The Abel map for surface singularities III: Elliptic germs

The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by ‘old’ techniques were not reachable. If (X~,E)→(X,o)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\widetilde{X}},E)\rightarrow (X,o)$$\end{document} is the resolution of a complex normal surface singularity and c1:Pic(X~)→H2(X~,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1:{\mathrm{Pic}}({\widetilde{X}})\rightarrow H^2({\widetilde{X}},{\mathbb {Z}})$$\end{document} is the Chern class map, then Picl′(X~):=c1-1(l′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{Pic}}^{l'}({\widetilde{X}}):= c_1^{-1}(l')$$\end{document} has a (Brill–Noether type) stratification Wl′,k:={L∈Picl′(X~):h1(L)=k}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{l', k}:= \{{\mathcal {L}}\in {\mathrm{Pic}}^{l'}({\widetilde{X}})\,:\, h^1({\mathcal {L}})=k\}$$\end{document}. In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any W(l′,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(l',k)$$\end{document} is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.


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Recall that the classical Brill-Noether problem for curves is the following. Let C be a smooth complex projective curve and let c 1 : Pic(C) → H 2 (C, Z) = Z be the first Chern class map. Set Pic l (C) := c −1 1 (l). Then one considers the stratification of Pic l (C) according to the h 1values, namely, W l,k := {L ∈ Pic l (C) : h 1 (L) = k}. The problem is to determine the values (l, k) for which W l,k is non-empty and in such cases to describe the different non-empty strata W l,k . (This depends heavily on the analytic structure of C.) For details see e.g. [1,7].

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For complex normal surface singularities the analogue question can be formulated as follows. Let (X , o) be such a singularity and let us fix a resolution φ : ( X , E) → (X , o). We will assume that the link is a rational homology sphere (RHS), equivalently, that the dual resolution graph is a tree of P 1 's. (In some parts of the presentation we will drop this assumption, however in the final key statements it is necessary, see 2.2.13 and the paragraph after Theorem 3.3.5 for further comments.) We consider the lattice L := H 2 ( X , Z), it is freely generated by the irreducible exceptional divisors and it is endowed with the negative definite intersection form.
Then one has the exponential exact sequence Here L might serve also as the dual lattice of L. Then for any possible Chern class l ∈ L set Pic l ( X ) := c −1 1 (l ). (Note that while Pic l (C) for a smooth curve is a compact complex torus, in the surface singularity case Pic l ( X ) is an affine space C p g , where p g is the geometric genus of (X , o).) Following [14,15] we consider the stratification W l ,k := {L ∈ Pic l ( X ) : h 1 (L) = k}. Again, the goal is to describe the spaces W l ,k . In general, they depend in a rather arithmetical way on the combinatorics of the resolution graph and also on the analytic structure of (X , o) supported on the topological type determined by . Usually the spaces W l ,k are neither open, nor closed, not even quasi-projective. Their closure might be nonlinear, or even singular. Though in the theory of singularities several results are known for the possible h 1 (L)-values (vanishing theorems, coarse topological bounds), a systematic study of the spaces W l ,k was missing. In the series of articles (starting with [14,15] and the present one) the authors aim to fill in this necessity.
In order to understand (and detect) general and peculiar properties of the W l ,k -stratification it is highly desirable to describe it completely for certain key non-trivial families of singularities. In this note this task is fulfilled for elliptic germs.

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The main tool in the study of the W -stratification (similarly as in the curve case) is the Abel map c l (Z ) : ECa l (Z ) → Pic l (Z ), where Z ∈ L is a nonzero effective cycle, and ECa l (Z ) is the space of effective Cartier divisors on Z , cf. [8][9][10]. (We emphasize again some major differences compared with the curve case: ECa l (Z ) is not compact, and c l is not proper, a fact which creates extra difficulties in the study of the fibers.) If Z 0 then Pic l (Z ) = Pic l ( X ), and the W -stratification can be analysed via the fiber structure of the Abel map c l . Besides the general theory, some concrete families of singularities were already analysed, e.g. superisolated and weighted homogeneous ones (partially) in [14], the generic analytic structure (as an extreme bound case of the theory) in [15]; see also [16]. In this manuscript we provide a complete description for elliptic singularities (with RHS link). Surprisingly, the new results and the needed developed machinery regarding the Abel map reshapes the 'classical' theory of elliptic singularities as well.
For the theory of elliptic singularities the reader might consult [13,18,28,33,[35][36][37][38][39]. The main technical machinery, which guides most of the properties of an elliptic singularity is the elliptic sequence defined by Laufer and Yau. In the numerical Gorenstein case, we will use this sequence; however, for the non-numerically Gorenstein case we introduce a new sequence, which mimics the numerical Gorenstein case better and it is more relevant for our purposes. (For the comparison of the old and new sequences see [17]).
The members of the elliptic sequences, and the Artin fundamental cycles and the canonical cycles on different supports satisfy several key compatibility properties; these relations are formulated (elegantly) in the minimal resolution. In any other resolution, they became uneasy and unpleasant. On the other hand, in [14] we developed several properties of the space of effective Cartier divisors and of the Abel map in the context of a good resolution: in several local verifications the normal crossing property of the exceptional curves was used. Therefore, strictly speaking, in this note we analyse only those elliptic singularities (with rational homology sphere links) whose minimal resolution is good. The interested reader is invited to extend the results for the remaining few cases (when the minimal resolution is not good, see e.g. [13] for their list in the minimally elliptic case, and also a model how one extends statements valid for the other general cases to these special germs). Hence, in the sequel, 'elliptic' means elliptic with these restrictions.

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In the body of the paper we prove that for elliptic singularities the Abel map has several very pleasant properties (see Theorem 6.1.1 below), which are not valid for arbitrary singularities.

Theorem
(a) The closure im(c l (Z )) of the image of the Abel map c l (Z ) is an affine subspace of Pic l (Z ). (b) h 1 (L) is uniform on im(c l (Z )) (whose value is computable).
This solves the Brill-Noether problem on the image im(c l (Z )) ⊂ Pic l (Z ). However, usually im(c l (Z )) = Pic l (Z ). Recall that the image of c l consists of line bundles without fixed components. Hence, to complete the picture, we need to analyse the possible cycles of fixed components, and twisting a certain bundle with the cycle of its fixed components (hence creating bundles without fixed components) we reduce the Brill-Noether problem to the study of several Abel map images.
The facts that the elliptic sequence imposes some 'total-ordered structure' of invariants, and that the closure of the Abel maps are affine subspaces, are inherited by the W -stratification as well: Theorem (Cf. Sect. 7) For elliptic Gorenstein singularities the W -stratification is organized in a flag of linear subspaces reflecting completely the concatenated structure of the elliptic sequence. Moreover, the corresponding dimensions can also be determined. ( The W -stratification of the Picard groups according to the h 1 -values (for different levels of generality) is completely described in Sect. 7. In Sect. 8 we provide even a finer stratification according to the cycles of fixed components.
of ECC and WECC elliptic singularities. We present two topological and two analytical characterizations of germs satisfying WECC.
The (anti)canonical cycle Z K ∈ L is defined by the adjunction formulae ( where g v and δ v denote the genus of E v and the sum of the delta invariants of the singularities of E v . (It is the first Chern class of the dual of the line bundle 2 X .) We write χ : L → Q for the (Riemann-Roch) expression χ(l ) := −(l , l − Z K )/2.
The singularity (or, its topological type) is called numerically Gorenstein if Z K ∈ L. (Since Z K ∈ L if and only if the line bundle 2 X \{o} of holomorphic 2-forms on X \{o} is topologically trivial, see e.g. [6], the Z K ∈ L property is independent of the resolution). (X , o) is called Gorenstein if Z K ∈ L and 2 X (the sheaf of holomorphic 2-forms) is isomorphic to O X (−Z K ) (or, equivalently, if the line bundle 2 X \{o} is holomorphically trivial). If X is a minimal resolution then (by the adjunction formulae) Z K ∈ S . In particular, L) for any L ∈ Pic( X ) with c 1 (L) ∈ −S . Proof By generalized Kodaira or Grauert-Riemenschneider type vanishing h 1 ( X , O X (− Z K )) = 0. Hence, if Z K = 0 then p g = 0. Otherwise, using the exact sequence 0 More generally, h 1 ( X , L) = h 1 (Z K − z i , L) for any i by similar argument.

Natural line bundles
Let φ : ( X , E) → (X , o) be as above. In the sequel we assume that the link is a rational homology sphere (required in the next construction and in 2.2). Consider the 'exponential' cohomology exact sequence (with H 1 ( X , O * X ) = Pic( 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 ). Then (based on the fact that the link is a rational homology sphere, hence Pic 0 ( X ) is torsion free), see e.g. [21,27], there exists a unique homomorphism (split) s 1 : L → Pic( X ) of c 1 such that c 1 • s 1 = id and 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 equivalent definitions of them see [21]. E.g., L is natural if and only if one of its powers 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 .

The Abel map [14]
As above, let Pic( X ) = H 1 ( X , O * X ) be the group of isomorphism classes of holomorphic line bundles on X . The first Chern class map c 1 : Similarly, if Z ∈ L >0 is an effective non-zero integral cycle supported by E, then Pic(Z ) = H 1 (Z , O * Z ) denotes the group of isomorphism classes of invertible sheaves on Z . Again, it appears in the exact sequence 0 → Pic 0 (Z ) → Pic(Z ) Here L(|Z |) denotes the sublattice of L generated by the base element E v ⊂ |Z |, and L (|Z |) is its dual.
Though for any effective cycle Z the Abel map might have its own peculiar properties, in this manuscript we always assume that all the E v -coefficients of Z are sufficiently large, denoted by Z 0. Under this assumption one has several stability properties, e.g.
For any Z 0 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 . By this definition (see (3.1.5) of [14]) ECa l (Z ) = ∅ if and only if −l ∈ S \{0}. 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 : Hence, the previous equivalence extends to this l = 0 case too and one has the uniform statement Sometimes (e.g. in Sect. 9) even for L ∈ Pic l ( X ) we write L ∈ im(c l ) whenever L| Z ∈ im(c l (Z )) for some Z 0. This is equivalent with the fact that L ∈ Pic( X ) has no fixed components.
It turns out that ECa l (Z ) 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 [14,15].

The modified Abel map
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 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

The monoid (or multiplicative) structure of divisors and of the modified Abel map
Consider the natural additive structure s l 1 ,l 2 (Z ) : ECa l 1 (Z ) × ECa l 2 (Z ) → ECa l 1 +l 2 (Z ) (l 1 , l 2 ∈ −S ) provided by the sum of the divisors. One verifies (see e.g. [14, Lemma 6.1.1]) that s l 1 ,l 2 (Z ) is dominant and quasi-finite. There is a parallel multiplication Pic . (e) For n 1 one has im( c nl ) = A(nl ), and h 1 (Z , L) = p g (X V\I (l ) , o V\I (l ) ) for any L ∈ im(c nl ).
For different other geometric reinterpretations of dim V Z (I ) see also [14,Sect. 9].

The linear subspace arrangement {V Z (I)} I ⊂ C p g and differential forms
The arrangement {V (I )} I transforms into a linear subspace arrangement of [14, 7.3]) as follows. Let (I ) be the subspace H 0 ( 2 (that is, the subspace generated by those forms which have no poles along generic points of any E v , v ∈ I ). Proposition 2.2.9 [14, 8.3] Via Laufer duality V (I ) = (I ) * .

The dim im(c l ) and differential forms
Next we recall a statement from [14,Sect. 10]. For simplicity we will assume that l = −E * v for some v ∈ V. This means that any divisor D ∈ ECa l ( X ) with Chern class l is a transversal cut (disc) of E v at a certain point p ∈ E v \ ∪ u =v E u . Let us fix some local coordinates (u, v) in some neighbourhood U of p such that {u = 0} = E v ∩ U , while D has local equation v. Any local section of 2 X (Z ) (Z 0 as above) near p has local form ω = i∈Z, j∈Z ≥0 a i, j u i v j du ∧ dv. We define the residue Res D (ω) = (w/dv)| v=0 := i a i,0 u i du.  . . . , a p g ) ∈ C p g : Res D ( α a α ω α ) has no pole along D}.
, and the number of independent relations between (a 1 , . . . , a p g ), In particular, dim(im(c l (Z ))) is the number of independent relations for D generic.
The following Corollary will be relevant in the case of elliptic germs.  13 In the definition of the natural line bundles and of the modified Abel map the rational homology sphere link assumption is necessary. Furthermore, the theory of Abel maps was also developed in [14] under this condition. Since our goal is to present an application of this theory, here we also impose the RHS restriction for our elliptic germs. Usually, in surface singularity theory, having this assumption one hopes that several invariants will be guided by the topology of the link. In the present situation this also happens: the final picture of the W -stratification follows the combinatorics of the link rather closely.

Elliptic singularities. The elliptic sequence.
Let (X , o) be a complex surface singularity as in 2.1 and let φ : X → X be a resolution. In the first part of our discussions (up to the second part of Theorem 3.3.5) we will not impose any restriction regarding the link.

Elliptic singularities
Let Z min ∈ L be the minimal (or fundamental) cycle of the resolution φ, that is, min{S\0} [2,3]. Recall that (X , o) is called elliptic if χ(Z min ) = 0, or equivalently, min l∈L >0 χ(l) = 0 [13,36]. It is known that if we decrease the decorations (Euler numbers), or we take a full subgraph of an elliptic graph, then we get either an elliptic or a rational graph.
Let C be the minimally elliptic cycle [13,18], that is, χ(C) = 0 and χ(l) > 0 for any 0 < l < C. There is a unique cycle with this property, and if χ(D) = 0 (D ∈ L) then necessarily C ≤ D. In particular, C ≤ Z min . In the sequel we assume that the resolution is minimal. Then Z K ∈ S \0, hence in the numerically Gorenstein case Z min ≤ Z K by the minimality of Z min in S\0.
The minimally elliptic singularities were introduced by Laufer in [13]. In a minimal resolution they are characterized (topologically) by Z min = Z K = C. Moreover, (X , o) is minimally elliptic if and only if p g (X , o) = 1 and (X , o) is Gorenstein. For details see [13,18,19].

Elliptic sequences
One of the most important tools in the study of elliptic singularities are the elliptic sequences. The elliptic sequence is a set of integral cycles associated with the topological type (graph). They were introduced by Laufer and Yau, for the definition in the general (non numerically Gorenstein) case see [37,39]. In the numerically Gorenstein case the construction is simpler, see also [18,19,28]. This second case will be recalled below. In fact, we will use an elliptic sequence even in the non numerically Gorenstein case, but not the 'classical' one defined by Laufer and Yau: we define a new one, whose structure is much closer to the structure of sequences associated with numerically Gorenstein graphs (and to the non-integral cycle Z K ). In fact, after the first step of the construction (which produces a rational cycle) we hit a numerically Gorenstein support, and the continuation of the sequence is the one imposed by the numerically Gorenstein case.
In the next discussions below φ is a minimal resolution.

The construction of the
If Z B 0 = Z K then we stop, m = 0, this situation corresponds to the minimally elliptic case.
Otherwise one takes B 1 := |Z K − Z B 0 |. One verifies that |C| ⊆ B 1 B 0 , B 1 is connected, and it supports a numerically Gorenstein elliptic topological type with canonical cycle The proof of all these facts are similar to the proof of Lemma 3.2.3 below.) In particular, C ≤ Z B 1 ≤ Z K − Z B 0 . Then we repeat the inductive argument. If Z B 1 = Z K − Z B 0 , then we stop, m = 1. Otherwise, we define B 2 := |Z K − Z B 0 − Z B 1 |. B 2 again is connected, |C| ⊆ B 2 B 1 , and supports a numerically Gorenstein elliptic topological type with canonical cycle Z K − Z B 0 − Z B 1 . After finitely many steps we get Z B m = Z K − Z B 0 − · · · − Z B m−1 , hence the minimal cycle and the canonical cycle on B m coincide. This means that B m supports a minimally elliptic singularity with Z B m = C.
We say that the length of the elliptic sequence {Z B j } m j=0 is m + 1.

The construction of the (new) elliptic sequence; the non-numerically Gorenstein case
We will use the following notations: (c) B 0 supports a numerically Gorenstein elliptic topological type with canonical cycle . Then by the previous inequalities the expressions from the right hand side are ≥ 0, hence necessarily χ(s [Z K ] ) = χ(l) = (l, s [Z K ] ) = 0. If l has more connected components, say ∪ i l i , then χ(l i ) = 0 for all i, hence each l i contains/dominates a minimally elliptic cycle (cf. [13]), a fact which contradicts the uniqueness of the minimally elliptic cycle. Hence |l| = B 0 is connected and |C| ⊂ B 0 . Furthermore, (l, s [Z K ] ) = 0 shows that |l| = E.
(c) C ⊆ B 0 E shows that min |l|⊂B 0 , l>0 χ(l) = 0, hence B 0 supports an elliptic topological type. Moreover, from (l, s [Z K ] ) = 0 we read that for any E v from the support of l one has (E v , Then, as a continuation of the sequence, starting from B 0 and its integral canonical class j=0 as in the numerically Gorenstein case. We say that the elliptic sequence {Z B j } m j=−1 has length m + 1 and 'pre-term' In order to have a uniform notation, in the numerically Gorenstein case we set Z B −1 := 0 (which, in fact, is s [Z k ] ). In both cases, in a unified notation (see also [18, 2.11]), Then we proceed inductively: the first support is E, and once B j is known then one sets B j+1 : We prefer to index them in such a way that B 0 is the first numerically Gorenstein support. The elliptic sequence imposes some kind of 'linearity' of the structure of the graph. E.g., the following statement holds (probably some parts of it are already known in the literature).
which contradicts the ellipticity of the large graph. By Laufer's algorithm [12] there exists a computation sequence of the fundamental cycle of the large graph such that one of its terms is Z min = Z min ( ) while the next one is Z min + E new . Since χ(Z min ) = χ(Z min + E new ) = 0, we get that the coefficient m E v (Z min ) of E v in Z min is 1. In the numerically Gorenstein case, since v / ∈ B 1 we get that m E v (Z K ) = 1 too. If g v = 0 then by the adjunction formula v is either an endvertex (as in the statement) or it has two neighbours both with multiplicity 1. But this last case would generate (by repeating the argument for the two neighbours) an infinite string, all with multiplicity one, which cannot happen. If g v = 1 then use again the adjunction formula.

The cycles C t and C t
The next Lemma generalizes [18,Lemma 2.13] valid in the numerically Gorenstein case.
, the statement reduces to the numerically Gorenstein case. (Alternatively, using the analogue of the previous line as an inductive step one can proceed also by induction on m. The first step is as follows. and it belongs to S(B 1 ). Then the induction runs.)

Remark 3.3.2 The cycles {C
, satisfy several other universal properties as well. E.g., assume that the graph is numerically Gorenstein, and let I ⊂ V, I = ∅, such that I supports a numerically Gorenstein (connected) subgraph. Then I is one of the supports {B i } m i=0 . Indeed, suppose, that I = B 0 . Then, by induction, it is enough to prove I ⊂ B 1 . Let the canonical cycle on I be Z ∈ L.
Next, assume that the graph is not numerically Gorenstein. Then we claim that the support of any numerically Gorenstein (connected) subgraph belongs again to {B i } m i=0 . First we show that the largest numerically Gorenstein subgraph is supported by B 0 . (Then the rest follows from the previous paragraph.) Indeed, if I is its support and Z is the canonical cycle on this support, then similarly as above,

Remark 3.3.3
Even if the graph is numerically Gorenstein, the list of antinef cycles l ∈ S with l Z K is much larger than the list given in Lemma 3.3.1. Indeed, take e.g l = 2Z min , which usually is Z K and Z K .

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is the first numerically Gorenstein germ in the sequence. The last one, (X m , o m ), is minimally elliptic (hence automatically Gorenstein).
Before we recall the next characterization of Gorenstein elliptic singularities we mention that any numerically Gorenstein topological type admits Gorenstein structure [32]. (But the generic analytic structure is Gorenstein only in the Klein and the minimally elliptic case [13,Th. 4.3], see also [15,Prop. 5 (a) p g = m + 1;

Additionally, if the link is a rational homology sphere then (a) is equivalent with any of the following conditions:
Since in the next discussions the equivalence (a) ⇔ (f) has an accentuated role (and in the application of the results regarding the Abel map from 2.2 and [14] we need a link restriction) in the sequel we assume that the link is a rational homology sphere. (For certain extensions see also [28].) The implication (f)⇒(e) from above says that if the top singularity of the tower {(X j , o j )} j is Gorenstein then all the others (automatically with smaller support) are necessarily Gorenstein. This fact applied for a fixed (X j , o j ) says that if one of the singularities (X j , o j ) is Gorenstein, then all the others {(X i , o i )} i> j with smaller support are Gorenstein too. In fact, one has the following statement of Okuma. Set Then by the above discussion A gor = { j | α ≤ j ≤ m} and the following facts hold as well.

Discussion
Assume that (X , o) is Gorenstein, and let us consider the function f j from Theorem 3.3.5(c).

Remark 3.3.9
The support condition of functions f j can be improved slightly more, but not too much. Indeed, in the Gorenstein case Z max = Z min [18,Sect. 5], that is, O X (−C 0 ) has no fixed components (so, f 0 can be chosen such that x 0 = 0). However, in general, a similar choice for f j (with x j = 0) is not possible. See e.g. the elliptic singularity {x 2 + y 3 + z 6m+7 = 0}. Nevertheless, using Theorem 4.2.1, if C 2 = −1 then there exists a function F with div E (F) = Z K , hence a general combination of f j and F has the property that div E ( f j + α F) = C j . (In general, the maximum what one can get via inductive steps using Gorenstein property is that the vanishing order of f j on E v is exactly the multiplicity of C j at E v for any E v ⊂ B j . See again {x 2 + y 3 + z 6m+7 = 0}.)

) for elliptic singularities
Assume that (X , o) is Gorenstein and that the link is RHS. Then each (X j , o j ) is Gorenstein, and, in fact, their Gorenstein forms are related. Indeed, let ω 0 ∈ H 0 ( 2 X (Z K )) be the Gorenstein form of (X , o) (that is, the section which trivializes 2 X \E ) and consider the function Then ω j+1 := f j ω 0 has pole C j+1 , and (by the discussion from 3.3.8) its restriction to a small neighbourhood . Next, assume an arbitrary numerically Gorenstein singularity with α = min{A gor } as in Theorem 3.3.7. Then the statement of the previous paragraph can be applied for (X α , o α ). In this way we get forms We claim that these forms (more precisely, some representatives of their classes modulo H 0 ( X , 2 X )) can be extended to forms {ω j } m j=α in H 0 ( X , 2 X (C α )), such that their classes generate the vector space H 0 ( X , 2 Indeed, set I := V\B α . Then, by Theorem 3.3.7, p g (X , o) = p g (X α , o α ), hence part (c) of Theorem 2.2.7 reads as V (I ) = 0. But this via Proposition 2.2.9 implies that (I ) , which is a priori injective; but since the two dimensions agree, it is necessarily bijective.
In fact, for any I the subspace (I ) is determined uniquely by j I :

Line bundles on X. Preliminary cohomological statements
Fix an elliptic singularity with RHS link and its minimal resolution X .
Hence, the statement holds for m = 0. Then we proceed by induction.
Since l ≥ Z min we can assume j ≥ 1. In the cohomology exact sequence of 0 This shows that L(−C j−1 )| C j is trivialized by the restriction of the generic section of L(−C j−1 ).

In the Gorenstein case use the fact that
Fix L ∈ Pic( X ) such that c 1 (L) ∈ −S . Recall that by Lemma 2.1.4 the computation of h 1 ( X , L) reduces to the numerically Gorenstein case: Proof Lemma 2.1.4 reduces the statements to the numerically Gorenstein case.
(a) Similar reductions were used in [13,18,19]. For the convenience of the reader we provide the details. By Lemma 2.1.4 the second equality follows, and also h 1 ( Chose u ∈ I ∩ B j−1 . We construct a computation sequence which connects 0 with is conveniently chosen. We construct the sequence as concatenated of several ones, each one being the (Laufer) computation sequence of a minimal cycle (cf. [19, 5.8]). Indeed, for each 0 ≤ k ≤ j − 1 let {z k,i } i be a computation sequence starting with z k,0 = E u and ending with z k,t k = Z B k , such that at every step [12]. Then we glue these sequences as follows. The first element is 0. Then we list all the elements of the sequence {z 0,i } i . This ends with Z B 0 . The next element is Then we repeat the procedure and continue with Z B 0 + Z B 1 + E u and all Z B 0 + Z B 1 + z 2,i . We call the steps 0 E u , . But analysing both cases (normal and gluing steps) we realize that Note also that L| X j I ∈ Pic 0 ( X j I ). Hence we need to show that for a numerically Gorenstein elliptic singularity if L ∈ Pic 0 ( X ) then , which is guaranteed whenever L is sufficiently positive along V\B. Note that in the above theorem, in the elliptic case, this reduction can be done with a 'minimal positivity requirement' of L along V\B. See the statement and the proof of Theorem 4.2.1 as well.
The fact that only such 'minimal positivity' is needed is a key additional property of elliptic singularities, which makes them special. (For another key special property, the 'distinct pole property' see Remarks 6.1.2-6.1.3.)

The cycle of fixed components of the line bundles
Assume that (X , o) is numerically Gorenstein and we fix L ∈ Pic 0 ( X ). We denote the cycle of fixed components of L by l.
Proof Assume first that (X , o) is minimally elliptic (and the resolution is minimal, hence Z K = Z min ). We recall the following facts, valid in this situation, cf. [13,Lemma 3.3].
Fix any pair [13]. Moreover, let E v be an irreducible component whose coefficient in Z min is strictly greater than one, then there exists a computation sequence for Z min which starts and ends with E v .
Next we prove that for any is not onto. We use similar arguments as in [13,Lemma 3.12].
Assume that there exists a computation sequence {z i } t i=1 with z 1 = E v and ends at some E u such that (E u , Z min ) < 0. Then consider the infinite sequence {x i } i : In this case we consider a computation sequence {z i } i which starts with E v and ends at some other E u , and a sequence {y i } i which starts with E u and ends at E v . Take the infinite sequence is onto, except when we pass from 2Z min − E u to 2Z min , in which case the corank is 1. Hence, α has corank at most one, Next, consider the case when (X , o) is numerically Gorenstein with m > 0. We will generalize the first argument presented above. Fix any v ∈ V. Let V m be the set of vertices {v : Recall that along any computation sequence of Z min one has We claim that for any v ∈ V there exists u ∈ V m and a computation sequence {z i } i which starts with E v and jumps at E u . Indeed, we construct the computation sequence as follows: it starts with E v and then we add consecutively the shortest string of E w 's connecting E v with B m . Let the last element of the string (the first one which is supported in and also, it can happen that v either belongs to V m , or not.) Then we continue starting from E v to construct the computation sequence of Z B m which jumps at E u . If V m = {v } this is possible. If V m = {v } and the multiplicity of Z B m at E u is ≥ 2 then again it is possible (for both cases see above). Otherwise Z 2 B m = −1 which case is excluded. Then, finally, after we completed Z B m , we continue (in an arbitrary way) Laufer's algorithm to complete Z min . If we concatenate this computation sequence as in the first part of minimally elliptic situation (that is, (b) l ∈ S and by (a) l ≤ Z K too. Hence the statement follows from Lemma 3.3.1.
It is instructive to compare this last theorem with the example from Sect. 5, which shows that without the required assumptions of the theorem l > Z K might happen.

.
Let us fix the following minimal resolution graph (where the (−2)-vertices are unmarked).
It is a numerically Gorenstein graph with L = L and m = 1, hence We show that any non-Gorenstein analytic type supported by this topological type admits a special unique line bundle L ∈ Pic 0 ( X ) such that the cycle of fixed components l of L is 2Z min , which is > Z K . (In any other situation l ≤ Z K , hence l ∈ {0, Z min , Z K }, cf. Lemma 3.3.1.) We will break the discussion into several steps.

The starting point
The cycle l of fixed components is zero if and only if L O X . Otherwise, since l ∈ S, necessarily l ≥ Z min . In the sequel we assume l ≥ Z min .

Inequalities for l
We claim that (a) if l > Z min then l ≥ Z K , and (b) if l > Z K then l ≥ 2Z min . (Here the only needed property of l is l ∈ S.) For (a) use Lemma 2.1.3. According to this algorithm, if let us denote by 8 the E 8 -subgraph of (obtained from by deleting E 1 and E 2 ). Assume that l > Z K but l 2Z min . Then l = Z K + x with x > 0 and x supported on the 8 subgraph. Then for any In particular, the coefficient of x at E 3 is ≥ 2. But then (l, E 2 ) ≤ 0 fails, which is a contradiction.

.
Using the exact sequence 0

Characterization of l = Z min
We claim that l > Z min if and only if h 1

Characterization of fixed components of L(−Z K )
Note first that Z K + E 1 = 2Z min . Using 5.1.2(b) one obtains that L(−Z K ) has a nontrivial fixed component if and only if E 1 is a fixed component. Then from the exact sequence

.
By 5.1.2 l is either Z min , or Z K , or it is > Z K . We claim that in the Gorenstein case l > Z K cannot happen. Indeed, if l > Z K then h 1 (L(−Z min )) = 0 (by 5.1.4) and h 1 (L(−2Z min )) = 0 (by 5.1.5). On the other hand, (X , o) is Gorenstein if and only if Z min = Z max (cf. Theorems 3.3.5 and 3.3.8-3.3.9). Hence O X (−Z min ) has no fixed components, let s be a generic section of it (that is, s is the generic linear section). Then consider the exact sequence 2Z min )), which is a contradiction.

.
Next assume that l > Z K . By the above discussion this means that l ≥ 2Z min , (X , o) is not Gorenstein and it has p g = 1, Clearly I = ∅. We consider two cases. If I = {E 1 }, then (X V\I , o V\I ) is necessarily rational with p g (X V\I , o V\I ) = 0, hence χ(l) = 0 too. We claim that this cannot happen, since l > Z K implies χ(l) > 0. Indeed, consider x := Z K − l ∈ L <0 , and the Laufer sequence from Lemma 2.1.3 connecting x with s(x) = 0. Along the sequence χ is nonincreasing and in the very last step before z t = 0 we have (The fact that l ∈ S and χ(l) = 0 imply l ∈ {C i } i can be deduced also from [35, Th. 6.3], or also from the structure of the graded root associated with elliptic singularities, cf. [20].) In particular, the only remaining possibility is the second case I = {E 1 }. This means that l = n E * 1 = n Z min for some n ≥ 2. In this case (X V\I , o V\I ) is the minimally elliptic singularity (X 1 , o 1 ) with p g (X 1 , o 1 ) = 1, hence form (5.1.8) we have χ(l) = 1. Since χ(n Z min ) = n(n − 1)/2, we get that n = 2 is the unique possibility.
So to sum up, if l > Z K then necessarily l = 2Z min (and (X , o) must satisfy all the cohomological restrictions listed at the beginning of this subsection).

.
We show that l = 2Z min can be realized for some special L indeed.
Fix any non-Gorenstein analytic type (X , o) and its resolution X with dual graph . First we consider the Abel map c −Z min . Since the E * -support I of Z min = E * 1 is E 1 , p g = 1 (cf. 3.3.5) and this p g is already supported on C, from Theorem 2.2.7 it follows that dim(V (I )) = 0. Hence im(c −Z min ) is a point, say B 1 ∈ Pic −Z min . Since Z min = Z max (the non-Gorenstein property, see again Theorem 3.3.5), O X (−Z min ) has nontrivial fixed components, that is, O X (−Z min ) / ∈ im(c −Z min ). In other words, By additivity, cf. 2.2.4, im(c −2Z min ) is a point too, say B 2 ∈ Pic −2Z min , and set L 2 := B 2 (2Z min ) = im( c −2Z min ) ∈ Pic 0 . By additivity again, L 2 = L 1 + L 1 (using additive notation of the group structure of Pic 0 = H 1 (O X ) = C), hence L 2 = 0 as well.
Consider next the bundle L(−Z min ) = B 2 (Z min ). Its restriction to C 1 = C is O C 1 (Z min ) (Indeed, the restriction of ECa −Z min to C 1 is the empty divisor, hence the restriction of B 2 to C 1 is the trivial bundle). Furthermore, by Theorem 3. [18,Sect. 3]. Therefore, h 1 (L(−Z min )) = 0. This combined with 5.1.3 shows that h 1 (L) = 0 too.

.
L constructed in 5.1.9 satisfies another uniqueness property as well. Recall that Z K = E * 2 . The image of c −Z K = c −E * 2 is 1-dimensional, and in fact (using the Laufer integration formula [14,Sect. 7] applied to the unique differential form of pole one along E 2 ) it is the bijective image of ECa −E * 2 (Z min ) = C * (the moving divisor/point along E 2 \(E 1 ∪ E 3 )). Since Z min is the cohomological cycle (or, for any Z ≥ Z min one has Pic 0 (Z ) = Pic 0 (Z min )), im(c −E * 2 (Z )) = im(c −E * 2 (Z min )), see also diagram (3.1.1) from [14]. Hence im(c −Z K ) = C * in Pic −Z K = C. In other words, Pic −Z K \im(c −Z K ) consists of one point. This is exactly L(−Z K ) (since this bundle has a nontrivial cycle of fixed components). In other words, This example suggests fully the subtlety of the cycle of fixed components l of a bundle L compared with h 1 (L). In Pic −Z K any line bundle has h 1 = 0 by the generalized Grauert-Riemenschneider vanishing. However, Pic −Z K might have a nontrivial interesting stratification according to l (and even the possible values of l are not evident at all).

.
Consider the situation from 5.
This shows that although for several different Chern classes l it can happen that they have the same I (l ) and V (I (l )), the corresponding affine spaces im( c l ) might be different, so each individual affine subspace preserves some information about l , more than just I (l ). The above computation shows that even the h 1 -behaviour along these subspaces might vary.
o V\I (l ) ) for every L ∈ im(c l (Z )). (6) In general, cf. 2.2.6, dim im(c l (Z )) ≤ dim V (l ). Assume that u ∈ B i \B i+1 . If i < α then by (1) and by the previous inequality we are done. Otherwise, by the general statement of Corollary 2.2.12 and from the structure of the poles of differential forms constructed in 3.4 it follows that dim im(c −E * u (Z )) = i +1−α. On the other hand, by (4), dim V (u) = i +1−α as well.
(7) We reduce the statement to (6) via the multiplicative structure from Sect. 2.2.4 and Theorem 2.2.7. Firstly, s l 1 ,l 2 (Z ) is dominant and quasi-finite. Therefore, for l = gen is a generic element of im(c l ) then h 1 (L im gen ) = p g − dim im(c l ) (cf. Theorem 2.2.7(d)), which equals p g − dim V (l ) by (7). Hence, by semicontinuity (see e.g.

Remark 6.1.2 (a) Parts (6)-(7)
can be compared with Theorem 2.2.7(e). Theorem 6.1.1 says that in the case of elliptic singularities there is no need to take any multiple nl in order to obtain the maximal stabilized dimension of im(c nl ), that is, dim im(c l ) = dim im(c nl ) for any l ∈ −S and n ≥ 1. As a consequence, the closure of any im(c l ) is an affine space.
Similarly, parts (8)- (9) can also be compared with Theorem 2.2.7(e): in order to have a uniform h 1 -behaviour along the (closure of the image), no stabilization is needed either.
By (6)-(7)-(9) the stabilization takes place from the very first term. The main property of elliptic singularities, which is responsible for this fact, is the existence of forms {ω j } p g j=1 , which form a basis of H 0 ( 2 X (Z ))/H 0 ( 2 X ) (Z 0), and which satisfies the assumption of Corollary 2.2.12. In the context of any singularity, we will call this property the 'distinct pole property'. It means the following: For any v ∈ V let J v be the index set of those forms ω j (from this list), which have nontrivial pole along E v . Then the poles along E v of all forms {ω j } j∈J v are pairwise distinct.
For elliptic germs this property is guaranteed by Corollary 3.4.1, since the pole of each ω j is C j .
The point is that if a normal surface singularity (with rational homology sphere link) admits a set of p g independent forms with the 'distinct pole property' then the above stabilization properties (6)-(7)- (9) hold. This follows from Propositions 2.2.9 and 2.2.11 proved in [14].
It is natural to ask whether the 'distinct pole property' is an idiosyncrasy merely of elliptic singularities. The answer is no, there are many germs with this property, see e.g. the next example.

Example 6.1.3 A singularity with 'distinct pole property'.
Consider the following resolution graph (the associated minimal one can be obtained by blowing down the two 'cusps'.) The graph is not elliptic, min χ = −1 (and it has two distinct candidates for the elliptic cyle).
It is realized e.g. by the hypersurface singularity with non-degenerate Newton boundary {z 3 + x 13 + y 13 + x 2 y 2 = 0}. This analytic structure has p g = 5 and it is clearly Gorenstein. Let ω be the Gorenstein form (with pole Z K ). Then the classes of the five forms ω, ωx, ωx 2 , ωy, ωy 2 constitute a basis of H 0 ( 2 X (Z ))/H 0 ( 2 X ) (Z 0), and they satisfy the 'distinct pole property' (the verification is left to the reader; the divisor of x is E * 1 , while the divisor of y is E * 2 ). In fact, even if we take the generic analytic structure on this graph (cf. [15]), the property survives. Indeed, in this case p g = 2 [15] and the two cycles of poles of the corresponding two differential forms have even distinct support. They are supported on the two minimally elliptic subgraphs obtained by deleting the two central (−3) vertices. Hence, again they satisfy the 'distinct pole property'.
(In fact we expect that the 'distinct pole property' is true for any analytic type supported on this graph. It is really amazing that for such graphs, when for any analytic type supported on them the 'distinct pole property' holds, the 'stability' analytic property 'h 1 (L ⊗n ) = h 1 (L) for n ≥ 1 and L without fixed components' is imposed by the combinatorics of the graph.)

The WECC and ECC properties and the set {im( c l )} l [14, Sect. 9]
The mutual position of the natural line bundle O Z (l ) and im(c l ) (or, equivalently, of 0 and im( c l )) is codified in the following submonoid of S . We set S im := {−l : O Z (l ) ∈ im(c l )} = {−l : 0 ∈ im( c l )}. In other words, l ∈ S im if and only if O Z (−l ) has no fixed components.
As usual, we define the saturation of a submonoid M ⊂ S as M := {l ∈ S : nl ∈ M for some n ≥ 1}. Accordingly, S im = {−l ∈ S : 0 ∈ im( c nl ) for some n ≥ 1}.
Recall also that we say that a resolution X satisfies the 'End Curve Condition' (ECC) if E * v ∈ S im for any end vertex v. The terminology was introduced by Neumann and Wahl in the context of splice quotient singularities [25]. By the 'End Curve Theorem' [26] X satisfies ECC if and only the analytic type is splice quotient associated with the dual graph of X . Furthermore, given a resolution graph , a singularity resolution X with dual graph and ECC exists if and only if the graph satisfies the 'semigroup and congruence conditions' of Neumann-Wahl [25], or, equivalently, the 'monomial condition' of Okuma [29].
We say that X satisfies the 'Weak End Curve Condition' (WECC) if E * v ∈ S im for any end vertex v. In fact, by [14, Proposition 9.2.2], X satisfies the WECC if and only if S im = S . In general, S im = S , for concrete examples see [14] (or below in Example 6.3.3).

The set {im( c l )} l ⊂ H 1 (O Z )
By Theorem 6.1.1 (7) for any end-vertex v, and also O(−n E * 2 ) ∈ im(c −nE * 2 ) for n 1). This can be proved as follows. Note that E * 1 and E * 2 cannot be realized as divisors of functions (restricted to E) simultaneously, since the linking number of their arrows is one, or equivalently, if f 1 , f 2 are some realizations then the degree of ( f 1 , f 2 ) : (X , o) → (C 2 , 0) would be one. Since in this hypersurface case (or any Gorenstein case) E * 1 is realized, E * 2 cannot be realized. (b) Note that WECC says that 0 ∈ im( c −nE * v ) for all end vertices v (and n 1). This in the elliptic case implies that im( (that is, the ECC property) in general cannot be guaranteed yet (by a general argument valid for any normal surface singularity). However, in the present situation it will hold and it will be proved later in Theorem 9.4.2.

The structure of the affine subspaces {im( c l )} l
of Pic 0 is the following. First, recall that the structure of the linear subspace arrangement {V (I )} I is very simple, it is a flag. Associated with a fixed V (I ) there are several (in general, infinitely many) associated parallel affine subspaces of type im( c l ). Indeed, as in Theorem 6.1.1(5), for any I ⊂ V let i be the maximal number, such that there exists a vertex u ∈ I with u ∈ B i \B i+1 . For all l , such that I (l ) has the same i, all the affine spaces im( c l ) have the same dimension, and are parallel to the same V (I (l )). (Their shifts have even an additional 'semigroup structure' in the sense that if A(l ) = a(l ) + V (l ) then A(nl ) = n · a(l ) + V (l ).) In particular, if two subspaces of type im( c l ) intersect each other nontrivially, then one of them should contain the other one.

Example 6.3.3
It can really happen that these parallel affine subspaces do not collapse into one vector space (namely into V (I )), see e.g. any elliptic singularity which does not satisfy ECC. For example, the points {L n } n≥1 in 5.1.11 are all parallel affine subspaces associated with V (Z min ) = 0.
Next we present a Gorenstein case as well.
Take This topological type does not support any analytic structure with ECC (it does not satisfy the semigroup or the monomial condition at the (−4)-node). In fact, later we will show that this graph does not admit any analytic type with WECC either. This will follow either from Theorem 9.2.7 directly, or from Theorem 9.4.2 using the nonexistence of the ECC structure. Hence, for the present Gorenstein structure WECC fails at least at one of the end-vertices.
On the other hand, one verifies that div (In fact if we denote div(z) by E * 0 + E * 2 + D 0 + D 2 , where D 0 and D 2 are two transversal cuts of E 0 and E 2 respectively, then div(u 2 − w) = 2E * 0 + 2D 0 and div(u 11 −w 3 ) = 2E * 2 +2D 2 .) Hence E * 0 , E * 1 , E * 2 ∈ S im . Therefore, the only obstruction for WECC can be caused by E 3 or E 4 . But, div E (u) = E * 3 + E * 4 . (The strict tranform is {u = z 2 − w 4 = 0}, whose two components are permuted by the Z 2 -Galois action of the double covering u → u, w → w, z → −z.) Hence there exists D j ∈ ECa −E * j ( j = 3, 4), so that O X (D 3 + D 4 + E * 3 + E * 4 ) = 0. This means that the two points L j : . All these 1-dimensional affine subspaces are distinct parallel ones in Pic 0 = C 2 , all associated with V . (The Galois action is n → −n.) V can also be realized as some im( c l ). Indeed, since

Definition of the strata W l ,k
Definition 7.1.1 [14, 5.8] We fix any singularity, one of its resolutions X , and l ∈ −S . We define W l ,k = {L ∈ Pic l ( X ) : h 1 ( X , L) = k}. Its closure in Pic l ( X ) will be denoted by W l ,k .
If it is necessary, when we handle several resolution spaces, we might also write W l ,k ( X ).
Hence, for each l ∈ −S , Pic l ( X ) has a stratification into constructible subsets according to L → h 1 (L). Consider again the setup of elliptic singularities as in the previous sections. We will describe the above stratification in several steps.

The general reduction to l = 0
For any fixed l ∈ −S let I = I (l ) be the E * -support of l , and let 0 ≤ i ≤ m + 1 be the maximal index with I ∩ B i−1 = ∅. (i = 0 happens when I = ∅.) Let K be the kernel of π i : Note that if L ∈ Pic l ( X ) then L| X i has trivial Chern class, hence the restriction induces a well-defined affine map π l i : Thus, the orbits are exactly the affine fibers of π l i . Proposition 7.2.1 (a) L → h 1 (L) is constant along the fibers of π l i .
In particular, the h 1 -stratification of Pic l ( X ) is completely determined (as a pull-back via an affine map) by the h 1 -stratification of Pic 0 ( X i ). This reduces its study to the l = 0 case.
In the next Sects. 7.3 and 7.4 we clarify the l = 0 case. Though the statements for the Gorenstein and non-Gorenstein cases can be formulated uniformly, we still decided to separate the two cases; in this way we can emphasize better the peculiarities of both situations.

The case (X, o) Gorenstein and l = 0
For any j ∈ {0, . . . , m + 1} we denote the natural linear projection by π j , and we also interpret it as the restriction Pic 0 ( X ) → Pic 0 ( X j ) = Pic 0 (C j ). (Here and below, by convention, We write W 0,k for W 0,k ( X ). Recall that m + 1 = p g .
(c)⇒(a) Fix some L ∈ W 0, p g − j ⊂ Pic 0 ( X ). This means that h 1 (L) = p g − j. Now, we know that l associated with L is C i−1 for some i, cf. Lemma 3.3.1. By Lemma 4.1.1(b) h 1 (L) = p g − i, hence using h 1 (L) = p g − j one gets i = j. If L ∈ W 0, p g − j \W 0, p g − j , then by semicontinuity of h 1 one has h 1 (L) = p g − j for some j < j, hence by the very same argument l = C j −1 .
If h 1 (L) = p g − j, then we are done. Next assume that h 1 (L) = p g − j > p g − j for some j < j. Then by the implication (c)⇒(b) already proved, from h 1 (L) = p g − j we get L ∈ ker(π j ). Consider a convergent sequence of line bundles {L n } n in ker(π j )\ ker(π j−1 ) with lim n→∞ L n = L. As above, but now for L n , However, here necessarily we should have equality (otherwise, if , which leads to a contradiction.) Hence L n ∈ W 0, p g − j and L = lim n→∞ L n ∈ W 0, p g − j .
The fine version follows directly from the coarse one.

Remark 7.3.2 The 'fine version' cannot be completed with
Indeed, (f-d) ⇒ (f-a,f-b,f-c) does not hold. Take for example a Gorenstein singularity with m = 1, and L := O. Then O(−Z min ) has no fixed components [18,Sect. 5], that is, O ∈ im( c −Z min ), hence (f-d) holds for j = 1. On the other hand, O / ∈ W 0,1 . However, the opposite implication (f-a,f-b,f-c) ⇒ (f-d) holds whenever l ≤ Z K (e.g. when either C 2 = −1, or when (X , o) is minimally elliptic, for details see Theorem 4.2.1). In such case l = l. Hence, if l = C j−1 then in fact l = C j−1 , or L(−C j−1 ) has no fixed components, L(−C j−1 ) ∈ im(c −C j−1 ) and L ∈ im( c −C j−1 ).

Question
Does W 0, p g − j ⊂ im( c −C j−1 ) hold for any Gorenstein singularity?

The case (X, o) non-Gorenstein and l = 0
Fix an elliptic numerically Gorenstein singularity with length m + 1 and minimal resolution X . Let α be defined as in (3.3.6). Recall that p g ( X ) = p g ( X j ) and Pic 0 ( X ) = Pic 0 ( X j ) = Pic 0 (C j ) for any 0 ≤ j ≤ α. Theorem 7.4. 1 We fix some L ∈ Pic 0 ( X ) and we denote by l the cycle of fixed components of L. We also set l := min{l, Z K }.
Fix any j ∈ {0, . . . , α}. Then for any such j, im( c −C j−1 ) consists of a single point and Moreover, one has the next inclusions as well: About the position of the other points corresponding to 1 ≤ j ≤ α we claim nothing (see also Remark 7.4 In particular, for any j > α, dimensional linear subspace of Pic 0 ( X ), and, in fact, it equals both im( c −C j−1 (Z )) and ker(π j ). E.g., W 0,0 = Pic 0 ( X ). Furthermore, W 0, p g ( X j−1 ) ⊂ W 0, p g ( X j ) whenever j ≥ α + 2 and the same is true for j = α+1 too if we disregard the points , then for any j ∈ {0, α + 1, α + 2, . . . , m + 1} the following facts are equivalent: Proof The proof of Theorem 7.3.1 can be adapted. Let us prove the statements valid for l ≤ α.
Since the restriction Pic −C j−1 ( X ) → Pic 0 (C j ) is an affine isomorphism, {L : the two sets (both of cardinality one) must agree. Furthermore, {L; l = C j−1 } ⊂ {L : 1.1(b) and Theorem 3.3.7. Finally, assume that L ∈ W 0, p g . Then, by definition, h 1 (L) = p g . On the other hand, assume that its l is some C i−1 . Then by Lemma 4.1.
The second case j ∈ {0, α + 1, . . . , m + 1} follows analogously as the proof of the Gorenstein case, once we replace the statement (d) of Theorem 3.3.5 with Theorem 3.3.7.

Remark 7.4.2 (a) If
)} j≥α+1 of linear vector spaces whose dimensions are increasing oneby-one from 0 to p g , and also by the 'wandering points' ∪ α j=1 im( c −C j−1 ), all of them being in W 0, p g ( X ) . All the irreducible components of W 0,k are affine subspaces of type im( c −C j−1 (Z )) (0 ≤ j ≤ m + 1), if k < p g then they are even linear subspaces. The non-linear ones are all points. (b) See Sect. 5 for a non-Gorenstein singularity with m = 1, α = 1, where there is a 'wandering point' L 1 = 0 (cf. 5.1.9-5.1.11). Since the position of the points ∪ α j=1 im( c −C j−1 ) is 'unpredictable', we called them 'wandering'.

Questions
(1) Are the wandering points all distinct? Are they all different than 0 ? (That is, is the cardinality of ∪ α j=0 im( c −C j−1 ) exactly α + 1?) (2) Are the wandering points in W 0, p g −1 ? Or, do they wander 'even more' ?

Question
Can those members of im( c l ), which serve as components of some W 0,k , be characterized by some universal property? Are they characterized by the maximality of h 1 ?

The case (X, o) arbitrary elliptic and l ∈ −S arbitrary
For any X and L with c 1 (L) ∈ −S , by Lemma 2.
In this way the h 1 -stratification of an arbitrary elliptic singularity is reduced to the case of numerically Gorenstein ones. Furthermore, if (X , o) is numerically Gorenstein, then by Proposition 7.2.1 the h 1stratification is reduced to the l = 0 case. If X i is Gorestein then one has to combine Proposition 7.2.1 with Theorem 7.3.1, otherwise Theorem 7.3.1 should be replaced by the more general Theorem 7.4.1.
We invite the reader to complete the details writing down the corresponding set-identities.

Remark 7.5.1
Recall that in the numerically Gorenstein case the identity W l ,k ( X ) = (π l i ) −1 (W 0,k ( X i )) holds (for the notation and the statement see 7.2 and Proposition 7.2.1). On the other hand, by Theorem 7.4.1, W 0,k ( X i ) equals im( c −C j−1 ( X i ) ) for some cycle C j−1 ( X i ) ∈ L( X i ) associated with the singularity (X i , o i ). We show that a similar structure statement is valid for W l ,k ( X ) too, that is, W l ,k ( X ) is the closure of the image of a certain Abel map (at the level of X ).
First, let us shift W l,k ( X ) into Pic 0 ( X ) (where the images of modified Abel maps c live).
That is, for each l , via identification Pic Its closure in Pic 0 ( X ) will be denoted by W 0 l ,k . Note also that W 0 0,k = W 0,k . Consider first the notations and situation from Proposition 7.2.1. First we analyze the Abel map im(c −l ) : ECa l ( X ) → Pic l ( X ). By Theorem 6.1.1 one gets dim im(c l ) = dim V (I (l )). On the other hand, the dimension of (π l i ) −1 (0) in Pic l ( X ) is p g ( X )− p g ( X i ) = dim V (I (l )) too, cf. the same Theorem 6.1.1. In particular, im(c l ) = (π l i ) −1 (0) in Pic l ( X ).
One sees that the restriction of C For another, more 'theoretical' presentation of W 0 l ,k ( X ) as im( c l −l ) (with certain additional properties) see Theorem 8.1.2.

Notations
Fix a resolution and l ∈ −S as above. In the previous section we considered the stratification of Pic l ( X ) provided by the value L → h 1 (L), namely by W l ,k = {L ∈ Pic l ( X ) : It is convenient to shift W l,k into Pic 0 ( X ) as follows (cf. (7.5.2)). For each l the Now we will consider a much 'finer' stratification. Again, it is convenient to shift the structure into Pic 0 ( X ), this has the advantage that the strata can be compared more naturally with subspaces of type im( c l ) ⊂ Pic 0 (Z ) = Pic 0 ( X ) (Z 0). The strata in W l ,k are defined as follows This stratification shifted (via identity L = L 0 (l )) provides a stratification of W 0 l ,k as well: We denote by I the set {im( c l )} l indexed by all possible l ∈ −S . Our goal is to describe for the fixed l ∈ −S the sets {F 0 l ,k (l)} k∈Z ≥0 ,l∈L ≥0 in terms of certain elements of I. This automatically will provide a new characterization of the sets {W 0 l ,k } k∈Z ≥0 as well, besides the one provided in the previous section. Though the next theorem has some overlaps with statements from the previous section regarding the W -stratification, we prefer this presentation since it provides a uniform presentation of the two types of stratification showing their interactions. (Even more, we deliberately use a formulation and proof independent of Sect. 7 with the hope that this version can serve as a prototype for arbitrary cycle Z , not necessarily Z 0, or for more general singularities.) For the fixed l ∈ −S and k ∈ Z ≥0 we define I l ,k by decreasing induction as follows. For k > p g we set I l ,k = ∅ (note that by Theorem 4.1.2 we know that W l ,k = ∅ for k > p g ; check also that the identity (8.1.1) from below has no solution in these cases). Assume next that I l ,k is already defined for any k > k. Then, by definition, I l ,k consists of the set of subspaces of type im( c l −l ) of I indexed by ⎧ ⎨ is not included in any subspace indexed from ∪ k >k I l ,k .    (c) imply (a). We start to prove (b). Note also that during the proof all the appeared cycles l sit in the bounded ellipsoid {l : χ(l) + (l, l ) ≤ p g }, hence we can assume that not only Z 0 but all the possible cycles of type Z − l are also 'large' (so, both Z and Z − l can be replaced by X in h 1 -computations, if we wish). Hence, sometimes we will omit Z or Z − l.
Since (8.1.1)(ii) has no solution for k > p g , we get that in such cases W 0 l ,k = ∅, and the choice of I l ,k = ∅ is also supported. Then we prove (b) by decreasing induction on k. Fix again k ≤ p g and assume that the statement is already proved for all k with k > k. Consider again the situation from the previous paragraph: let S be an irreducible component of W 0 l ,k , L 0 ∈ S ∩ W 0 l ,k and L = L 0 (l ), h 1 (L) = k. Then we verified that there exists l ∈ L ≥0 so that L 0 ∈ im( c l −l ), l − l ∈ −S and it satisfies (8.1.1)(ii). Note that the subspace im( c l −l ) cannot be included in any subspace index by any I l ,k with k > k since by inductive step all the subspaces indexed by I l ,k belong to ∪ k >k W 0 l ,k , hence all their elements K 0 satisfy h 1 (K 0 (l )) > k; however L 0 ∈ im( c l −l ) with h 1 (L 0 (l )) = k. Therefore, l belongs to I l ,k .
(II) Now, by taking L 0 generic in S, the inclusion L 0 ∈ im( c l −l ) implies S ⊂ im( c l −l ), where the subspace im( c l −l ) is indexed from I l ,k .
(III) Conversely, consider somel ∈ L ≥0 such that the subspace im( c l −l ) is indexed from I l ,k and S ⊂ im( c l −l ). Let K 0 be a generic bundle of im( c l −l ), and write K := K 0 (l ). From Theorem 6.1.1 (8) h 1 (K(−l)) = p g − dim V (I (l −l)). This combined with (8.1.1) and (8.1.4) give We claim that necessarily h 1 (K 0 (l )) = k. Indeed, since im( c l −l ) contains the bundle L 0 with h 1 (L 0 (l )) = k (cf. (I)-(II)), its generic bundle K 0 cannot satisfy h 1 (K 0 (l )) > k by the semicontinuity of h 1 . Hence, the generic element of im( c l −l ) belongs to W 0 l ,k , which implies that im( c l −l ) = S. This in particular also shows, cf. (8.1.5), that the cycle of fixed components of K isl. Finally note that l is maximal in I l ,k . Indeed, assume that there exists an overset of type im( c l −l ), then by the above discussion im( c l −l ) equals S too, hence it must equal im( c l −l ) as well.
For the irreducibility see part (V) as well. This ends the proof of part (b). Next we prove (c). We will repeat several steps of the proof of (b), but now applied for the irreducible components of F 0 l ,k (l). (IV) Let S be an irreducible component of F 0 l ,k (l), and choose L 0 ∈ S ∩ F 0 l ,k (l). Set L := L 0 (l ) ∈ W l ,k , hence h 1 (L) = k. Next, assume that l ∈ L ≥0 is the cycle of fixed components of L. Then, similarly as in (I), L 0 ∈ im( c l −l ), l − l ∈ −S , l satisfies (8.1.1), and im( c l −l ) belongs to I l ,k .
By taking L 0 generic in S ∩ F 0 l ,k (l) we get S ⊂ im( c l −l ). Conversely, as in (III) forl = l, one shows that im( c l −l ) ⊂ S too, hence necessarily S = im( c l −l ). Since for fixed l , k and l there is a unique affine subspace of type im( c l −l ) with these data, F 0 l ,k (l) should only have one irreducible component, which equals im( c l −l ). Note that from L 0 ∈ im( c l −l ) we also have F 0 l ,k (l) ⊂ W 0 l ,k ∩ im( c l −l ). The opposite inequality also follows as above (or as in (III)) since in the presence of W 0 l ,k we automatically have h 1 (K 0 (l )) = k. This shows (8.1.3) as well.
(V) In (IV) we proved that each F 0 l ,k (l) is irreducible and equals some im( c l −l ) from I l ,k . Next we plan to show that any subspace from I l ,k is realized in this way by some Fix im( c l −l ) from I l ,k . Let K 0 be a generic bundle from im( c l −l ), set K := K 0 (l ) ∈ Pic l . Then (8.1.4) is still valid, and as in (III) one also has We claim that h 1 (K) = k. Assume that this is not the case, that is, k := h 1 (K) > k. Letl be the cycle of fixed components of K. Then, as above, K ∈ im( c l −l ) with h 1 (K) = k . By the inductive step, we can assume that im( c l −l ) is indexed from I l ,k . Since K 0 was chosen generically from im( c l −l ), we get that im( c l −l ) is included in a space of type im( c l −l ) from I l ,k , a contradiction.
Hence Furthermore, the description/characterization of the non-closed sets im( c l −l ) (indexed by I l ,k ), respectively of F 0 l ,k (l), is even harder.

Problem
Is it true that F 0 l ,k (l) = im( c l −l ) (with the notations of Theorem 8.1.2) ?

.
In Example 8.1.9 we show that the F-stratification of a certain W can be non-trivial, while Example 8.1.12 presents a different case when the F-stratification is trivial (based on an additional geometric argument).

Example 8.1.9
Consider the elliptic graph from Sect. 5. It has m = 1, hence p g ≤ 2. The maximal value p g = 2 can be realized e.g. by the hypersurface singularity {x 2 +y 3 +z 17 = 0}; see also Remark 6.3.1. Assume in the sequel that p g = 2. Furthermore, assume also that l = −Z K . In this case by Kodaira type or Grauert-Riemenschneider vanishing h 1 In fact, from the point of view of Proposition 7.2.1 the situation is also trivial: p g ( X i ) = 0, hence Pic −Z K ( X ) consists of a unique stratum , namely On the other hand, we will see that the 'fixed component' stratification is not trivial. First note that im(c l ) is 2-dimensional, hence it is Pic l , and along it h 1 = 0, hence im(c l ) = W l ,0 . To find the F-stratification we have to find the solutions for l ∈ L ≥0 of the system Z K + l ∈ S and One solution is l = 0 which provides im(c l ). The other solution is l = E 1 (see Sect. 5 for notation). In this case Z K + E 1 = 2Z min ∈ S, and χ(l) − (l, Z K ) = dim V (I (2Z min )) = 1, hence (8.1.10) is satisfied. We will show that these are the only solutions. Indeed, assume that l > 0 is such a solution. Then χ(l) − (l, Z K ) = χ(−l) > 0 (see the third paragraph in 5.1.7), hence dim V (I (−Z K − l)) ≤ 1. But, since Z K + l > 0 and the singularity is This shows that necessarily n = 2.
In conclusion,

Remark 8.1.11
Though the set of subspaces of type {F 0 l ,k (l)} l is in bijection with I l ,k (completely defined/described above), sometimes, in order to reduce the possible candidate solutions of (8.1.1) we can use some additional geometric restrictions as well (which, by Theorem 8.1.2, are automatically satisfied, but this fact might not be so transparent from (8.1.1)). E.g., if l is a solution, hence {F 0 l ,k (l)} l is a non-empty stratum, then necessarily dim V (I (l − l)) = dim im(c l −l ) ≤ dim W l ,k , and equality dim V (I (l − l)) < dim W l ,k whenever im(c l −l ) is a proper subspace of W l ,k . See the next Example for such an argument.

WECC for arbitrary singularities
Recall, that by definition, a minimal resolution X of a normal surface singularity satisfies WECC if and only if E * v ∈ S im for any end-vertex, that is, if for some n > 0 one has , see also 6.3. By [14,Prop. 9.2.2] this happens exactly when S im = S , that is, for any l ∈ S there is a certain n > 0 so that As a comparison, ECC for X , by definition, is given by the (This can also be compared with the criterion (9.2.2) valid for elliptic singularities.) It is known (see e.g. [29, (2.15)] or [23, 5.27]) that ECC is closed by taking 'sub-singularities'. This means the following. For any connected union E I := ∪ w∈I E w , where I ⊂ V, take X I a convenient small neighbourhood of E I in X , then X I -as the resolution of (X I , o I ) := ( X I /E I , E I /E I )-satisfies ECC as well. The very same proof gives the following.

Lemma 9.1.1 Fix any (not necessarily elliptic) singularity and one of its resolutions X. Then
WECC of X is closed by taking 'sub-singularities'.
We will use the same notations even if E I is not connected, in such cases (X I , o I ) is a multigerm. We set {I j } j for the connected components of I .

.
Before we state the next result, we warn the reader that, in general, the restriction to a certain X I of a natural line bundle of X is not natural (that is, O X (l )| X I = O X I (R(l )), where R is the Chern class restriction). By the next statements we prove that an analytic structure is free from this pathology if and only if it satisfies WECC.
In order to test the fact that the restriction of any natural line bundle is natural it is enough to verify that O X (E v )| X I j is natural for any vertex v and j, where I := V\v and I j is a component of I . Indeed, first note that it is enough to test only integral cycles. Next, for any u = v, hence by additivity applied for O X (l) (l ∈ L) the claim follows. Amazingly, this property fits perfectly with the WECC.
Lemma 9.1.3 As above, consider any singularity and one of its resolutions X . Fix any vertex v ∈ V and set I := V\v, I = ∪ j I j . Then This means that for a convenient large n, such that n E * v has the form j l j +m v E v , we get that O X (−n E * v )| X I is trivial. Consider next the restriction π I : Pic −nE * v ( X ) → Pic 0 ( X I ). Then (π I ) −1 (0) is an affine subspace of dimension h 1 (O X ) − h 1 (O X I ). The same is true for the affine subspace im(c −nE * v ) for n 0. Since im(c −nE * v ) ⊂ (π I ) −1 (0), the two subspaces should agree. Thus, O X (−n E * v ) ∈ im(c −nE * v ) for n 0. (a) WECC for X; is trivial for any l ∈ S with E * -support I ; (e) The restriction to any X I of any natural line bundle of X is natural.
Proof Use Lemma 9.1.3 and its proof and the comment from 9.1.2.

WECC for elliptic singularities. First consequences.
In the elliptic case, cf. Theorem 6.1.
for n 0, we get the following. We invite the reader to review the definition of the analytic multivariable Poincaré series P(t), associated with a fixed resolution of a normal surface singularity e.g. from [22,23], or [4,5], see also [14, 2.3.6]. Usually P(t) is not topological. However, for singularities, which satisfy ECC P(t) equals the topological series Z (t), cf. [23].  part (a)). In particular, p g is topological. Furthermore, it is known, see e.g. [22, 4.2], that P(t) can be recovered from the dual resolution graph of X combined with the knowledge of the cohomology groups {h 1 (O X (l ))} l ∈−S of natural line bundles indexed by −S . However, for each such l one has O X (l ) ∈ im(c l ) and h 1 along im(c l ) is topological (apply Theorem 6.1.1 for a Gorenstein singularity).  that (X , o) Hence, from ellipticity, there exists exactly one step when (z i , E v(i) ) = 2 and in all other steps it is 1.
Next, assume that X (hence, by Lemma 9.1.1 X too) satisfies WECC. Then O X (−E * v ) has no fixed components at all (cf. 9.2.5). Since (E u , −E * v ) = δ uv , it can have a base point only along E v , where it really has one by (a). Let the disc E w ∩ X be D w . Having the WECC for X , we can choose a divisor D ∈ ECa( X ), which intersects E( X ) only along E w \E v , and an integer m . Therefore, for these Gorenstein but not WECC singularities P(t) = Z (t) also fails.

First topological characterization of the existence of WECC structure
In the following we will give a topological characterization in terms of the combinatorics of the minimal resolution graph for the existence of a WECC analytic type supported on . Proof The extension obstruction from Theorem 9.2.7 (applied via Corollary 9.2.3) shows that the combinatorial restriction is necessary. Now, we fix a graph , which satisfies the gluing obstruction of the statement, and we wish to construct a WECC analytic type supported on it. The construction builds a resolution space X by analytic plumbing based on induction on m. If m = 0 then the graph is minimally elliptic, hence any analytic realization satisfies ECC [25], hence WECC too.
Next, we assume that X i was already constructed, and it satisfies WECC. Fix v ∈ B i , which has a neighbour w in B i−1 . By assumption, v admits only one such w. By Lemma 3.2.7 v / ∈ B i+1 , and by Lemma 9.2.
The Chern class shows that D w is smooth and it intersects E v transversally and it intersects no other exceptional curve. [The 'Extension Theorem' 9.2.7 and its proof show that D w ∩ E v is uniquely determined by the analytic type of X i , it is the base point of O X i (−E * v ).] Then let T w be an analytic disc bundle over E w with Chern number E 2 w , and we analytically glue T w to X i in such a way that X i ∩ E w = D w . We proceed similarly for all other such w ∈ B i−1 \B i vertices, which have a neighbour in B i . (We call such w a contact vertex.) The other disc bundles (corresponding to vertices w ∈ B i−1 \B i , which have no neighbours in B i ) are glued arbitrarily. The obtained resolution space will be denoted by X i−1 .
We claim that X i−1 supports a singularity with WECC. In the proof we use Corollary 9.1.4(c)⇒(a). According to this, we need to verify that First we prove a lemma. In order to formulate it, let us fix a connected subgraph supported on B with B i ⊂ B B i−1 . Note that the maximal numerical Gorenstein support in B is B i . [Indeed, B has a unique maximal numerically Gorenstein subgraph with length m + 1 − i by Remark 3.3.2, but B i satisfies this requirement.] is natural by construction.
'⇐' Note that the restriction Pic l ( X B ) → Pic R(l ) ( X i ) is an isomorphism (here l ∈ L ( X B ), and R(l ) is its restriction). Now, if the restriction of L ∈ Pic( X B ) is natural, then L n | X i = O X i (l) for some n ∈ Z and l ∈ L( X i ).
is the lattice embedding. Then O X B (ι(l))| X i = O X i (l) = L n | X i , hence by the injectivity of the restriction L n = O X B (ι(l)).
Now we verify (9.3.2). If u ∈ B i−1 \B i then B i−1 \u has a connected component B with B i ⊂ B B i−1 , and maybe some other components, all of them supporting rational graphs. Along the rational components any bundle is automatically natural. Then O X i−1 (E u )| X B i−1 \u is natural by Lemma 9.3.3, since its restriction to X i is natural (this last statement can be proved as the part '⇒' of Lemma 9.3.3).
Next, assume that u ∈ B i . Let j (where m + 1 ≥ j > i) be maximal so that u ∈ B j−1 . Then, similarly as in the previous case, O X i−1 (E u )| X B i−1 \u is natural whenever its restriction to X j is natural. (For j = m + 1 this reads as follows: all the components are rational, hence the restricted bundle is natural.) This follows from the WECC of X i .

Further topological/analytical characterizations of the WECC structure
In this subsection we will prove the following two statements: if a minimal elliptic graph supports an analytic structure with WECC then it necessarily supports also one with ECC. Even more, any analytic structure with WECC satisfies in fact ECC too.
We wish to separate sharply these two statements for the following reason. Recall that the existence of an analytic structure with ECC is topological: it exists if and only if the graph either satisfies the semigroup and congruence conditions of Neumann-Wahl [25], or the monomial condition of Okuma [29]. In this article we will use the monomial condition (for definition see below). Hence, the first statement basically is equivalent with the fact that a WECC elliptic singularity necessarily must satisfy the combinatorial monomial condition. (The other direction is already in the literature: the monomial condition assures the existence of a splice quotient analytic type [29], while splice quotients by their construction satisfy ECC, hence WECC too.) An immediate consequence of this is that the 'old' combinatorial criteria, namely the semigroup-congruence condition, or the monomial condition, for elliptic graph are equivalent with the existence of the extension criterion from Theorem 9.3.1 (which is much easier to test!).
The second part is analytical in nature, it says that in the elliptic case for any analytic structure already the WECC itself guarantees ECC. Recall that by the 'End Curve Theorem' [26,30] the ECC is equivalent with splice quotient analytic type. Hence, a consequence of our next theorem is that in the elliptic case the three notions-splice quotient, WECC, ECC-are equivalent.
Definition 9.4.1 [29] satisfies the monomial condition (MC) if for any node (rupture vertex) v and any connected full subgraph i of \v there exists an effective cycle C i supported on i such that (E * v + C i , E u ) = 0 for any u ∈ V( i ) ∪ {v}, which is not an end-vertex of sitting in V( i ).
[In fact, below, we will use only the 'melody' of this definition: MC is satisfied iff any node v and any i satisfy some combinatorial property, which not necessarily should be specified.]  Proof We will prove the two statements by simultaneous induction on the number of vertices |V|. For minimally elliptic or rational graphs the statements are true, because any minimally elliptic or rational singularity is splice quotient. Thus, assume that the statements are valid for graphs with less than k vertices, and assume, that |V| = k. We claim that it is enough to prove (1), because (1) implies (2). Indeed, if P(t) = l ∈S p(l )t l is the analytic multivariable Poincaré series then an analytic structure satisfies ECC if and only if p(E * v ) = 1 for every end vertex v (this follows basically from the definition of P). On the other hand, if a WECC analytic structure exists, then all of them have the same P(t) determined topologically, cf. Corollary 9.2.3(b). By part (1) a structure with ECC also exists, for which p(E * v ) = 1. Since ECC is WECC too, for all WECC structures p(E * v ) = 1. Hence any WECC is ECC. In the sequel we focus on part (1), where is an elliptic minimal resolution graph with |V| = k. We assume the existence of an analytic structure X with WECC (on ) and we wish to prove MC.
Assume that MC fails at a certain node v and branch 1 of \v. Denote by v 1 , . . . , v δ the adjacent vertices of v in with v 1 ∈ V( 1 ), δ ≥ 3. [In fact, by the inductive step, we can even assume that V( ) = V( 1 ) ∪ {v, v 2 , v 3 }, otherwise we take the subgraph with these vertices, it is WECC by restriction, cf. 9.1.1, it is ECC by induction, hence it satisfies MC at 1 , which is a contradiction. But this reduction does not really help in the next proof.] Besides 1 we will consider several graphs. v 1 denotes the full subgraph 1 ∪ {v} of . m 1 is obtained from v 1 by modifying the decoration of v by a very negative integer N 0. Furthermore, the 'extended-modified' me 1 is obtained from the full subgraph 1 ∪ {v, v 2 , . . . , v δ } of by replacing all decorations of {v, v 2 , . . . , v δ } by N .
We claim that me 1 is elliptic. Indeed, if we take subgraphs or we decrease decorations of an elliptic graph we get an elliptic or rational graph. However, me 1 has a node and branch Step 4. We fix (generic) sections s i ∈ H 0 (O X m 1 (−E * v )), 2 ≤ i ≤ δ. Write div(s i ) = C i , they are transversal cuts of E v in X m 1 with O X m in Pic( X m 1 ). Then we make along each C i an analytic plumbing by gluing a disc bundle over E v i with Euler number N to X m 1 such that E v i ∩ X m 1 = C i . In this way we get a resolution space X me 1 associated with me 1 .
In fact, for these analytic gluings, any transversal cut C works (instead of C i 's considered above). Indeed, by Step 2 we have v / ∈ B 0 ( X m 1 ). In particular, the image of the Abel map c −E * v (associated with a large cycle Z ⊂ L( X m 1 )) is a point, hence any C ∈ ECa −E * v ( X m 1 ) satisfies . This together with the construction from the proof of Theorem 9.3.1 also shows that X me 1 satisfies WECC. Therefore Corollary 9.1.4 applies and (R(l ))) (R is the restriction). (9.4.4) We claim that X me 1 satisfies ECC as well. Let us focus first on an end vertex u of me 1 different than v i (i ≥ 2). By Lemma 9.4.3 we know that the restriction H 1 (O X me is an isomorphism, hence we have a bijection Pic −E * u ( X me 1 ) → Pic E * u ( X m 1 ) too. This bijection together with O X m 1 (−E * u ) ∈ im(c −E * u ( X m 1 )) (ECC for X m 1 ) and (9.4.4) gives that O X me 1 (−E * u ) ∈ im(c −E * u ( X me 1 )), hence it gives ECC for X me 1 and for the vertex u. Finally, take the end vertex u = v i (i ≥ 2) of me 1 . Since X me 1 satisfied WECC and u / ∈ B 0 ( me 1 ), by Lemma 9.2.5 we get O X me This analytic structure satisfies ECC (e.g., since it is weighted homogeneous). The point is that it has a (positive weight, topological constant and p g -constant) hypersurface deformation (X t , o) = {x 11 = z 2 − y 4 + t yx 9 }, such that for t = 0 the germ (X t , o) is not a splice quotient. In other words, the topological type admits a splice quotient analytic structure, however, the hypersurface/Gorenstein analytic type (X t =0 , o) does not satisfy ECC. This can be verified similarly as in the case treated in [34, 3.2.12] and [26,Ex. 10.4] by checking that the deformation monomial cannot be realized by splice quotient equations. By our results, this analytic type does not satisfy WECC either. However, we do not know any 'elementary' method to verify this statement, not even for this particular example. (This shows that usually the 'direct' verification of WECC can be very hard.) Funding Open access funding provided by ELKH Alfréd Rényi Institute of Mathematics.
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