranging from 1 to r − 1 such that α+r 2r ∈ 1 gcd(b,n) Z. ii) If the singularities of B are of type A 1 or A 2m , i.e. p = 2, q = 2m + 1 and d = 1, then n and b are even, and α ranges from 1 to m such that α 2m+1 ∈ 1 gcd(b,n) Z.

Applications
We shall now apply the results in the previous sections to illustrate how to compute in an uniform way, the irregularity for some examples of cyclic multiple planes.

Zariski's example
The curve B is irreducible, of degree 6 and has six cusps as singularities. In the formula for the irregularity of the 6-cyclic multiple plane in Remark 3.11 ii), since m = 1, α may only be 1. Hence q(S) = h 1 (P 2 , I Z (2)), where Z is the support of the cusps. So either the cusps lie on a conic and the irregularity is 1, or they do not, and the irregularity is 0. Notice that the result is the same for every n-cyclic multiple plane, provided that 6 divides n.
Artal-Bartolo's first example in [1] Let C ⊂ P 2 be a smooth elliptic curve and let P 1 , P 2 , P 3 be three inflexion points of C, with L i the tangent lines at P i to C. Taking B = C + L 1 + L 2 + L 3 we construct the multiple cyclic plane with three sheets S 0 associated to B and H ∞ . The curve B has three points of type A 5 at the P i 's, hence n = 3, b = 6 and r = 3 in Remark 3.11 i). We have q(S) = h 1 (P 2 , I {P 1 ,P 2 ,P 3 } (1)).
So, if the three inflexion points are chosen on a line, then the irregularity is 1. If the points are not aligned, then the irregularity is 0. These two configurations give an example of a Zariski pair.
Artal-Bartolo's second example in [1] Let P be a fixed point and K = {P 1 , . . . , P 9 } a cluster centered at P , all its points being free. It represents a curvilinear subscheme Z = Z K . In [1], Artal-Bartolo considers sextics with an A 17 type singularity at P , with P 2 , . . . , P 9 the infinitely near points of the minimal resolution.
1) If P 3 lies on the line L determined by P 1 and P 2 and if K does not impose independent conditions on cubics, then all sextics are reducible. Let B be the union of two smooth cubics from |I Z (3)|, and let H ∞ be a line transverse to B. If S 0 is the 3-cyclic multiple plane associated to B and H ∞ , then by Remark 3.11 i), q(S) = h 1 (P 2 , I Z [3] (1)) = 1.
2) If P 3 / ∈ L and P 6 ∈ Γ, the conic through P 1 , . . . , P 5 , then there exists an irreducible sextic with an A 17 type singularity at P , such that the intersection with Γ is supported only at P . If S 0 is the n-cyclic multiple plane associated to B and to a transverse line to it, then q(S) = h 1 (P 2 , I Z [6] (2)) = 1 when n is divisible by 6, and q(S) = 0 otherwise.
3) If P 3 / ∈ L and P 6 / ∈ Γ, the conic through P 1 , . . . , P 5 , then, for every reduced sextic B with an A 17 type singularity at P , if S 0 is the n-cyclic multiple plane associated to B and to a transverse line to it, then q(S) = 0.
Remark. In [1] it is shown that in the third case above, two configurations may appear: either P 1 , . . . , P 9 do not impose independent conditions on cubics and B is the union of two smooth cubics, or the points impose independent conditions on cubics and B is irreducible. Using these and the two configurations in 1) and 2), two more Zariski couples are thus produced there.
Oka's example in [17] In [17], when p and q are relatively prime integers, Oka constructs the curve C p,q of degree pq enjoying the following property: C p,q has pq cusp singularities each of which is locally defined by the equation x p + y q = 0. We shall show that the pq-multiple plane associated to C p,q is irregular, the irregularity being equal to (p − 1)(q − 1)/2.
We start with the particular case p = 2, since all ideas of the general computation are already present in this situation. The construction of the branching curve B = C 2,2m+1 is as follows. Let C ⊂ P 2 be a curve of degree 2m + 1 and let Γ be a conic transverse to C. If f = 0 and g = 0 are homogeneous equations for C and Γ respectively, then the curve B is defined by f 2 + g 2m+1 = 0. It is a curve of degree 4m + 2 with 4m + 2 singular points of type A 2m . Let S 0 be the (4m + 2)-cyclic multiple plane associated to B and let S be the normal cyclic covering constructed in Section 2.
Remark. The irregularity of the n-cyclic multiple plane associated to B and to a line H ∞ transversal to B, n being an arbitrary positive integer, may be computed by the same argument. Of course, if 2m + 1 is a prime number, then q(S) = 0 unless 4m + 2 divides n, see Corollary 2.6. But if 2m + 1 is not a prime number, then irregular cyclic multiple planes exist for other values of n. For example, if 2m + 1 = 15 and n = 40, then q(S) = h 1 (P 2 , I Z [3] (18)) + h 1 (P 2 , I Z [6] (24)) = 2.
In the general case, if p < q, let B = C p,q and C p and C q be the smooth curves of degree p and respectively q used in the construction of B. C p and C q intersect transversely and if P is an intersection point, then B has a singularity at P given locally by an x p + y q = 0 type equation.
Remark 4.1 (The trace-residual exact sequence). Let X be a projective variety, D be a Cartier divisor on X and Z be a a closed subscheme of X. The schematic intersection Tr D Z = D ∩ Z defined by the ideal sheaf (I D + I Z )/I D is called the trace of Z on D. The closed subscheme Res D Z ⊂ X defined by the conductor ideal (I Z : I D ) is called the residual of Z with respect to D. Following [8], the canonical exact sequence is called the trace-residual exact sequence of Z with respect to D.
A specialization of Oka's example when p = 2 Keeping the notation from the first part of the previous paragraph, the conic Γ is now the union of two distinct lines that intersect at O and C is a smooth curve of degree 2m + 1 passing through O and intersecting transversely the lines of Γ at this point. The curve B has 4m points of type A 2m and one singular point at O of type A 4m+1 . It can be shown that the irregularity of the (4m + 2)-cyclic multiple plane associated to B is again m. We develop the computation for m = 2. In this case, B is a curve of degree 10 with 8 points of type A 4 and one point of type A 9 . By Theorem 2.1 and using the notation from Remark 3.11, the irregularity is given by h 1 (P 2 , I ξ [1] ∪Z [2] O (4)) + h 1 (P 2 , I ξ [2] ∪Z [4] O (6)), where ξ [1] is the support of the points of type A 4 and ξ [2] = ∪ P of type A 4 Z [2] P is the support plus the tangent directions. Now, 10 points on a conic do not impose independent conditions on quartics, hence the first term is 1. The second term is seen to be equal to the first after applying the trace-residual exact sequence with respect to the two lines of Γ. So the irregularity is 2.
The computations for m = 1 lead to a branching curve of degree 6 with 4 cusps and an A 5 singularity at O. The irregularity of a 6-cyclic multiple plane is 1, given by h 1 (P 2 , I ξ [1] ∪Z [2] O (2)). If in addition, the two lines of the degenerate conic Γ are brought together such that the cusps collapse two by two, the branching curve has 3 A 5 singularities. For a 6-multiple plane, q = 2, with the contributions of the superabundance of the singularities with respect to the lines and the conics both equal to 1. Necessarily, by Corollary 2.6, the branching curve is reducible; it is Artal-Bartolo's first example.
Line arrangements following [5] In this example we consider as branch curve a line arrangement B = ∪ b i=1 L i ⊂ P 2 that has only nodes and ordinary triple points as singularities. For an ordinary triple point 2/3 is the only subunitary jumping number. By Corollary 3.1, if H ∞ is a line transverse to B = ∪ b i=1 L i , then the normal n-cyclic covering S corresponding to the n-cyclic multiple plane associated to B and H ∞ is irregular if and only if 3 divides both b and n, and |I Z (−3 + 2b 3 )| is superabundant, in which case q(S) = h 1 (P 2 , I Z (−3 + 2b 3 )).
In case S is irregular, it can be shown that the irregularity is bounded by a constant depending on the arrangement B. More precisely, we have Proposition 4.2. Let B = ∪ b i=1 L i , H ∞ and S be as above with b and n divisible by 3. If t i is the number of triple points lying on the line L i for each i, then For the proof (see [5] for a different argument), we will start with a preliminary lemma. Proof. Let B ′ be the arrangement of the b − 1 lines of B except the one one with no triple point. If S ′ is the normal n-cyclic covering corresponding to the n-cyclic multiple plane associated to B ′ and H ∞ , then q(S ′ ) = 0 since 3 does not divide deg B ′ . Taking k = 2n/3 and denoting by Z the support of the triple points, we obtain 0 = h 1 (P 2 , I Z (−3 + 2(b − 1) 3 ) = h 1 (P 2 , I Z (−3 + 2b 3 ) = q(S).

O
It is easy to see that there are two cubics-each the union of three lines-through the 9 triple points, i.e. the system of cubics through the points is superabundant. It follows that the irregularity of the n-cyclic multiple plane associated to B and to a line H ∞ transverse to B, is 1 if and only if 3 divides n.
If s = 2, then the arrangement specialize to an arrangement with 10 triple points, 4 of them lying on the line x + y = 0. But these points lie on a cubic, the union of three of the lines of B, and again h 1 (P 2 , I Z (3)) = 1, hence the irregularity is 1 in this case too. To see this, let B be the line arragement below of 5 lines with 2 triple points.

Q P
If H ∞ is transverse to B, then the irregularity of the 6-cyclic multiple plane is 0. But if H ∞ is the line through the double points P and Q then the irregularity jumps to 1.