THE EMBEDDING DIMENSION OF WEIGHTED HOMOGENEOUS SURFACE SINGULARITIES

ANDRÁS NÉMETHI AND TOMOHIRO OKUMA

Abstract. We analyze the embedding dimension of a normal weighted homogeneous surface singularity, and more generally, the Poincaré series of the minimal set of generators of the graded algebra of regular functions, provided that the link of the germs is a rational homology sphere. In the case of several sub-families we provide explicit formulas in terms of the Seifert invariants (generalizing results of Wagreich and VanDyke), and we also provide key examples showing that, in general, these invariants are not topological. We extend the discussion to the case of splice–quotient singularities with star–shaped graph as well.

1. Introduction

Let (X, o) be a normal weighted homogeneous surface singularity defined over the complex numbers ℂ. Then (X, o) is the germ at the origin of an affine variety X with a good ℂ^{∗}–action. Our goal is to determine the minimal set of generators of the graded algebra G_{X} = ∑_{l≥0}(G_{X})_{l} of regular functions on X. This numerically is codified in the Poincaré series

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = ∑_{l≥0} dim (𝔪_{X}/𝔪_{X}^{2})_{l} t^{l},

where 𝔪_{X} is the homogeneous maximal ideal of G_{X}, and dim(𝔪_{X}/𝔪_{X}^{2})_{l} is exactly the number of generators of degree l (in a minimal set of generators). Notice that e.d.(X, o) := P_{𝔪_{X}/𝔪_{X}^{2}} (1) is the embedding dimension of (X, o).

We will assume that the link of (X, o) is a rational homology sphere. This means that the graph of the minimal good resolution (or, equivalently, the minimal plumbing graph of the link) is star–shaped, and all the irreducible exceptional divisors are rational. In particular, the link is a 3–dimensional Seifert manifold (of genus zero). Usually, it is characterized by its Seifert invariants (b_{0}, (α_{i}, ω_{i})_{i=1}^{ν}). Here −b_{0} is the self–intersection number of the central curve, ν is the number of legs.

In fact, our effort to understand P_{𝔪_{X}/𝔪_{X}^{2}} is part of a rather intense activity which targets the topological characterization of several analytic invariants of weighted homogeneous (or more generally, the splice–quotient) singularities. For example, if the link is a rational homology sphere, then the following invariants can be recovered from the link (i.e from the Seifert invariants): the Poincaré series of G_{X} and the geometric genus by [18], the equisingularity type of the universal abelian cover [14], the multiplicity of (X, o) [12]. In fact, by [12], the Poincaré series of the multi–variable filtration of G_{X} associated with the valuations of the irreducible components of the minimal good resolution is also determined topologically (proving the so–called Campillo–Delgado–Gusein-Zade identity). This basically says that a numerical invariant is topological, provided that it can be expressed as dimension of a vector space identified by divisors supported on the exceptional set.

2000 Mathematics Subject Classification. Primary 14B05, 32S25; Secondary 14J17.

Key words and phrases. Surface singularities, weighted homogeneous singularities, Poincaré series, embedding dimension, Seifert invariants, rational singularities, automorphic forms.

The first author is partially supported by Marie Curie grant and OTKA grants; the second author by KAKENHI 20540060.

To our surprise, P_{𝔪_{X}/𝔪_{X}^{2}} (t), in general, does not have a divisorial description, and, even more, it is not topological: it depends on the chosen analytic structure (a fact which makes its computation rather subtle).

In the present article we try to find the boundaries of this phenomenon.

First, we search for sub-families when P_{𝔪_{X}/𝔪_{X}^{2}} (t) is topological. Recall that by ‘classical’ results of Artin and Laufer, 𝔪_{X}/𝔪_{X}^{2} has a divisorial description (via the fundamental cycle) and it is topological; hence it is natural to recover and extend these results for more general families. The (topological) restrictions we found (and which guarantee the topological nature of P_{𝔪_{X}/𝔪_{X}^{2}} (t)) are grouped in three classes: restrictions regarding the structure of the first homology of the link (e.g. the order 𝔬 of the generic S^{1}–fibre in the link is ‘small’), regarding b_{0} (e.g. b_{0} − ν is positive, or negative with small absolute value), or target (global) shape of the graph (e.g. either ν ≤ 5 or ω_{i} = 1 for all i). Additionally, we provide explicit formulas for P_{𝔪_{X}/𝔪_{X}^{2}} (t) including the case of minimal rational germs (b_{0} ≥ ν) generalizing results of Wagreich and VanDyke [23, 24, 25], and when G_{X} is the graded ring of automorphic forms relative to a Fuchsian group of first kind (completing the list of Wagreich [25]).

On the other hand, we provide key examples (sitting at the ‘boundary’ of the positive results) when P_{𝔪_{X}/𝔪_{X}^{2}} (t) depends on the analytic parameters, even computing the ‘discriminants’ when the embedding dimension jumps. In the last section we extend the discussion to the case of splice–quotients (associated with star–shaped graphs) as well. In some of the computations we used the SINGULAR program [4].

The second author thanks the BudAlgGeo Marie Curie Fellowship for Transfer of Knowledge for supporting his visit at the Rényi Institute of Mathematics, Budapest, Hungary, where the most part of this paper was done. He is also grateful to the members of the Rényi Institute for the warm hospitality.

2. Preliminaries

In this section we introduce our notations and review Neumann’s theorem on the universal abelian covers of weighted homogeneous surface singularities with ℚ-homology sphere links.

Let (X, o) be a normal surface singularity with a ℚ-homology sphere link Σ. Let π: X̃ → X be the minimal good resolution with the exceptional divisor E. Then E is a tree of rational curves. Let {E_{v}}_{v∈V} denote the set of irreducible components of E and −b_{v} the self-intersection number E_{v}^{2}. The set V is regarded as the set of vertices of the dual graph Γ of E (i.e., the resolution graph associated with π). Let L denote the group of divisors supported on E, namely L = ∑_{v∈V} ℤE_{v}. We call an element of L (resp. L ⊗ ℚ) a cycle (resp. ℚ-cycle). Since the intersection matrix I(E) := (E_{v} · E_{w}) is negative definite, for each v ∈ V there exists an effective ℚ-cycle E_{v}^{∗} such that E_{v}^{∗} · E_{w} = −δ_{vw} for every w ∈ V, where δ_{vw} denotes the Kronecker delta. Set L^{∗} = ∑_{v∈V} ℤE_{v}^{∗}. Note that the finite group H := L^{∗}/L is naturally isomorphic to H_{1}(Σ, ℤ).

2.1. The Seifert invariants. Suppose that Γ is a star-shaped graph with central vertex 0 ∈ V and it has ν ≥ 3 branches. It is well-known that Γ is determined by the so-called Seifert invariants {(α_{i}, ω_{i})}_{i=1}^{ν} and the orbifold Euler number e defined as follows. If the i-th branch has the form

[figure: …; −b_{0}; −b_{i1}; −b_{ir_{i}}]

(where the far left–vertex corresponds to the ‘central curve’ E_{0}) then the positive integers α_{i} and ω_{i} are defined by the (negative) continued fraction expansion :

\frac{α_{i}}{ω_{i}} = cf [b_{i1}, …, b_{ir_{i}}] := b_{i1} − \frac{1}{b_{i2} − \frac{1}{... − \frac{1}{b_{ir_{i}}}}}, gcd(α_{i}, ω_{i}) = 1, 0 < ω_{i} < α_{i}.

Moreover, we define e := −b_{0} + ∑_{i} ω_{i}/α_{i}. Note that e < 0 if and only if the intersection matrix I(E) is negative definite. By [10, (11.1)]

(2.1.1)  \begin{aligned}(E_{0}^{∗} · E_{0}^{∗}) & = e^{−1}; \\ (E_{0}^{∗} · E_{ir_{i}}^{∗}) & = (eα_{i})^{−1} for 1 ≤ i ≤ ν; \\ (E_{ir_{i}}^{∗} · E_{jr_{j}}^{∗}) & = \{ \begin{matrix}(eα_{i}α_{j})^{−1} & if i ≠ j, 1 ≤ i, j ≤ ν; \\ (eα_{i}^{2})^{−1} − ω′_{i}/α_{i} & if i = j, 1 ≤ i, j ≤ ν, \end{matrix} \end{aligned}

where 0 < ω′_{i} < α_{i} and ω_{i}ω′_{i} ≡ 1 modulo α_{i}.

If (X, o) is a weighted homogeneous surface singularity (i.e. a surface singularity with a good ℂ^{∗}-action), then Γ is automatically star-shaped and the complex structure is completely recovered from the Seifert invariants and the configuration of the points {P_{i} := E_{0} ∩ E_{i1}}_{i=1}^{ν} ⊂ E_{0}. In fact, cf. [18], the graded affine coordinate ring G_{X} = ⨁_{l≥0}(G_{X})_{l} is given by

(2.1.2) (G_{X})_{l} = H^{0}(E_{0}, O_{E_{0}}(D^{(l)})), with D^{(l)} = l(−E_{0}|_{E_{0}}) − ∑_{i} ⌈lω_{i}/α_{i}⌉ P_{i},

where for r ∈ ℝ, ⌈r⌉ denotes the smallest integer greater than or equal to r.

In the sequel we write α := lcm{α_{1}, …, α_{ν}} and the following notation (with abbreviation \underline{1} = (1, …, 1) ∈ ℤ^{n}) for the Seifert invariants:

Sf := (b_{0}, \underline{α}, \underline{ω}), \underline{α} := (α_{1}, …, α_{ν}), \underline{ω} := (ω_{1}, …, ω_{ν}).

2.2. Neumann’s theorem and some consequences. Assume that (X, o) is a weighted homogeneous surface singularity and ν ≥ 3. Then there exists a universal abelian cover (X^{ab}, o) → (X, o) of normal surface singularities that induces an unramified Galois covering X^{ab} \ {o} → X \ {o} with Galois group H_{1}(Σ, ℤ); if Σ′ denotes the link of X^{ab}, then the natural covering Σ′ → Σ is the universal abelian cover in the topological sense.

Theorem 2.2.1. [14] The universal abelian cover X^{ab} is a Brieskorn complete intersection singularity

{(z_{i}) ∈ ℂ^{ν} | f_{j} := a_{j1}z_{1}^{α_{1}} + ⋯ a_{jν}z_{ν}^{α_{ν}} = 0, j = 1, …, ν − 2},

where every maximal minor of the matrix (a_{ji}) does not vanish (i.e. (a_{ji}) has ‘full rank’).

In fact, by row operations, the matrix (a_{ji}) can be transformed into

(2.2.2)  (\begin{matrix}p_{1} & q_{1} & 1 & 0 & ⋯ & 0 \\ p_{2} & q_{2} & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \\ p_{ν−2} & q_{ν−2} & 0 & 0 & ⋯ & 1 \end{matrix})

such that all the entries p_{i} and q_{i} are nonzero, and p_{i}q_{j} − p_{j}q_{i} ≠ 0 for any i ≠ j; moreover, [p_{i} : q_{i}] are the projective coordinates of the points {P_{3}, …, P_{ν}} ⊂ ℙ^{1}, while the remaining two points are P_{1} = [1 : 0] and P_{2} = [0 : 1].

We define the weights wt(z_{i}) = (|e|α_{i})^{−1} ∈ ℚ. This induces a grading of the polynomial ring R = ℂ[z_{1}, …, z_{ν}] by z\underline{k} := z_{1}^{k_{1}} ⋯ z_{ν}^{k_{ν}} ∈ R_{k} if and only if (∑ k_{i}E_{ir_{i}}^{∗} · E_{0}^{∗}) = −k. Since all the splice polynomials f_{j} are weighted homogeneous of degree |e|^{−1}, there is a naturally induced grading on the affine coordinate ring of X^{ab} too.

The group H = L^{∗}/L is generated by the classes of {E_{ir_{i}}^{∗}}_{i=1}^{ν} and acts on the polynomial ring R (and/or on ℂ^{ν}) as follows. The class [E_{ir_{i}}^{∗}] acts by the ν × ν diagonal matrix [e_{i1}, …, e_{iν}], where e_{ij} = exp(2π\sqrt{−1}E_{ir_{i}}^{∗} · E_{jr_{j}}^{∗}), i = 1, …, ν, cf. [15, §5]. By this action X = X^{ab}/H; the invariant subring of R is denoted by R^{H}.

Let Ĥ = Hom(H, ℂ^{∗}). For any character λ ∈ Ĥ, the λ–eigenspace R^{λ} is

{f ∈ R | h · f = λ(h)f for all h ∈ H }.

A computation shows that for the character μ ∈ Ĥ defined by μ([E_{ir_{i}}^{∗}]) = exp(2π\sqrt{−1}E_{0}^{∗} · E_{ir_{i}}^{∗}) one has \{f_{1}, …, f_{ν−2}\} ⊂ R^{μ}. We have the following facts:

Lemma 2.2.3. Let I_{X} ⊂ R^{H} be the ideal generated by R^{μ^{−1}} · {f_{1}, …, f_{ν−2}}. Then the affine coordinate ring G_{X} of X is isomorphic to R^{H}/I_{X}.

Set 𝔬 := |e|α, which is the order of [E_{0}^{∗}] in L^{∗}/L (cf. [14]). Then 𝔬 = 1 if and only if the splice functions f_{j} are in R^{H} (i.e. μ is the trivial character).

3. Properties of the graded ring G_{X}

3.1. The Hilbert/Poincaré series of G_{X}. For any graded vector space V = ⊕_{l≥0}V_{l} we define (as usual) its Poincaré series P_{V} (t) := ∑_{l≥0} dim V_{l} t^{l}. Since E_{0} = ℙ^{1}, Pinkham’s result (2.1.2) reads as

(3.1.1)  P_{G_{X}} (t) = ∑_{l≥0} max(0, s_{l} + 1) t^{l},

where (for any l ≥ 0) we set

(3.1.2)  s_{l} := deg D^{(l)} = lb_{0} − ∑_{i} ⌈lω_{i}/α_{i}⌉ ∈ ℤ.

Although P_{G_{X}} contains considerably less information than the graded ring G_{X} itself, still it determines rather strong numerical analytic invariants. (E.g., it determines the geometric genus p_{g} of X, or the log discrepancy of E_{0} as well, facts which are less obvious and which will be clear from the next discussion.)

Definitely, the series

P (t) := ∑_{l≥0} χ(O_{E_{0}}(D^{(l)}))t^{l} = ∑_{l≥0}(s_{l} + 1)t^{l}

is more ‘regular’ (e.g., it is polynomial periodic), and it determines both P_{G_{X}} and

P_{H^{1}}(t) := ∑_{l≥0} dim H^{1}(E_{0}, O_{E_{0}}(D^{(l)})t^{l} = ∑_{l≥0} max(0, −s_{l} − 1) t^{l}.

By definition, P = P_{G_{X}} − P_{H^{1}}. Since e < 0, lim_{l→∞} s_{l} = ∞, and P_{H^{1}}(t) is a polynomial (and by [18], p_{g} = P_{H^{1}}(1)). In fact,

−(α − 1)|e| − ν ≤ s_{l} − ⌈l/α⌉ α|e| ≤ −1.

Indeed, if one writes l as ⌈l/α⌉ α − d, and s̃_{d} = db_{0} − ∑ ⌊dω_{i}/α_{i}⌋, then s_{l} = ⌈l/α⌉ α|e| − s̃_{d}, while s̃_{d} ≥ 1 by [10, 11.5] and s̃_{d} ≤ (α − 1)|e| + ν by a computation.

Remark 3.1.3. From P_{G_{X}} (t) one can recover P (t), hence P_{H^{1}}(t) too. Indeed, one can show (see e.g. [21, Corollary 1.5]) that P (t) can be written as a rational function p(t)/q(t) with p, q ∈ ℂ[t] and deg p < deg q. Then, if one writes (in a unique way) P_{G_{X}} (t) as p(t)/q(t) + r(t) with p, q, r ∈ ℂ[t] and deg p < deg q, then p/q = P and r = P_{H^{1}}.

The next subsection provides a (topological) upper bound for the degree of the polynomial P_{H^{1}}.

3.2. The a–invariant of the universal abelian cover. First recall that any Cohen–Macaulay (positively) graded ℂ–algebra G admits the so–called a–invariant a(G), for details see the article of Goto and Watanabe [3, (3.1.4)], or [2, (3.6.13)]. Let G_{X^{ab}} denote the affine coordinate ring of X^{ab}. Since X^{ab} is a complete intersection, cf. (2.2.1), its a-invariant a(G_{X^{ab}}) is determined by [2, (3.6.14-15)]. This, in terms of Seifert invariants, reads as follows (see also [17, §3]):

a(G_{X^{ab}}) = (ν − 2)α − ∑_{i=1}^{ν} \frac{α}{α_{i}} .

In order to make it compatible with the rational weights of the coordinates (and with the grading of X^{ab}, see the proof of (3.2.2)), we divide it by the order 𝔬 = |e|α of [E_{0}^{∗}]:

(3.2.1)  γ := \frac{a(G_{X^{ab}})}{𝔬} = \frac{1}{|e|} (ν − 2 − ∑_{i=1}^{ν} \frac{1}{α_{i}} ) ∈ ℚ.

Proposition 3.2.2. If l > γ then H^{1}(O_{E_{0}}(D^{(l)})) = 0, i.e. s_{l} ≥ −1. Hence deg P_{H^{1}}(t) ≤ max(0, γ).

Proof. Let π′: X̃ → X′ be the morphism that contracts all the branches (this is denoted by τ in [22, §6]). This can be inserted in the following commutative diagram, which is used in [17, §3], where Y := X^{ab}, Ỹ is the pull–back of π by the universal abelian cover q, and p′ is finite:

\xymatrix{Ỹ \ar[rrr]^{p} \ar[dr]^{ρ′} \ar[dd]_{ρ} &  &  & X̃ \ar[dl]_{π′} \ar[dd]^{π} \\  & Y′ \ar[r]^{p′} \ar[dl]^{ρ_{1}} & X′ \ar[dr]_{π_{1}} \\ Y \ar[rrr]_{q} &  &  & X}

The exceptional sets of π_{1} and ρ_{1} are E′ := π′(E_{0}) and F′ := ρ′(p^{−1}(E_{0})), respectively. Then p^{′∗}E′ = 𝔬F′, and by [17, (3.4)], based on (1.13)(ii) and (1.18)(i) of [22], H^{1}(O_{Y′}(−l𝔬F′)) = 0 for l𝔬 > a(G_{X^{ab}}). Taking the H–invariants (and using the corresponding eigenspace decompositions, see e.g. [17, (4.3)]), we get that

(3.2.3)  H^{1}(X′, π′_{∗}OX̃(⌊−lπ^{′∗}E′⌋)) = 0.

Next, recall that by Sakai’s result [20, (2.1)], we have

(3.2.4)  π′_{∗}OX̃(⌊−lπ^{′∗}E′⌋) = O_{X′}(−lE′).

These sheaves can be put in the exact sequence of sheaves (cf. [22, (6.4)]):

(3.2.5)  0 → O_{X′}(−(l + 1)E′) → O_{X′}(−lE′) → π′_{∗}O_{E_{0}}(⌊−π^{′∗}(lE′)⌋) → 0.

The associated cohomological long exact sequence together with (3.2.3) and (3.2.4) provides the vanishing

H^{1}(X′, π′_{∗}O_{E_{0}}(⌊−π^{′∗}(lE′)⌋)) = 0.

Since the restriction π′|_{E_{0}} is an isomorphism, this is equivalent with the vanishing H^{1}(X̃, O_{E_{0}}(⌊−π^{′∗}(lE′)⌋)) = 0. Since the (rational) Pinkham-Demazure divisor on E_{0} is given by π^{′∗}E′|_{E_{0}} = E_{0}|_{E_{0}} + ∑_{i} \frac{ω_{i}}{α_{i}} P_{i}, and

O_{E_{0}}(D^{(l)}) = O_{E_{0}}(⌊−π^{′∗}(lE′)⌋),

cf. [22, (6.10.1) and (6.15)], the vanishing H^{1}(O_{E_{0}}(D^{(l)})) = 0 follows. □

Remark 3.2.6. (1) s_{α} = α(b_{0} − ∑ ω_{i}/α_{i}) = α|e| = 𝔬 is a positive integer. (2) For any l ≥ 0, s_{l+α} = s_{l} + s_{α} = s_{l} + 𝔬 > s_{l}.

Corollary 3.2.7. Assume l > α + γ. Then s_{l} ≥ 0, hence (G_{X})_{l} is non–trivial.

3.3. −γ as log discrepancy. The rational number γ, cf. (3.2.1), has a different interpretation as well. Let −Z_{K} be the canonical cycle associated with the canonical line bundle of X̃, i.e. Z_{K} ∈ L^{∗} satisfies the adjunction relations (Z_{K} · E_{v}) = E_{v}^{2} + 2 for all v ∈ V. Then (see e.g. [10, (11.1)]) the coefficient of E_{0} in Z_{K} is exactly 1 + γ, hence −γ is the log discrepancy of E_{0}. In particular, the above proof can be explained in a more transparent way via ‘computation sequences’ and Grauert– Riemenschneider type vanishing theorems (and without reclining upon the universal abelian cover), at least in the numerical Gorenstein case. (In fact, the main argument of Tomari and Watanabe in the proof of (6.4) [22] is also in this flavour.)

Second proof of (3.2.2) under the assumption that (X, o) is numerically Gorenstein (i.e. Z_{K} ∈ L). (Notice that in this case γ ∈ ℤ.) If Γ is an A-D-E graph then (X, o) is rational, hence P_{H^{1}} ≡ 0, and γ = −1. Therefore, we will assume that Γ is not A-D-E; in particular, all the coefficients of Z_{K} are positive.

Let us assume at the beginning that (X, o) is a (not necessarily weighted homogeneous) singularity with the star–shaped resolution graph Γ. By [10, (7.6)], for any l ≥ 0 there exists a unique effective cycle x(l) ∈ L such that

(3.3.1)

(a) the coefficient of E_{0} in x(l) is l;
(b) (x(l) · E_{v}) ≤ 0 for all v ∈ V \ {0};
(c) x(l) is minimal with properties (a)–(b).

For the precise form of x(l) (which is rather arithmetical in terms of Seifert invariants), see [10, (11.11)]. This formula also shows that

(3.3.2)  −(x(l) · E_{0}) = s_{l}.

By [10, (9.1(c))], x(l)+E_{0} and x(l+1) can be connected by a computation sequence (i.e. a sequence of cycles {z_{n}}_{n=0}^{s} with z_{0} = x(l) + E_{0}, z_{s} = x(l + 1), z_{n+1} = z_{n} + E_{v(n)} for some v(n) ∈ V \ {0} such that (z_{n} · E_{v(n)}) = 1 for all n). Hence, by the cohomological exact sequences associated with O_{z_{n+1}} → O_{z_{n}} we get that h^{∗}(O_{x(l)+E_{0}}) = h^{∗}(O_{x(l+1)}). On the other hand, the long cohomological exact sequence associated with O_{x(l)+E_{0}} → O_{x(l)} gives

⟶^{δ_{l}} ℂ^{max(0,−1−s_{l})} → H^{1}(O_{x(l)+E_{0}}) → H^{1}(O_{x(l)}) → 0.

In particular,

(3.3.3)  h^{1}(O_{x(l+1)}) − h^{1}(O_{x(l)}) ≤ max(0, −1 − s_{l}),

and equality holds if and only if δ_{l} is trivial. Clearly, for l ≫ 0, h^{1}(O_{x(l)}) = p_{g} and the left hand side of (3.3.3) is zero. Hence, summing up (3.3.3), we get

(3.3.4)  p_{g} ≤ ∑_{l≥0} max(0, −1 − s_{l}).

Now, assume that (X, o) is a weighted homogeneous singularity. Then, by [18], in (3.3.4) we have equality, hence all δ_{l}’s are trivial. Hence, it is enough to show that h^{1}(O_{x(l)}) = p_{g} for all l > γ. Clearly, since h^{1}(O_{x(l)}) ≤ h^{1}(O_{x(l+1)}) ≤ p_{g}, it is enough to verify this only for l = γ + 1.

Notice that Z_{K} satisfies the conditions (a)–(b) of (3.3.1) for l = γ + 1, hence Z_{K} ≥ x(γ + 1). Then one has the exact sequence

H^{1}(O_{Z_{K}−x(γ+1)}(−x(γ + 1))) → H^{1}(O_{Z_{K}}) → H^{1}(O_{x(γ+1)}) → 0.

Since Z_{K} − x(γ + 1) is supported only on the legs (hence the support is the graph of rational singularities), and −x(γ + 1) is a numerical effective cycle on this support, we get the vanishing of the far left term (see [6] or [11, (4.3.2)]). Hence h^{1}(O_{x(γ+1)}) = h^{1}(O_{Z_{K}}) = p_{g}.

In particular, max(0, −1 − s_{l}) = 0 for all l ≥ γ + 1. This ends the proof.

The above proof also shows that the bound in (3.2.2) is sharp:

Proposition 3.3.5. Assume that (X, o) is Gorenstein, but not of type A-D-E (i.e. it is not rational). Then the degree of P_{H^{1}}(t) is exactly γ. Moreover, the coefficient of t^{γ} is 1. In particular, if (X, o) is minimally elliptic, then P_{H^{1}}(t) = t^{γ}.

Proof. Consider the notations of the second proof of (3.2.2). By that discussion, it is enough to show that h^{1}(O_{x(γ)}) = p_{g} − 1.

Since Z_{K} − E_{0} satisfies the conditions (a)–(b) of (3.3.1) for l = γ, by part (c) we get that Z_{K} −E_{0} ≥ x(γ). By similar argument as above (for the pair (Z_{k}, x(γ+1))), we have that h^{1}(O_{Z_{K}−E_{0}}) = h^{1}(O_{x(γ)}). Finally, consider the exact sequence

ℂ = H^{1}(O_{E_{0}}(−Z_{K} + E_{0})) ⟶^{i} H^{1}(O_{Z_{K}}) → H^{1}(O_{Z_{K}−E_{0}}) → 0.

It is known (see e.g. [19, (4.20)]) that if (X, o) is Gorenstein then i is injective. □

Remark 3.3.6. In fact, γ can be recovered already from the asymptotic behaviour of the coefficients of P (t) or P_{G_{X}} (t). Indeed, by a result of Dolgachev, the Laurent expansions of both P (t) and P_{G_{X}} (t) at t = 1 have the form (cf. [14, (4.7)], or [13, Proposition 3.4] for one more term of the expansion):

|e| · ( \frac{1}{(t − 1)^{2}} + \frac{1 + γ/2}{t − 1} + regular part ).

3.4. The homogeneous parts (G_{X})_{l} reinterpreted. Assume that (X, o) is weighted homogeneous with graded ring ⊕_{l≥0}(G_{X})_{l} and graph Γ as above.

Let M = \{ z\underline{k} | k_{i} ∈ ℤ_{≥0}, i = 1, …, ν\} ⊂ R = ℂ[z_{1}, …, z_{ν}] be the set of all monomials. For any \underline{k} = (k_{1}, …, k_{ν}), set d\underline{k} := (∑_{i} k_{i}/α_{i})/|e| ∈ ℚ, the degree of z\underline{k}. We define a homomorphism of semigroups φ: (ℤ_{≥0})^{ν} → ℚ^{ν} by

φ(\underline{k}) = (l_{1}, …, l_{ν}), where l_{i} := (k_{i} + d\underline{k}ω_{i})/α_{i}.

Let M^{H} denote the set of invariant monomials (with respect to the action of H).

Lemma 3.4.1. z\underline{k} ∈ M^{H} if and only if φ(\underline{k}) ∈ (ℤ_{≥0})^{ν} and d\underline{k} ∈ ℤ_{≥0}.

Proof. We first recall the description of H from [14, §1] (see also [13, §2.5]). Let h_{i} denote the class [E_{ir_{i}}^{∗}] (i = 1, …, ν) and h_{0} the class [E_{0}^{∗}] in H. Then

(3.4.2)  H = ⟨h_{0}, h_{1}, …, h_{ν} | h_{0}^{b_{0}} ∏_{i=1}^{ν} h_{i}^{−ω_{i}} = 1, h_{0}^{−1}h_{i}^{α_{i}} = 1 (i = 1, …, ν)⟩.

Moreover, all the relations among {h_{i}}_{i=0}^{ν} have the form

h_{0}^{lb_{0}−∑ l_{i}} ∏ h_{i}^{−lω_{i}+l_{i}α_{i}} = 1, for some l, l_{1}, …, l_{ν} ∈ ℤ.

Next, notice that z\underline{k} ∈ M^{H} if and only if ∏_{i=1}^{ν} h_{i}^{k_{i}} = 1. If this is happening, then there exist l, l_{1}, …, l_{ν} ∈ ℤ such that lb_{0} − ∑ l_{i} = 0 and −lω_{i} + l_{i}α_{i} = k_{i} for i = 1, …, ν. Then we have l_{i} = (k_{i} + lω_{i})/α_{i} and

∑ k_{i}/α_{i} = ∑(l_{i} − lω_{i}/α_{i}) = lb_{0} − l ∑ ω_{i}/α_{i} = l|e|.

Hence l = d\underline{k} and φ(\underline{k}) = (l_{i}). The converse is now easy. □

Corollary 3.4.3. For any l ≥ 0, the linear space (R^{H})_{l} ⊂ R^{H} of forms of degree l is spanned by \{z\underline{k} | ∑_{i} k_{i}/α_{i} = l|e|, k_{i} + lω_{i} ≡ 0 (mod α_{i}), i = 1, …, ν \}.

Definition 3.4.4. For each 1 ≤ i ≤ ν and l ≥ 0 define:

• β_{i} := α_{i} − ω_{i};

• a_{i} := z_{i}^{α_{i}};

• M_{l} := ∏ z_{i}^{{lβ_{i}/α_{i}}α_{i}}, where for r ∈ ℝ, {r} denotes the fractional part of r.

Note that M_{l} = M_{l′} if l ≡ l′ (mod α).

Consider z\underline{k} ∈ (M^{H})_{l} and take (l_{i}) = φ(\underline{k}). Then n_{i} := l_{i} − ⌈lω_{i}/α_{i}⌉ is a non–negative integer which satisfy ∑ n_{i} = s_{l} and k_{i} − n_{i}α_{i} = {lβ_{i}/α_{i}}α_{i}. Hence:

Proposition 3.4.5. For any l ≥ 0 one has

(M^{H})_{l} = \begin{cases}M_{l} · \{∏ a_{i}^{n_{i}} | ∑ n_{i} = s_{l}, \underline{n} = (n_{i}) ∈ (ℤ_{≥0})^{ν} \} & if s_{l} ≥ 0, \\ ∅ & if s_{l} < 0. \end{cases}

Definition 3.4.6. Let A = ℂ[a_{1}, …, a_{ν}] be the polynomial ring graded by wt(a_{i}) = 1. For any l, define the map ψ_{l}: (R^{H})_{l} → A_{s_{l}} by ψ(f) = f/M_{l}.

Then Proposition (3.4.5) implies that ψ_{l} is an isomorphism of ℂ-linear spaces. By this correspondence, the splice polynomials f_{j} = ∑_{i} a_{ji}z_{i}^{α_{i}} (cf. 2.2.1) transform into the linear forms ℓ_{j} = ∑_{i} a_{ji}a_{i} of A (j = 1, …, ν). Hence, every element of ψ_{l}(I_{X}) has the form ∑_{j} q_{j}ℓ_{j}, where q_{j} are arbitrary (s_{l} − 1)–forms of A. In particular, if I denotes the ideal generated by the linear forms {ℓ_{j}}_{j=1}^{ν−2}, then

(3.4.7)  (G_{X})_{l} = A_{s_{l}}/⟨_{∑j}q_{j}ℓ_{j}⟩ = (A/I)_{s_{l}}.

Notice that via this representation we easily can recover Pinkham’s formula (3.1.1). Indeed, if s_{l} ≥ 0, then using the linear forms {ℓ_{j}}, the variables a_{3}, …, a_{ν} can be eliminated; hence dim(G_{X})_{l} = dim ℂ[a_{1}, a_{2}]_{s_{l}} = s_{l} + 1.

4. The square of the maximal ideal

4.1. The general picture. Let 𝔪 ⊂ R^{H} denote the homogeneous maximal ideal and 𝔪_{X} the homogeneous maximal ideal of G_{X} = R^{H}/I_{X}. The aim of this section is to compute the dimension Q(l) of the ℂ-linear space

(𝔪_{X}/𝔪_{X}^{2})_{l} = (𝔪/I_{X} + 𝔪^{2})_{l}.

These numbers can be inserted in the Poincaré series of 𝔪_{X}/𝔪_{X}^{2}:

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = ∑_{l≥1} Q(l)t^{l}.

This is a polynomial with P_{𝔪_{X}/𝔪_{X}^{2}} (1) = e.d.(X, o), the embedding dimension of (X, o). Indeed, if one considers a minimal set of homogeneous generators of the ℂ–algebra G_{X}, then Q(l) is exactly the number of generators of degree l.

Our method relies on the description developed in the subsection (3.4).

First we describe the structure of 𝔪^{2} in terms of the monomials of M^{H}.

Definition 4.1.1. Let ℕ^{∗} denote the set of positive integers. For l ∈ ℕ^{∗} we set

Λ_{l} := {(l_{1}, l_{2}) ∈ (ℕ^{∗})^{2} | l_{1} + l_{2} = l, s_{l_{1}} ≥ 0, s_{l_{2}} ≥ 0}.

The monomial z\underline{k} ∈ (M^{H})_{l} is called linear if either Λ_{l} = ∅, or z\underline{k} ∉ (M^{H})_{l_{1}} ·(M^{H})_{l_{2}} for any (l_{1}, l_{2}) ∈ Λ_{l}.

The linear monomials form a set of minimal generators of 𝔪.

Next, we transport this structure on the polynomial ring A. In order not to make confusions between the degrees of the monomials from R and A, we emphasize the corresponding degrees by writing deg_{R} and deg_{A} respectively.

Definition 4.1.2. (a) For \underline{l} = (l_{1}, l_{2}) ∈ Λ_{l} and 1 ≤ i ≤ ν, set

ϵ_{i,\underline{l}} = \{l_{1}β_{i}/α_{i}\} + \{l_{2}β_{i}/α_{i}\} − \{lβ_{i}/α_{i}\} ∈ \{0, 1\}.

(b) X_{l} will denote the set of (automatically reduced) monomials of A defined by

X_{l} := \{M_{l_{1}}M_{l_{2}}/M_{l} | (l_{1}, l_{2}) ∈ Λ_{l} \} = \{ ∏_{i=1}^{ν} a_{i}^{ϵ_{i},\underline{l}} | \underline{l} ∈ Λ_{l} \}.

(c) Let J(l) be the ideal generated by X_{l} in A (J(l) = (0) if X_{l} = ∅).

From the definition of the ideals J(l) and the map ψ_{l} one has:

Proposition 4.1.3. ψ_{l}((𝔪^{2})_{l}) = J(l)_{s_{l}}. In particular, Q(l) = dim(A/I + J(l))_{s_{l}}.

For the convenience of the reader, we list some properties of the monomials M_{l} and of the generators X_{l} which might be helpful in the concrete computations.

Lemma 4.1.4. (1) deg_{R} M_{l} + s_{l}/|e| = l; (2) ∑_{i} ϵ_{i,\underline{l}} = deg_{A} (M_{l_{1}}M_{l_{2}}/M_{l}) = s_{l}−s_{l_{1}} −s_{l_{2}} ≤ min\{ν, s_{l}\} for any \underline{l} = (l_{1}, l_{2});

(3) for any m, n ∈ ℕ, if (l_{1}, l_{2}) ∈ Λ_{l}, then (l_{1} + mα, l_{2} + nα) ∈ Λ_{l+(m+n)α} too, and ϵ_{i,(l_{1},l_{2})} = ϵ_{i,(l_{1}+mα,l_{2}+nα)}; hence X_{l} ⊂ X_{l+α};

(4) if s_{l_{2}} ≥ 0 then ϵ_{i,(α,l_{2})} = 0 for every i; hence 1 ∈ X_{α+l_{2}};

(5) if Λ_{l} ≠ ∅ and s_{l} = 0, then 1 ∈ X_{l} (cf. (2));

(6) if l > 2α + γ, then 1 ∈ X_{l} (cf. (3.2.7) and (5)).

4.2. Is Q(l) topological ? Let us recall/comment the formula Q(l) = dim(A/I + J(l))_{s_{l}} from (4.1.3). Here, for any l ∈ ℕ^{∗}, the ideal J(l) is combinatorial depending only on the Seifert invariants of the legs {(α_{i}, ω_{i})}_{i}. The integer s_{l} is also combinatorial (for it, one also needs the integer b_{0}). On the other hand, the ideal I is generated by the ‘generic’ linear forms {ℓ_{j} = ∑_{i} a_{ji}a_{i}}_{j=1}^{ν−2}, where the ‘genericity’ means that the matrix (a_{ji}) has full rank.

By a superficial argument, one might conclude that Q(l) is topological, i.e. for all matrices (a_{ji}) of full rank, Q(l) is the same. But, surprisingly, and this is the main point of the present article, this is not the case: In the space of full rank matrices, there are some sub–varieties along which the value of Q(l) might jump.

Therefore, in the sequel our investigation bifurcates into two directions:

1. Find equisingular families of weighted homogeneous surface singularities in which the embedding dimension is not constant (i.e. the embedding dimension cannot be determined from the Seifert invariants). Analyze the ‘discriminant’ (the ‘jump– loci’), and write the corresponding equations, deformations.

2. Find topologically identified families of weighted homogeneous (or even splice quotient) singularities, which are characterized by special properties of the Seifert invariants, for which P_{𝔪_{X}/𝔪_{X}^{2}} (t), hence the embedding dimension too, is topological. Then, determine them from the Seifert invariants.

The remaining part of the present article deals with these two directions, providing key positive results and examples in both directions, and trying to find the boundary limit between the two categories. Let us provide as a warm up, some intuitive easy explanation for both directions how they might appear.

Lemma 4.2.1. (‘Easy cases’ when Q(l) is topological)

I. If s_{l} ≥ 0 and X_{l} = ∅, then Q(l) = s_{l} + 1.

II. Assume that there exists an i ∈ {1, …, ν} such that a_{i} ∈ X_{l}. Let (a_{i}) the ideal in A generated by a_{i}. Then

(1) if J(l) ⊂ (a_{i}), then Q(l) = 1.

(2) if J(l) ⊄ (a_{i}), then Q(l) = 0.

In particular, Q(l) is topological, provided that s_{l} ≤ 1: Q(l) is the number of variables appearing in all the monomials of X_{l} whenever X_{l} ≠ ∅.

Proof. In the case (I): Q(l) = dim ℂ[a_{1}, a_{2}]_{s_{l}}. In (II) we may assume i = 1. In case (1), J(l) = (a_{1}) and A/I+J(l) ≅ ℂ[a_{2}]. For (2), let δ = min {deg_{A} m | m ∈ X_{l} \ (a_{1})}. Then A/I + J(l) ≅ ℂ[a_{2}]/(a_{2}^{δ}). Since δ ≤ s_{l}, cf. (4.1.4)(2), (A/I + J(l))_{s_{l}} = 0. □

Example 4.2.2. (How can Q(l) be non–topological?) Assume that in some situation ν = 6, X_{l} = {a_{1}a_{2}, a_{3}a_{4}, a_{5}a_{6}} and s_{l} = 2 (cf. (7.1.1)). Consider the linear forms ℓ_{j} = ∑_{i} a_{ji}a_{i}, where the matrix (a_{ji}) is from (2.2.2). Since all q_{i}’s are non–zero, we may assume q_{i} = 1 for all i. The full rank condition is equivalent with the fact that all p_{i}’s are non–zero and distinct.

Then, by eliminating the variables a_{3}, …, a_{ν} using the linear forms, Q(l) = dim(ℂ[a_{1}, a_{2}]/J′)_{2}, where J′ is generated by a_{1}a_{2}, (p_{1}a_{1}+a_{2})(p_{2}a_{1}+a_{2}) and (p_{3}a_{1}+ a_{2})(p_{4}a_{1} + a_{2}). Therefore,

Q(l) = \begin{cases}0 & if p_{1}p_{2} − p_{3}p_{4} ≠ 0, \\ 1 & if p_{1}p_{2} − p_{3}p_{4} = 0. \end{cases}

Hence, along the ‘non–topological discriminant’ p_{1}p_{2} − p_{3}p_{4} = 0, the embedding dimension increases.

In the next subsections we provide our first examples when e.d.(X, o) is topological.

4.3. The case 𝔬 = 1. Assume that 𝔬 = 1, i.e. E_{0}^{∗} ∈ L. In this case the splice functions belong to R^{H}, cf. (2.2.3), and their degree is exactly α. We will proceed in several steps. First, we consider the exact sequence

(4.3.1)  0 → \frac{I_{X}}{I_{X} ∩ 𝔪^{2}} → \frac{𝔪}{𝔪^{2}} → \frac{𝔪_{X}}{𝔪_{X}^{2}} → 0.

This is compatible with the weight–decomposition. Denote by P_{\overline{I}_{X}} (t) = ∑_{l≥1} i_{l}t^{l} the Poincaré polynomial of I_{X}/(I_{X} ∩ 𝔪^{2}). Then

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = P_{𝔪/𝔪^{2}}(t) − P_{\overline{I}_{X}} (t).

Clearly, P_{𝔪/𝔪^{2}}(t) is topological, its l^{th} coefficient is dim(A/J(l))_{s_{l}}, i.e. it is the Poincaré polynomial of the linear monomials of 𝔪. (In particular, P_{𝔪/𝔪^{2}}(1) is the embedding dimension of the quotient singularity ℂ^{ν}/H.)

The degree l = α (of the splice equations) is of special interest. We claim that the set of invariant monomials of degree α is {z_{1}^{α_{1}}, …, z_{ν}^{α_{ν}} }. Indeed, by (3.4.5), for l = α one has M_{l} = 1 and s_{l} = 𝔬 = 1; hence the invariant monomials of degree α correspond to the monomials of A_{1}.

In the next Proposition, h_{i} = [E_{ir_{i}}^{∗}] ∈ H as in (3.4).

Proposition 4.3.2.

(1) Set α̌_{i} = lcm({α_{1}, …, α_{ν}} \ {α_{i}}). Then ord(h_{i}) = α_{i}α̌_{i}/α.

(2) z_{i}^{α_{i}} is linear ⇔ ord(h_{i}) = α_{i} ⇔ α = α̌_{i}.

Proof. From (3.4.2), H = {h_{1}, …, h_{ν} | ∏_{j} h_{j}^{ω_{j}} = 1, h_{j}^{α_{j}} = 1 (1 ≤ j ≤ ν)}. Since

(α_{j}, ω_{j}) = 1, h_{j}^{ω_{j}} generates ⟨h_{j}⟩ ≅ ℤ_{α_{j}}. Hence H ≅ ∏_{j} ℤ_{α_{j}}/(1, …, 1). Hence (1) follows from the exact sequence

1 → ⟨h_{i}⟩ → ∏_{j} ℤ_{α_{j}}/(1, …, 1) → ∏_{j≠i} ℤ_{α_{j}}/(1, …, 1) → 1.

For (2), let us first determine those integers k_{i} for which z_{i}^{k_{i}} is invariant. By (3.4.1), this is equivalent to the facts: ord(h_{i})|k_{i} and \frac{k_{i}}{α_{i}} (1+ \frac{α}{α_{i}} ω_{i}) ∈ ℤ. This implies (2). □

Next, we analyze P_{\overline{I}_{X}}. Since \{f_{j}\}𝔪 ⊂ 𝔪^{2}, the linear space ∑_{j} ℂf_{j} generates I_{X}/I_{X} ∩ 𝔪^{2}. In particular, P_{\overline{I}_{X}} (t) = i_{α}t^{α}, where i_{α} = dim I_{X}/I_{X} ∩ 𝔪^{2}. Hence P_{𝔪_{X}/𝔪_{X}^{2}} (t) = ∑ Q(l)t^{l} is P_{𝔪/𝔪^{2}}(t) − i_{α}t^{α}, and it is determined completely by P_{𝔪/𝔪^{2}}(t) and Q(α).

Proposition 4.3.3. Q(α) = max(0, 2 − #X_{α}). In particular, P_{𝔪_{X}/𝔪_{X}^{2}} (t) is a topological invariant in this case.

Proof. First notice that s_{α} = 𝔬 = 1 and M_{α} = 1. Moreover, α is the smallest integer l with M_{l} = 1, hence 1 ∉ X_{α}. Hence, by (4.1.4)(2) X_{α} ⊂ {a_{1}, …, a_{ν}}, and X_{α} corresponds bijectively to the non–linear monomials of type z_{i}^{α_{i}} (which are characterized in (4.3.2)). The dimension of the linear space generated by I and X_{α} is r := min(ν, ν − 2 + #X_{α}). Hence Q(α) = ν − r. □

Example 4.3.4. Consider the case of the E_{6} singularity. In this case H = ℤ_{3} = {ξ|ξ^{3} = 1}, and the universal abelian cover has equations z_{1}^{3} + z_{2}^{3} + z_{3}^{2} = 0. The coordinates (z_{1}, z_{2}, z_{3}) have degrees (2, 2, 3), H acts on them by (ξ, \overline{ξ}, 1). The linear invariant monomials are z_{1}^{3}, z_{2}^{3}, z_{1}z_{2} and z_{3}, of degree 6,6,4,3. Hence P_{𝔪/𝔪^{2}}(t) = t^{3} + t^{4} + 2t^{6}. f_{1} kills a 1–dimensional space of degree α = 6, hence P_{𝔪_{X}/𝔪_{X}^{2}} (t) = t^{3} + t^{4} + t^{6}. This is compatible with the identity X_{6} = {a_{3}} (cf. (4.3.3)) as well.

Example 4.3.5. If Sf = (1, (14, 21, 5), (5, 5, 2)) then 𝔬 = 1 and f_{1} ∈ 𝔪^{2}. Hence P_{𝔪_{X}/𝔪_{X}^{2}} = P_{𝔪/𝔪^{2}}.

Example 4.3.6. Surprisingly, the degree of P_{𝔪/𝔪^{2}} (hence of P_{𝔪_{X}/𝔪_{X}^{2}} too) can be larger than α (i.e. there may exist linear monomials of R^{H} of degree larger than the degree of the splice equations).

Consider e.g. the graph with Seifert invariants

b_{0} = 1, \underline{α} = (3, 4, 5, 6, 21), \underline{ω} = \underline{1}.

Then e = −1/420, α = 420, 𝔬 = 1. The degrees of the variables of R = ℂ[z_{1}, …, z_{5}] are 140, 105, 84, 70, 20. The group H acts on R via diagonal matrices

[ζ^{4}, −1, 1, ζ, ζ^{4}], [ζ^{2}, 1, 1, ζ^{4}, 1],

where ζ denotes a primitive 6-th root of unity. There are 9 linear monomials in R^{H}, namely:

z_{5}^{3}, z_{3} , z_{2}^{2}, z_{2}z_{4}^{3}, z_{1}z_{2}z_{4}z_{5}, z_{1}^{3}, z_{4}^{6}, z_{1}z_{4}^{4}z_{5}, z_{1}^{2}z_{4}^{2}z_{5}^{2}.

Their degrees are: 60, 84, 210, 315, 335, 420, 420, 440, 460 respectively. The two terms of degree 420 are eliminated by the (three) splice equations. Hence

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = t^{60} + t^{84} + t^{210} + t^{315} + t^{335} + t^{440} + t^{460}.

Notice that in this case #X_{420} = 3, hence (4.3.3) gives Q(420) = 0 as well. (For further properties of this singularity, see (8.2.2).)

Remark 4.3.7. Notice that if H is trivial then e.d.(X, o) = ν by (2.2.1), if 𝔬 = 1 then in (4.3.6) we characterize topologically the embedding dimension; hence, apparently, the structure of the group H has subtle influence on the topological nature of e.d.(X, o). Nevertheless, at least for the authors, this connection is still rather hidden and mysterious.

On the other hand, in order to show that the above two cases are not accidental and isolated, in the next subsection we will treat one more situation.

4.4. The case 𝔬 ‘small’. We will start with a graphs Γ with the property that for each i ∈ {1, …, ν}, there exists an integer k_{i}, 1 ≤ k_{i} ≤ α_{i}, such that z_{i}^{k_{i}} ∈ R^{H}.

Since z_{i}^{k_{i}} ∈ R^{H} if and only if h_{i}^{k_{i}} = 1 in H, the above property is equivalent with ord(h_{i}) ≤ α_{i} for all i. Since H/⟨h_{i}⟩ = ∏_{j≠i} ℤ_{α_{j}}/(1, …, 1), one gets that ord(h_{i}) = α_{i}α̃_{i}𝔬/α. Therefore, ord(h_{i}) ≤ α_{i} reads as

(4.4.1)  𝔬 ≤ α/α̃_{i} for all i.

Assume now that 𝔬 satisfies (4.4.1), but 𝔬 > 1. We claim that

(4.4.2)  P_{𝔪_{X}/𝔪_{X}^{2}} (t) ≡ P_{𝔪/𝔪^{2}}(t),

i.e. the splice equations have absolutely no effect in P_{𝔪_{X}/𝔪_{X}^{2}} (t). By (4.3.1), this is equivalent with the inclusion I_{X} ⊂ 𝔪^{2}. Recall that I_{X} is generated by expressions of form f_{j} · m, where m ∈ R^{μ^{−1}}, where μ is not the trivial character (since 𝔬 ≠ 1), hence 1 ∉ R^{μ^{−1}}. Let us analyze a product of type z_{i}^{α_{i}}m. It can be rewritten as z_{i}^{k_{i}} · (z_{i}^{α_{i}−k_{i}} · m), where the element in the parenthesis is also invariant. Hence this expression is in 𝔪^{2}, and (4.4.2) is proved.

The above case can really appear. Indeed, consider e.g. the Seifert invariants Sf = (1, (3, 5, 11), (1, 1, 5)) (see (4.5) for the notation). Then 𝔬 = |H| = 2 and α/α̃_{i} ∈ {3, 5, 11}. (Clearly, if the α_{i}’s are pairwise relative prime, and 𝔬 = 2, or more generally, 𝔬 ≤ min_{i}(α_{i}), then the above assumptions are still satisfied.)

On the other hand, merely the identity 𝔬 = 2 is not enough, see e.g. (7.1.1).

4.5. Via the next example we show how the general procedure (presented in the previous theoretical parts) runs. In this example P_{𝔪_{X}/𝔪_{X}^{2}} will again be topological.

We denote the monomial ∏ a_{i}^{k_{i}} by (k_{1}, …, k_{ν}). For example, a_{1}^{2} = (2, 0, …, 0).

Example 4.5.1. The case Sf = (2, (2, 3, 4, 5), (1, 1, 1, 4)).

Suppose that the numbering of E_{i}’s satisfy

\begin{aligned}& & & & E_{1} \\ E_{4} & E_{5} & E_{6} & E_{7} & E_{8} & E_{2} \\ & & & & E_{3} \end{aligned},

where E_{8} is the central curve and E_{i} is the end corresponding to (α_{i}, ω_{i}) for i = 1, 2, 3, 4. For ∑_{i=1}^{8} a_{i}E_{i}, we write {a_{1}, …, a_{8}}. The fundamental invariants are listed below:

(1) e = −7/60, α = 60, 𝔬 = 7, γ = 43/7 = 6\frac{1}{7}.

(2) |H| = 14.

(3) The fundamental cycle is Z = {3, 2, 2, 2, 3, 4, 5, 6}, p_{a}(Z) = 1.

(4) The canonical cycle is Z_{K} = \{\frac{25}{7}, \frac{19}{7}, \frac{16}{7}, \frac{10}{7}, \frac{20}{7}, \frac{30}{7}, \frac{40}{7}, \frac{50}{7}\}.

(5) The Hilbert series is

P_{G_{X}} (t) = 1 + t^{6} + t^{8} + t^{10} + t^{11} + 2t^{12} + t^{14} + 2t^{15} + 2t^{16} + t^{17} + 2t^{18} + t^{19} + 3t^{20}+

2t^{21} + 2t^{22} + 2t^{23} + 3t^{24} + 2t^{25} + 3t^{26} + 3t^{27} + 3t^{28} + 2t^{29} + 4t^{30} + O (t^{31}) .

(6) P_{H^{1}}(t) = t and p_{g} = 1.
(7) The degrees of z_{1}, z_{2}, z_{3}, z_{4} are \frac{1}{7}(30, 20, 15, 12).

Lemma 4.5.2.

(1) For α − l ≥ 0, s_{α−l} ≥ s_{α} − s_{l} − ν.

(2) s_{l} ≥ 6 for l ≥ 62, s_{61} = 5, s_{60} = 7.

(3) s_{l} ≥ 0 for l ≥ 14.

(4) X_{l+14} ∋ 1 for l ≥ α = 60.

Proof. (1) is elementary. For (2) write s_{l+α} = s_{l} + s_{α} = s_{l} + 7. Since P_{H^{1}}(t) = t, s_{1} = −2 and s_{l} ≥ −1 for l ≥ 2. For (3), consider the formula for the Hilbert series (see above). Hence s_{l} ≤ 3 for l ≤ 29. By (1), s_{60−l} ≥ 7 − s_{l} − 4 ≥ 0. On the other hand, from the same formula of P_{G_{X}}, one has s_{l} ≥ 0 for 14 ≤ l ≤ 30. Now, (4) follows from (4.1.4)(4). □

The following is the list of (l, s_{l}, X_{l}) with s_{l} ≥ 0, X_{l} ∌ 1, and l ≤ 74 (cf. (4.5.2)), where a = (1, 0, 0, 0), b = (0, 1, 0, 0), c = (0, 0, 1, 0), d = (0, 0, 0, 1):

6 0  ∅
8 0  ∅
10 0  ∅
11 0  ∅
12 1  {c}
15 1  ∅
16 1  {d, c}
20 2  {d, c + d, b + c}
30 3  {d, b, b + d, a, a + b + d}
36 4  {d, c, c + d, b, b + d, b + c, b + c + d,
a + c, a + b + c, a + b + c + d}
60 7  {d, c, c + d, b, b + d, b + c, b + c + d,
a + c, a + c + d, a + b + c, a + b + c + d}

Using Lemma (4.2.1) we verify

Q(12) = 1, Q(15) = 2, Q(j) = 0 for j = 16, 20, 30, 36, 60.

Therefore, the Hilbert series is

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = t^{6} + t^{8} + t^{10} + t^{11} + t^{12} + 2t^{15}.

One can check that in this case the difference P_{𝔪/𝔪^{2}}(t) − P_{𝔪_{X}/𝔪_{X}^{2}} is not concentrated only in one degree. In order to see this, notice that the action of the group H on the variables (z_{1}, …, z_{4}) is given by the following diagonal matrices [ζ^{13}, ζ^{4}, ζ^{3}, ζ^{8}], where ζ denotes a primitive 14-th root of unity. The invariant subring of ℂ[z_{1}, …, z_{4}] is generated by the following 21 monomials (we used [4]):

\begin{matrix}z_{3}^{2}z_{4} & z_{2}z_{4}^{3} & z_{1}^{2}z_{4}^{2} & z_{1}z_{2}z_{3}z_{4} & z_{2}^{2}z_{3}^{2} & z_{1}^{3}z_{3} & z_{2}^{3}z_{4}^{2} \\ z_{1}^{2}z_{2}^{2}z_{4} & z_{1}z_{2}^{3}z_{3} & z_{1}^{4}z_{2} & z_{2}^{5}z_{4} & z_{1}z_{3}^{5} & z_{1}^{2}z_{2}^{4} & z_{4}^{7} \\ z_{1}z_{3}z_{4}^{5} & z_{1}^{2}z_{2}z_{3}^{4} & z_{2}^{7} & z_{1}^{8}z_{4} & z_{2}z_{3}^{8} & z_{3}^{14} & z_{1}^{14} \end{matrix}

5. More combinatorics. The case ν ≤ 5.

5.1. A key combinatorial lemma. Recall that for each l > 0 the ideal J(l) is generated by the square–free monomials of X_{l}. In the literature such an ideal is called Stanley–Reisner ideal, A/J(l) is a Stanley–Reisner (graded) ring. Their literature is very large, see e.g. [8] and references therein. In particular, their Hilbert/Poincaré series are determined combinatorially and other beautiful combinatorial connections are provided.

Still, in the literature we were not able to find how this ideal behaves with respect to intersection with a generic hyperplane section, or even more generally, with respect to a 2–codimensional generic linear section (as it is the case in our situation, cf. the ideal I in (3.4)). As this article shows, these questions outgrow the combinatorial commutative algebra, in general. Nevertheless, the goal of the present section is to establish combinatorial results about the dimension of some graded parts (A/J(l) + I))_{s_{l}}, at least under some restriction; facts which will guarantee the topological nature of the embedding dimension and of P_{𝔪_{X}/𝔪_{X}^{2}} .

We keep all the previous notations; additionally, if X_{l} ≠ ∅ we write n_{l} := #{i | 1 ≤ i ≤ ν, X_{l} ⊂ (a_{i})}, the number of variables appearing in all the monomials of X_{l}, and m_{l} for the number of variables not appearing at all in the set of monomials which form a minimal set of generators of J(l). If n_{l} > 0 then by reordering the variables we may assume that J(l) ⊂ (u), where u = ∏_{i=ν−n_{l}+1}^{ν} a_{i}. Then J(l) has the form (u)J′(l) for some ideal J′(l) (generated by monomials which do not contain the last n_{l} variables).

Notice that by (4.1.4) the A–degree of any monomial of X_{l} is not larger than s_{l}, hence n_{l} ≤ s_{l}.

Lemma 5.1.1. (i)

dim (\frac{A}{J(l) + I} )_{s_{l}} = dim (\frac{A}{J′(l) + I} )_{s_{l}−n_{l}} + n_{l}.

In particular, Q(l) ≥ n_{l}.

(ii) If s_{l} ≥ ν − m_{l} − 1, then Q(l) = n_{l}.

Proof. First we prove (ii) in several steps.

(a) First step: If J ⊂ A is an ideal generated by reduced monomials such that J ⊄ (a_{i}) for any i ∈ {1, …, ν}, and if k ≥ ν − 1, then (A/J + I)_{k} = 0.

Consider a monomial a = ∏_{i=1}^{ν} a_{i}^{k_{i}} ∈ A_{k} for which we wish to prove that a ∈ J + I. Set y = y(a) := #{i | k_{i} = 0}. If y = 0 then a ∈ J since J is generated by reduced monomials. If y = 1, say k_{j} = 0, then by the assumption there exists a monomial m ∈ J with a_{j} ∤ m. Then a ∈ (m) ⊂ J. Suppose y = 2 and k_{1} = k_{2} = 0. Since ∑_{i=3}^{ν} k_{i} = k ≥ ν − 1, we may assume that k_{3} ≥ 2. Let a′ = a/a_{3}. Since a′a_{1}, …, a′a_{ν} are related by ν − 2 generic linear equations (induced by I), and a′a_{1}, a′a_{2} ∈ J by the argument above, we get that a′a_{1}, …, a′a_{ν} ∈ J + I as well. By induction on y, we obtain that a ∈ J + I for all a ∈ A_{k} and y(a).

(b) Next, start with the same assumptions (i.e., J is generated by reduced monomials, and there is no variable a_{i} appearing in all these monomials), and write m for the number of variables not appearing at all in the set of monomials which form a minimal set of generators of J. Then if k ≥ ν − m − 1 then (A/J + I)_{k} = 0.

Indeed, assume that a_{ν}, …, a_{ν−m+1} are not appearing in the monomials (here we may assume that m ≤ ν − 2). Assume that in the equations l_{j} = ∑_{i} a_{ji}a_{i} the matrix (a_{ji}) has the form (2.2.2); hence by the last m equations l_{j} one can eliminate the variables a_{ν}, …, a_{ν−m+1}. Doing this, we find ourself in the situation when we have ν − m variables, the same J (satisfying the same assumption), hence the statement follows from part (a).

(c) Now we prove the statement of the lemma for arbitrary n_{l}. Let φ denote the natural isomorphism A/I → A′ := ℂ[a_{1}, a_{2}]. Let J′_{l} ⊂ A′ denote the ideal such that φ(J(l)) = φ(u)J′_{l} (or, take J′_{l} := φ(J′(l))). Then we have the following exact sequence:

0 → (A′/J′_{l})_{s_{l}−n_{l}} ⟶^{×φ(u)} (A′/φ(u)J′_{l})_{s_{l}} → (A′/φ(u)A′)_{s_{l}} → 0.

(A′/J′_{l})_{s_{l}−n_{l}} corresponds to the situation covered by (b) (with ν variables, degree s_{l} − n_{l} and n_{l} + m_{l} of variables not appearing in the generators of J′(l)), hence it is zero. Therefore, Q(l) = dim (A′/φ(u)A′)_{s_{l}} = n_{l}. This proves (ii). Notice that the exact sequence used in this step (c) proves part (i) as well, since A′/J′_{l} ≈ A/(J′(l) + I). □

The ‘decomposition’ (5.1.1)(i) can be exploited more:

Theorem 5.1.2. Assume that ν − n_{l} ≤ 5, then Q(l) is topological. Consequently, P_{𝔪_{X}/𝔪_{X}^{2}} is topological provided that ν ≤ 5.

Proof. By the proof of (5.1.1) we only have to verify that dim(A/(J′(l)+I))_{s_{l}−n_{l}} is topological. If ν − n_{l} = 0 or 1, then J′(l) = (1). On the other hand, if a_{i} ∈ J′(l) for some 1 ≤ i ≤ ν, then Q(l) is topological by Lemma (4.2.1). Hence we may assume that J′(l) contains no generator a_{i} of A–degree one.

Assume that J′(l) contains a monomial of degree 2, say a_{1}a_{2}. Since J′(l) is not principal, there exists a monomial m ∈ J′(l) such that (m) ⊄ (a_{1}a_{2}). Assume that m has minimal degree among such monomials. If a_{1} | m, then a_{1}a_{2}, a_{1}^{deg m} ∈ J′_{l}. Since the image of other monomials of J′(l) in J′_{l} is of the form pa_{1}^{k} + qa_{2}^{k} modulo (a_{1}a_{2}) with k ≥ deg m, Q(l) is topological. After this discussion (up to a permutation of variables) only the following two cases remain uncovered:

m = a_{3}a_{4}, or m = a_{3}a_{4}a_{5}

For these case, we have

a_{1}a_{2}, a_{1}^{2} + ca_{2}^{2} ∈ J′_{l}, or a_{1}a_{2}, a_{1}^{3} + ca_{2}^{3} ∈ J′_{l}, c ≠ 0,

respectively. Then clearly Q(l) is topological.

Next, we may assume that J′(l) contains no monomial of degree ≤ 2. Then, automatically ν − n_{l} ≥ 4. If ν − n_{l} = 4, since non of the variables a_{1}, …, a_{4} divide all the monomials of J′(l), J′(l) should contain all the monomials of degree 3 in a_{1}, …, a_{4}. Then Q(l) is again topological.

Finally, assume that ν − n_{l} = 5. If the minimal degree of monomials of J′(l) is 4, then by a similar argument as above, Q(l) is topological. Assume that a_{1}a_{2}a_{3} ∈ J′(l) and let c denote the number of monomials of degree 3 in the minimal set G of generators of J′(l) which consists of monomials.

It is not hard to verify (using again the definition of J′(l)) that J′(l) contains all the monomials of degree 4. Hence, the only unclarified rank is 3, i.e., dim(A′/J′_{l})_{3}.

If c = 1, then clearly dim(A′/J′_{l})_{3} is topological, and G contains at least 3 monomials of degree 4. If c = 2, then we have the following two possibilities:

a_{1}a_{2}a_{3}, a_{3}a_{4}a_{5} ∈ G; a_{1}a_{2}a_{3}, a_{2}a_{3}a_{4} ∈ G.

Then clearly dim(A′/J′_{l})_{3} is topological, and G contains at least 1 (resp. 2) monomials of degree 4. The case c = 3 reduces to:

a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{3}a_{4}a_{5} ∈ G.

The image of 3 monomials in J′_{l} is:

(\begin{matrix}0 & p_{3} & 1 & 0 \\ 0 & p_{4} & 1 & 0 \\ σ_{3} & σ_{2} & σ_{1} & 1 \end{matrix}) (\begin{matrix}a_{1}^{3} \\ a_{1}^{2}a_{2} \\ a_{1}a_{2}^{2} \\ a_{2}^{3} \end{matrix}) ,

where a_{i} = p_{i}a_{1} + a_{2} (i = 3, 4, 5) and σ_{i} denotes the elementary symmetric polynomial of degree i in p_{3}, p_{4}, p_{5}. We easily see that this matrix has rank 3, which is independent of the parameters, and hence dim(A′/J′_{l})_{3} is topological.

Suppose c ≥ 4. Then we have following 6 cases:

(1) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{1}a_{2}a_{5}, a_{3}a_{4}a_{5} ∈ G,

(2) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{1}a_{3}a_{4}, a_{2}a_{3}a_{4} ∈ G,

(3) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{1}a_{3}a_{4}, a_{2}a_{3}a_{5} ∈ G,

(4) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{1}a_{3}a_{5}, a_{2}a_{3}a_{4} ∈ G,

(5) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{5}, a_{1}a_{3}a_{4}, a_{2}a_{3}a_{4} ∈ G,

(6) a_{1}a_{2}a_{3}, a_{1}a_{2}a_{4}, a_{1}a_{3}a_{5}, a_{2}a_{4}a_{5} ∈ G. For (1), the corresponding matrix is the matrix on the left:

(5.1.3)  (\begin{matrix}0 & p_{3} & 1 & 0 \\ 0 & p_{4} & 1 & 0 \\ 0 & p_{5} & 1 & 0 \\ σ_{3} & σ_{2} & σ_{1} & 1 \end{matrix}) ,  (\begin{matrix}0 & p_{3} & 1 & 0 \\ 0 & p_{4} & 1 & 0 \\ σ_{2} & σ_{1} & 1 & 0 \\ 0 & σ_{2} & σ_{1} & 1 \end{matrix}) .

Since this matrix has rank 3, dim(A′/J′_{l})_{3} is topological. For case (2), the matrix is the second one in (5.1.3). Here σ_{i} denotes the elementary symmetric polynomial of degree i in p_{3}, p_{4}. Since this matrix has rank 4, dim(A′/J′_{l})_{3} is topological. In (3)–(6), we have the same type matrices as in the case (2), and their ranks are 4. □

Remark 5.1.4. In the statement (5.1.2), the case ν = 3 is not surprising, since in this case the analytic structure has no moduli (the position of the three points P_{1}, P_{2}, P_{3} in ℙ^{1} has no moduli, cf. (2.1)). On the other hand, for the cases ν = 4 and 5 we do not see any explanation other than the (case by case combinatorial) discussion of the above proof based on the geometry of low–dimensional forms.

Remark 5.1.5. (Cf. (4.2.2).) The above proof really shows the limits of the topological nature of Q(l) from the point of view of the invariants involved in Theorem (5.1.2). E.g., if J′(l) is one of the ideals from the following list, then Q(l) is not topological (below ∗ indicates the empty set or sequence of monomials of higher degrees).

(1) s_{l} = 2, J′(l) = (a_{1}a_{2}, a_{3}a_{4}, a_{5}a_{6}, ∗); s_{l} = 3, J′(l) = (a_{1}a_{2}a_{3}, a_{4}a_{5}a_{6}, a_{7}a_{8}a_{9}, ∗); ...

(2) s_{l} = 3, J′(l) = (a_{1}a_{2}, a_{3}a_{4}a_{5}, a_{6}a_{7}a_{8}, ∗).

6. In the ‘neighborhood’ of rational singularities

6.1. The topological nature of P_{𝔪_{X}/𝔪_{X}^{2}} for rational and minimally elliptic germs. Recall that for rational and minimally elliptic singularities, by results of Artin and Laufer [1, 5], the embedding dimension is topological (see also [9] for the Gorenstein elliptic case). Therefore, it is natural to expect the topological nature of P_{𝔪_{X}/𝔪_{X}^{2}} for those weighted homogeneous singularities which belong to these families. The next result shows that this is indeed the case:

Proposition 6.1.1. Assume that the star–shaped graph Γ is either rational or numerically Gorenstein elliptic. Then for any normal weighted homogeneous singularity (X, o) with graph Γ, the polynomial P_{𝔪_{X}/𝔪_{X}^{2}} is independent of the analytic structure, it depends only on Γ.

Proof. First note that a weighted homogeneous singularity with Z_{K} ∈ L is Gorenstein. By the above description of the coefficients Q(l) of P_{𝔪_{X}/𝔪_{X}^{2}} , they are upper semi–continuous with respect to the parameter space (i.e. with respect to the full rank matrices (a_{ji})). Since P_{𝔪_{X}/𝔪_{X}^{2}} (1) = e.d.(X, o) is independent on the choice of the parameter, all these coefficients have the same property too. □

In the case of minimal rational and automorphic case we provide explicit formulae in terms of Seifert invariants.

6.2. The minimal rational case. Recall that (X, o) is minimal rational if b_{0} ≥ ν, i.e. if the fundamental cycle is reduced.

Lemma 6.2.1. Assume that b_{0} ≥ ν. Then the following facts hold:

(a) s_{l} ≥ 0 for any l, hence X_{l} ≠ ∅ for any l > 0. (In particular, the integer m_{l} is well–defined for any l > 0, cf. (5.1).)

(b) s_{l} ≥ ν − m_{l} for any l > 0.

Proof. (a) follows from l ≥ ⌈lω_{i}/α_{i}⌉ (†). For (b), notice that

s_{l} − ν ≥ _{∑i}( l − 1 − ⌈lω_{i}/α_{i}⌉ ).

For any fixed i, l − 1 − ⌈lω_{i}/α_{i}⌉ ≥ −1 by (†). Moreover, it is −1 if and only if l = ⌈lω_{i}/α_{i}⌉, or lβ_{i} < α_{i}. This implies ϵ_{i,\underline{l}} = 0 for any \underline{l}, hence a_{i} is not present in the monomials of X_{l}. □

(6.2.1)(b) combined with the combinatorial lemma (5.1.1)(ii), we get

Corollary 6.2.2. If Γ is minimal rational then Q(l) = n_{l} for any l > 0.

Let us reinterpret the integer n_{l} in terms of Seifert invariants {(α_{i}, ω_{i})}_{i}. The next result connects the arithmetical properties of continued fractions with our construction (namely with the construction of the monomials from X_{l}).

We consider a pair 0 < β < α, with gcd(α, β) = 1, and the continued fraction expansion α/β = cf [u_{1}, …, u_{𝔯}] (with all u_{i} ≥ 2, cf. (2.1)). For any 1 ≤ k ≤ 𝔯, we consider r_{k}/t_{k} := cf [u_{1}, …, u_{k}], with gcd(r_{k}, t_{k}) = 1 and r_{k} > 0.

Proposition 6.2.3. Fix (α, β) as above and l ≥ 2. Then l ∈ {r_{1}, …, r_{𝔯}} if and only if

ϵ_{j} := {jβ/α} + {(l − j)β/α} − {lβ/α} = 1

for all j ∈ {1, …, l − 1}.

Proof. First assume that l > α. Then ϵ_{α} = 0 and r_{k} ≤ α for all k, hence the equivalence follows. If l = α then l = r_{𝔯} and clearly ϵ_{j} = 1 for all j (ϵ_{j} = 0 can happen only if α | jβ which is impossible). Hence, in the sequel we assume l < α.

The proof is based on lattice point count in planar domains. If D is a closed integral polyhedron in the plane, we denote by LP (D^{∘}) the number of lattice points in its interior. We denote by Δ_{P,Q,R} the closed triangle with vertices P, Q, R.

Let Δ_{l} be the closed triangle determined by y ≤ βx/α, x ≤ l and y ≥ 0. Then

∑_{1≤j≤l−1}⌊jβ/α⌋ = LP (Δ_{l}^{∘}).

If we replace j by l − j and we add it to the previous identity, we get that

∑_{1≤j≤l−1} ϵ_{j} = (l − 1)⌊lβ/α⌋ − 2LP (Δ_{l}^{∘}).

Hence, ϵ_{j} = 1 for all j if and only if 2LP (Δ_{l}^{∘}) = (l − 1)(⌊lβ/α⌋ − 1). Hence, we have to prove that

(6.2.4)  2LP (Δ_{l}^{∘}) ≥ (l − 1)(⌊lβ/α⌋ − 1)

with equality if and only if l ∈ {r_{1}, …, r_{𝔯}}.

Assume that l = r_{k} for some k. Recall that the convex closure of the points {(r_{k}, t_{k})}_{k} together with (α, 0) contains all the lattice points of Δ_{(0,0),(α,0),(α,β)}^{∘} (†), see e.g. [16, (1.6)]. Moreover, by Pick theorem, for any triangle Δ, 2LP (Δ^{∘}) = 2+2· Area(Δ) − number of lattice points on the boundary of Δ. Therefore, LP (Δ_{r_{k}}^{∘}) = LP (Δ_{(0,0),(r_{k},0),(r_{k},t_{k})}); this by Pick theorem is (r_{k} − 1)(t_{k} − 1). Since l = r_{k} and ⌊lβ/α⌋ = t_{k} by (†), (6.2.4) follows.

Next, we show that in (6.2.4) one has strict inequality whenever r_{k} < l < r_{k+1}. Let Δ be the triangle with vertices (r_{k}, t_{k}), (r_{k+1}, t_{k+1}) and (the non–lattice

point) (r_{k+1}, r_{k+1}t_{k}/r_{k}) (with two vertices on the line y = t_{k}y/r_{k}). Notice that (r_{k+1}, t_{k+1}) is above the line y = t_{k}x/r_{k}, cf. also with (6.2.6).

Lemma 6.2.5. Δ^{∘} contains no lattice points. All the lattice points on the boundary of Δ, except (r_{k+1}, t_{k+1}), are sitting on the line y = t_{k}y/r_{k}.

Proof. This basically follows from the following identity of continued fractions:

(6.2.6)  \frac{r_{k}}{t_{k}} − \frac{r_{k+1}}{t_{k+1}} = \frac{1}{t_{k}t_{k+1}} .

Indeed, considering the slope of the segment with ends (x, y) and (r_{k}, t_{k}), the lattice points in Δ with y > t_{k}x/r_{k} and x < r_{k+1} are characterized by r_{k} < x < r_{k+1} and

\frac{t_{k}}{r_{k}} < \frac{y − t_{k}}{x − r_{k}} ≤ \frac{t_{k+1} − t_{k}}{r_{k+1} − r_{k}} .

By a computation and using (6.2.6) this transforms into

0 < yr_{k} − xt_{k} ≤ \frac{x − r_{k}}{r_{k+1} − r_{k}} ,

which has no integral solution. Since t_{k+1} − r_{k+1}t_{k}/r_{k} = 1/r_{k} ≤ 1/2, there is only one lattice point on the vertical edge of Δ, namely (r_{k+1}, t_{k+1}). □

Set P := (l, lt_{k}/r_{k}), the intersection point of {x = l} with {y = t_{k}x/r_{k}}; and let Q (resp. R) be the intersection of {x = l} with the segment [(r_{k}, t_{k}), (r_{k+1}, t_{k+1})] (resp. with y = βx/α). Let M be the number of lattice points on the open segment with ends (r_{k}, t_{k}) and P. Then, by (†) and (6.2.5),

(6.2.7)  LP (Δ_{l}^{∘}) = LP (Δ_{(0,0),(l,0),P}^{∘}) + M + 1.

(The last ‘1’ counts (r_{k}, t_{k}).) Notice that R is not a lattice point (lβ/α ∉ ℤ), on the line x = l there is no lattice point strict between R and Q (by †), Q is not a lattice point and there is no lattice point in the interior of [P Q] (by (6.2.5)). Hence

(6.2.8)  lt_{k}/r_{k} ≥ ⌊lβ/α⌋.

Assume that in (6.2.8) one has equality, i.e., P is a lattice point. Then, by Pick theorem, 2LP (Δ_{(0,0),(l,0),P}^{∘}) = (l − 1)(⌊lβ/α⌋ − 1) − 1 − M. This together with (6.2.7) provides the needed strict inequality.

Finally, assume that P is not a lattice point. Let N be the number of lattice points on the interior of the segment with ends (0, 0) and P′ := (l, ⌊lβ/α⌋). Then comparing the triangles Δ_{(0,0),(l,0),P} and Δ_{(0,0),(l,0),P′}, and by Pick theorem we get

2LP (Δ_{l}^{∘}) ≥ (l − 1)(⌊lβ/α⌋ − 1) + N + M + 1. □

Now we are ready to formulate the main result of this subsection. For any (α, β) as above, consider the sequence {r_{1}, …, r_{𝔯}} as in (6.2.3) and define (following Wagreich and VanDyke (cf. [24]))

f_{α,β}(t) = ∑_{k=1}^{𝔯} t^{r_{k}}.

The next theorem generalizes the result of VanDyke valid for singularities satisfying b_{0} ≥ ν + 1, [23].

Theorem 6.2.9. Assume that Γ has Seifert invariants b_{0} and {(α_{i}, ω_{i})}_{i=1}^{ν} with b_{0} ≥ ν. Set β_{i} = α_{i} − ω_{i}. Then

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = (b_{0} − ν + 1)t + ∑_{i=1}^{ν} f_{α_{i},β_{i}}(t).

Proof. Clearly, Q(1) = s_{1} + 1 = b_{0} − ν + 1. For l > 1, Q(l) is given by (6.2.2) and (6.2.3). □

6.3. The case of automorphic forms. Assume that G is a Fuchsian group (of the first kind) with signature (g = 0; s; α_{1}, …, α_{ν}), where s ≥ 0. Let A(G) be the graded ring of automorphic forms relative to G; for details see [25]. Then by [25, (5.4.2)], X = Spec(A(G)) is a normal weighted homogeneous singularity whose star–shaped graph has Seifert invariants b_{0} = ν + s − 2, and {(α_{i}, ω_{i})}_{i=1}^{ν} with all ω_{i} = 1. The values s ≥ 2, s = 1 resp. s = 0 correspond exactly to the fact that X is minimal rational, non–minimal rational or minimally elliptic, cf. ([25, (5.5.1)]).

Wagreich in [25, Theorem (3.3)] provides an incomplete list for P_{𝔪_{X}/𝔪_{X}^{2}} (t) for all these singularities. By our method not only we can reprove his formulae, but we also complete his result clarifying all the possible cases.

Notice that if β = 1 then f_{α,β}(t) = f_{α,1}(t) = ∑_{k=2}^{α} t^{k}.

Theorem 6.3.1. Assume that the Seifert invariants of the graph Γ satisfy ω_{i} = 1 for all i and s = b_{0} − ν + 2 ≥ 0 and ν ≥ 3. Then P_{𝔪_{X}/𝔪_{X}^{2}} (t) has the next forms: (I) The cases considered by Wagreich [25]:

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = f(t) + ∑_{i=1}^{ν} f_{α_{i},1}(t),

where f(t) is given by the following list

\begin{matrix}s ≥ 2 & f(t) = (b_{0} − ν + 1)t \\ s = 1 & f(t) = −t^{2} + (ν − 2)t^{3} \\ s = 0, ν ≥ 4, ∑_{i} α_{i} ≥ 11 & f(t) = −3t^{2} + (ν − 5)t^{3} \\ s = 0, ν = 3, α_{i} ≥ 3 for all i, ∑_{i} α_{i} ≥ 12 & f(t) = −3t^{2} − 2t^{3} − t^{4} \\ s = 0, ν = 3, α_{1} = 2, α_{2}, α_{3} ≥ 4, ∑_{i} α_{i} ≥ 13 & f(t) = −3t^{2} − 2t^{3} − t^{4} − t^{5} \\ s = 0, ν = 3, α_{1} = 2, α_{2} = 3, α_{3} ≥ 9 & f(t) = −3t^{2} − 2t^{3} − t^{4} − t^{5} − t^{7} \end{matrix}

(II). In all the remaining cases s = 0 (and X is a hypersurface with γ = 1), and P_{𝔪_{X}/𝔪_{X}^{2}} (t) is:

Sf = (1, (2, 3, 7), \underline{1})  t^{6} + t^{14} + t^{21},
Sf = (1, (2, 3, 8), \underline{1})  t^{6} + t^{8} + t^{15},
Sf = (1, (2, 4, 5), \underline{1})  t^{4} + t^{10} + t^{15},
Sf = (1, (2, 4, 6), \underline{1})  t^{4} + t^{6} + t^{11},
Sf = (1, (2, 5, 5), \underline{1})  t^{4} + t^{5} + t^{10},
Sf = (1, (3, 3, 4), \underline{1})  t^{3} + t^{8} + t^{12},
Sf = (1, (3, 3, 5), \underline{1})  t^{3} + t^{5} + t^{9},
Sf = (1, (3, 4, 4), \underline{1})  t^{3} + t^{4} + t^{8},
Sf = (2, (2, 2, 2, 3), \underline{1}) t^{2} + t^{6} + t^{9},
Sf = (2, (2, 2, 2, 4), \underline{1}) t^{2} + t^{4} + t^{7},
Sf = (2, (2, 2, 3, 3), \underline{1}) t^{2} + t^{3} + t^{6},
Sf = (3, (2, 2, 2, 2, 2), \underline{1}) 2t^{2} + t^{5}.

Proof. The proof is a case by case verification based on the results of (4.1). □

7. Examples when P_{𝔪_{X}/𝔪_{X}^{2}} (t) is not topological

7.1. Our goal is to present an equisingular family of weighted homogeneous surface singularities in which the embedding dimension is not constant. As a consequence, the embedding dimension of weighted homogeneous singularities, in general, is not a topological invariant (i.e., cannot be determined by the Seifert invariants).

Example 7.1.1. Consider the Seifert invariants Sf = (2, (2, 2, 3, 3, 7, 7), \underline{1}).

Suppose that the numbering of E_{i}’s satisfies (−b_{1}, …, −b_{6}) = \underline{α}, and we will use similar notations as in (4.5.1). Here is the list of the fundamental invariants:

(1) e = −1/21, α = 42, 𝔬 = 2, γ = 43.

(2) |H| = 84.

(3) The fundamental cycle is Z = {3, 3, 2, 2, 1, 1, 6}, p_{a}(Z) = 7.

(4) The canonical cycle is Z_{K} = {22, 22, 15, 15, 7, 7, 44}.

(5) The Hilbert series is

P_{G_{X}} (t) = \frac{1 − 2t − 4t^{2} − 3t^{3} + 2t^{5} + 2t^{6} + t^{7} + 2t^{8} + 2t^{9} + t^{10}}{1 + t − t^{3} − t^{4} − t^{7} − t^{8} + t^{10} + t^{11}} + P_{H^{1}}(t)

= 1 + t^{6} + t^{12} + t^{14} + t^{18} + t^{20} + t^{21} + t^{24} + t^{26} + t^{27} + t^{28} + t^{30}
+ t^{32} + t^{33} + t^{34} + t^{35} + t^{36} + t^{38} + t^{39} + t^{40}

+ t^{41} + 3t^{42} + t^{44} + t^{45} + t^{46} + t^{47} + 3t^{48} + t^{49} + t^{50} + O (t^{51}) .

(6) p_{g} = 24 and

P_{H^{1}}(t) = 3t + t^{2} + t^{3} + t^{4} + t^{5} + t^{7} + t^{8} + t^{9} + t^{10}

+ t^{11} + t^{13} + t^{15} + t^{16} + t^{17} + t^{19}

+ t^{22} + t^{23} + t^{25} + t^{29} + t^{31} + t^{37} + t^{43}.

(7) The degrees of z_{1}, … z_{6} are (\frac{21}{2}, \frac{21}{2}, \frac{21}{3}, \frac{21}{3}, \frac{21}{7}, \frac{21}{7}). Claim. s_{l} ≥ 0 for l ≥ 44. In particular, X_{l+44} ∋ 1 for l ≥ α = 42.

Proof. From the expression of P_{H^{1}}(t), we read s_{1} = −4 and s_{l} ≥ −2 otherwise. Then use s_{l+α} = s_{l} + 𝔬. For the second statement use (4.1.4)(4). □

Then, by a computation (for l ≤ 86) one verifies that the only values l for which 1 ∉ X_{l} are l ∈ {6, 14, 21, 42}. In the first three cases s_{l} = 0 and X_{l} = ∅. For l = 42 one has s_{42} = 2 and X_{42} = {a_{1}a_{2}, a_{3}a_{4}, a_{5}a_{6}}. It depends essentially on the choice of the full rank matrix (a_{ji}), cf. (4.2.2). In particular,

e.d.(X, o) = \begin{cases}3 & if p_{1}p_{2} − p_{3}p_{4} ≠ 0, \\ 4 & if p_{1}p_{2} − p_{3}p_{4} = 0; \end{cases}

where for the notations {p_{k}}, see (4.2.2).

Next, let us find a system of equations for X. We take the following splice diagram equations:

p_{1}x_{1}^{2} + x_{2}^{2} + x_{3}^{3}, p_{2}x_{1}^{2} + x_{2}^{2} + x_{4}^{3}, p_{3}x_{1}^{2} + x_{2}^{2} + x_{5}^{7}, p_{4}x_{1}^{2} + x_{2}^{2} + x_{6}^{7}.

The action of the group H is given by the following diagonal matrix:

[−i, i, e^{\frac{iπ}{3}} , e^{− \frac{iπ}{3}} , e^{\frac{iπ}{7}} , e^{− \frac{iπ}{7}} ].

Then R^{H} is generated by the following 21 monomials (in the next computations we used SINGULAR [4]):

z_{5}z_{6}, z_{3}z_{4}, z_{1}z_{2}, z_{2}^{4}, z_{1}^{4}, z_{2}^{2}z_{4}^{3}, z_{1}^{2}z_{4}^{3}, z_{2}^{2}z_{3}^{3}, z_{1}^{2}z_{3}^{3}, z_{4}^{6}, z_{3}^{6},
z_{2}^{2}z_{6}^{7}, z_{1}^{2}z_{6}^{7}, z_{2}^{2}z_{5}^{7}, z_{1}^{2}z_{5}^{7}, z_{4}^{3}z_{6}^{7}, z_{3}^{3}z_{6}^{7}, z_{4}^{3}z_{5}^{7}, z_{3}^{3}z_{5}^{7}, z_{6}^{14}, z_{5}^{14}.

These are denoted by y_{1}, …, y_{21}. Then X is defined by the following ideal:

y_{9} + y_{3}^{2} + p_{1}y_{5},  y_{11} − 2p_{1}y_{3}^{2} − y_{4} − p_{1}^{2}y_{5},
y_{8} + p_{1}y_{3}^{2} + y_{4},  y_{10} − 2p_{2}y_{3}^{2} − y_{4} − p_{2}^{2}y_{5},
y_{7} + y_{3}^{2} + p_{2}y_{5},  y_{2}^{3} − (p_{1} + p_{2})y_{3}^{2} − y_{4} − p_{1}p_{2}y_{5},
y_{6} + p_{2}y_{3}^{2} + y_{4},  y_{19} − (p_{1} + p_{3})y_{3}^{2} − y_{4} − p_{1}p_{3}y_{5},
y_{3}^{4} − y_{4}y_{5},  y_{18} − (p_{2} + p_{3})y_{3}^{2} − y_{4} − p_{2}p_{3}y_{5},
y_{15} + y_{3}^{2} + p_{3}y_{5},  y_{17} − (p_{1} + p_{4})y_{3}^{2} − y_{4} − p_{1}p_{4}y_{5},
y_{14} + p_{3}y_{3}^{2} + y_{4},  y_{16} − (p_{2} + p_{4})y_{3}^{2} − y_{4} − p_{2}p_{4}y_{5},
y_{13} + y_{3}^{2} + p_{4}y_{5},  y_{21} − 2p_{3}y_{3}^{2} − y_{4} − p_{3}^{2}y_{5},
y_{12} + p_{4}y_{3}^{2} + y_{4},  y_{20} − 2p_{4}y_{3}^{2} − y_{4} − p_{4}^{2}y_{5},
y_{1}^{7} − (p_{3} + p_{4})y_{3}^{2} − y_{4} − p_{3}p_{4}y_{5}

By eliminating the variables except for y_{1}, y_{2}, y_{3}, and y_{5}, we obtain

\begin{matrix}y_{3}^{4} − y_{2}^{3}y_{5} + (p_{1} + p_{2})y_{3}^{2}y_{5} + p_{1}p_{2}y_{5}^{2}, \\ y_{1}^{7} − y_{2}^{3} + (p_{1} + p_{2} − p_{3} − p_{4})y_{3}^{2} + (p_{1}p_{2} − p_{3}p_{4})y_{5} \end{matrix}

Notice that if p_{1}p_{2} − p_{3}p_{4} ≠ 0, then y_{5} can also be eliminated.

Example 7.1.2. It is instructive to consider the graph with Seifert invariants Sf = (3, (2, 2, 3, 3, 7, 7), \underline{1}) too; it has the same legs as (7.1.1), but its b_{0} is −3 (instead of −2). By this ‘move’, we modify the graph into the ‘direction of rational graphs’. The consequence is that the ambiguity of the example (7.1.1) disappears, and P_{𝔪_{X}/𝔪_{X}^{2}} (t) becomes topological (and the embedding dimension increases).

In this case e = −22/21, α = 42, γ = 43/22, P_{H^{1}}(t) = 2t. Since γ < 2, s_{l} ≥ −1 for l ≥ 2. Since s_{3} = 1, s_{l} ≥ 0 for l ≥ 5. Then X_{l} ∋ 1 for l ≥ 5 + α = 47.

The following is the list of (l, s_{l}, X_{l}) with s_{l} ≥ 0 and 1 ∉ X_{l} (and where l = (1, 1, 0, 0, 0, 0), m = (0, 0, 1, 1, 0, 0), n = (0, 0, 0, 0, 1, 1)).

\begin{matrix}2 & 0 & ∅ \\ 3 & 1 & ∅ \\ 4 & 2 & \{n\} \\ 5 & 3 & \{n\} \\ 6 & 6 & \{m + n, l + n\} \\ 7 & 5 & \{n\} \\ 14 & 14 & \{n, m + n, l + n, l + m\} \\ 21 & 21 & \{n, m, m + n\} \\ 42 & 44 & \{n, m, m + n, l, l + n, l + m, l + m + n\} \end{matrix}

By a computation one gets

P_{𝔪_{X}/𝔪_{X}^{2}} (t) = t^{2} + 2t^{3} + 2t^{4} + 2t^{5} + 2t^{6} + 2t^{7}.

8. Splice–quotients with star–shaped graphs

8.1. General discussion. In this section we extend our study to the case of splice–quotient singularities. We briefly recall their definition under the assumption that their graphs is star–shaped.

Let Γ as above, and let {f_{j}} be the set of Brieskorn polynomials considered in subsection (2.2), and let X be the quotient (weighted homogeneous singularity) of their zero–set by the action of H. Let {g_{j}(z_{1}, …, z_{ν})}_{j=1}^{ν−2} be a set of power series satisfying the following conditions:

• with respect to the weights wt(z_{i}) = (|e|α_{i})^{−1}, the degree of the leading form of g_{j} is bigger than deg f_{j} for any j;

• all monomials in g_{j} are elements of μ-eigenspace R^{μ} (recall that {f_{j}} ⊂ R^{μ}).

Then the singularity Y := {(z_{i}) ∈ ℂ^{ν−2} | f_{j} + g_{j} = 0, j = 1, …, ν − 2}/H is a normal surface singularity; it is called a splice–quotient (see [15]). All these singularities belong to an equisingular family sharing the same resolution graph Γ.

Here is the main question: is P_{𝔪_{X}/𝔪_{X}^{2}} (t) topological (i.e. independent of the choice of the matrix (a_{ji}) and of the power series g_{j}) — at least when for the weighted homogeneous case with the corresponding graph the answer is positive ?

The answer splits in two parts. First, if the graph is rational, or minimally elliptic, then by a semi–continuity argument, the discussion of subsection (6.1) remain true for splice quotients as well. Moreover, the combinatorial formulas of theorems (6.2.9) (valid for b_{0} ≥ ν) and of (6.3.1) (valid for ω_{i} = 1 and b_{0} − ν ≥ −2) are true for splice quotients too (as far as the corresponding combinatorial assumptions on the graph are satisfied).

On the other hand, we do not expect that the positive results proved for weighted homogeneous singularities listed in (4.3) (the case 𝔬 = 1), in (4.4) (𝔬 ‘small’), or in (5.1.2) (ν ≤ 5) will remain valid for splice quotients too. A counterexample in the case 𝔬 = 1 is analyzed in the next subsection. This example also emphasizes that in the next inequality

Proposition 8.1.1. e.d.(Y, o) ≤ e.d.(X, o),

— a consequence of the upper-semicontinuity —, the strict inequality might occur.

8.2. In [7, (4.5)], it is shown that the embedding dimension is not constant in an equisingular deformation of a weighted homogeneous singularity. However, in that example, the general fibre is not a splice–quotient. The aim of this section is to show the following:

Proposition 8.2.1. There exists an equisingular deformation of a weighted homogeneous singularity in which the embedding dimension is not constant, and it satisfies the following:

(1) The embedding dimension of the central fibre is determined by the Seifert invariants (i.e. all the weighted homogeneous singularities with the same link have the same embedding dimension).

(2) Every fibre is a splice-quotient.

We will analyze with more details the case 𝔬 = 1, i.e. when {f_{j}} ⊂ R^{H} and deg(f_{j}) = α. In this case the ideals I_{X}, I_{Y} ⊂ R^{H} of X and Y are generated by {f_{j}}, {f_{j} + g_{j}}, respectively. Let l_{X} (resp. l_{Y}) be the rank of the image of {f_{j}}_{j} (resp. {f_{j} + g_{j}}_{j}) in 𝔪/𝔪^{2}. Then e.d.(X, o) = dim(𝔪/𝔪^{2}) − l_{X} (and similarly for Y). Hence, e.d.(X, o) − e.d.(Y, o) = l_{Y} − l_{X}.

If l denotes the number of linear monomials of degree > α, then by letting g_{j} be general linear combinations of those monomials, we obtain l_{Y} = min(ν − 2, l_{X} + l), which in special situations can be larger than l_{X}. Taking special linear combinations, we get for l_{Y}:

l_{X} ≤ l_{Y} ≤ min(ν − 2, l_{X} + l)

and any value between the two combinatorial bounds can be realized. The following example shows the existence of a deformation Y which verifies (8.2.1).

Example 8.2.2. Let X be the weighted homogeneous singularity considered in (4.3.6). Recall that there are 9 linear monomials in R^{H}, l_{X} = 2 and the embedding dimension is topological, it is 7. Moreover, α = 420, and there are two linear monomials of degree greater than 420: deg(z_{1}^{2}z_{4}^{2}z_{5}^{2}) = 460, deg(z_{1}z_{4}^{4}z_{5}) = 440. Let us take the following equations for Y (with c, d ∈ ℂ):

z_{1}^{3} + z_{3}^{5} + z_{5}^{21} = 0
z_{4}^{6} + z_{3}^{5} + z_{5}^{21} = 0
z_{2}^{4} + z_{3}^{5} + z_{5}^{21} + cz_{1}z_{4}^{4}z_{5} + dz_{1}^{2}z_{4}^{2}z_{5}^{2} = 0.

Then l_{Y} = 2 if c = d = 0, but l_{Y} = 3 (hence e.d.(Y, o) = 6) otherwise.

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Rényi Institute of Mathematics, Budapest, Hungary.

E-mail address: nemethi@renyi.hu

Department of Education, Yamagata University, Yamagata 990-8560, Japan.

E-mail address: okuma@e.yamagata-u.ac.jp