The dimension of the image of the Abel map associated with normal surface singularities

Let $(X,o)$ be a complex normal surface singularity with rational homology sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an effective cycle $Z$ supported on the exceptional curve and also a possible Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. Define ${\rm Eca}^{l'}(Z)$ as the space of effective Cartier divisors on $Z$ and $c^{l'}(Z):{\rm Eca}^{l'}(Z)\to {\rm Pic}^{l'}(Z)$, the corresponding Abel map. In this note we provide two algorithms, which provide the dimension of the image of the Abel map. Usually, $\dim {\rm Pic}^{l'}(Z)=p_g$, $\dim\,{\rm Im} (c^{l'}(Z))$ and ${\rm codim}\,{\rm Im} (c^{l'}(Z))$ are not topological, they are in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial formulae for them whenever the analytic structure on $\widetilde{X}$ is generic. The ${\rm codim}\,{\rm Im} (c^{l'}(Z))$ is related with $\{h^1(\widetilde{X},\mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$; in order to treat the `twisted' family $\{h^1(\widetilde{X},\mathcal{L}_0\otimes \mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$ we need to elaborate a generalization of the Picard group and of the Abel map. The above algorithms are also generalized.

1. Introduction 1.1. Fix a complex normal surface singularity (X, o) and let X be one of its good resolutions. We assume that the link of (X, o) is a rational homology sphere. Denote by L the lattice H 2 ( X, Z) (endowed with its negative definite intersection form), by L ′ its dual lattice H 2 ( X, Z) and by S ′ ⊂ L ′ the Lipman cone of antinef cycles. The irreducible exceptional curves are denoted by {E v } v∈V , their duals in L ′ by {E * v } v∈V , E := ∪ v E v . (For details see section 2). In [NN18a] for any effective cycle Z ≥ E and Chern class l ′ ∈ −S ′ the authors introduced (based on [Gro62,Kl05,Kl13]) and investigated the set of effective Cartier divisors ECa l ′ (Z) and the corresponding Abel maps c l ′ (Z) : ECa l ′ (Z) → Pic l ′ (Z), where Pic l ′ (Z) is the affine subspace of the Picard group of line bundles over Z with Chern class l ′ . The image of the Abel map consists of line bundles without fixed components. [NN18a] and follow-up articles contain several properties of the Abel map, e.g. characterisation when it is dominant, or its relationship with cohomological properties of line bundles. See [NN18b] and [NN19a] for the study in the case of generic and elliptic singularities. In all these treatments the investigation of the image Im(c l ′ (Z)) was extremely useful. The main goal of the present article is the computation of dim Im(c l ′ (Z)) and the deduction of several new consequences. We consider these as necessary steps towards a long-term final goal: the development of the Brill-Noether theory of normal surface singularities.
Though the dimension (l ′ , Z) (and the homotopy type) of the connected complex manifold ECa l ′ (Z) is topological (i.e. it depends only on the link, or on the lattice L), the dimension h 1 (O Z ) of the target affine space Pic l ′ (Z) depends essentially on the analytic structure: if we fix the topological Such type of formulas already appeared in the computation of d Z (l ′ ) for weighted homogeneous singularities (and specific l ′ ) in [NN18a], case which lead us to the present general case. (The type of formula, and also the conceptual approach behind, can also be compared e.g. with Pflueger's formula regarding the dimension of the Brill-Noether varieties of a generic smooth projective curve C with fixed gonality, cf. [P16,JR17].) Nevertheless, the approach of the testing function (and the corresponding min-type close formulae) is the novelty of the present manuscript.
1.3. The testing functions for d s . Obviously, the above theorem is valuable only if τ s is essentially different than d s and also if it is computable from other different geometrical behaviours. It is also clear that not any upper bound d s ≤ τ s satisfies the testing property (2): this is satisfied only for bounds τ (s) with very structural relationship, symbiosis with the original d s . Hence it is not easy to find testing functions, they must 'testify' about some deep geometric property: even the existence of computable testing function(s) is really remarkable. Our first test function is defined as follows. Consider again Z ≥ E, l ′ ∈ −S ′ associated with a resolution X, as above. Then, besides the Abel map c l ′ (Z) one can consider its 'multiples' {c nl ′ (Z)} n≥1 . It turns out that n → dim Im(c nl ′ (Z)) is a non-decreasing sequence, Im(c nl ′ (Z)) is an affine subspace for n ≫ 1, whose dimension e Z (l ′ ) is independent of n ≫ 0, and essentially it depends only on the E * -support of l ′ (i.e., on I ⊂ V, where −l ′ = v∈I a v E * v with all {a v } v∈I nonzero). From construction d Z (l ′ ) ≤ e Z (l ′ ), however they usually are not the same. Furthermore, e Z (l ′ ) = e Z (I) plays a crucial role in different analytic properties of X (surgery formula, h 1 (L)computations, base point freeness properties). For details see [NN18a] or subsections 2.2 and 2.4 here, especially definition 3.1.1 and Theorem 2.2.5 (and also the proof of Theorem 3.2.2). Now, at any step of the tower X s one can consider this invariant e Zs (l ′ s ), an integer denoted by e s .
Theorem 3.2.2 (the 'first algorithm') guarantees that e s is a testing function for d s .
The invariants {e s } s are still hard to compute (cf. 4.1). However, the first algorithm is a necessary intermediate step for the second algorithm, valid for another testing function.
The advantage of the second testing function is that it is defined at the level of X only. It is based on Laufer's perfect pairing H 1 (O Z ) ⊗ G Z → C, where G Z denoted the space of classes of forms H 0 ( X, Ω 2 X (Z))/H 0 ( X, Ω 2 X ). G Z has a natural divisorial filtration {G l } 0≤l≤Z , where G l is generated by forms with pole ≤ l. Its dimension (via Laufer duality) is h 1 (O l ). (For more see [NN18a] and 2.4 here.) Next, for any s define the cycle l s ∈ L of X by l s := min v∈V min 1≤kv ≤av Set also g s := dim G ls as well. It turns out (see 4.1) that d s ≤ e s ≤ h 1 (O Z )−g s . Usually, the equality e s = h 1 (O Z ) − g s rarely happens, however, it happens whenever the testing property requires it! Theorem 4.1.2 (the 'second algorithm') says that h 1 (O Z ) − g s is a testing function for d s indeed.
The cases of superisolated singularities is exemplified.
The second algorithm has several consequences. E.g., a 'numerical' one, cf. (4.1.6): The cycles Z 1 for which the above minimum is realized have several additional geometric properties (cf. Lemma 4.1.14 and 4.2). In particular, such a Z 1 imposes the following conceptual consequence: Structure Theorem for the image of the Abel map. Fix a resolution X, a cycle Z ≥ E and a Chern class l ′ ∈ −S ′ as above. Then there exists an effective cycle Z 1 ≤ Z, such that: (i) the map ECa l ′ (Z) → H 1 (Z 1 ) is birational onto its image, and (ii) the generic fibres of the restriction of r, r im : Im(c l ′ (Z)) → Im(c l ′ (Z 1 )), have dimension h 1 (O Z ) − h 1 (O Z1 ). In particular, for any such Z 1 , the space Im(c l ′ (Z)) is birationally equivalent with an affine fibration over ECa l ′ (Z 1 ) with affine fibers of dimension h 1 (O Z ) − h 1 (O Z1 ).
The maximum at the right hand side is realized e.g. for the cohomology cycle of L im gen ∈ Im(c l ′ (Z)) ⊂ Pic l ′ (Z). Furthermore, for any L ∈ Im(c l ′ (Z)) and equality holds for generic L im gen ∈ Im(c l ′ (Z)). The identity (1.4.1), valid for a generic analytic structure of X, extends to an optimal inequality valid for any analytic structure.
Theorem 1.4.2. Consider an arbitrary normal surface singularity (X, o), its resolution X, Z ≥ E and l ′ ∈ −S ′ . Then codim Im(c l ′ (Z)) = h 1 (Z, L im gen ) satisfies In particular, for any L ∈ Im(c l ′ (Z)) one also has The image of the Abel map 5 The right hand side of (1.4.3) is a sharp topological lower bound for codim Im(c l ′ (Z)). The inequality (1.4.3) can also be interpreted as the semi-continuity statement codim Im(c l ′ (Z))(arbitrary analytic structure) ≥ codim Im(c l ′ (Z))(generic analytic structure).
1.5. Generalization. Sections 7 and 8 target generalizations of the previous parts, valid for {h 1 (Z, L)} L∈Imc l ′ (Z) , to the shifted case, valid for {h 1 (Z, L 0 ⊗L)} L∈Imc l ′ (Z) , where L 0 ∈ Pic l ′ 0 (Z) is a fixed bundle without fixed components. In order to run a parallel theory based on Abel maps, we have to create the new (Z) appears also as an affine quotient of the classical Pic l ′ (Z) as well.) Section 7 contains the definitions and the needed exact sequences. Section 8 contains the extension of the two algorithms to this situation.

Preliminaries
2.1. Notations regarding a good resolution. [N99b,N07,N12,NN18a] Let (X, o) be the germ of a complex analytic normal surface singularity, and let us fix a good resolution φ : X → X of (X, o). Let E be the exceptional curve φ −1 (0) and ∪ v∈V E v be its irreducible decomposition. Define E I := v∈I E v for any subset I ⊂ V.
We will assume that each E v is rational, and the dual graph is a tree. This happens exactly when the link M of (X, o) is a rational homology sphere.
L := H 2 ( X, Z), endowed with a negative definite intersection form ( , ), is a lattice. It is freely generated by the classes of where the first Chern classes live.
All the E v -coordinates of any E * u are strict positive. We define the Lipman cone as S ′ := {l ′ ∈ L ′ : (l ′ , E v ) ≤ 0 for all v}. As a monoid it is generated over There is a natural (partial) ordering of L ′ and L: 2.1.1. Natural line bundles. Let φ : ( X, E) → (X, o) be as above. Consider the 'exponential' cohomology exact sequence (with H 1 ( X, O * X ) = Pic( X), the group of isomorphic classes of holomorphic line bundles on X, and H 1 ( X, O X ) = Pic 0 ( X)) Here c 1 (L) ∈ H 2 ( X, Z) = L ′ is the first Chern class of L ∈ Pic( X). Since H 1 (M, Q) = 0, Pic 0 ( X) ≃ H 1 ( X, O X ) ≃ C pg , where p g is the geometric genus. Write also Pic l ′ ( X) = c −1 1 (l ′ ). Furthermore, see e.g. [O04, N07], there exists a unique homomorphism (split) s 1 : L ′ → Pic( X) of c 1 , that is c 1 • s 1 = id, such that s 1 restricted to L is l → O X (l). The line bundles s 1 (l ′ ) are called natural line bundles of X. For several definitions of them see [N07]. E.g., L is natural if and only if one of its power has the form O X (l) for some integral cycle l ∈ L supported on E. In order to have a uniform notation we write O X (l ′ ) for s 1 (l ′ ) for any l ′ ∈ L ′ . For any Z ≥ E let O Z (l ′ ) be the restriction of the natural line bundle O X (l ′ ) to Z. In fact, O Z (l ′ ) can be defined in an identical way as O X (l ′ ) starting from the exponential cohomological sequence 0 → Pic 0 (Z) → Pic(Z) → H 2 ( X, Z) → 0 as well. Set also Pic l ′ (Z) = c −1 1,Z (l ′ ).
2.2. The Abel map [NN18a]. For any Z ≥ E let ECa(Z) be the space of (analytic) effective Cartier divisors on Z. Their supports are zero-dimensional in E. Taking the line bundle of a Cartier divisor provides the Abel map c = c(Z) : ECa(Z) → Pic(Z). Let ECa l ′ (Z) be the set of effective Cartier divisors with Chern class l ′ ∈ L ′ , i.e. ECa l ′ (Z) := c −1 (Pic l ′ (Z)). The restriction of c is denoted by It is advantageous to have a similar statement for l ′ = 0 too, hence we redefine ECa 0 (Z) as {∅}, a set/space with one element (the empty divisor), and c 0 : ECa 0 (Z) → Pic 0 (Z) by c 0 (∅) = O Z . In particular, Hence, the extended statement valid for any l ′ is: Sometimes even for L ∈ Pic l ′ ( X) we write L ∈ Im(c l ′ ) whenever L| Z ∈ Im(c l ′ (Z)) for some Z ≫ 0. This happens if and only if L ∈ Pic( X) has no fixed components. It turns out that ECa l ′ (Z) (−l ′ ∈ S ′ ) is a smooth complex algebraic variety of dimension (l ′ , Z) and the Abel map is an algebraic regular map. For more properties and applications see [NN18a,NN18b].
2.2.3. The modified Abel map. Multiplication by O Z (−l ′ ) gives an isomorphism of the affine spaces Pic l ′ (Z) → Pic 0 (Z). Furthermore, we identify (via the exponential exact sequence) Pic 0 (Z) with the vector space H 1 (Z, O Z ).
It is convenient to replace the Abel map c l ′ with the composition The advantage of this new set of maps is that all the images sit in the same vector space H 1 (O Z ). Consider the natural additive structure s l ′ provided by the sum of the divisors. One verifies (see e.g. [NN18a, Lemma 6.1.1]) that s l ′ 1 ,l ′ 2 (Z) is dominant and quasi-finite. There is a parallel multiplication For different geometric reinterpretations of dim V Z (I) see also [NN18a,§9].
2.3. Theorem 4.1.1 of [NN18a] says that c l ′ (Z) is dominant if and only if χ(−l ′ ) < χ(−l ′ + l) for any 0 < l ≤ Z. In particular, the dominance of c l ′ (Z) is a topological property. If c l ′ (Z) is dominant then c l ′ (Z ′ ) is dominant for any 0 < Z ′ ≤ Z. [La72], [La77, p. 1281]. Following Laufer, we identify the dual space H 1 ( X, O X ) * with the space of global holomorphic 2-forms on X \ E up to the subspace of those forms which can be extended holomorphically over X.

Review of Laufer Duality
For this, use first Serre duality H 1 ( X, O X ) * ≃ H 1 c ( X, Ω 2 X ). Then, in the exact sequence

2.4.2.
Above H 0 ( X \ E, Ω 2 X ) can be replaced by H 0 ( X, Ω 2 X (Z)) for a large cycle Z (e.g. for Z ≥ ⌊Z K ⌋). Indeed, for any cycle Z > 0 from the exacts sequence of sheaves 0 → Ω 2 X → Ω 2 X (Z) → O Z (Z + K X ) → 0 and from the vanishing h 1 (Ω 2 X ) = 0 and Serre duality one has (2.4.3) This pairing reduces to a perfect pairing at the level of an arbitrary Z > 0, cf. [NN18a,7.4]. Indeed, consider the above perfect pairing ·, · : ). Since A, B = 0, the pairing factorizes to a perfect pairing H 1 (O Z ) ⊗ H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ) → C. It can be described by the very same integral form of the corresponding class representatives.
2.4.5. The linear subspace arrangement {V Z (I)} I ⊂ H 1 (O Z ) and differential forms. The arrangement {V Z (I)} I transforms into a linear subspace arrangement of H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ) via the (Laufer) non-degenerate pairing H 1 (O Z ) ⊗ H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ) → C as follows. Let Ω Z (I) be the subspace H 0 (Ω 2 X (Z| V\I ))/H 0 (Ω 2 X ) in H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ), that is, the subspace generated by those forms which have no poles along generic points of any E v , v ∈ I. 2.4.7. Furthermore, for any l ′ ∈ −S ′ \ {0} consider a divisor D ∈ ECa l ′ (Z), which is a union of We introduce a subsheaf Ω 2 X (Z) regRes D of Ω 2 X (Z) consisting of those forms ω which have the property that the residue Res Di (ω) has no poles along D i for all i. This means that the restrictions of Ω 2 X (Z) regRes D and Ω 2 X (Z) on the complement of the support of D coincide, however along D one has the following local picture. Introduce near p = E ∩ D i = E vi ∩ D i local coordinates (u, v) such that {u = 0} = E and D i has local equation v. Then a local section of Ω 2 X (Z) in this system has the form ω = k≥−rv i ,j≥0 a k,j u k v j du ∧ dv. Then, by definition, the residue Res Di (ω) is (ω/dv)| v=0 = k a k,0 u k du, hence the pole-vanishing reads as a k,0 = 0 for all k < 0. Note that Ω 2 X (Z − D) and the sheaf of regular forms Ω 2 X are subsheaves of Ω 2 . This can be regarded as a subspace of  Let us fix a point p ∈ E and a local coordinate system (u, v) around p such that E = {u = 0}, cf. 2.4.7. Fix also some ω ∈ H 0 ( X, Ω 2 X (Z)) which has pole of order o > 0 at the exceptional divisor in E containing p. We say that (the divisor of) ω has no support point at p if it can be represented locally as (ϕ(u, v)/u o )du ∧ dv with ϕ holomorphic and ϕ(0, 0) = 0. The other points are the support points denoted by supp(ω).
Lemma 2.4.10. Fix ω ∈ H 0 ( X, Ω 2 X (Z)) such that there exists a point p ∈ E v , a local divisor D 1 in X with the following properties: (a) D 1 is part of certain g produces a tangent vector in T D ECa l ′ (Z) and the action of ω on it is given by (for details see Hence if we realize a deformation g t for which the expression from (2.4.11) is non-zero, we get a contradiction. Note that g necessarily has the form cv k + n>k c n v n + uh(u, v) = cv k + h ′ for some k ≥ 1, c n ∈ C and c ∈ C * . Then set we consider a more subtle object, a filtration indexed by l ∈ L, 0 ≤ l ≤ Z as well, called the multivariable divisorial filtration of forms. Indeed, for any such l we define G l : 3. The first algorithm for the computation of dim Im(c l ′ (Z)) 3.1. We fix Z ≥ E and l ′ ∈ −S ′ as above.
From definitions and Propositions 2.2.5 and 2.4.6 (see also (2.4.9)) . Next statement provides a criterion for the validity of the equality.
Proof. Since L is a regular value, L is a smooth point of Im( c l ′ ) and T L Im( . We have to prove that T L Im( c l ′ ) = A Z (l ′ ); we prove the dual identity in the space of forms, namely, (T L Im( c l ′ ) ⊥ = Ω Z (I) (see (2.4.9)).
In this section we provide an algorithm, valid for any analytic structure, which determines d Z (l ′ ) in terms of a finite collection of invariants of type e Z (l ′ ), associated with a finite sequence of resolutions obtained via certain extra blowing ups from X.

Preparation for the algorithm. Fix some resolution
In the next construction we will consider a finite sequence of blowing ups starting from X. In order to find a bound for the number of blowing ups recall that for any Starting from each p v,kv we consider a sequence of blowing ups of length m v : first we blow up p v,kv and we create the exceptional curve F v,kv ,1 , then we blow up a generic point of F v,kv ,1 and we create F v,kv ,2 , and we do this all together m v times. We proceed in this way with all points p v,kv , hence we get v a v chains of modifications. If a v m v = 0 we do no modification along E v . A set of integers s = {s v,kv } v∈V, 1≤kv≤av with 0 ≤ s v,k ≤ m v provides an intermediate step of the tower: in the (v, k v ) tower we do exactly s v,kv blowing ups; s v,kv = 0 means that we do not blow up p v,kv at all. (In the sequel, in order to avoid aggregation of indices, we simplify k v into k.) Let us denote this modification by π s : X s → X. In X s we find the exceptional curves At each level s we set the next objects: s (Z s ) and e s := e Zz (I s ) (both considered in X s ). By similar argument as in (3.1.2) one has again d s ≤ e s for any s. From definitions, for s = 0 one has There is a natural partial ordering on the set of s-tuples. Some of the above invariants are constant with respect to s, some of them are only monotonous. E.g., by Leray spectral sequence one has h 1 (O Zs ) = h 1 (O Z ) for all s. One the other hand, because Ω Zs 1 (I s1 ) ⊂ Ω Zs 2 (I s2 ). In fact, for any ω, the pole-order along F v,k,s v,k +1 of its pullback is one less than the pole-order of ω along F v,k,s v,k . Hence, for s = m (that is, when s v,k = m v for all v and k, hence all the possible pole-orders along I m automatically vanish) one has Ω Zm ( . Hence e m = 0. In particular, necessarily d m = 0 too. More generally, for any s and (v, k) let s v,k denote that tuple which is obtained from s by increasing s v,k by one. By the above discussion if no form has pole along F v,k,s then Ω Zs ( , hence e s = e s v,k . Furthermore, by Laufer duality (or, integral presentation of the Abel map as in [NN18a,§7]), under such condition d s = d s v,k as well.
Therefore, we can redefine e s and d s for tuples s = {s v,k } v,k even for arbitrary s v,k ≥ 0: e s = e min{s,m} and d s = d min{s,m} (and these values agree with the ones which might be obtained by the first original construction applied for larger chains of blow ups).
The next theorem relates the invariants {d s } s and {e s } s .
Theorem 3.2.2. (First algorithm) With the above notations the following facts hold. ( (2) If for some fixed s the numbers {d s v,k } v,k are not the same, then In the case when all the numbers {d s v,k } v,k are the same, then if this common value d s v,k equals e s , then The proof of Theorem 3.2.2 together with the proof of Theorem 4.1.2 (the 'Second algorithm') from the next section will be given in a more general context in section 8.
3.2.3. Theorem 3.2.2 is suitable to run a decreasing induction over the entries of s in order to determine {d s } s from {e s } s . In fact we can obtain even a closed-form expression.  Proof. By Theorem 3.2.2(1) for any s ≥ s one has d s − d s ≤ | s − s|, and by (3.1.2) d s ≤ e s . These two imply d s ≤ | s − s| + e s , hence d s ≤ min s≤ s≤m {| s − s| + e s }. Next we show that d s in fact equals | s − s| + e s for some s. The wished s is the last term of the sequence {s i } t i=0 constructed as follows. Set s 0 := s. Then, assume that s i is already constructed, and that there exists (v, k) such that d si = d (si) v,k + 1. Then set s i+1 := (s i ) v,k (for one of the choices of such possible (v, k)). This inductive construction will stop after finitely many steps (since each 4. The second algorithm for the computation of dim Im(c l ′ (Z)) 4.1. Preparation. The algorithm from the previous section determines the dimensions of the Abel maps d Z (l ′ ) in terms of a finite collection of invariants of type e Z (l ′ ) associated with a finite sequence of resolutions obtained via certain extra blowing ups from X. Though, in principle, e Z (l ′ ) is much simpler than d Z (l ′ ) (it is the 'stabilizer' of d Z (l ′ )), the algorithm is still slightly cumbersome, it is more theoretical, it is not easy to apply in concrete examples: one needs to know all the integers {e s } s , that is, cf. Proposition 2.2.5, all the integers {h 1 (O Zs| Vs\Is } s associated with the tower of blowing ups. (However, it is a necessary intermediate step in the proof of the new algorithm).
The new algorithm is considerably simpler, e.g. it can be formulated in terms of the resolution X (see also the comments below). It provides d Z (l ′ ) in terms of the filtration {G l } l of 2-forms.
As a starting point, consider the construction from 3.2. For any s define the cycle l s ∈ L of X by Set G s := G ls and g s : 1.2)). In particular, However, in principle it can happen that for a certain ω with even higher pole than l s its pullback is in Ω Zs (I s ). E.g., if ω in some local coordinates (u, v) of an open set U is vdu ∧ dv/u o (and U ∩ E = {u = 0}) then its pullback via blowing up (once) at u = v = 0 has pole order o − 2. This phenomenon can happen even if we blow up a generic point: imagine a family of forms ω t with 'moving divisor', parametrized by t given by (v − t)du ∧ dv/u o . Then, even if we blow up E at a generic point u = v − t 0 = 0, in the family {ω t } t there is a form ω t0 whose pole along E v is o while its pullback has pole o − 2. Hence the equality of subspaces G s ⊂ Ω Zs (I s ), or of the equality e s = h 1 (O Z ) − g s in principle is subtle and it is hard to test. Note also that the invariant h 1 (O Z ) − g s conceptually (and technically) is much simpler than e s . E.g., it depends only on v → min kv ≤av {s v,kv }, and it can be described via a cycle of X (namely l s ) instead of the geometry of the tower X s . Nevertheless, via the next theorem, it still contains sufficient information to determine d s , in particular d Z (l ′ ). In order to emphasize the parallelism between the two algorithms we formulate them in a completely symmetric way (in particular, the first parts are completely identical). ( In the case when all the numbers {d s v,k } v,k are the same, then if this common value d For the proof see section 8. The proof runs similarly as the proof of Corollary 3.2.4. The formula (4.1.4) can be rewritten in a different flavour.
Corollary 4.1.5. For l ′ ∈ −S ′ and Z ≥ E one has The opposite inequality is also true since any such Z 1 can be represented as a certain l s with |s| = (l ′ , l s ). The converse statement is not true: take e.g. a Gorenstein elliptic singularity with length of elliptic sequence m + 1. (For elliptic singularities consult [N99,NN19a,NN19b]. For more on the Abel map of elliptic singularities see [NN19a].) Set Z ≫ 0 and −l ′ = Z min , the fundamental (minimal) cycle. Then Im(c l ′ (Z)) = 1 and h 1 (Z) = p g = m + 1. However, χ(Z min ) = min 0≤l≤Z χ(Z min + l) = 0. Therefore, if m = 1 then Im(c l ′ ) is a hypersurface, but for m ≥ 2 it is not. It is instructive to consider with the same topological data (elliptic numerically Gorenstein singularity with m ≥ 1, Z ≫ 0, −l ′ = Z min ) the generic analytic structure. Then p g = 1 (cf. [La77,NN18b]) but Im(c l ′ (Z)) is a point (this follows from part (1) too). Hence Im(c l ′ (Z)) is a hypersurface for any m ≥ 1. In particular, the property that Im(c l ′ (Z)) is a hypersurface is not a topological property.
In [NN18a,11.2] d Z (−kE * 0 ) was computed in a different way as  } 0≤kv ≤av , or, on a v . However, the final output, namely d s (and the right hand side of (4.1.4) and the algorithm itself) do depend on l ′ . We encourage the reader to work out the algorithm for an example when a v ≥ 2 (say, for −l ′ = 2E * v ).
(2) Notice that the formulas min s (|s| + h 1 (Z) − g s ) and min s (|s| + e s ) can be defined without any restriction on the numbers g s and e s , however in our case these numbers are restricted. For example we have min s≥s1 (|s| − |s , 1} for all v, k, s 1 . Or, g s ≤ |s| for all s if and only if χ(−l ′ ) < χ(−l ′ + l) for all Z ≥ l > 0 (cf. Example 4.1.7(2)).
(3) (Bounds for codim Im c l ′ (Z)) In some expression the codimension of Im(c l ′ (Z)) appears more naturally. E.g., we have the following two general statements from [NN18a, Prop. 5.6.1] (under the conditions of Corollary 4.1.5): (a) h 1 (Z, L) ≥ codim Im(c l ′ (Z)) for any L ∈ Im(c l ′ (Z)). Equality holds whenever L is generic in Im(c l ′ (Z)).
(b) codim Im c l ′ (Z) ≥ χ(−l ′ ) − min 0≤l≤Z χ(−l ′ + l), and this inequality is strict whenever c l ′ (Z) is not dominant. (This can be compared with the discussion from Example 4.1.7(3).) Note that Corollary 4.1.5 reads as: (4.1.11) codim Im(c l ′ (Z)) = max 4.1.12. Before we state the next theorem let us emphasise the obvious fact that for any 0 ≤ Z 1 ≤ Z the natural restriction (linear projection) r : However, though the restriction of r to Im(c l ′ (Z)) → Im(c l ′ (Z 1 )) is dominant, in general dim Im(c l ′ (Z)) can be smaller than dim r −1 (Im(c l ′ (Z 1 ))).

4.1.13.
It is instructive to see that certain extremal geometric phenomenons (indexed by effective cycles) are realized by the very same set of cycles.
Lemma 4.1.14. The following three sets of cycles coincide (for fixed Z ≥ E and l ′ ∈ −S ′ as above): (I) the set of cycles Z 1 with 0 ≤ Z 1 ≤ Z realizing the minimality in (4.1.6), that is: (II) the set of cycles Z 1 with 0 ≤ Z 1 ≤ Z such that (i) the map ECa l ′ (Z) → H 1 (Z 1 ) is birational onto its image, and (ii) the generic fibres of the restriction of r, r im : Im(c l ′ (Z)) → Im(c l ′ (Z 1 )), (That is, the fibers of r im have maximal possible dimension.) (III) the set of cycles Z 1 with 0 ≤ Z 1 ≤ Z such that for the generic element L im gen ∈ Im(c l ′ (Z)) and arbitrary section s ∈ H 0 (Z 1 , L im gen ) reg with divisor D (i) in the (analogue of the Mittag-Lefler sequence associated with the exact sequence 0 → O Z1 Proof. For (I)⇒(II) use the following. First recall that dim ECa l ′ (Z ′ ) = (l ′ , Z ′ ) for any effective cycle Z ′ . Next, from (4.1.6), there exists an effective cycle Z 1 ≤ Z, such that dim Im(c l ′ (Z)) = (l ′ ,  (a) There exists an effective cycle Z 1 ≤ Z, such that: (i) the map ECa l ′ (Z) → H 1 (Z 1 ) is birational onto its image, and (ii) the generic fibres of the restriction of r, r im : Im(c l ′ (Z)) → Im(c l ′ (Z 1 )), Lemma 4.1.14(II).) (b) In particular, for any such Z 1 , the space Im(c l ′ (Z)) is birationally equivalent with an affine fibration with affine fibers of dimension h 1 (O Z ) − h 1 (O Z1 ) over ECa l ′ (Z 1 ).
(c) The set of effective cycles Z 1 with property as in (a) has a unique minimal and a unique maximal element denoted by C min (Z, l ′ ) and C max (Z, l ′ ). Furthermore, C min (Z, l ′ ) coincides with the cohomology cycle of the pair (Z, L im gen ) (the unique minimal element of the set {0 ≤ Z 1 ≤ Z : h 1 (Z, L im gen ) = h 1 (Z 1 , L im gen )) for the generic L im gen ∈ Im(c l ′ (Z)).
(c) Assume that two cycles Z 1 and Z 2 satisfy (a). We claim that Z ′ := max{Z 1 , Z 2 } satisfies too. First, for any cycle Z ′′ with Z 1 ≤ Z ′′ ≤ Z, if Z 1 satisfies (a)(ii) then Z ′′ satisfies too. This applies for Z ′ too. To prove (a)(i) for Z ′ , let us denote by ECa l ′ (Z ′′ ) 0 ⊂ ECa l ′ (Z ′′ ) the set of divisors whose support is disjoint from the singular points of E.
In order to prove the existence of C min (Z, l ′ ), first we claim that the set of cycles Z ii , which satisfy (a)(ii) has a unique minimal element Z ii min . This fact via Remark 4.1.10(3)(a) is equivalent with the existence of the (unique) cohomological cycle for the pair (Z, L im gen ). This was proved in [NN18a, 5.5], see also [Re97,4.8]. Next, we claim that the map ECa l ′ (Z ii min ) → H 1 (Z ii min ) is BioIm as well. From the existence of the cycle C max (·, l ′ ) (already proved above), applied for Z ii min , there exists a cycle C max (Z ii min , l ′ ) ≤ Z ii min , which satisfies (a). In particular, (a)(ii) is valid for the pair C max (Z ii min , l ′ ) ≤ Z ii min . By the definition of Z ii min the condition (a)(ii) is valid for the pair Z ii min ≤ Z too. Hence, (a)(ii) is valid for the pair C max (Z ii min , l ′ ) ≤ Z as well. Therefore, by the definition of Z ii min necessarily C max (Z ii min , l ′ ) = Z ii min , hence Z ii min satisfies (a).

5.
Example. The case of generic analytic structure 5.1. Let us fix the topological type of a good resolution of a normal surface singularity, and we assume that the analytic type on X is generic (in the sense of [NN18b], see [La73] as well). Recall that in such a situation, if Z ′ = n v E v is a non-zero effective cycle, whose support |Z ′ | = ∪ nv =0 E v is connected, then by [NN18b, Corollary 6.1.7] one has Corollary 5.1.1. Assume that X has a generic analytic type, Z ≥ E an integral cycle and l ′ ∈ −S ′ .
Let us concentrate again on the codimension h 1 (O Z ) − d Z (l ′ ) of Im(c l ′ (Z)) ⊂ Pic l ′ (Z) instead of the dimension. Then, (5.1.2) reads as This is a rather complicated combinatorial expression in terms of the intersection lattice L. The next lemma aims to simplify it.
Proposition 5.1.4. Consider the assumptions of Corollary 5.1.1. Let Z 1 be minimal such that the maximum in (5.1.3) is realized for it. Then min E |Z 1 | ≤l≤Z1 {χ(l)} = χ(Z 1 ). In particular, The maximum at the right hand side is realized e.g. for the cohomology cycle of L im gen ∈ Im(c l ′ (Z)) ⊂ Pic l ′ (Z). Furthermore, for any L ∈ Im(c l ′ (Z)) and equality holds for generic L im gen ∈ Im(c l ′ (Z)).

Proof. Assume that the minimum min
Since the maximality in (5.1.3) is realized by Z 1 , which is minimal with this property, necessarily Z 1 = l 1 . Next, But the maximum at the left hand side is realized by a term from the right. For the last statement use again Remark 4.1.10(3)(a).
5.2. The identity (5.1.5), valid for a generic analytic structure of X, extends to an optimal inequality valid for any analytic structure.
Theorem 5.2.1. Consider an arbitrary normal surface singularity (X, o), its resolution X, Z ≥ E and l ′ ∈ −S ′ . Then codim Im(c l ′ (Z)) = h 1 (Z, L im gen ) (cf. Remark 4.1.10(3)(a)) satisfies In particular, for any L ∈ Im(c l ′ (Z)) one also has (everything computed in X) Note that the right hand side of (5.2.2) is a sharp topological lower bound for codim Im(c l ′ (Z)). The inequality (5.2.2) can also be interpreted as the semi-continuity statement codim Im(c l ′ (Z))(arbitrary analytic structure) ≥ codim Im(c l ′ (Z))(generic analytic structure).
Proof. Consider the identity (4.1.11) applied for an arbitrary X and for the generic X, denoted by X gen . Then, by semi-continuity of h 1 (O Z1 ) with respect to the analytic structure as parameter space (see e.g. [NN18b, 3.6]), for any fixed effective cycle Z 1 > 0, h 1 (O Z1 ) computed in X is greater than or equal to h 1 (O Z1 ) computed in X gen . Therefore, by (4.1.11) one has codim Im(c l ′ (Z))(in X) ≥ codim Im(c l ′ (Z))(in X gen ). Then for X gen apply (5.1.5).
Remark 5.2.4. Certain upper bounds for {h 1 (Z, L)} L∈Pic l ′ (Z) , valid for any analytic structure, were established in [NN18a,Prop. 5.7.1] (see alo Remark 5.3.3). However, an optimal upper bound is not known (see [NO17] for a particular case). Large h 1 -values are realized by special strata, whose existence and study is extremely hard. 5.3. The cohomology of L im gen (l). Assume that Z ≥ E, l ′ ∈ −S ′ and let L im gen be a generic element of Im(c l ′ (Z)). If the analytic structure of (X, o) is generic, then by Proposition 5.1.4 h 1 (Z, L im gen ) = t Z (l ′ ), where t Z (l ′ ) is the topological expression from the right hand side of (5.1.5). Our goal is to give a topological lower bound for h 1 (Z, L), where L := L im gen (l) = L im gen ⊗ O(l) ∈ Pic l ′ +l (Z) whenever l ∈ L >0 . In this way we will control the generic element of the 'new' strata O(l) ⊗ (Im(c l ′ (Z))) of Pic l ′ +l (Z), unreachable directly by the previous result. Our hidden goal is to construct in this way line bundles with 'high' h 1 .
For simplicity we will assume that all the coefficients of Z are sufficiently large (even compared with l, hence the coefficients of Z − l are large as well). The monomorphism of sheaves L im gen | Z−l ֒→ L im gen (l) gives h 0 (Z − l, L im gen ) ≤ h 0 (Z, L im gen (l)), hence By a computation regarding χ this transforms into E.g., with the choice l = −l ′ ∈ S ′ ∩ L >0 we get that L im gen (−l ′ ) ∈ Pic 0 (Z) and Remark 5.3.3. By [NN18a, Prop. 5.7.1] for Z ≫ 0, L ∈ Pic(Z) with c 1 (L) ∈ −S ′ one has h 1 (Z, L) ≤ p g whenever either H 0 (Z, L) = 0 or L ∈ Im(c l ′ (Z)). For other line bundles a weaker bound is established (see [loc. cit.]), which does not guarantee h 1 (L) ≤ p g . However, it is not so easy to find singularities and bundles with h 1 (L) > p g in order to show that such cases indeed might appear. In the next 5.3.4 we provide such an examples (with a recipe to find many others as well) based partly on (5.3.2).
Example 5.3.4. Assume that we can construct a nonrational resolution graph which satisfies the following (combinatorial) properties, valid for certain Z ≫ 0 and l ′ ∈ −S ′ ∩ L:  The wished graph Γ consists of Γ 1 , Γ 2 and a new vertex v, which has two adjacent edges connecting v to the (−13)-vertices of Γ 1 and Γ 2 . Let the decoration of [NN18b,5.7]) we get that −l ′ = Z min ≤ max M. One verifies that χ(Z min ) = −3 (e.g. by Laufer's criterion), and also that min χ = −5 (realized e.g. for 2Z min − E v ). Therefore χ(−l ′ ) − min l≥0 χ(−l ′ + l) + 2 = −3 + 5 + 2 = 4. On the other hand, the expression (under max) in (5.1.5) for Z 1 = Z min (Γ 1 ) + Z min (Γ 2 ) supported on Γ \ v is 4, hence t Z (l ′ ) ≥ 4. 6. Appendix. Geometrical aspects behind the lower bound Theorem 5.2.1 6.1. Let us discuss with more details the geometry behind the inequality (5.2.2). Along the discussion we will provide a second independent proof of it and we also provide several examples, which show its sharpness/weakness in several situations. Similar construction (with similar philosophy) will appear in forthcoming manuscripts on the subject as well. The construction of the present section shows also in a conceptual way how one can produce different sharp lower bounds for sheaf cohomologies (for another case see e.g. subsection 7.2).
We provide the new proof in several steps. First, we define a topological lower bound for codim Im(c l ′ (Z)), which (a priori) will have a more elaborated form then the right hand side t Z (l ′ ) of (5.2.2). Then via several steps we will simplify it and we show that in fact it is exactly t Z (l ′ ). By [NN18a,Prop. 5.6.1], see also 4.1.10(3), for any Z ≥ E and for any l ′ ∈ −S ′ , if L im gen is a generic element of Im(c l ′ (Z)), then h 1 (Z, L im gen ) = codim Im(c l ′ (Z)) satisfies (the semicontinuity) Remark 6.1.5. Assume that Z > 0 is a nonzero cycle with connected support |Z|, but with Z ≥ E. Then the statements from (6.1.4) remain valid for such Z once we replace l ′ by its restriction R(l ′ ), where R : L ′ → L ′ (|Z|) is the natural cohomological operator dual to the natural homological inclusion L(|Z|) ֒→ L. (For this apply the statement for the singularity supported on |Z|.) On the other hand, for l ∈ L(|Z|) one has Hence, in fact, (6.1.4) remains valid in its original form for any such Z > 0 with |Z| connected.
Example 6.1.6. The difference h 1 (Z, L im gen ) − h 1 (Z, L gen ) can be arbitrary large. Indeed, let us start with a singularity with an arbitrary analytic structure, we fix a resolution X with dual graph Γ, and we distinguish a vertex, say v 0 , associated with the irreducible divisor E 0 . Let k (k > 0) be the number of connected components of Γ \ v 0 , and we assume that each of them is non-rational.
6.1.7. We wish to estimate h 1 (Z, L im gen ). Note that the estimate given by (6.1.4), that is, h 1 (Z, L im gen ) ≥ T (Z, l ′ ), sometimes is week, see the previous example. However, surprisingly, if we replace Z by a smaller cycle Z ′ ≤ Z, then we might get a better bound. More precisely, first note that if L im gen is a generic element of Im(c l ′ (Z)), and 0 < Z ′ ≤ Z, then its restriction r(L im gen ) (via r : Pic l ′ (Z) → Pic R(l ′ ) (Z ′ )) is a generic element of Im(c l ′ (Z ′ )). If Z ′ has more connected components, we can apply (6.1.4). Therefore, we get Define (6.1.9) t(Z, l ′ ) := max (Here there is no need to restrict l ′ , cf. Remark 6.1.5.) Hence (6.1.8) reads as (6.1.10) h 1 (Z, L im gen ) ≥ t(Z, l ′ ).
Example 6.1.11. (Continuation of Examle 6.1.6) The last computation of Example 6.1.6 shows that the maximum of χ(nE * 0 ) − min l≥0 χ(nE * 0 + l) is obtained for l 0 = 0 and T (Z, l ′ ) = 1 + i (− min χ(Γ i )). Hence, taking Z ′ = i Z ′ i , each Z ′ i supported on Γ i and large, we get that the restriction of l ′ is zero and i T (Z ′ i , l ′ ) = i (1 − min χ(Γ i )) = T (Z, l ′ ) + k − 1. Summarized (also from Example 6.1.6), for any analytic type one has i p g, . However, if X is generic then p g,i = 1 − min χ(Γ i ) (cf. [NN18b]), hence, all the inequalities transform into equalities. Hence, for generic analytic structure h 1 (Z, L im gen ) = t(Z, l ′ ), that is, (6.1.10) provides the optimal sharp topological lower bound. Note also that both t(Z, l ′ ) and i (1 − min χ(Γ i )) are topological, hence if they agree for X generic, then they are in fact equal. Since p g,i − 1 + min χ(Γ i ) for arbitrary analytic type can be considerably large, for arbitrary analytic types the inequality (6.1.10) can be rather week.
6.2. Our goal is to simplify the expression (6.1.9) of t(Z, l ′ ).
First we analyse the set of cycles Z ′ for which the maximum in the right hand side of (6.1.9) can be realized. E.g., if c l ′ (Z) is dominant (equivalently, t(Z, l ′ ) = 0, cf. 2.3) then any 0 ≤ Z ′ ≤ Z realizes the maximum 0 (with all l i = 0). (Indeed, use the fact that D(Z 2 , l ′ ) ≥ D(Z 1 , l ′ ) for Z 2 ≥ Z 1 and |Z i | connected.) In the next Lemmas 6.2.1 and 6.2.4 we will assume that c l ′ (Z) is not dominant.
Lemma 6.2.1. (a) Assume that Z ′ is a minimal cycle (or a cycle with minimal number of connected components) among those cycles which realize the maximum in the right hand side of (6.1.9). Then D(Z ′ i , l ′ ) = 1 for all i. Proof. (a) Otherwise, c l ′ (Z ′ i ) is dominant, and by 2.3 χ(−l ′ ) − min 0≤li≤Z ′ i χ(−l ′ + l i ) = 0 (realized for l i = 0). Hence T (Z ′ i , l ′ ) = 0, that is, the right hand side of (6.1.9) is realized by Z ′ − Z ′ i too, contradicting the minimality of Z ′ . (b) If the wished minimum is realized by l i = 0, and only by l i = 0, then by 2.3 c l ′ (Z ′ i ) is dominant, contradicting D(Z ′ i , l ′ ) = 1.
Example 6.2.2. Though in Example 6.1.6 we have shown that h 1 (Z, L im gen ) = t(Z, l ′ ) can be much larger than T (Z, l ′ ) (that is, the maximizing Z ′ usually should be necessarily strict smaller than Z), in some cases Z ′ = Z still works. Indeed, we claim that if the E * -support I of l ′ is included in the set of end vertices of Γ, then t(Z, l ′ ) = T (Z, l ′ ).
On the other hand, if Z ′ is connected, then T (Z ′ , l ′ ) ≤ T (Z, l ′ ), hence the maximal value in the right hand side of (6.1.10) is realized for Z as well (and maybe by several other smaller cycles too; here we minimalized #|Z ′ | by increasing Z ′ ).
The present example together with Examples 6.1.6 and 6.1.11 show that the structure of possible cycles Z ′ for which the maximality in (6.1.9) realizes can be rather subtle.
Lemma 6.2.4. Assume that Z ′ is a minimal cycle among those cycle which realizes the maximum in the right hand side of (6.1.9). Then the following facts hold: hence by the criterion from 2.3 the Abel map c l ′ (l i,k ) must be non-dominant. Thus (using also D(Z ′ i , l ′ ) = 1 from Lemma 6.2.1(a)) (6.2.5) In particular, by the minimality of Part (c) follows from (6.1.9) and (a).
Proof. If c l ′ (Z) is dominant then both sides are zero. Otherwise, by Lemma 6.2.4(c) (with its On the other hand, let us fix some Z ′ = ∪ i Z ′ i for which the maximum in t Z (l ′ ) is realized. Then we can assume that each Remark 6.2.7. The second proof of Theorem 5.2.1 follows from (6.1.10) and Corolary 6.2.6. 7. The L 0 -projected Abel map In this section we introduce a new object, a modification of the Picard group Pic(Z), which will play a key role in the cohomology computation of the shifted line bundles of type {L 0 ⊗L} L∈Im(c l ′ (Z)) . 7.1. The L 0 -projected Picard group. Let (X, o) be a normal surface singularity. For simplicity we assume (as always in this manuscript) that the link is a rational homology sphere. Let X be one of its good resolutions and Z ≥ E an effective cycle. Fix also L 0 ∈ Pic(Z) such that H 0 (Z, L 0 ) reg = ∅ (cf. 2.2). Choose s 0 ∈ H 0 (Z, L 0 ) reg arbitrarily, and write div( where the second morphism is the multiplication by (restrictions of) s 0 . Then we have the following commutative diagram of sheaves: . Therefore, if L = L im gen is a generic element of Im(c l ′ (Z)) then codim Im(c l ′ (Z)) = dim H 1 (O Z )/Im(δ 0 L ) = h 1 (Z, L). Similarly, consider the composition We call it the L 0 -projection of the Abel map c l ′ (Z). Using the previous paragraph we obtain that the tangent linear map . Therefore, if L is a generic element of Im(c l ′ L0 (Z)) (or, it is the image by s L0 of a generic element L im gen of Im(c l ′ (Z))) then This fact fully motivates the next point of view: if one wishes to study h 1 (Z, L 0 ⊗ L) with L 0 fixed and L ∈ Pic l ′ (Z) then -as a tool -the right Abel map is the L 0 -projected c l ′ L0 (Z).
7.2. The cohomology h 1 (Z, L 0 ⊗L). Using the exact sequence Usually it is not sharp, since δ 0 L might not be injective. However, as in the prototype construction from section 6 (and even in its preceding sections), if we consider any Z 1 ≤ Z then we also have h 1 (Z, , and, remarkably, this for the generic L im gen ∈ Im(c l ′ (Z)) is an equality (cf. (4.1.11)). Similarly, using the exact sequence Z). Again, this usually is not sharp. However, by the same procedure, In the next section (cf. Corollary 8.2.4) we will prove that this is again an equality for the generic L = L im gen ∈ Im(c l ′ L0 (Z)). (The above inequality (7.2.1) can be compared with (5.3.1) as well.) 7.3. Compatibility with Laufer duality and differential forms. Consider the perfect pairing , : H 1 (O Z ) ⊗ H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ) → C from 2.4.2, see alo [NN18a]. Once we fix D 0 = div(s 0 ) of certain s 0 ∈ H 0 (Z, L 0 ) reg , we can define Ω Z (D 0 ) := (Im(δ 0 L0 )) ⊥ ⊂ H 0 (Ω 2 X (Z))/H 0 (Ω 2 X ). It is generated by forms which vanish on the image of the tangent map T D0 c l ′ 0 (Z), identified with δ 0 L0 , cf. 2.4.7 and (2.4.9). The pairing , induces a perfect pairing , L0 : H 1 (Z, L 0 ) ⊗ Ω Z (D 0 ) → C, see also Theorem 2.4.8.
In the next section we provide two algorithms for the computation of d L0,Z (l ′ ), the analogues of the algorithms from Theorems 3.2.2 and 4.1.2.
8. L 0 -projected versions of the algorithms 8.1. The setup. Let us fix (X, o), a good resolution X, Z ≥ E and l ′ ∈ −S ′ . We also fix a line bundle L 0 as in section 7, whose notations we will adopt. In order to estimate d L0,Z (l ′ ) we proceed as in sections 3 and 4. In particular, we perform the modificatiosn π s : X s → X, and we adopt the notations of 3.2 as well. By the generic choice of the centers of blow ups we can assume that they differ from the support of D 0 . Notice that we have a natural identification between H 1 (O Z ) and H 1 (O Zs ), and also between H 1 (O * Z ) and H 1 (O * Zs ). Furthermore, we denote the divisor π −1 s (D 0 ) on X s still by Proof. (1) Assume first that either s v,k ≥ 1 or a v = 1. Then divisors from ECa l ′ s (Z s ) intersect F v,k,s v,k by multiplicity one, hence the intersection (supporting) point gives a map q : ECa l ′ s (Z s ) → F v,k,s v,k , which is dominant. Moreover, ECa l ′ s v,k (Z s v,k ) is birational with a generic fiber of q (the fiber over the point which was blown up), hence the first statement follows. Note also that d L0,s = d L0,s v,k if and only if the generic fiber of the L 0 -projected Abel map c l ′ s L0 is not included in a q-fiber. This implies the second part of (1).
If s v,k = 0 and a v > 1 then write l ′ − := l ′ s − E * v and consider the 'addition map' s : Next, assume that the numbers {d L0,s v,k } are the same, say d. Let 1 ≤ i ≤ a v be an integer such that s v,i = (l s ) v (abridged in the sequel by t) and we denote the order of vanishing of ω on an arbitrary exceptional divisor E u by b u , where u is an arbitrary vertex along the blowing up procedure. Next we focus on the string between v and w v,i,sv,i and we denote them by v 0 = v, . . . , v t = w v,i,sv,i . Set r := min{0 ≤ s ≤ t : b vs + t − s ≥ 0}. Since for s = t one has b vt ≥ 0 (since ω has no pole along F v,i,sv,i ) r is well-defined. On the other hand we have r ≥ 1. Indeed, b v0 + t < 0, since pole order of ω along E v is higher than (l s ) v = t. Note that b vr−1 + t − r + 1 < 0 and b vr + t − r ≥ 0 imply b vr − b vr−1 ≥ 2 ( †).
Let X ′ be that resolution obtained from X, as an intermediate step of the tower between X and X s , when in the (v, i) sequence of blow ups we do not proceed all s v,i of them, but we create only the divisors {F v,i,k } k≤r−1 . Let V ′ be its vertex set and {E u } u∈V ′ its exceptional divisors. On X ′ consider the line bundle L := Ω 2 X ′ (− u∈V ′ b u E u ). Since F v,i,vr was created by blowing up a generic point p of E vr−1 = F v,i,vr−1 , the existence of ω guarantees the existence of a section s ∈ H 0 ( X ′ , L), which does not vanish along E vr−1 and it has multiplicity m := b vr − b vr−1 − 1 at the generic point p ∈ E vr1 . By ( †) m ≥ 1. By construction, ω (or s) belongs also to the subvectorspace Ω X (D 0 ) after certain identifications. Now by the technical Lemma 9.1.1 (valid for general line bundles, and separated in section 9) for any 0 ≤ k < m and a generic point p ∈ E vr−1 there exists a section s ′ ∈ H 0 ( X ′ , L), which does not vanish along the exceptional divisor E vr−1 , and the divisor of s ′ has multiplicity k at p. We apply for k = −(b vr−1 + t − r + 1) − 1. (Note that 0 ≤ k < m.) The section s ′ gives a differential form ω ′ ∈ Ω X (D 0 ), such that if we blow up E vr−1 in the generic point p and we denote the new exceptional divisor by E vr,new , then ω ′ has wanishing order −(t − r + 1) on E vr,new . This means, that if we blow up it in generic points t − r + 1 times, then ω ′ has a pole on E vt,new , but has no pole on E vt+1,new . This means that e L0,s v,i = e L0,s , which is a contradiction. This combined with (7.1.1) gives for a generic L im gen ∈ Im(c l ′ (Z)): h 1 (Z, L 0 ⊗ L im gen ) = max 0≤Z1≤Z { h 1 (Z 1 , L 0 ) − (l ′ , Z 1 )}.
9. Appendix 2. A technical lemma 9.1. The next lemma is used in the body of the article, however, it might have also an independent general interest.
Lemma 9.1.1. Let X be an arbitrary resolution of a normal surface singularity (X, 0). Let us fix an arbitrary line bundle L ∈ Pic( X) with c 1 (L) = l ′ ∈ −S ′ , an irreducible exceptional curve E v , and an integer m > 0.
Assume that there exists a sub-vectorspace V ⊂ H 0 ( X, L) with the following property: for a generic point p ∈ E v there exists a section s ∈ V such that s does not vanish along E v and the multiplicity of the divisor of s at p ∈ E v is m. Then for any number 0 ≤ k ≤ m and a generic point p ∈ E v there exists a section s ∈ V such that s does not vanish along E v and the multiplicity of the divisor of s at p ∈ E v is k.
Proof. By induction we need to prove the statement only for k = m − 1.
First we fix a very large integer N ≫ m, and consider the restriction r : H 0 ( X, L) → H 0 (N E v , L). Then r induces a map from H 0 ( X, L) reg := H 0 ( X, L) \ H 0 ( X, L(−E v )) to H 0 (N E v , L) reg := H 0 (N E v , L) \ H 0 ((N − 1)E v , L(−E v )). Denote its restriction H 0 ( X, L) reg ∩ V → H 0 (N E v , L) reg ∩ r(V ) by r V . Consider also the natural map div : H 0 (N E v , L) reg → ECa l ′ (N E v ), and the composition map div • r V = g : H 0 ( X, L) reg ∩ V → ECa l ′ (N E v ), which sends a section to its divisor restricted to the cycle N E v .
Next, for any p ∈ E 0 v := E v \∪ u =v E u set D m,p ⊂ ECa l ′ (N E v ), the set of divisors with multiplicity m at p. (Since N ≫ m this notion is well-defined). Set also D m := ∪ p D m,p .
By the assumption, the image of g intersects D m,p for any generic p. Since D m is constructible subvariety of ECa l ′ (N E v ), g −1 (D m ) is a nonempty constructible subset of H 0 ( X, L) reg ∩ V . Define an analytic curve h 0 : (−ǫ, ǫ) → g −1 (D m ) such that its image is not a subset of some g −1 (D m,p ). Let us denote the zeros of the section h 0 (0) along E 0 v by {p 1 , . . . , p r }. Then there exists a small neighborhood U of one of the points p i and a restriction of h 0 to some smaller (−ǫ ′ , ǫ ′ ), such that for any t ∈ (−ǫ ′ , ǫ ′ ) the restriction of h 0 (t) to U has a unique zero, say p(t), and its multiplicity is m. Furthermore, t → p(t), (−ǫ ′ , ǫ ′ ) → U ∩ E 0 v is not constant, hence taking further restrictions to some interval we can assume that t → p(t) is locally invertible. Reparametrising h 0 by the inverse of this map, we obtain an analytic map U ∩ E 0 v → g −1 (D m ), t → h(t) such that the restriction of the section h(t) to some local chart U has only one zero, namely t, and the multiplicity of the section at t is m. In some local coordinates (x, y) of U (with U ∩ E v = {y = 0}) the equation of h(t) has the form (modulo y N ) where by the multiplicity condition c j,i ≡ 0, if j +i < m and, there is a pair (j, i), such that j +i = m and c j,i (t) ≡ 0. Moreover, by the non-vanishing condition y |h(t), or, c j,0 (t) ≡ 0 for some j. We claim that there is a generic choice of t 1 , . . . , t r (for some large r) of t-values, and a convenient choice of the coefficients {α l } r l=1 such that s := r l=1 α l h(t l ) satisfies the requirements. Indeed, first