“Weakly” Elliptic Gorenstein Singularities of Surfaces

András Némethi^{1}

Department of Mathematics, The Ohio State University, Columbus OH 43210, USA, (e-mail: nemethi@math.ohio-state.edu)

1. Introduction.

Let p be a singularity of a normal two–dimensional analytic surface X. In general π : M → X will denote a resolution of (X, p), and A = π^{−1}(p). It is well–known that the dual resolution graph of the germ (X, p) determines the topology of the germ (X, p). Indeed, the (real) link of (X, p) can be reconstructed from the resolution graph as a plumbing. Conversely, by a result of W. Neumann [15], the topology of (X, p) determines the (minimal) resolution graph. Therefore, if an invariant of (X, p) depends only on the dual resolution graph, we say that it is a “topological invariant”. For example, the self– intersection Z_{num}^{2} of Artin’s (fundamental) cycle Z_{num}, or the Euler–characteristic χ(D) of any cycle D supported by A, can be determined from the graph.

Now, it is fascinating to investigate if an invariant, a priori defined from the analytic structure of (X, p), is topological or not. In this article, we ask this question for the geometric genus p_{g} = h^{1}(M, O_{M}) and the Hilbert–Samuel function of (X, p), in particular for the multiplicity mult(X, p) and the embedding dimension emb dim(X, p) = dim m_{p}/m_{p}^{2}. In general, these invariants are not topological, but if we restrict our study to some special classes, then they can be determined from the graph.

The first result of this type was obtained by M. Artin [1, 2] for rational singularities. He proved that they can be characterized topologically:

(X, p) is rational, i.e. p_{g} = 0 ⇔ χ(Z_{num}) = 1 ⇔ min_{D>0} χ(D) ≥ 1.

Moreover, for these singularities, the Hilbert–Samuel function (hence mult(X, p) and emb dim(X, p) too) can also be computed from the resolution graph.

The elliptic singularities were introduced by P. Wagreich in [24] (cf. “Terminology” at the end of the introduction). They are defined topologically:

(X, p) is elliptic ⇔ χ(Z_{num}) = 0 ⇔ min_{D>0} χ(D) = 0.

The class of elliptic singularities contains all the singularities with p_{g} = 1, and all the Gorenstein singularities with p_{g} = 2; but an elliptic singularity can have arbitrary high geometric genus (see e.g. the examples after Theorem C). The next step in the above program was obtained by H. Laufer [12]. He proved that the Gorenstein singularities with p_{g} = 1 (he called them minimally elliptic singularities) can be characterized topologically (cf. 2.7), and for these singularities all the above (a priori) analytical invariants are topological.

1Partially supported by NSF Grant No. DMS-9622724

Moreover, he noticed that singularities with p_{g} = 1 (without Gorenstein assumption), or even Gorenstein singularities with p_{g} = 2 (or p_{g} ≥ 2) cannot be characterized topologically. In this second case, one can easily construct pairs of hypersurface singularities with the same resolution graph but different p_{g} (cf. 2.22). But in all these examples (known by the author) either there is a non-rational exceptional divisor in the resolution, or the graph is not a tree, i.e. H^{1}(A, Z) ≠ 0. (Notice that H^{1}(A, Z) = 0 if and only if the link of (X, p) is a rational homology sphere.)

The main message of the present paper is that for Gorenstein singularities with H^{1}(A, Z) = 0 the Artin–Laufer program can be continued. Here we give the complete answer in the case of elliptic singularities.

Elliptic singularities were intensively studied by Wagreich [24], Laufer [12], S. S.-T. Yau [26, 27, 28, 29, 30, 31, 32] and others. In Yau’s papers, the fundamental topological invariant is the “elliptic sequence”. In the numerical Gorenstein case its definition is the following. Let π : M → X be the minimal resolution of (X, p), and A = π^{−1}(p) as above. Let Z_{K} be the canonical cycle (cf. §2). The elliptic sequence consists of a sequence {Z_{B_{j}}}_{j=0}^{m}, where Z_{B_{j}} is Artin’s (fundamental) cycle of B_{j} ⊂ A. We define {B_{j}}_{j} inductively as follows. For j = 0 take B_{0} = A hence Z_{B_{0}} = Z_{num}. Then Z_{K} ≥ Z_{B_{0}}. If Z_{K} > Z_{B_{0}} then we set B_{1} := |Z_{K} − Z_{B_{0}}|. Similarly, if B_{i} is already defined for any i ≤ j, then Z_{K} ≥ Z_{B_{0}} + ⋯ + Z_{B_{j}} (for details, see 2.10). If the inequality is strict then we define B_{j+1} := |Z_{K} − Z_{B_{0}} − ⋯ − Z_{B_{j}}|, otherwise we stop. The length of the elliptic sequence {Z_{B_{j}}}_{j=0}^{m} is m+1. The case m = 0 corresponds exactly to the minimally elliptic singularities of Laufer [12].

S. S.-T. Yau proved that for a numerical Gorenstein elliptic singularity p_{g} ≤ m + 1 ([32, (3.9)], cf. also 2.19). Particular examples show that strict inequality can occur.

In the first sections we give several characterizations of the “extremal property” p_{g} = m + 1. The most important characterization, from the point of view of the present manuscript, is the following:

Theorem A. Let (X, p) be an elliptic numerical Gorenstein singularity and set C′_{j} = ∑_{i≥j} Z_{B_{i}}. Then p_{g} = m + 1 if and only if the “obstruction line bundles” O_{C′_{j}}(−Z_{B_{j−1}}) are trivial for all 1 ≤ j ≤ m.

Now, the main technical result of the paper (see section 4) says:

Theorem B. Let (X, p) be an elliptic Gorenstein singularity. Fix an integer 1 ≤ k ≤ m and assume that for k + 1 ≤ j ≤ m the “obstruction line bundles” O_{C′_{j}}(−Z_{B_{j−1}}) are trivial. (If k = m then this assumption is vacuous.) Then O_{C′_{k}}(−Z_{B_{k−1}}) has finite order. (Actually, its order is not greater than k!.)

Now, if H^{1}(A, Z) = 0 then P ic(D) is torsion free for any positive cycle D. Hence, we obtain the main result of the paper:

Theorem C. Assume that (X, p) is an elliptic Gorenstein singularity with H^{1}(A, Z) = 0. Then p_{g} is a topological invariant. In fact p_{g} = m + 1 = the length of the elliptic sequence in the minimal resolution of (X, p).

Actually, one of the starting points of our investigation was S. S.-T. Yau’s result in [30], which says that hypersurface singularities with H^{1}(A, Z) = 0 and p_{g} = 2 satisfies m = 1 (cf. Remark 4.14). (In Yau’s terminology, a numerical Gorenstein elliptic singularity with p_{g} = m + 1 is called “maximally elliptic”.)

Examples. [24, 32] Set (X_{i}, 0) ⊂ (C^{3}, 0) given by (X_{1}, 0) = {z^{2} + y^{3} + x^{9+6m} = 0}, (X_{2}, 0) = {z^{2} + y^{3} + x^{11+6m} = 0}, (X_{3}, 0) = {z^{3} + y^{3} + x^{4+3m} = 0}, (X_{4}, 0) = {z^{3} + y^{3} + x^{5+3m} = 0} (where m ≥ 0). Then, in all these cases, H^{1}(A_{i}, Z) = 0, (X_{i}, 0) is elliptic, and p_{g} = m + 1 = the length of the elliptic sequence (1 ≤ i ≤ 4) (cf. also with 6.4.c).

In sections 5 and 6, we generalize Laufer’s results about the minimally elliptic singularities [12] (which corresponds to m = 0). Namely, we prove:

Theorem D. Assume that (X, p) is an elliptic Gorenstein singularity with p_{g} = m + 1. (If m = 0 or H^{1}(A, Z) = 0 then the last assumption is satisfied.) Let Z_{num} be Artin’s cycle in the minimal resolution (M, A) → (X, p). Then:

(a) If Z_{num}^{2} ≤ −2, then m_{p}O_{M} = O(−Z_{num}), hence mult(X, p) = −Z_{num}^{2};

(b) If Z_{num}^{2} = −1, then m_{p}O_{M} = m_{Q}O(−Z_{num}) for some smooth point Q of A, and mult(X, p) = 2;

(c) emb dim(X, p) = max(3, −Z_{num}^{2});

(d) H^{0}(M, O(−kZ_{num})) = m_{p}^{k} for any k ≥ 0 provided that Z_{num}^{2} ≤ −3;

(e) dim O_{(X,p)}/m_{p}^{k} = χ(O_{kZ_{num}}) + 1, and dim m_{p}^{k}/m_{p}^{k+1} = −kZ_{num}^{2} for any k ≥ 1 provided that Z_{num}^{2} ≤ −3; (for the cases Z_{num}^{2} = −1 or −2, see (6.4)).

Notice that Theorem D is not true if χ(Z_{num}) < 0. E.g. for (X, 0) = {x^{3} +y^{4} +z^{7} = 0} one can verify that χ(Z_{num}) = −1, H^{1}(A, Z) = 0, and Z_{num}^{2} = −2, but mult(X, 0) = 3.

Some partial results about elliptic singularities were obtained by S. S.-T. Yau in his series of papers (we will cite them in the body of the paper at the corresponding places). In order to have a self–contained presentation, we reprove those facts what we use from his work. In some cases we follow Yau’s original arguments, in other cases we give different proofs (original ones or arguments in the spirit of [19]). The article of M. Reid [19] was extremely helpful for the author. Actually, the proofs in sections 5 and 6 have their origins in [19]. One of the ideas of the proof of Theorem B was borrowed from J. Wahl’s paper [25]. On the other hand, we emphasize that almost all the classical arguments, used in the case of rational or minimally elliptic singularities, and based on some vanishing theorems or on the numerically 1– or 2–connectivity of Z_{num}, in our general situation fail, and we had to replace them by different arguments.

Some of the results of the present article about multiplicity and embedding dimension computations can be compared with the results of U. Karras and J. Stevens proved for Kodaira, respectively Kulikov singularities (for details, see [6, 7] and [21, 22]), and also with some results of M. Tomari [23].

Finally, we mention a consequence of a result of W. Neumann and J. Wahl [16]: the geometric genus of the hypersurface singularity {f(x, y)+z^{n} = 0} is topological, provided that its link is an integer homology sphere; moreover it can be related to the Casson invariant of the link. This paper influenced our research interest significantly.

The results of the present paper were presented at the Summer School organized in Budapest, Hungary, 1998 August. The written version of the lectures will appear in [18], where the interested reader can find a more detailed presentation and more examples.

Terminology: Singularities characterized by χ(Z_{num}) = 0 sometimes are called “weakly” elliptic. By this terminology one wants to emphasize the difference between these singularities and the “strongly” elliptic singularities, defined by p_{g} = 1; cf. also Yau’s papers. Notice also that in the terminology of M. Reid [19], “elliptic” means “minimally elliptic” in the sense of Laufer (cf. 2.7). In this article we will adopt the terminology used by Wagreich and Laufer.

The author wishes to thank École Polytechnique at Palaiseau, University of Nice and University of Nantes (especially Professors C. Sabbah, M. Merle and F. Elzein) for their hospitality and excellent working atmosphere.

2. Preliminaries.

We fix a normal surface singularity (X, p). Let π : M → X be its minimal resolution, and let A = ∪_{i}A_{i} be the decomposition of the exceptional set A into irreducible components. Let K be the canonical divisor on M, hence:

(2.1)  A_{i}K = −A_{i}^{2} − 2 + 2g_{i} + 2δ_{i}  for all i,

where g_{i} is the genus of A_{i}, and δ_{i} is the sum of all delta-invariants of the singular points (the “number of nodes and cusps”) on A_{i} [20].

In this paper all the cycles will be integer combinations of the A_{i}’s. If D = ∑_{i} n_{i}A_{i}, we write n_{i} = m_{A_{i}}(D). |D| denotes the support of D.

We denote Artin’s numerical (fundamental) cycle by Z_{num}, i.e. Z_{num} is the minimal positive cycle Z with ZA_{i} ≤ 0 for all i [1, 2]. (X, p) is called numerical Gorenstein if there is a cycle Z_{K} (called canonical cycle) which satisfies Z_{K}A_{i} = −KA_{i} for all i. If Z_{K} exists, then (by 2.1) Z_{K} is trivial if and only if (X, p) is a Du Val singularity. Otherwise m_{A_{i}}(Z_{K}) > 0 for all A_{i} (cf. [14], page 490). Moreover, if Z_{K} ≠ 0, then from (2.1) one has Z_{K}A_{i} ≤ 0 for all i, hence Z_{num} ≤ Z_{K}.

By a (Kodaira type) vanishing theorem (cf. [19], Exercise 15, page 119), for any cycle D ≥ 0 which satisfies A_{i}D ≤ 0 for all A_{i}, and any line bundle L with deg_{A_{i}}L ≥ KA_{i} for all i:

(2.2) \{ \begin{matrix}h^{1}(M, O(−D) ⊗ L) = 0, hence in the numerical Gorenstein case: \\ H^{0}(M, O(−D)) → H^{0}(O_{Z_{K}}(−D)) is onto. \end{matrix}

For any cycle D, we denote the Euler–characteristic h^{0}(O_{D}) − h^{1}(O_{D}) by χ(D). By Riemann–Roch theorem χ(D) = −D(D + K)/2. By [12], (2.6):

(2.3) H^{0}(O_{Z_{num}}) = C = {constants}, hence χ(Z_{num}) ≤ 1.

By a result of Artin [1, 2], the following facts are equivalent:

(2.4) (X, p) is rational (i.e. p_{g} = 0) ⇔ χ(Z_{num}) = 1 ⇔ min_{D>0} χ(D) ≥ 1.

The dual resolution graph of a rational singularity is a tree, and all the vertices corresponds to smooth rational curves.

In [24], Wagreich introduced the elliptic singularities. They are defined by the property min_{D>0} χ(D) = 0. In particular, by (2.3) and (2.4), χ(Z_{num}) = 0. The inverse implication is also true, see e.g. [12] (4.2), hence:

(2.5) (X, p) is elliptic ⇔ χ(Z_{num}) = 0 ⇔ min_{D>0} χ(D) = 0.

Elliptic singularities include all the singularities with p_{g} = 1 and all the Gorenstein singularities with p_{g} = 2 (cf. the proof of 4.13); but elliptic singularities can have arbitrary high geometric genus.

Following Laufer (Definition 3.1 [12]), we say that a cycle E > 0 is minimally elliptic if χ(E) = 0 and χ(D) > 0 for all cycles 0 < D < E. Laufer in [12] (3.2) proved that if χ(Z_{num}) = 0 then there exists a unique minimally elliptic cycle E. Hence, in the elliptic numerical Gorenstein case:

(2.6) E ≤ Z_{num} ≤ Z_{K}.

The minimally elliptic singularities are characterized by the following equivalent properties (cf. Laufer’s paper [12]) (notice that (b), (c) and (d) are topological properties; (d) provides the name of the singularity):

(2.7)  \{ \begin{matrix}(a) (X, p) is Gorenstein and p_{g} = 1; \\ (b) Z_{num} is a minimally elliptic cycle; \\ (c) (X, p) is numerical Gorenstein with Z_{K} = Z_{num}; \\ (d) χ(Z_{num}) = 0 and any connected proper subvariety of A \\ supports a rational singularity. \end{matrix}

2.8. If (X, p) is elliptic, then the topology of the irreducible exceptional divisors A_{i} cannot be very complicated. Wagreich in [24], page 428, proved that precisely one of the followings hold: (a) Precisely one component A_{i_{0}} satisfies χ(A_{i_{0}}) = 0 (which is either a smooth elliptic curve or a rational curve with δ_{i_{0}} = 1) and the other irreducible exceptional divisors are smooth rational curves. (In this case E = A_{i_{0}}). Or (b): all the exceptional divisors are smooth rational curves.

Actually, from the uniqueness of the minimally elliptic cycle it follows that all the connected components of A \ |E| support the exceptional set of rational singularities, hence the dual graph of (X, p) can be obtained from the dual graph of |E| by gluing trees whose vertices correspond to smooth rational curves. In particular, the restriction map

(2.9) H^{1}(A, Z) → H^{1}(|E|, Z) is an isomorphism.

In a series of papers S. S.-T. Yau investigated the properties of elliptic singularities (he called them “weakly elliptic”), cf. [26, 27, 28, 29, 30, 31, 32]. His main tools are the “computation sequence” introduced by Laufer in [10] and [12], and the “elliptic sequence”. The interested reader can find in Yau’s papers the definition of the elliptic sequence for an arbitrary elliptic singularity. In the non–numerical Gorenstein case the sequence has a lot of anomalies (see e.g. some examples on the page 881 of [31]), but in the numerical Gorenstein case it is a powerful tool. Since the definition in this later case is easier and more natural, and it is sufficient for our goals, we adopt the general definition for this situation. 2.10. The construction of the elliptic sequence. Consider the minimal resolution of an elliptic numerical Gorenstein singularity. The elliptic sequence consists of the sequence {Z_{B_{j}}}_{j=0}^{m}, where Z_{B_{j}} is the numerical (fundamental) cycle of B_{j} ⊂ A. We define {B_{j}}_{j} inductively as follows. For j = 0 take B_{0} = A hence Z_{B_{0}} = Z_{num}. By (2.6) E ≤ Z_{num} ≤ Z_{K}.

If Z_{B_{0}} < Z_{K}, we define B_{1} := |Z_{K} − Z_{num}|. Then B_{1} is connected. Indeed, assume that B_{1} has more connected components {B_{1,t}}_{t}; then write Z_{K} − Z_{num} = ∑_{t} D_{t} with |D_{t}| = B_{1,t}. Then the vanishing χ(Z_{K} − Z_{num}) = 0 and (2.5) imply that χ(D_{t}) = 0 for all t. Then by [12] (3.2), each B_{1,t} supports a minimally elliptic cycle. This contradicts the uniqueness of the minimally elliptic cycle E. Now, using again χ(Z_{num}) = 0, one gets Z_{num}(Z_{K} − Z_{num}) = 0. Since Z_{num}A_{i} ≤ 0 for all A_{i}, it follows that for any A_{i} ⊂ B_{1} the equality Z_{num}A_{i} = 0 holds. In particular, by the non-degeneracy of the intersection form, B_{1} ≠ B_{0}. Moreover, for any A_{i} ⊂ B_{1} one has: A_{i}(Z_{K} − Z_{B_{0}}) = A_{i}Z_{K}, hence B_{1} supports a numerical Gorenstein singularity with canonical cycle Z_{K} − Z_{B_{0}}. This singularity is again elliptic. Indeed, min_{D⊂B_{1}} χ(D) ≥ min_{D⊂B_{0}} χ(D) = 0, but χ(Z_{K} − Z_{num}) = 0, hence min_{D⊂B_{1}} χ(D) = 0. In particular, (cf. 2.6), E ≤ Z_{B_{1}} ≤ Z_{K} − Z_{B_{0}}.

Now we repeat the above arguments. If Z_{B_{1}} < Z_{K} − Z_{B_{0}}, we define B_{2} := |Z_{K} − Z_{B_{0}} − Z_{B_{1}}| and we verify that it supports an elliptic singularity with canonical cycle Z_{K} − Z_{B_{0}} − Z_{B_{1}} and for any A_{i} ⊂ B_{2} the vanishing A_{i}Z_{B_{1}} = 0 holds. If we repeat this procedure, after a finite step we will obtain Z_{B_{m}} = Z_{K} − Z_{B_{0}} − ⋯ − Z_{B_{m−1}}, i.e. the numerical cycle and the canonical cycle of B_{m} coincides, hence by (2.7) B_{m} supports a minimally elliptic singularity with E = Z_{B_{m}}. In particular B_{m} = |E|.

If we contract the connected exceptional curve B_{j} ⊂ M, we obtain a unique singular point; this will be denoted by (M/B_{j}, p_{j}) (0 ≤ j ≤ m). It is convenient to introduce the notations C_{t} = ∑_{i=0}^{t} Z_{B_{i}} and C′_{t} = ∑_{i=t}^{m} Z_{B_{i}} (0 ≤ t ≤ m) too. 2.11. Definition/first properties of the elliptic sequence:

(a) B_{0} = A, B_{1} = |Z_{K} − Z_{B_{0}}|, B_{2} = |Z_{K} − Z_{B_{0}} − Z_{B_{1}}|, …, B_{m} = |E|; each B_{j} is connected and the inclusions B_{j+1} ⊂ B_{j} are strict. Moreover, Z_{num} = Z_{B_{0}} ⊃ Z_{B_{1}} ⊃ ⋯ ⊃ Z_{B_{m}} = E.

(b) If A_{i} ⊂ B_{j+1} then A_{i}Z_{B_{j}} = 0 for all i and j. In particular, Z_{B_{i}}Z_{B_{j}} = 0 for all 0 ≤ i < j ≤ m.

(c) Z_{K} = ∑_{i=0}^{m} Z_{B_{i}} (cf. also [32, (3.7)]).

(d) A_{i}C′_{t} = A_{i}Z_{K} for any A_{i} ⊂ |C′_{t}|. In other words, C′_{t} is the canonical cycle of |C′_{t}| = B_{t} (i.e. of (M/B_{t}, p_{t})).

(e) For any A_{i} ⊂ A the inequality A_{i} · C_{t} ≤ 0 holds. Proof. a, b, c, d follows from the above construction. The proof of (e) is as follows. If A_{i} ⊂ B_{t} then A_{i}Z_{B_{j}} ≤ 0 for any j ≤ t by the definition of the numerical cycle, hence A_{i}C_{t} ≤ 0. If A_{i} ⊄ B_{t} then A_{i}C_{t} = A_{i}(Z_{K} − C′_{t+1}). Now, A_{i}Z_{K} ≤ 0 (by the minimality of the resolution) and A_{i}C′_{t+1} ≥ 0 (because |C′_{t+1}| ⊂ B_{t}). □

2.12. The next result shows that the properties (d) and (e) characterize the elliptic sequence. 2.13. Lemma. Assume that (X, p) is a numerical Gorenstein elliptic singularity.

(a) If a cycle Z ≥ 0 satisfies A_{i}(Z−Z_{K}) ≥ 0 for all A_{i} ⊂ |Z|, then in fact A_{i}(Z−Z_{K}) = 0 for all A_{i} ⊂ |Z|, and Z ∈ {C′_{0}, C′_{1}, …, C′_{m}, 0}.

(b) If a cycle 0 ≤ Z ≤ Z_{K} satisfies A_{i}Z ≤ 0 for all A_{i} ⊂ A, then Z ∈ {0, C_{0}, C_{1}, …, C_{m}}.

Proof. If A_{i}(Z − Z_{K}) ≥ 0, then Z(Z − Z_{K}) ≥ 0, hence χ(Z) ≤ 0. This together with (2.5) guarantees that χ(Z) = 0 and A_{i}(Z − Z_{K}) = 0 for all A_{i} ⊂ |Z|. Now, if |Z| = B_{0}, then by the non–degeneracy of the intersection form Z = Z_{K}. If |Z| ≠ B_{0}, then for A_{i} ⊂ |Z| obviously A_{i}(Z_{K} − Z) = 0; for A_{i} ⊄ |Z| one has A_{i}(Z_{K} − Z) ≤ 0 (because A_{i}Z_{K} ≤ 0 and A_{i}Z ≥ 0). Hence A_{i}(Z_{K} − Z) ≤ 0 for any A_{i}, therefore Z_{K} − Z ≥ Z_{B_{0}}, hence Z ≤ Z_{K} − Z_{B_{0}}. In particular, |Z| ⊂ B_{1}. If |Z| = B_{1}, then Z is the canonical cycle C′_{1} of B_{1}, otherwise, by the same argument as above Z ≤ Z_{K} − Z_{B_{0}} − Z_{B_{1}}, hence |Z| ⊂ B_{2}. Continuing the precess, (a) follows. For the second part, apply (a) for Z_{K} − Z. □

Since E ≤ Z_{B_{j}} ≤ Z_{num},

(2.14) h^{1}(O_{Z_{B_{j}}}) = 1 hence χ(Z_{B_{j}}) = χ(C_{j}) = χ(C′_{j}) = 0 for all 0 ≤ j ≤ m.

2.15. Since any Z_{B_{i}} is a numerical fundamental cycle, by [10] (proof of 4.1) or [12] (cf. also [32]), there is a “computation sequence” which can start with any of the irreducible exceptional divisors A_{j} ⊂ B_{i} and ends with Z_{B_{i}}. This means that there exists a sequence of cycles Z_{0}, …, Z_{k} with Z_{0} = A_{j}, Z_{k} = Z_{B_{i}}, Z_{l+1} = Z_{l} +A_{i_{l}} and A_{i_{l}}Z_{l} > 0 for 0 ≤ l ≤ k −1. Moreover, for any two integers 0 ≤ i < j ≤ m, since Z_{B_{i}} is a numerical cycle with Z_{B_{j}} < Z_{B_{i}}, there is a computation sequence which starts with Z_{B_{j}} and ends with Z_{B_{i}}. More precisely, there is a sequence Z_{0}, …, Z_{k} with Z_{0} = Z_{B_{j}}, Z_{k} = Z_{B_{i}}, Z_{l+1} = Z_{l} + A_{i_{l}} and A_{i_{l}}Z_{l} > 0 for 0 ≤ l ≤ k − 1. Actually in this second case, since h^{1}(O_{Z_{B_{i}}}) = h^{1}(O_{Z_{B_{j}}}) = 1 (cf. 2.14), for any 0 ≤ l ≤ k − 1 the curve A_{i_{l}} is smooth rational and A_{i_{l}}Z_{l} = 1 (see also [12] (2.7)). We will call a sequence like this a “computation sequence which connects Z_{B_{j}} and Z_{B_{i}}”.

Let D be a cycle with DA_{k} = 0 for any A_{k} ⊂ B_{i}, and consider a computation sequence {Z_{l}}_{l} which connects Z_{B_{j}} and Z_{B_{i}} (where j > i). In the exact sequence 0 → O_{A_{i_{l}}}(−D − Z_{l}) → O_{D+Z_{l+1}} → O_{D+Z_{l}} → 0 (0 ≤ l ≤ k − 1) the Chern number is A_{i_{l}}(−D − Z_{l}) = −1, hence the natural map

(2.16) H^{k}(O_{D+Z_{B_{i}}}) → H^{k}(O_{D+Z_{B_{j}}}) (k = 1, 2) is an isomorphism.

For example, if D = C_{i−1} and j = m, then

(2.17) H^{k}(O_{C_{i}}) → H^{k}(O_{C_{i−1}+E}) (1 ≤ i ≤ m; k = 1, 2) is an isomorphism.

Now consider the exact sequence

(2.18) 0 → O_{E}(−C_{i−1}) → O_{C_{i−1}+E} → O_{C_{i−1}} → 0.

Since E is 2–connected (see e.g. [19, (4.21)]), O_{E}(−C_{i−1}) is trivial if and only if it has a non-zero section (see e.g. [19], page 82), and in this case h^{k}(O_{E}(−C_{i−1})) = 1 for k = 1, 2. Therefore (2.17) and (2.18) imply h^{1}(O_{C_{i−1}}) ≤ h^{1}(O_{C_{i}}) ≤ h^{1}(O_{C_{i−1}}) + 1 for 1 ≤ i ≤ m. Since h^{1}(O_{C_{0}}) = 1, one obtains S. S.-T. Yau ’s result ([32, (3.9)]):

(2.19) h^{1}(O_{C_{i}}) ≤ i + 1 (for 0 ≤ i ≤ m), hence p_{g} = h^{1}(O_{C_{m}}) ≤ m + 1.

The next theorem gives some characterizations of the “extremal property” p_{g} = m+1.

2.20. First characterization of p_{g} = m + 1. Assume that (X, p) is a numerical Gorenstein elliptic singularity. Then the following facts are equivalent:

(a) p_{g} = m + 1;

(b) h^{1}(O_{C_{i}}) = i + 1 for all 0 ≤ i ≤ m;

(c) The line bundles O_{E}(−C_{j}) are trivial (in P ic(E)), and H^{0}(O_{C_{j}+E}) → H^{0}(O_{C_{j}}) is surjective for any 0 ≤ j ≤ m − 1;

(d) h^{1}(O(−C_{j})) = m − j for all 0 ≤ j ≤ m;

(e) h^{1}(O_{C′_{j}}) = m − j + 1 for all 0 ≤ j ≤ m;

(f) For any 0 ≤ j ≤ m − 1, there exists f_{j} ∈ H^{0}(M, O(−C_{j})), such that for any A_{l} ⊂ B_{j+1} the vanishing order of f_{j} on A_{l} is exactly m_{A_{l}}(C_{j});

(g) The line bundles O_{C′_{j+1}}(−C_{j}) are trivial for 0 ≤ j ≤ m − 1;

(h) The line bundles O_{C′_{j+1}}(−Z_{B_{j}}) are trivial for 0 ≤ j ≤ m − 1. Proof. a ⇔ b ⇔ c is just a reformulation of the above discussion (2.17-18-19). (Actually a ⇔ c can be found in [28] (2.4).)

d ⇒ a is easy: for j = 0 one has m = h^{1}(O(−Z_{num})) =^{(2.3)} h^{1}(O) − h^{1}(O_{Z_{num}}) =^{(2.5)}

p_{g} − 1. Conversely, for (a, b, c) ⇒ d, notice that H^{0}(O) → H^{0}(O_{C_{m}}) is onto by (2.2), and H^{0}(O_{C_{m}}) → H^{0}(O_{C_{j}}) is onto by part (c) and (2.17) applied several times. Therefore H^{0}(O) → H^{0}(O_{C_{j}}) is onto, hence h^{1}(O(−C_{j})) = h^{1}(O) − h^{1}(O_{C_{j}}) = m + 1 − j − 1.

e ⇒ a again is trivial (take j = 0). In order to prove a ⇒ e, we construct a sequence {Z_{l}}_{l}. We start with Z_{0} = Z_{K}. If Z_{l} > 0 is already constructed, then we define Z_{l+1} = Z_{l} − A_{i_{l}} as follows. If there is at least one A_{i} ⊂ |Z_{l}| with A_{i}(Z_{l} − Z_{K}) < 0 then take for A_{i_{l}} one of these A_{i}’s. Otherwise A_{i_{l}} is an arbitrary A_{i} ⊂ |Z_{l}|. In this second case, by (2.13), A_{i_{l}}(Z_{l} − Z_{K}) = 0 and Z_{l} ∈ {C′_{0}, …, C′_{m}}. If Z_{l} = 0 then we stop.

Now, from the exact sequence 0 → O_{A_{i_{l}}}(−Z_{l+1}) → O_{Z_{l}} → O_{Z_{l+1}} → 0 and Serre duality one gets that h^{1}(O_{Z_{l}}) = h^{1}(O_{Z_{l+1}}) in the first case, and in second case h^{1}(O_{Z_{l+1}}) = h^{1}(O_{Z_{l}}) − ϵ_{l}, where ϵ_{l} ∈ {0, 1}. Since h^{1} must drop exactly m + 1 times, we obtain that in the above sequence we reach all the cycles C′_{j} and at every time ϵ_{l} = 1.

a ⇒ f is proved in [32] (3.13). We present a short proof of it. Fix an arbitrary A_{k} ⊂ B_{j+1}. Consider a computation sequence {Z_{l}}_{l} of Z_{B_{j+1}} which starts with A_{k} (cf. 2.15). Since the Chern numbers A_{i_{l}}(−Z_{l} − C_{j}) < 0, using the sequences 0 → O_{A_{i_{l}}}(−Z_{l} − C_{j}) → O_{Z_{l+1}}(−C_{j}) → O_{Z_{l}}(−C_{j}) → 0, we obtain that β : H^{0}(O_{Z_{B_{j+1}}}(−C_{j})) → H^{0}(O_{A_{k}}(−C_{j})) is injective. Now, H^{0}(O_{A_{k}}(−C_{j})) = C. Indeed, if A_{k} ≈ P^{1} then H^{0}(O_{A_{k}}(−C_{j})) = H^{0}(O_{P^{1}}). If A_{k} ≉ P^{1} then A_{k} = E (by 2.8), and O_{A_{k}}(−C_{j}) is trivial by (c). Now, considers the diagram:

[figure: H^{0}(O(−C_{j})) ⟶^{α_{k}} H^{0}(O_{A_{k}}(−C_{j})); H^{0}(O(−C_{j})) ⟶^{α} H^{0}(O_{Z_{B_{j+1}}}(−C_{j})) → H^{1}(O(−C_{j+1})) →; H^{1}(O(−C_{j})) → H^{1}(O_{Z_{B_{j+1}}}(−C_{j})) → 0; =; β]

By part (d): h^{1}(O(−C_{t})) = m−t, hence by dimension argument h^{1}(O_{Z_{B_{j+1}}}(−C_{j})) ≠ 0. Since χ(O_{Z_{B_{j+1}}}(−C_{j})) = 0 and β is injective, we obtain that, in fact, β is an isomorphism, and α is onto; hence α_{k} is onto as well. Finally, the surjectivity of α_{k} for all A_{k} ⊂ B_{j+1} implies (f).

For f ⇒ b notice that the image of the elements 1, f_{0}, …, f_{i−1} are linearly independent in H^{0}(O_{C_{i}}), this together with (2.19) proves (b).

f ⇔ g. For any A_{l} ⊂ B_{j+1} consider the composed map

H^{0}(M, O(−C_{j})) →^{u} H^{0}(O_{C′_{j+1}}(−C_{j})) →^{v} H^{0}(O_{A_{l}}(−C_{j})).

By (2.2) u is onto. Hence v∘u is onto if and only if v is onto. Notice that h^{0}(O_{A_{l}}(−C_{j})) = 1 always (if (f) is true then see the proof of a ⇒ f; if (g) is true, then O_{A_{l}}(−C_{j}) is a trivial line bundle).

g ⇔ h is easy: O_{C′_{j+1}}(−C_{j}) = ⊗_{i≤j}O_{C′_{i+1}}(−Z_{B_{i}})|_{C′_{j+1}}, and O_{C′_{j+1}}(−Z_{B_{j}}) = O_{C′_{j+1}}(−C_{j})⊗ O_{C′_{j}}(−C_{j−1})|_{C′_{j+1}}^{−1}. □

2.21. Corollary. Assume that (X, p) is an elliptic Gorenstein singularity.

a) Then p_{g} = m + 1 if and only if the line bundles O_{C′_{j+1}}(−Z_{B_{j}}) are trivial for all 1 ≤ j ≤ m − 1.

b) If p_{g} = m + 1, then h^{1}(M, O(−lZ_{num})) = h^{1}(Z_{K}, O_{Z_{K}}(−lZ_{num})) = p_{g} − 1 for any l ≥ 1. Proof of (a). We have to prove that in the Gorenstein case the triviality of O_{C′_{1}}(−Z_{B_{0}}) follows from the triviality of the other line bundles. Indeed, if all the other line bundles are trivial, then (2.20.a ⇐ h), applied for (M/B_{1}, p_{1}), gives h^{1}(O_{C′_{1}}) = m. But the Gorenstein property implies that for any positive cycle D < Z_{K} the strict inequality h^{1}(O_{D}) < h^{1}(O_{Z_{K}}) = p_{g} holds (see e.g. [19], page 109). Hence h^{1}(O_{Z_{K}}) > h^{1}(O_{C′_{1}}) = m. Therefore p_{g} = m + 1. Now, we prove (b). Using (2.2) h^{1}(O(−lZ_{num})) = h^{1}(O_{Z_{K}}(−lZ_{num})) = h^{1}(O_{C′_{1}}(−lZ_{num})) for l ≥ 1. But the line bundle O_{C′_{1}}(−lZ_{num}) is trivial by (2.20.g), hence its first Betti–number is p_{g} − 1 by (2.20.e). □

In the next sections, (2.21.b) will replace some vanishing theorems which were used in the classical case of minimally elliptic singularities (i.e. when p_{g} − 1 = 0). 2.22. Definition. In the sequel, we call the line bundles O_{C′_{j+1}}(−Z_{B_{j}}) (0 ≤ j ≤ m − 1) “obstruction line bundles”.

The next example shows that the property p_{g} = m + 1 is not always true (even if we deal with Gorenstein singularities). 2.23. Example. Consider (X_{1}, 0) = {x^{2} + y^{3} + z^{18} = 0} ⊂ (C^{3}, 0) and (X_{2}, 0) = {z^{2} = y(x^{4} + y^{6})} ⊂ (C^{3}, 0) (cf. also [14, 9] and [32], page 291). Using the method of [9], Chapter II, (or the Appendix of [17]) one can verify that the minimal resolution graph in both cases is:

[figure: −1; −2; −2; [1]]

On the other hand, the geometric genus in the first case is p_{g} = 3, and in the second case is p_{g} = 2 (this can be verified using e.g. [13] or [17]).

Denote by A_{0}, A_{1}, A_{2} the irreducible exceptional divisors (starting from left). Then A_{0} = E = Z_{B_{2}}, A_{0} + A_{1} = Z_{B_{1}}, A_{0} + A_{1} + A_{2} = Z_{num}, and Z_{K} = 3A_{0} + 2A_{1} + A_{2} (hence m = 2). Moreover χ(Z_{num}) = 0, hence the singularities are elliptic. It is very instructive to check property (2.20.f) for these examples. For a function f : (X, 0) → (C, 0) we denote its divisor supported by A by (f)_{A} := ∑_{i} m_{A_{i}}(f ∘ π)A_{i}.

In the first case, Z_{num} = (z)_{A}. This shows that in (2.20.f) (a possible choice is) f_{0} = z and f_{1} = z^{2}. In the second case there is no function f_{0} with m_{A_{0}}(f_{0}) = m_{A_{0}}(Z_{num}) = 1, hence (2.20.f) fails (in particular, there is no function f with (f)_{A} = Z_{num}.)

3. Characterization of the trivial line bundles in P ic(C′_{j}).

As usual, for any positive cycle D, we denote the isomorphism classes of invertible sheaves over O_{D} by P ic(D) (= H^{1}(O_{D}^{∗})). The kernel of the degree map deg : P ic(D) → Z^{#{A_{i}|A_{i}⊂|D|}} defined by L → {deg_{A_{i}}L}_{i} is denoted by P ic^{0}(D).

If D = E is a minimally elliptic cycle, then

(3.1) \{ \begin{matrix}L ∈ P ic^{0}(E) is trivial ⇔ H^{0}(E, L) ≠ 0; \\ moreover, if L is trivial then h^{0}(E, L) = 1. \end{matrix}

This basically follows from the 2–connectivity of E (see e.g. [19] page 82). But, in general, it is not really easy to provide similar result for P ic^{0}(D). Fortunately, in the case of an elliptic singularity, for any 0 ≤ j ≤ m, P ic^{0}(Z_{B_{j}}) is as simple as P ic^{0}(E). Indeed, consider the “exponential cohomology sequences” (see, e.g. [5] page 49) of the cycles Z_{B_{j}} and E. Then the natural maps H^{1}(|Z_{B_{j}}|, Z) → H^{1}(|E|, Z) and H^{1}(O_{Z_{B_{j}}}) → H^{1}(O_{E}) are isomorphism (the first from 2.9, or by a similar argument; for the second use 2.16 for D = 0). Hence P ic^{0}(Z_{B_{j}}) → P ic^{0}(E) is an isomorphism as well. Moreover, if we consider a computation sequence {Z_{l}}_{l} which connects E = Z_{B_{m}} and Z_{B_{j}} (cf. 2.15), then from the exact sequences 0 → O_{A_{i_{l}}}(−Z_{l}) ⊗ L → L|Z_{l+1} → L|Z_{l} → 0 follows that H^{0}(Z_{B_{j}}, L) = H^{0}(E, L|_{E}). Therefore, for any line bundle L ∈ P ic^{0}(Z_{B_{j}}):

(3.2) \{ \begin{matrix}h^{0}(Z_{B_{j}}, L) ≤ 1, and \\ h^{0}(Z_{B_{j}}, L) = 1 if and only if L is trivial. \end{matrix}

3.3. Theorem. Assume that the elliptic numerical Gorenstein singularity (X, p) satisfies p_{g} = m + 1; and fix an integer 0 ≤ j ≤ m. Then the followings are equivalent.

a) L ∈ P ic^{0}(C′_{j}) is trivial;

b) h^{0}(C′_{j}, L) ≥ m − j + 1. Actually, if L is trivial, then h^{0}(C′_{j}, L) is exactly m − j + 1. Proof. a ⇒ b and the equality follow from (2.20; a ⇒ e) and (2.14). We prove b ⇒ a by descending induction over j. If j = m, then C′_{j} = E, hence (a) follows from (3.1). Assume that b ⇒ a is true for j + 1, and h^{0}(C′_{j}, L) ≥ m − j + 1. Consider the exact sequence

0 → O_{C′_{j+1}}(−Z_{B_{j}}) ⊗ L → L → L|Z_{B_{j}} → 0.

Then h^{0}(L|Z_{B_{j}}) ≤ 1 by (3.2), hence h^{0} := h^{0}(O_{C′_{j+1}}(−Z_{B_{j}}) ⊗ L) ≥ m − j. Hence, by the inductive step O_{C′_{j+1}}(−Z_{B_{j}}) ⊗ L is trivial in P ic^{0}(C′_{j+1}), and h^{0} = m − j. Therefore h^{0}(L|Z_{B_{j}}) must be 1, L|Z_{B_{j}} must be trivial, and r : H^{0}(L) → H^{0}(L|Z_{B_{j}}) onto. In particular, there exists s ∈ H^{0}(L) such that r(s) generates O(L|Z_{B_{j}}). Since |Z_{B_{j}}| = |C′_{j}|, it follows that s generates the sections of L as well. □ 3.4. Corollary. Assume that (X, p) is an elliptic Gorenstein singularity, and we fix an integer 0 ≤ j ≤ m. Assume that h^{1}(O_{C′_{j}}) = m − j + 1. Then O_{C′_{j}}(−C_{j−1}) is trivial in P ic^{0}(C′_{j}).

For example, for m = j we obtain that O_{E}(−C_{m−1}) is trivial in P ic^{0}(E) (the assumption h^{1}(O_{E}) = 1 is automatically satisfied). For j = m − 1, if contracting B_{m−1} one has p_{g}((M/B_{m−1}, p_{m−1})) = 2, then O_{C′_{m−1}}(−C_{m−2}) is trivial in P ic^{0}(C′_{m−1}). Proof. First notice that contracting B_{j} we obtain a numerical Gorenstein singularity with canonical cycle C′_{j} (cf. 2.12), and with the “extremal property” p_{g}(M/B_{j}, p_{j}) = length of the elliptic sequence of B_{j} = m − j + 1. Therefore, (3.3) for this singularity gives: L ∈ P ic^{0}(C′_{j}) is trivial ⇔ h^{0}(C′_{j}, L) = m − j + 1. But by Serre duality h^{0}(O_{C′_{j}}(−C_{j−1}) = h^{1}(O_{C′_{j}}) = m − j + 1. □

3.5. Second characterization of p_{g} = m + 1. Assume that (X, p) is a numerical Gorenstein elliptic singularity. Consider its minimal resolution M → X and the singularity (M/B_{j}, p_{j}) obtained by the contraction of B_{j} (0 ≤ j ≤ m). Then (X, p) satisfies the extremal property p_{g} = m + 1 if and only if the singularities (M/B_{j}, p_{j}) are Gorenstein for all 0 ≤ j ≤ m − 1. (Notice that (M/B_{m}, p_{m}) is automatically Gorenstein by 2.7.) Proof. “⇒” First we prove that a numerical Gorenstein singularity (X, p) with p_{g} = m+1 is Gorenstein. (This fact was proved by S. S.-T. Yau in [32, (3.11)]; we present here a very short proof in the spirit of [19, (4.21)]; cf. also with the papers of L. Bădescu [3, 4].)

Consider the line bundle L := O_{Z_{K}}(Z_{K} + K) ∈ P ic^{0}(Z_{K}). By Serre duality h^{0}(L) = h^{1}(O_{Z_{K}}), hence h^{0}(L) = p_{g} = m + 1. Therefore, by (3.3), L is trivial. Moreover, by (2.2) h^{1}(O(K)) = 0, hence H^{0}(M, O(Z_{K} + K)) → H^{0}(L) ≈ H^{0}(O_{Z_{K}}) is onto. Therefore, there is a global section of O(Z_{K} + K) which has no zeros in the neighbourhood of A, hence Z_{K} + K is linearly equivalent to the zero cycle. But this is one of the characterizations of the Gorenstein property.

Now, M/B_{j} is an elliptic numarical Gorenstein singularity with numerical cycle C′_{j} and the length of its elliptic sequence= h^{1}(O_{C′_{j}}) = m − j + 1 (cf. 2.11 and 2.20.e). Therefore, the above fact applied for the singularities M/B_{j} ends the proof.

“⇐” By (2.20.h), it is enough to prove that the obstruction line bundles L_{j} := O_{C′_{j}}(−Z_{B_{j−1}}) are trivial for 1 ≤ j ≤ m. We will prove this by induction. Assume that for a fixed 1 ≤ j_{0} ≤ m the line bundles L_{j} are trivial for j > j_{0} (if j_{0} = m then this assumption is vacuous). Since the singularity M/B_{j_{0}−1} is Gorenstein, by (2.20.h) and (2.21.a) (applied for M/B_{j_{0}−1}), the triviality of {L_{j}}_{j>j_{0}} implies the triviality of L_{j_{0}}. □

3.6. Examples. Assume that (X, p) is a numerical Gorenstein elliptic singularity. (a) If m = 1 then p_{g} ≤ 2, and:

p_{g} = 2 ⇔ O_{E}(−Z_{num}) is trivial ⇔ (X, p) is Gorenstein.

Examples with p_{g} = 1 exists (actually this is the generic case!). E.g. take the minimal resolution of any Gorenstein singularity with p_{g} = 2 and m = 1. Then deform its analytic structure. Generically we obtain a non–Gorenstein singularity with p_{g} = 1 (cf. [12], page 1279).

(b) Assume that m = 2. Then in the numerical Gorenstein case we have two obstruction line bundles. For the generic analytic structure p_{g} = 1 as above. If (X, p) is Gorenstein, then p_{g} = h^{1}(O_{Z_{K}}) > h^{1}(O_{E}) = 1, hence p_{g} ≥ 2. From (2.21.a), p_{g} = 3 if and only if O_{E}(−Z_{B_{1}}) is trivial. From (3.4) O_{E}(−Z_{num} − Z_{B_{1}}) is trivial, hence if (X, p) is Gorenstein, then:

p_{g} = 3 ⇔ O_{E}(−Z_{num}) is trivial ⇔ (M/B_{1}, p_{1}) is Gorenstein.

4. Torsion properties of the “obstruction line bundles”.

The main result of this section is the following: 4.1. Theorem. Consider the minimal resolution of an elliptic Gorenstein singularity (X, p) with elliptic sequence {Z_{B_{j}}}_{j=0}^{m} (m ≥ 1), and fix an integer 1 ≤ k ≤ m.

Assume that for k + 1 ≤ j ≤ m the “obstruction line bundles” O_{C′_{j}}(−Z_{B_{j−1}}) ∈ P ic^{0}(O_{C′_{j}}) are trivial. (If k = m then this assumption is vacuous.) Then O_{C′_{k}}(−Z_{B_{k−1}}) ∈ P ic^{0}(O_{C′_{k}}) has finite order.

Actually, there exist integers 1 ≤ l_{r} ≤ k − r (0 ≤ r ≤ k − 1) such that for any 0 ≤ r ≤ k − 1 the line bundle O_{C′_{k}}(l_{0} ⋯ l_{r} Z_{B_{r}}) is trivial. In particular, the order of O_{C′_{k}}(−Z_{B_{k−1}}) in P ic^{0}(O_{C′_{k}}) is not greater than k!.

Proof. First assume that k = 1. Then O_{C′_{1}}(−Z_{B_{0}}) is trivial by (2.21.a) and (2.20.a ⇔ h). So, in the sequel we will assume that k ≥ 2.

Since (M/B_{k}, p_{k}) is a numerical Gorenstein singularity with canonical class C′_{k} (cf. 2.11.d), the assumption together with (2.20) (applied for this singularity) provide that:

(4.2) h^{1}(O_{C′_{k}}) = m − k + 1.

4.3. Lemma. For any 0 ≤ r ≤ t ≤ k − 2 one has h^{1}(O(−C_{t} − Z_{B_{r}})) ≥ m − k + 1. Proof. Using (2.2), Serre duality, (2.14), and in the last step C′_{t+2} ≥ C′_{k}, one has:

h^{1}(O(−C_{t+1})) =^{2.2} h^{1}(O_{C′_{t+2}}(−C_{t+1})) =^{SD} h^{0}(O_{C′_{t+2}}) =^{2.14} h^{1}(O_{C′_{t+2}}) ≥ h^{1}(O_{C′_{k}}). Hence, by

(4.2):

(4.4) h^{1}(O(−C_{t+1}) ≥ m − k + 1.

Now, consider a computation sequence {Z_{l}}_{l} which connects Z_{B_{t+1}} with Z_{B_{r}} (cf. 2.15) (i.e. Z_{l+1} = Z_{l} + A_{i_{l}} where A_{i_{l}} is smooth rational curve with A_{i_{l}}Z_{l} = 1). In the exact sequence 0 → O(−C_{t} − Z_{l+1}) → O(−C_{t} − Z_{l}) → O_{A_{i_{l}}}(−C_{t} − Z_{l}) → 0 the Chern number A_{i_{l}}(−C_{t} −Z_{l}) is −A_{i_{l}}C_{t} −1 ≥ −1 (cf. 2.11.e). Hence H^{1}(O(−C_{t} −Z_{l+1})) → H^{1}(O(−C_{t} − Z_{l})) is onto for any l, in particular, h^{1}(O(−C_{t} −Z_{B_{r}})) ≥ h^{1}(O(−C_{t} −Z_{B_{t+1}})). The lemma follows from this and (4.4). □ Now consider the exact sequence:

(4.5) 0 → O(−Z_{K} − Z_{B_{r}}) → O(−C_{t} − Z_{B_{r}}) → O_{C′_{t+1}}(−C_{t} − Z_{B_{r}}) → 0.

4.6. Lemma. H^{1}(O(−Z_{K} − Z_{B_{r}}) = 0. Proof. Consider a computation sequence {Z_{l}}_{l} which connects Z_{B_{r}} and Z_{B_{0}} = Z_{num} (cf. 2.15). Consider the exact sequences:

0 → O(−Z_{K} − Z_{l+1}) → O(−Z_{K} − Z_{l}) → O_{A_{i_{l}}}(−Z_{K} − Z_{l}) → 0

and notice that A_{i_{l}}(−Z_{K} − Z_{l}) = −A_{i_{l}}Z_{K} − 1 ≥ −1 (cf. 2.1). Hence H^{1}(O(−Z_{K} − Z_{num})) → H^{1}(O(−Z_{K} − Z_{B_{r}})) is onto. But H^{1}(O(−Z_{K} − Z_{num})) = 0 (by 2.2). □ Now, from (4.5) and duality:

H^{1}(O(−C_{t} − Z_{B_{r}})) = H^{1}(O_{C′_{t+1}}(−C_{t} − Z_{B_{r}})) = H^{0}(O_{C′_{t+1}}(C′_{t+1} + C_{t} + Z_{B_{r}} − Z_{K})).

Hence (4.3) reads as follows: for 0 ≤ r ≤ t ≤ k − 2:

h^{0}(O_{C′_{t+1}}(C′_{t+1} + C_{t} + Z_{B_{r}} − Z_{K})) ≥ m − k + 1.

Now, for any positive cycle 0 < D ≤ C′_{t+1} write o(D) := O_{D}(D + C_{t} + Z_{B_{r}} − Z_{K}). For any irreducible exceptional divisor A_{i} ⊂ |D|, one has the exact sequence:

0 → o(D − A_{i}) → o(D) → O_{A_{i}}(D + C_{t} + Z_{B_{r}} − Z_{K}) → 0.

The Chern number A_{i}(D + C_{t} + Z_{B_{r}} − Z_{K}) = A_{i}(D − Z_{K}), for A_{i}Z_{B_{r}} = A_{i}C_{t} = 0. Therefore, if A_{i}(D − Z_{K}) < 0, then H^{0}(o(D − A_{i})) → H^{0}(o(D)) is an isomorphism. In the sequel, we will construct a decreasing sequence of cycles as follows. The starting cycle is D = C′_{t+1}. If D is already constructed, and there exists A_{i} ⊂ |D| with A_{i}(D − Z_{K}) < 0, then the next term is D − A_{i}. During this step h^{0}(o(D)) will stay constant.

If at a moment we reach a cycle D > 0 with the property A_{i}(D − Z_{K}) ≥ 0 for all A_{i} ⊂ |D|, then by (2.13) in fact A_{i}(D − Z_{K}) = 0 for all A_{i} ⊂ |D| and D must be one of the cycles C′_{s}. If there is a A_{i} ⊂ |D| such that

r_{i} : H^{0}(O_{D}(D + C_{t} + Z_{B_{r}} − Z_{K})) → H^{0}(O_{A_{i}}(D + C_{t} + Z_{B_{r}} − Z_{K}))

is the trivial map, then the term following D is D − A_{i} for this A_{i}. Hence, in this case again, h^{0}(o(D)) = h^{0}(o(D − A_{i})). If the above map r_{i} is nontrivial for all A_{i} ⊂ |D|, then the term following D is D − A_{i}, where A_{i} arbitrary with A_{i} ⊂ |D|. In this later case r_{i} has rank one, hence h^{0}(o(D)) will decrease by one. If D = 0 we stop.

Since finally h^{0}(o(D)) must decrease to zero, and h^{0}(o(C′_{t+1})) ≥ m − k + 1, h^{0}(o(D)) must decrease at least m − k + 1 times. Therefore, the first time when we decrease h^{0} must happen for a cycle D = C′_{s} with s ≤ k (i.e. the length of the sequence {C′_{s}, …, C′_{m}} must be at least m−k+1). Obviously C′_{s} ≤ C′_{t+1} (because we deal with cycles D ≤ C′_{t+1}), hence: s ≥ t + 1.

This shows that there exists t + 1 ≤ s ≤ k such that

H^{0}(O_{C′_{s}}(C′_{s} + C_{t} + Z_{B_{r}} − Z_{K})) → H^{0}(O_{A_{i}}(C′_{s} + C_{t} + Z_{B_{r}} − Z_{K})) = C

is onto for all A_{i} ⊂ |C′_{s}|. In other words, O_{C′_{s}}(C′_{s} + C_{t} + Z_{B_{r}} − Z_{K}) is a trivial line bundle. Taking its restriction to C′_{k}, we obtain:

(4.7)  \{ \begin{matrix}for any pair (r, t) with 0 ≤ r ≤ t ≤ k − 2, \\ there exists s with t + 1 ≤ s ≤ k, such that \\ O_{C′_{k}}(C′_{s} + C_{t} + Z_{B_{r}} − Z_{K}) is a trivial line bundle in P ic^{0}(C′_{k}). \end{matrix}

4.8. Lemma. For any 0 ≤ r ≤ k − 1, there exists l_{r} with 1 ≤ l_{r} ≤ k − r, such that

O_{C′_{k}}(l_{0} ⋯ l_{r} Z_{B_{r}}) is trivial in P ic^{0}(C′_{k}).

Proof. First we prove for r = 0. If there is at least one s in (4.7) with s = t + 1, then C′_{s} + C_{t} − Z_{K} = 0, hence O_{C′_{k}}(Z_{B_{0}}) is trivial. Now, assume that s ≥ t + 2 for any t. This means that for any 0 ≤ t ≤ k − 2, there exists t + 2 ≤ s(t) ≤ k such that:

(∗_{t}) O_{C′_{k}}(Z_{B_{0}}) ≈ O(Z_{B_{t+1}} + ⋯ + Z_{B_{s(t)−1}}).

Now, we set t_{1} := 0. If s(t_{1}) = k we stop; otherwise consider t_{2} := s(t_{1}) − 1. If s(t_{2}) = k we stop, otherwise we continue until we obtain s(t_{u}) = k for some u ≥ 1. If we multiply the identities (∗_{t}) for t_{1}, …, t_{u}, we obtain:

O_{C′_{k}}(uZ_{B_{0}}) ≈ O_{C′_{k}}(Z_{B_{1}} + ⋯ + Z_{B_{k−1}}).

Here u satisfies 1 ≤ u ≤ k − 1 (u = k − 1 exactly when s(t) = t + 2 for every t, i.e. when we have to consider all the isomorphisms (∗_{t})). But from (3.4) and (4.2):

(4.9) O_{C′_{k}}(Z_{B_{0}} + ⋯ + Z_{B_{k−1}}) is trivial,

hence with the notation l_{0} := u + 1 one has O_{C′_{t}}(l_{0}Z_{B_{0}}) is trivial.

Now, take r = 1 and consider again (4.7). If s = t+1 for at least one t, then O_{C′_{k}}(Z_{B_{1}}) is trivial. If s ≥ t + 2 for all 1 ≤ t ≤ k − 2, by a similar argument as above one obtains that O_{C′_{k}}(uZ_{B_{1}}) ≈ O_{C′_{t}}(Z_{B_{2}} + ⋯ + Z_{B_{k−1}}) for some 1 ≤ u ≤ k − 2 (u = k − 2 occurs exactly when we have to consider all the identities corresponding to 1 ≤ t ≤ k − 2). This together with (4.9) gives that O_{C′_{k}}(Z_{B_{0}} + (u + 1)Z_{B_{1}}) is trivial. Now consider its l_{0}^{th} power and take l_{1} := u + 1.

Similar argument can be used for all 0 ≤ r ≤ k − 2. For r = k − 1, just take (4.9) again and take its l_{0} ⋯ l_{k−2}-th power. □

4.10. Example. (cf. 3.6.b) Assume that (X, p) is elliptic Gorenstein, and m = 2. Recall that O_{E}(−Z_{B_{1}} − Z_{num}) ≈ O_{E}. The above theorem (for m = k = 2) says that the order of O_{E}(−Z_{num}) divides 2, hence:

O_{E}(−Z_{num}) is trivial ⇔ p_{g} = 3; O_{E}(−Z_{num}) has order 2 ⇔ p_{g} = 2.

One of the main results of the present paper is the following:

4.11. Theorem. Assume that (X, p) is an elliptic Gorenstein singularity with H^{1}(A, Z) = 0. Then p_{g} = m + 1 = the length of the elliptic sequence in the minimal resolution of (X, p). In particular p_{g} is a topological invariant. Proof. If H^{1}(A, Z) = 0, then by the exponential exact sequence ([5], page 49) P ic^{0}(D) ≈ H^{1}(O_{D}) for any positive cycle D, in particular, it is torsion free. Therefore, (4.1) (as an inductive step) proves that all the obstruction line bundles are trivial. Hence p_{g} = m + 1 by (2.20). □

4.12. Remark. If m ≥ 1, then in (4.11) both assumptions are necessary. Without Gorenstein condition p_{g} generically is one ([12, page 1279]). Moreover, Example (2.23) shows that p_{g} < m + 1 can occur if H^{1}(A, Z) ≠ 0. 4.13. Corollary. For a Gorenstein singularity (X, p) with H^{1}(A, Z) = 0 the following facts are equivalent:

p_{g} = 2 ⇔ χ(Z_{num}) = 0 and Z_{K} = Z_{num} + E.

(Notice that the right hand side is completely topological.) Proof. Indeed, if (X, p) is Gorenstein with p_{g} = 2, then h^{1}(Z_{num}) cannot be zero (because of 2.4), cannot be greater than or equal to two (that whould imply that Z_{num} = Z_{K}, which characterise the minimally elliptic singularities, cf. 2.7), hence it is one. Therefore (X, p) is elliptic. The rest follows from the above results. □ 4.14. Remark. S. S.-T. Yau in [32], Theorem B, proved that a Gorenstein singularity with p_{g} = 2 is elliptic. Moreover, in [30], he proved also that a hypersurface singularity with H^{1}(A, Z) = 0 and p_{g} = 2 satisfies m = 1. His proof (of this second fact) is based on the classification of all possible dual resolution graphs of hypersurface singularities with p_{g} = 2 (250 cases).

Notice that the “second characterization” (3.5) gives: 4.15. Corollary. If (X, p) is an elliptic Gorenstein singularity with H^{1}(A, Z) = 0 then the singularities (M/B_{j}, p_{j}) are Gorenstein for all 0 ≤ j ≤ m.

5. The multiplicity.

5.1. Assume that (X, p) is an elliptic Gorenstein singularity with p_{g} = m + 1. In this section we will prove that its multiplicity mult(X, p) is a topological invariant, in fact depends only on Z_{num}^{2}.

First notice that by (2.20.f) there exists f_{0} ∈ H^{0}(M, O(−Z_{num})) such that for any A_{l} ⊂ B_{1}, the order of vanishing of f_{0} on A_{l} is exactly m_{A_{l}}(Z_{num}). In particular:

(5.2) H^{0}(M, O(−Z_{num})) → H^{0}(O_{A_{l}}(−Z_{num})) = C is onto for all A_{l} ⊂ B_{1}.

By (2.2), if |O_{Z_{K}}(−Z_{num})| is free, then |O(−Z_{num})| is free too. By [24, page 426], if |O(−Z_{num})| is free, then m_{p}O_{M} = O(−Z_{num}), in which case mult(X, p) = −Z_{num}^{2}. In particular:

(5.3) Z_{num}^{2} = −1 ⇒ −Z_{num}^{2} ≠ mult(X, p) ⇒ |O_{Z_{K}}(−Z_{num})| is not free.

If |O(−Z_{num})| is not free, then mult(X, p) > −Z_{num}^{2}. E.g. if m_{p}O_{M} = m_{Q}O(−Z_{num}), where m_{Q} is the maximal ideal of a smooth point Q of A, then mult(X, p) = −Z_{num}^{2} + 1.

The next theorem generalizes Laufer’s result about the multiplicity of minimally elliptic singularities [12].

5.4. Theorem. Assume that (X, p) is an elliptic Gorenstein singularity with p_{g} = m + 1. (If m = 0 or H^{1}(A, Z) = 0 then the last assumption is satisfied.) Then:

If Z_{num}^{2} ≤ −2, then m_{p}O_{M} = O(−Z_{num}), hence mult(X, p) = −Z_{num}^{2};

If Z_{num}^{2} = −1, then m_{p}O_{M} = m_{Q}O(−Z_{num}) for some smooth point Q of A, and mult(X, p) = 2.

We prove theorem (5.4) in several steps. First we prove the converse of (5.3). For simplicity, in the sequel we will write L := O_{Z_{K}}(−Z_{num}).

5.5. Proposition. Assume that |L| is not free. Then Z_{num}^{2} = −1.

Proof. By (5.2), all the basepoints must lie outside of B_{1}. Since all the connected components of \overline{A \ B_{1}} are trees whose vertices correspond to smooth rational curves (cf. 2.8), we can fix a basepoint Q such that Q is “as close to B_{1} as possible”. More precisely, we will fix a basepoint Q such that there exist irreducible exceptional divisors {A′_{i}}_{i=0}^{l} with

(5.6)  \{ \begin{matrix}A′_{i} ⊄ B_{1} and A′_{i} is smooth rational curve (0 ≤ i ≤ l); \\ Q ∈ A′_{0}, A′_{i}A′_{i+1} = 1 for 0 ≤ i ≤ l − 1, and A′_{l}B_{1} = 1; \\ there is no basepoint of |L| on ∪_{i>0}A′_{i}. \end{matrix}

With this choice of Q, A′_{0}Z_{num} cannot be zero. Indeed, since Q is a basepoint, the evaluation map H^{0}(L) ⟶^{α} H^{0}(O_{A′_{0}}(−Z_{num})) ⟶^{e} C_{Q} is trivial. If, by assumption, A′_{0}Z_{num} = 0 then e must be an isomorphism, hence α must be trivial as well. This shows that A′_{0} ∩ A′_{1} is also a basepoint, which contradicts the above choice of Q in (5.6). For later reference:

(5.7) A′_{0}Z_{num} < 0.

Since the evaluation map H^{0}(Z_{K}, L) → C_{Q} is zero, from the exact sequence 0 → m_{Q}L → L → C_{Q} → 0 and (2.21.b), one gets h^{1}(m_{Q}L) = h^{1}(L) + 1 = p_{g}.

In the next paragraph, we follow the beginning of the proof of (4.23) of [19]. Take 0 < D ≤ Z_{K} minimal with the property h^{1}(D, m_{Q}L|_{D}) = p_{g}. Minimality means:

h^{1}(D − A_{i}, m_{Q}L|_{D−A_{i}}) ≤ p_{g} − 1 for any A_{i} ⊂ |D|.

Let K_{i} be the kernel of m_{Q}L|_{D} → m_{Q}L|_{D−A_{i}}. Outside Q, clearly K_{i} = L⊗O_{A_{i}}(−D+A_{i}). Actually, if Q ∈ |D − A_{i}|, then even at Q one has K_{i} = L ⊗ O_{A_{i}}(−D + A_{i}). If Q ∈ |D| but Q ∉ |D − A_{i}| (i.e. if Q ∈ A_{i} \ {other components of D} and m_{A_{i}}(D) = 1), then K_{i} = L ⊗ m_{Q}O_{A_{i}}(−D + A_{i}). We define δ_{i} := 0 in the first two cases, and δ_{i} := 1 in the third case.

Now, the minimality of D implies that for all i: h^{1}(D, K_{i}) ≠ 0, in other words:

(5.8) A_{i}L + A_{i}(Z_{K} − D) ≤ δ_{i} for all A_{i} ⊂ |D|.

5.9. Lemma. The situation when A_{i}L + A_{i}(Z_{K} − D) = 0 for all A_{i} ⊂ |D| cannot occur. Proof. Assume the contrary. Since A_{i}L ≥ 0 for all A_{i}, and χ(D) ≥ 0, the relation

0 ≤ D(Z_{K} − D) = ∑ m_{A_{i}}(D)A_{i}(Z_{K} − D) = − ∑ m_{A_{i}}(D)A_{i}L ≤ 0

says that A_{i}L = A_{i}(Z_{K} − D) = 0 for all A_{i} ⊂ |D|. In particular, D = C′_{s} for some 0 ≤ s ≤ m (by 2.13), and DL = C′_{s}(−Z_{num}) = 0. Since C′_{0}(−Z_{num}) = −Z_{num}^{2} ≠ 0, we must have s ≥ 1, i.e. |D| ⊂ B_{1}. On the other hand, (5.2) guarantees that Q ∉ B_{1}, hence Q ∉ |D|. Therefore, m_{Q}L|_{D} = O_{C′_{s}}(−Z_{num}) for some s ≥ 1. Using (3.3) we obtain h^{1}(O_{C′_{s}}(−Z_{num})) = h^{0}(O_{C′_{s}}(−Z_{num})) ≤ m − s + 1 ≤ p_{g} − 1, which contradicts h^{1}(D, m_{Q}L|_{D}) = p_{g}. □

The above lemma implies that there exists A_{i_{0}} ⊂ |D| with A_{i_{0}}L + A_{i_{0}}(Z_{K} − D) = δ_{i_{0}} = 1. (In particular, either Q = A_{i_{0}} ∩ A′_{0} and A′_{0} ⊄ |D|, or A_{i_{0}} = A′_{0}, cf. 5.6. The first case will be eliminated later.) Summing (5.8) over all components of D (recall m_{A_{i_{0}}}(D) = 1) we obtain DL + D(Z_{K} − D) = 1. Since DL ≥ 0 (by the definition of Z_{num}) and D(Z_{K} − D) is an even non–negative integer (by Riemann–Roch), we get:

(5.10) DL = 1 and D(Z_{K} − D) = 0.

If we write D = ∑ D_{i}, where {|D_{i}|}_{i} are the the connected components of |D|, since χ(D)=0 and χ(D_{i}) ≥ 0 for all i (cf. 2.5), we must have χ(D_{i}) for all i. This shows that no D_{i} lies in \overline{A \ B_{1}} (for all the components of \overline{A \ B_{1}} support rational singularities whose positive cycles have χ > 0, cf. 2.8 and 2.4). Therefore, all the exceptional divisors {A′_{i}}_{i=0}^{l} defined in (5.6) must be components of |D| and A′_{0} = A_{i_{0}}. Then (5.7) reads as

(5.11) A_{i_{0}}Z_{num} < 0.

Now, A_{j}L ≥ 0 for all A_{j} ⊂ |D| and DL = 1 (cf. 5.10), hence there is exactly one index j_{0} with A_{j_{0}}Z_{num} ≤ −1. (5.11) guarantees that this j_{0} is exactly i_{0}. Therefore, we get:

\begin{matrix}A_{i_{0}}Z_{num} = −1, & A_{i_{0}}(Z_{K} − D) = 0; \\ A_{j}Z_{num} = 0, & A_{j}(Z_{K} − D) = 0 & for j ≠ i_{0}; A_{j} ⊂ |D|. \end{matrix}

Again by (2.13): D = C′_{s} for some 0 ≤ s ≤ m. Since Q ∈ |D| but Q ∉ B_{1}, the only possibility is D = Z_{K}. This means that 1 = DL = Z_{K}(−Z_{num}) = −Z_{num}^{2}, which ends the proof of (5.5). □

5.12. The special case Z_{num}^{2} = −1. All the possible dual (minimal) resolution graphs of elliptic numerical Gorenstein singularities with Z_{num}^{2} = −1 are classified. This is done by S. S.-T. Yau (in [29], (2.3) and Table 1) based on Laufer’s classification of the minimal resolution graphs of the minimally elliptic singularities with E^{2} = −1 ([12], Table 1.)

5.13. Proposition. [29, 12] The minimal resolution graph of a numerical Gorenstein elliptic singularity with Z_{num}^{2} = −1 has the form A = B_{1} ∪ A_{0}, where A_{0} is a smooth rational curve with self–intersection −2, and it is unique with A_{0}Z_{num} = −1. Moreover, m_{A_{0}}(Z_{num}) = m_{A_{0}}(Z_{K}) = 1. B_{1} supports a singularity with Z_{B_{1}}^{2} = −1. The curve A_{0} intersects only one irreducible exceptional divisor Ã of B_{1}, and it intersects this exactly in one point. The curve Ã is the unique component of B_{1} with the property ÃZ_{B_{1}} = −1.

Therefore, by induction, A = |E| ∪ A_{m−1} ∪ ⋯ ∪ A_{0}, where all curves {A_{i}}_{i=0}^{m−1} are smooth rational curves with self–intersection −2, E^{2} = −1, Z_{B_{j}} = E + ∑_{i≥j} A_{i}, and the dual graph has the following form:

[figure: minimal resolution graph of a; minimally elliptic singularity; with E^{2} = −1 and ÃE = −1.; Ã; A_{m−1}; A_{m−2}; A_{1}; A_{0}; −2; −2; −2; −2; ⋯]

Now, Theorem 5.4 follows from the above discussions and the next proposition. 5.14. Proposition. Assume that (X, p) is numerical Gorenstein elliptic singularity with p_{g} = m + 1 (m > 0) and Z_{num}^{2} = −1. Consider its minimal resolution graph and write A = B_{1} ∪ A_{0} (cf. 5.13). Then |O(−Z_{num})| has a unique basepoint Q ∈ A_{0} \ B_{1}, and m_{p}O_{M} = m_{Q}O_{M}(−Z_{num}). The proof is similar to [12] (3.13), and it is left to the reader.

6. The Hilbert–Samuel function and the embedding dimension.

Assume that (X, p) is an elliptic Gorenstein singularity with p_{g} = m + 1. Consider its minimal resolution and set L := O_{Z_{K}}(−Z_{num}) and d := −Z_{num}^{2}.

6.1. Theorem. If d ≥ 3 then ⊕_{k≥0}H^{0}(Z_{K}, L^{⊗k}) is generated by elements of degree k = 1. Proof. We will follow – and modify – the proof of the corresponding statement for the minimally elliptic singularities (as it is presented in [19], page 114; i.e. “Castelnuovo’s free pencil trick”).

By (5.4) there exists s_{0} ∈ H^{0}(L) with m_{A_{i}}(s_{0}) = m_{A_{i}}(Z_{num}) for all A_{i}. Its divisor has form Z_{num}+D, where D is an effective Cartier divisor with |D|∩A finite, and A_{i}D = A_{i}L. Obviously O_{Z_{K}}(D) → L (given by f ↦ fs_{0}) is an isomorphism. We use another generic section s ∈ H^{0}(L) to identify L with O_{Z_{K}} near D, hence we will write O_{D} instead of L|_{D} = O_{D}(D). By construction, O_{D} is an Artinian scheme of degree d. Consider the exact sequence 0 → O_{Z_{K}} → L → O_{D} → 0, where the second arrow is the multiplication by s_{0}, (and the third is the “division by s near D”). Its long cohomology exact sequence is:

0 → H^{0}(O_{Z_{K}}) ⟶^{s_{0}} H^{0}(L) ⟶^{r_{1}} O_{D} → H^{1}(O_{Z_{K}}) → H^{1}(L) → 0.

Since h^{1}(O_{Z_{K}}) = p_{g} and h^{1}(L) = p_{g} − 1 (cf. 2.21.b), the rank of r_{1} is d − 1.

In the following lemmas D′ ⊂ D is an arbitrary subscheme of length d − 1. Let I_{D′} ⊂ O_{Z_{K}} be its ideal sheaf. If I_{D} ⊂ O_{Z_{K}} is the ideal sheaf of D, then I_{D} ⊂ I_{D′} and I_{D′}/I_{D} = C_{Q} for some point Q ∈ A. 6.2. Lemma. h^{1}(I_{D′}L) = p_{g} − 1. Proof. The inclusion I_{D′}O_{Z_{K}}(D) ⊂ O_{Z_{K}}(D) = L guarantee that I_{D′}L is torsion free. Hence the exact sequence 0 → O_{Z_{K}} →^{s_{0}} I_{D′}L → C_{Q} → 0 does not split. Hence in the

exact sequence 0 → Hom(I_{D′}L, O_{Z_{K}}) → Hom(O_{Z_{K}}, O_{Z_{K}}) →^{α} Ext^{1}(C_{Q}, O_{Z_{K}}) = C the

map α is not trivial. Hence by Serre duality h^{1}(I_{D′}L) = h^{1}(O_{Z_{K}}) − 1. □ 6.3. Lemma. H^{0}(L) → O_{D′} is surjective for any subscheme D′ ⊂ D of length d − 1. Proof. Use the cohomology exact sequence of 0 → I_{D′}L → L → O_{D′} → 0 and h^{1}(I_{D′}L) = h^{1}(L) (by 6.2. and 2.21.b). □ Now, by similar arguments as in [19], pages 114-115, Theorem 6.1 and Theorem D (of the introduction) follows. □ 6.4. Remarks.

(a) The cases d = −Z_{num}^{2} = 1, 2. Similarly to the minimally elliptic singularities (cf. e.g. [19], page 116), one can write a generator set of the O_{X}–algebra R := ⊕_{k≥0}H^{0}(O(−kZ_{num})) in the case d ≤ 2 as well. If d = 2, then R is generated by m_{p} in degree 1 and an element y ∈ m_{p} \ m_{p}^{2} in degree two. If d = 1, then R is generated by m_{p} in degree 1, by an element y ∈ m_{p} \ m_{p}^{2} in degree 2, and z ∈ m_{p} \ (m_{p}^{2}, y) in degree 3.

(b) Moreover (cf. [12]), if Z_{num}^{2} = −4, then (X, p) is a (tangential) complete intersection; if Z_{num}^{2} ≤ −5, then (X, p) is not a complete intersection.

(c) Using similar arguments as in this section, one can verify that for any normal surface singularity emb dim(X, p) ≥ mult(X, p) + min_{D>0}χ(D). Actually, for any elliptic Gorenstein singularity emb dim(X, p) = mult(X, p) provided that mult(X, p) ≥ 3, and (obviously) emb dim(X, p) = 3 if mult(X, p) = 2. In particular, for elliptic hypersurface singularities mult(X, p) ≤ 3.

(d) S. S.-T. Yau computed the multiplicity and the Hilbert–Samuel function for many different situations. His basic assumption was the much stronger Z_{E}^{2} ≤ −2 for the first case and Z_{E}^{2} ≤ −3 for the second case. (For details see his series of papers listed in the References.)

(e) If Z_{num}^{2} = −1, then (5.13) shows that the dual resolution graph is a “Kodaira graph” (cf. e.g. [6], 2.7). Moreover, by (5.14), the maximal ideal cycle is exactly Z_{num}. Hence, by [6] (2.9.1), (X, p) is a Kodaira singularity.

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