Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000

ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

J. FERNÁNDEZ DE BOBADILLA, I. LUENGO-VELASCO, A. MELLE-HERNÁNDEZ and A. NÉMETHI

1. Introduction

Let C be an irreducible projective plane curve in the complex projective space ℙ^{2} with singular points {p_{i}}_{i=1}^{ν}. From C one can extract the following information: its degree d, and the local embedded topological types T_{i} of the local singular germs (C, p_{i}) ⊂ (ℙ^{2}, p_{i}). It is a very interesting, and still open problem, to characterize those collections of local embedded topological types {T_{i}}_{i=1}^{ν} (without fixing the positions of the points p_{i}) which can be realized by such a projective curve C of degree d. This remarkable problem is not only important for its own sake, but it is also connected with crucial properties, problems and conjectures in the theory of open surfaces.

For instance, the open surface ℙ^{2} \ C is ℚ-acyclic if and only if C is a rational cuspidal curve. On the other hand, the rigidity conjecture proposed by Flenner and Zaidenberg in [13] (and supported by many examples, see [9], [13], [14], [15]) says that every ℚ-acyclic affine surfaces Y with logarithmic Kodaira dimension κ̄(Y) = 2 must be rigid. (E.g., if C has at least three cusps then κ̄(ℙ^{2} \ C) = 2, cf. [46].) This conjecture for Y = ℙ^{2} \ C would imply the projective rigidity of the curve C in the sense that every equisingular deformation of C in ℙ^{2} would be projectively equivalent to C.

Among other interesting related open problems we mention: every rational cuspidal curve can be transformed by a Cremona transformation into a line (this is the Coolidge-Nagata problem, see [6],[26]); or, the determination of the maximal number of cusps among all the rational cuspidal plane curves (proposed by F. Sakai in [16]) – this number is expected to be small (the maximal known by the authors is four). In a recent paper by K. Tono [40] it is proved that the maximal number is strictly less than nine. Finally, we also list the conjectured numerical inequality (2.3) [34] of S.Yu. Orevkov.

The first-mentioned ‘characterization problem’ (on the realization of prescribed topological types of singularities) has a long and rich history providing many interesting compatibility properties connecting local invariants of the germs {(C, p_{i})}_{i} with some global invariants of C (like its degree, or the log-Kodaira dimension of ℙ^{2} \C, etc.). Most of the compatibility properties are mere identities or inequalities (see e.g. the Matsuoka-Sakai inequality, or Orevkov’s sharp inequality, both being consequences of the logarithmic version of the Bogomolov-Miyaoka-Yau inequality).

2000 Mathematics Subject Classification 14B05,14J17,32S25,57M27,57R57 (primary), 14E15,32S45,57M25 (secondary).

The first author is supported by the Netherlands Organization for Scientific Research (NWO). The first three authors are partially supported by BFM2001-1488-C02-01. The forth author is partially supported by NSF grant DMS-0304759 and OTKA 42769/46878.

2 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

The more complex Varchenko’s compatibility property is provided by the semicontinuity of the spectral numbers of isolated hypersurface singularities, and it consists of a finite set of inequalities.

Our goal is to propose a new compatibility property – valid for rational cuspidal curves C – which seems to be rather powerful (see section 3 for some comparison with some classical criterions). Its formulation is surprisingly very elementary. Consider a collection (C, p_{i})_{i=1}^{ν} of locally irreducible plane curve singularities (i.e. cusps), let Δ_{i}(t) be the characteristic polynomials of the monodromy action associated with (C, p_{i}), and Δ(t) := ∏_{i} Δ_{i}(t). Its degree is 2δ, where δ is the sum of the delta-invariants of the singular points. Then Δ(t) can be written as 1 + (t − 1)δ + (t − 1)^{2}Q(t) for some polynomial Q(t). Let c_{l} be the coefficient of t^{(d−3−l)d} in Q(t) for any l = 0, …, d − 3.

Conjecture 1. Let (C, p_{i})_{i=1}^{ν} be a collection of local plane curve singularities, all of them locally irreducible, such that 2δ = (d−1)(d−2) for some integer d. Then if (C, p_{i})_{i=1}^{ν} can be realized as the local singularities of a degree d (automatically rational and cuspidal) projective plane curve of degree d then

c_{l} ≤ (l + 1)(l + 2)/2 for all l = 0, …, d − 3. (∗_{l})

In fact, the integers N_{l} := c_{l} − (l + 1)(l + 2)/2 are symmetric: N_{l} = N_{d−3−l}; and N_{0} = N_{d−3} = 0 automatically. Moreover, there is a surprising phenomenon in the above conjecture:

If ν = 1, then the conjecture is true if and only if in all the inequalities (∗_{l}), in fact, one has equality (cf. Theorem 3).

The conjecture can be reformulated in the language of the semigroups of the germs (C, p_{i}) (and the degree d) as well. E.g., if ν = 1, then collection of vanishings of all the coefficients N_{l} can be described by a very precise and mysterious distribution of the elements of the semigroup of the unique singular point with respect to the intervals I_{l} := ( (l − 1)d, ld] (cf. section 3):

If ν = 1, the conjecture predicts that there are exactly min{l + 1, d} elements of the semigroup in I_{l} for any l > 0.

The idea (and the main motivation) of the above conjecture came from the Seiberg-Witten invariant conjecture formulated for normal surface singularities by Nicolaescu and the forth author [29], respectively from the set of counterexamples for this conjecture provided by superisolated singularities [22] found by the last three authors. For the completeness of the presentation, this is described in short in section 2, but we emphasize that the present article is independent of the techniques of [29], in particular involves no Seiberg-Witten theory. (On the other hand, we are witnesses of a mysterious connection whose deeper understanding would be a wonderful mathematical goal.)

As supporting evidence, in the body of the paper we prove the following result:

Theorem 1. If the logarithmic Kodaira dimension κ̄ := κ̄(ℙ^{2} \ C) is ≤ 1, then the above conjecture is true. In fact, in all these cases N_{l} = 0 for any l = 0, …, d−3 (regardless of ν).

on rational cuspidal projective plane curves 3

The proof of Theorem 1 consists of many steps. Its structure is the following.

(a) If κ̄ = −∞ then ν = 1 by [46]. Moreover, all these curves are classified by H. Kashiwara [19]. The family contains as an important subfamily the Abhyankar-Moh-Suzuki (AMS) curves.

In this case, our proof runs as follows: first we verify the vanishing of the coefficients N_{l} for the AMS curves (section 4) – this corresponds to the F_{I} family in Kashiwara’s classification. The case of the other subfamily (the F_{II} curves) is treated in section 6. The proof is based on an induction where the starting point is given by some unicuspidal curves which have only one Puiseux pair. This case is completely resolved in section 5.

(b) The case κ̄ = 0 cannot occur by a result of Sh. Tsunoda [41], see also the paper by Orevkov [34].

(c) If κ̄ = 1 then by a result of Wakabayashi [46] one has ν ≤ 2. In the case ν = 1, K. Tono writes the possible equations of the curves [39]. (Notice that Tsunoda’ classification in [42] is incomplete.) We verify the conjecture for them in section 7. On the other hand, by another result of Tono [38], the case ν = 2 corresponds exactly to the Lin-Zaidenberg bicuspidal rational plane curves. For them we verify the conjecture in section 8.

We wish to emphasize that in the process of the verification of the conjecture, in fact, we list all the possible local topological types (e.g. Eisenbud-Neumann splice diagrams) of local plane curve singularities, together with the degrees d, which can be geometrically realized with κ̄ ≤ 1. But, from the point of view of the proof of Theorem 1, this information is far to be enough to verify the conjecture. Even if one knows the local topological types (e.g. the local resolution graphs, or even better, the generators of the corresponding semigroups), one needs sometimes additional rather involved arithmetical arguments to complete the proof.

It is important to notice that in the κ̄ = 2 case there are (infinitely many, and for arbitrarily large d) examples when not all the coefficients N_{l} vanish. The complete picture for curves with d ≤ 6 is provided in section 2, together with some additional introductory examples with larger d. The following list provides those cases which are verified in the present article.

Theorem 2. If κ̄ = 2, then in the following cases the conjecture is true:

(a) d ≤ 6;

(b) C is unicuspidal with one Puiseux pair (see section 5);

(c) Orevkov’s curves C_{4k} and C_{4k}^{∗} [34] (see section 9).

(d) d is even, ν = 2, and the multiplicity sequence is [d − 2], [2_{d−2}] (see 2.4).

See also Remark 1, valid for ν = 1.

2. The Main Conjecture. Motivation and first comments.

For the convenience of the reader, we recall some notations and classical properties of plane curve singularities, which will be intensively used in the sequel.

4 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

2.1. Invariants of germs of irreducible plane curve singularities.

To encode the topology of a germ of an irreducible plane curve singularity (C, 0) ⊂ (ℂ^{2}, 0) several sets of invariants can be used: Puiseux pairs, Newton pairs, (minimal) embedded resolution graph, Eisenbud-Neumann splice diagram, semigroup, etc. We mainly use the Eisenbud-Neumann splice diagram (cf. [7], page 49). If the germ (C, 0) has g Newton pairs {(p_{k}, q_{k})}_{k=1}^{g} with gcd(p_{k}, q_{k}) = 1 and p_{k} ≥ 2 and q_{k} ≥ 2 (and by convention, q_{1} > p_{1}), define the integers {a_{k}}_{k=1}^{g} by a_{1} := q_{1} and a_{k+1} := q_{k+1} + p_{k+1}p_{k}a_{k} for k ≥ 1. Then its Eisenbud-Neumann splice diagram decorated by the numerical data {(p_{k}, a_{k})}_{k=1}^{g} has the following shape:

[figure: C̃; a_{1}; a_{2}; a_{g−1}; a_{g}; p_{1}; p_{2}; p_{g−1}; p_{g}; …]

The characteristic polynomial Δ_{(C,0)}(t) of the monodromy acting on the first homology of the Milnor fiber of the singularity can be computed by A’Campo’s formula [2] from the splice diagram. If we define

β_{k} := a_{k}p_{k}p_{k+1} … p_{g} for 1 ≤ k ≤ g;
β̄_{0} := p_{1}p_{2} … p_{g};
β̄_{k} := a_{k}p_{k+1} … p_{g}  for 1 ≤ k ≤ g,

then Δ_{(C,0)} is given by:

Δ_{(C,0)}(t) = \frac{(t − 1) ∏_{1≤k≤g}(t^{β_{k}} − 1)}{∏_{0≤k≤g} (tβ̄_{k} − 1)}.

The polynomial Δ_{(C,0)} is a complete (embedded) topological invariant of the germ (C, 0) ⊂ (ℂ^{2}, 0), similarly as the semigroup Γ_{(C,0)} ⊂ ℕ generated by all the possible intersection multiplicities i({h = 0}, C) at 0 for all h ∈ O_{(ℂ^{2},0)}. The degree 2δ_{(C,0)} of Δ_{(C,0)} is the conductor of the singularity, where the delta-invariant δ_{(C,0)} is the cardinality of the finite set ℕ \ Γ_{(C,0)}.

By [17], Δ_{(C,0)}(t) = (1 − t) · L(t), where L(t) := ∑_{k∈Γ_{(C,0)}} t^{k} is the Poincaré series of Γ_{(C,0)}. In fact, the minimal set of generators of Γ_{(C,0)} consists of the g + 1 elements β̄_{i} (0 ≤ i ≤ g). It is also known that each element γ ∈ Γ_{(C,0)} can be represented in a unique way in the form γ = k_{0}β̄_{0} + ∑_{1≤j≤g} k_{j}β̄_{j} with k_{0} ≥ 0 and 0 ≤ k_{j} ≤ p_{j} − 1 for 1 ≤ j ≤ g, see [36].

2.2. Motivation. The Seiberg-Witten invariant conjecture (SW C).

In [29] L. Nicolaescu and the forth author formulated the following conjecture (as a generalization of the “Casson invariant conjecture” of Neumann and Wahl [32]):

If the link of a normal surface singularity (X, 0) is a rational homology sphere then the geometric genus p_{g} of (X, 0) has an “optimal” topological upper bound. Namely,

p_{g} ≤ sw(M) − (K^{2} + s)/8. (SW C)

on rational cuspidal projective plane curves 5

Moreover, if (X, 0) is a ℚ-Gorenstein (e.g. hypersurface) singularity then in (SW C) the equality holds.

Here, sw(M) is the (topological, or ‘modified’) Seiberg-Witten invariant of the link M of (X, 0) associated with its canonical spin^{c} structure, K is the canonical cycle associated with a fixed resolution graph G of (X, 0), and s is the number of vertices of G (see [29] for more details).

The (SW C)-conjecture was verified successfully for many different families, see e.g. [28, 29, 30, 31]. But the last three authors in [22] found some counterexamples based on superisolated singularities. This class “contains” in a canonical way the theory of complex projective plane curves, a fact which is crucial in the next discussion.

Hypersurface superisolated singularities were introduced in [21] by the second author and achieved the reputation of being a distinguished class of singularities and source of interesting examples and counterexamples. A hypersurface singularity f : (ℂ^{3}, 0) → (ℂ, 0), f = f_{d} + f_{d+1} + … (where f_{j} is homogeneous of degree j) is superisolated if the projective plane curve C := {f_{d} = 0} ⊂ ℙ^{2} is reduced with isolated singularities {p_{i}}_{i=1}^{ν}, and these points are not situated on the projective curve {f_{d+1} = 0}. In this case the embedded topological type (and the equisingular type) of f depends only on the curve C. Notice also that the link of f is a rational homology sphere if the curve C is rational and cuspidal (i.e. if all the germs (C, p_{i}) are locally irreducible). In the sequel we also will assume these two facts.

In [22] the authors have shown that some hypersurface superisolated singularities with ν = #Sing(C) ≥ 2 do not satisfy the above Seiberg-Witten invariant conjecture. Moreover, in all the counterexamples p_{g} > sw(M) − (K^{2} + s)/8 (contrary to the inequality predicted by the general conjecture 2.2 !). On the other hand, even after an intense search of the existing cases, the authors were not able to find any counterexample with ν = 1. (In fact, one of the goal of the present article is to explain, at least partially, these phenomenons, cf. Theorem 3.)

In the next paragraphs, we plan to reformulate (SW C) relating with an even deeper conjectural property. We denote by Δ_{i} the characteristic polynomial of (C, p_{i}) ⊂ (ℙ^{2}, p_{i}), set Δ(t) := ∏_{i} Δ_{i}(t) and 2δ := deg Δ(t). By the rationality of C one has (d − 1)(d − 2) = 2δ. Clearly, δ is the sum of the delta-invariants of the germs (C, p_{i})_{i=1}^{ν}. Notice also that (see e.g. [30] (4.3)):

Δ(1) = 1 and Δ′(1) = δ.  (1)

In the next discussion it is convenient to introduce the polynomials P and Q by

Δ(t) = 1 + (t − 1)P (t) = 1 + (t − 1)δ + (t − 1)^{2}Q(t).

The corresponding coefficients are denoted by

P (t) = ∑_{l=0}^{2δ−1} a_{k}t^{k}, and Q(t) = ∑_{l∤d} b_{l}t^{l} + ∑_{l=0}^{d−3} c_{l}t^{(d−3−l)d}.  (2)

The following definition helps to describe the properties of these coefficients.

Definition/Lemma 1. We say that a polynomial D(t) = ∑_{l} α_{l}t^{l} has the negative distribution property if ∑_{0≤l≤k} α_{l} ≤ 0 for any k ≥ 0. If D(t) = N(t)(1 − t) for some other polynomial N(t), then D(t) has the negative distribution property if and only if all the coefficients of N(t) are nonpositive.

6 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

For the superisolated singularity (X, 0) := {f = 0} those invariants which appear in the formula (SW C) (cf. 2.2) are computed in [22] as follows (here sw(M) is computed by the Reidemeister-Turaev torsion normalized by the Casson-Walker invariant) :

\{ \begin{matrix}K^{2} + s = −(d − 1)(d^{2} − 3d + 1); p_{g} = d(d − 1)(d − 2)/6; and \\ sw(M) = \frac{1}{d} ∑_{ξ^{d}=1≠ξ} \frac{Δ(ξ)}{(ξ − 1)^{2}} + \frac{1}{2d}Δ(t)′′(1) − \frac{δ(6δ − 5)}{12d}. \end{matrix}  (3)

Consider now the (a priori) rational function

R(t) := \frac{1}{d} ∑_{ξ^{d}=1} \frac{Δ(ξt)}{(1 − ξt)^{2}} − \frac{1 − t^{d^{2}}}{(1 − t^{d})^{3}} .  (4)

Proposition 1. With the above notations one has:

(a) R(t) can be written in the form D(t^{d})/(1 − t^{d}), where

D(t) = (d − 2)t + (d − 3)t^{2} + … + t^{d−2} − ∑_{k=0}^{2δ−1} a_{k}t^{⌈k/d⌉}

= ∑_{k≥0}(1 − a_{k})t^{⌈k/d⌉} − \frac{1 − t^{d}}{(1 − t)^{2}} .

(b) In (a), some combinations of the coefficients {a_{k}}_{k} give no contribution in R(t). Indeed, if one writes P (t) = δ + (t − 1)Q(t) as above (cf. (2)), then the {b_{l}}_{l} coefficients have no effect in R. More precisely, R(t) = N(t^{d}), where

N(t) = ∑_{l=0}^{d−3} (c_{l} − \frac{(l + 1)(l + 2)}{2}) t^{d−3−l}.

(c) N(t) is a symmetric polynomial (i.e. N(t) = t^{d−3} · N(1/t)) with integral coefficients and with N(0) = 0.

(d)

R(1) = sw(M) − \frac{K^{2} + s}{8} − p_{g}.

In particular, the following facts also hold:

(e) R(t) ≡ 0 if and only if D(t) ≡ 0 if and only if N(t) ≡ 0. In this case evidently (SW C) is true (with equality).

(f) D(t) has the negative distribution property if and only if all the coefficients of R(t) (or, equivalently, of N(t)) are nonpositive. In this case p_{g} ≥ sw(M) − (K^{2} + s)/8.

Proof. Write Δ(t)/(1 − t)^{2} = 1/(1 − t)^{2} − P (t)/(1 − t). By an elementary computation

\frac{1}{d} ∑_{ξ^{d}=1} \frac{1}{(1 − ξt)^{2}} = \frac{1 + (d − 1)t^{d}}{(1 − t^{d})^{2}} .

For the second contribution consider the monomials a_{k}t^{k} of P (t). Express any fixed k in the form k = dn + r with r = 1, 2, …, d. Then

\frac{1}{d} ∑_{ξ^{d}=1} \frac{ξ^{k}t^{k}}{1 − ξt} = \frac{t^{k}}{d} ∑_{ξ^{d}=1} \frac{ξ^{r}}{1 − ξt} = \frac{t^{k}}{d(1 − t^{d})} ∑_{ξ^{d}=1}( 1 + ξt + … + (ξt)^{d−1})ξ^{r} = \frac{t^{d(n+1)}}{1 − t^{d}} .

Notice also that

\frac{1 − t^{d}}{(1 − t)^{2}} = 1 + 2t + … + (d − 1)t^{d−2} + d(t^{d−1} + t^{d} + …),

and

\frac{1 + (d − 1)t}{(1 − t)} = 1 + d(t + t^{2} + …) = ∑_{k≥0} t^{⌈k/d⌉},

which ends the proof of (a). The proof of (b) is similar. (c) follows from (b) and from (4) using the fact that Δ is monic and symmetric. For (d) write

R(t) = \frac{1}{d} ∑_{ξ^{d}=1≠ξ} \frac{Δ(ξt)}{(1 − ξt)^{2}} + \frac{(1 + t + … + t^{d−1})^{3}Δ(t) − d(1 + t + … + t^{d^{2}−1})}{d(1 − t)^{2}(1 + t + … + t^{d−1})^{3}}.

Then compute lim_{t→1} R(t) by the L’Hospital rule, then use (1) and (3).

The starting point of the present article is the following conjecture.

Conjecture 2. Assume that C is a rational cuspidal projective plane curve as above. Then the property (f) of Proposition 1 is always true. Namely, N(t) (or equivalently, R(t)) has only nonpositive coefficients. In other words, D(t) has the negative distribution property.

In fact, one can ask about the stronger property (e) in Proposition 1, i.e. about the vanishing of N(t) (or equivalently, of R(t)). Obviously, this does not hold in general: all the examples of [22] with p_{g} > sw(M) − (K^{2} + s)/8 have R(t) ≢ 0 (cf. also with 2.3). On the other hand, using Proposition 1(d) one gets the following: if one can verify independently the inequality p_{g} ≤ sw(M) − (K^{2} + s)/8 then the above Conjecture 2 is true if and only if, in fact, R(t) ≡ 0. This is the case, e.g., if ν = 1:

Theorem 3. Assume that C is an unicuspidal rational curve. Then Conjecture 2 is true if and only if R(t) is identical zero.

We will give two proofs of this theorem. The first one, given below, is based on a geometrical result which reveals the connection of Conjecture 2 with links of surface singularities and topological invariants of 3-manifolds. The second (see section 3.4) is rather elementary and surprising.

Proof. Recall that a good resolution graph of a normal surface singularity is called AR (almost rational) if the graph has at least one vertex with the following property: if one replaces the decoration (self-intersection) of the corresponding vertex by a smaller (more negative) integer then one gets a rational graph (in the sense of Artin); for more details see [28]. The point is that if a normal surface singularity 8 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

with rational homology sphere link admits an AR good resolution graph then it satisfies the inequality p_{g} ≤ sw(M) − (K^{2} + s)/8, cf. [28] (9.6). [Here we have to mention that [28] verifies the inequality for the Ozsváth-Szabó ‘Euler-characteristic’ sw^{OSZ}(M), but, by [35], this agrees with the Reidemeister-Turaev sign refined torsion normalized by the Casson-Walker invariant, used here and in [29].] Therefore, it is enough to verify that {f = 0} admits an AR good resolution graph. Recall that the minimal good resolution graph G of {f = 0} can be obtained from the minimal good embedded resolution graph Γ of the unique singular point (C, p) of C when gluing an extra vertex w by a unique edge to the unique (−1) vertex v of Γ. The decoration of w (as any decoration) is negative, say (−r), its precise value is not essential here – for details see e.g. [22]. Now, let G_{s} be the graph obtained from G by changing the decoration (−1) of v by (−2). Then G_{s} is the resolution graph of a sandwiched singularity. Indeed, consider again Γ. Blow up a generic point on the curve corresponding to v, and then blow up r − 1 generic points on the new curve. In this way one gets a non-minimal resolution graph of (C, p) which contains the graph G_{s} as a subgraph.

See also Example 3 for some additional comments about Theorem 3.

2.3. Examples (d small; ν ≥ 1).

Below, any singular point (C, p_{i}) will be identified by its multiplicity sequence. Since the number of occurrences of the multiplicity 1 in the multiplicity sequence equals the last multiplicity greater than 1, we omit the multiplicity 1: we denote such a sequence by [m_{1}, …, m_{l}] where m_{1} ≥ m_{2} ≥ … ≥ m_{l} > m_{l+1} = 1 for a suitable l ≥ 1. In fact, we will write [m̂_{1_{r_{1}}} , …, m̂_{k_{r_{k}}}] for a multiplicity sequence which means that the multiplicity m̂_{i} occurs r_{i} times for i = 1, …, k. For example, [4_{2}, 2_{3}] means [4, 4, 2, 2, 2, 1, 1].

For the classification of the cuspidal rational curves with degree d ≤ 5, see e.g. the book of Namba [27]; for the classification of multiplicity sequences of rational cuspidal plane curves of degree 6 see e.g. Fenske’s paper [8].

If d = 3, then C has a unique singularity of type [2]. If d = 4, then there are four possibilities; the corresponding multiplicity sequences of the singular points are [3]; [2_{3}]; [2_{2}], [2] and [2], [2], [2]. By a verification (or by Proposition 1(c)), in all these cases N(t) ≡ 0, hence the conjecture is true.

For d = 5 and d = 6 Table 1 shows all the possible multiplicity sequences together with the polynomials N(t). For all these cases the conjecture is also true.

d  ν  type of cusp  −N(t)

d  ν  type of cusp  −N(t)

5  1  [4]  0 5  1  [2_{6}]  0 5  2  [3, 2] , [2_{2}]  0 5  2  [3] , [2_{3}]  2t 5  2  [2_{2}] , [2_{4}]  2t 5  3  [3] , [2_{2}] , [2]  2t 5  3  [2_{2}] , [2_{2}] , [2_{2}]  6t 5  4  [2_{3}] , [2] , [2], [2]  8t

6  1  [5]  0 6  1  [4, 2_{4}]  0 6  1  [3_{3}, 2]  0 6  2  [3_{3}] , [2]  0 6  2  [3_{2}, 2] , [3]  0 6  2  [3_{2}] , [3, 2]  0 6  2  [4, 2_{3}] , [2]  0 6  2  [4, 2_{2}] , [2_{2}]  0 6  2  [4] , [2_{4}]  t + t^{2} 6  3  [4] , [2_{3}] , [2]  0 6  3  [4] , [2_{2}] , [2_{2}]  0 Table 1. Rational cuspidal curves of degrees 5 and 6

Table 2. Rational cuspidal curves of degree 7

reference  d  ν  type of cusp  −N(t)
x^{6}z + y^{7}  7  1  [6]  0
x^{5}z^{2} + y^{7}  7  2  [5, 2_{2}] , [2_{3}]  0
x^{4}z^{3} + y^{7}  7  2  [4, 3] , [3_{2}]  0
[8]  7  2  [5] , [2_{5}]  2t + 2t^{2} + 2t^{3}
[8]  7  2  [4, 2_{3}] , [3_{2}]  t + 2t^{2} + t^{3}
[8]  7  2  [4, 2_{2}] , [3_{2}, 2]  2t + 2t^{2} + 2t^{3}
[8]  7  2  [4] , [3_{3}]  2t + 2t^{2} + 2t^{3}
[15]  7  3  [4, 2_{2}] , [3_{2}] , [2]  3t + 4t^{2} + 3t^{3}
[14]  7  3  [5] , [2_{4}] , [2]  3t + 3t^{3}
[14]  7  3  [5] , [2_{3}] , [2_{2}]  3t + 3t^{3}

For d = 7 there is no complete classification (known by the authors). Table 2 shows some examples (the first column provides either a possible equation, or the reference where the corresponding curve has been constructed).

Here the first example is of Abhyankar-Moh-Suzuki type, while the next two of Lin-Zaidenberg type, cf. with the next sections.

2.4. Example (d large).

Examples with arbitrarily large d and with non-vanishing R(t) (but still satisfying the conjecture) exist. E.g., if C has two cusps of types [d − 2], [2_{d−2}] (see e.g. [8]), and d is even, then

−N(t) = ∑_{k=1}^{\frac{d}{2} −2} k(t^{k} + t^{d−3−k}).

It is instructive (and sometimes rather mysterious) to verify our conjecture for the other families listed e.g. in Fenske’s article [8].

Theorem 1 has the following immediate consequence:

Corollary 1. Let f = f_{d} + f_{d+1} + …: (ℂ^{3}, 0) → (ℂ, 0) be a hypersurface superisolated singularity with κ̄(ℙ^{2} \{f_{d} = 0}) < 2. Then the Seiberg-Witten invariant conjecture (cf. [29], see 2.2 here) is true for (X, 0) = ({f = 0}, 0).

It is a big question for the authors: what is the (conceptual) connection between the log Kodaira dimension (of what?) with the general conjecture 2.2 ?

3. ν = 1 revisited. Comparison with other criterions.

Assume now that ν = 1, and write Sing(C) = {p}. Recall that the characteristic polynomial Δ of (C, p) ⊂ (ℙ^{2}, p) is a complete (embedded) topological invariant of this germ, similarly as the semigroup Γ_{(C,p)} ⊂ ℕ, see 2.1. In the next discussion we will replace Δ by Γ_{(C,p)}.

Recall, that by [17], Δ(t) = (1−t)·L(t), where L(t) = ∑_{k∈Γ_{(C,p)}} t^{k} is the Poincaré series of Γ_{(C,p)}. Since P (t) was defined by the identity Δ(t) = 1 − P (t)(1 − t), one gets L(t) + P (t) = 1/(1 − t) = ∑_{k≥0} t^{k}. In particular, P (t) = ∑_{k∈ℕ\Γ_{(C,p)}} t^{k}.

10 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

We let the reader to verify that Proposition 1 implies the following facts (cf. also with Theorem 3).

3.1. Fact I.

If ν = 1 then

sw(M) − \frac{K^{2} + s}{8} = ∑_{k∉Γ(C,p)}⌈k/d⌉.

In particular, p_{g} = sw(M) − (K^{2} + s)/8 if and only if D′(1) = 0, or equivalently, if:

∑_{k∉Γ_{(C,0)}}⌈k/d⌉ = d(d − 1)(d − 2)/6.

3.2. Fact II.

Q(t) = \frac{(∑_{k∉Γ_{(C,p)}} t^{k}) − δ}{t − 1} .

Then Conjecture 2 (in the light of Theorem 3) predicts that the coefficient c_{l} of the monomial t^{d(d−3−l)} of Q(t) is exactly (l + 1)(l + 2)/2 (for any 0 ≤ l ≤ d − 3).

3.3. Fact III.

D(t) = ∑_{k∈Γ_{(C,p)}} t^{⌈k/d⌉} − \frac{1 − t^{d}}{(1 − t)^{2}} .

In particular, the second reformulation of the Conjecture 2 (and Theorem 3) is the identity

∑_{k∈Γ_{(C,p)}} t^{⌈k/d⌉} = 1 + 2t + … + (d − 1)t^{d−2} + d(t^{d−1} + t^{d} + t^{d+1} + …). (CP)

Property (CP) (similarly as the property of Fact II) connects the local invariant Γ_{(C,p)} with the degree d of C. It predicts a very precise distribution law for the elements of the semigroup Γ_{(C,p)} with respect to the intervals I_{l} := ( (l − 1)d, ld] (l ∈ ℕ). It says that for any l ≥ 0 one has

#Γ_{(C,p)} ∩ I_{l} = min{l + 1, d}. (CP_{l})

By the symmetry of the semigroup (namely, k ∈ Γ_{(C,p)} if and only if 2δ − 1 − k ∉ Γ_{(C,p)}), one has that (CP_{l}) is true if and only if (CP_{d−2−l}) is true. In fact, (CP_{l}) is automatically true for l = 0 and any l ≥ d − 2. But for 1 ≤ l ≤ d − 3 it combines a lot of restrictions.

3.4. A property of semigroups of plane curves

Now we will provide an alternative and very elementary proof of Theorem 3. It is based on a rather surprising and general property about the semigroups of a (not necessarily rational) plane curve at any collection of points with the sole assumption that the curve is locally irreducible at them.

Let C be a (not necessarily rational) irreducible plane curve of degree d. Let p_{1}, ..., p_{r} be a set of points of C such that the germ (C, p_{i}) is irreducible for any i. Denote by Γ_{i} the semigroup of (C, p_{i}).

Proposition 2. Suppose l < d. Let (n_{1}, ..., n_{r}) be an r-uple of positive numbers such that n_{1} + ... + n_{r} ≥ ld. Then

∑_{i=1}^{r} #(Γ_{i} ∩ [0, n_{i}]) ≥ (l + 1)(l + 2)/2.

Proof. Since the ideas of the proof appear when r = 1, and this is the case what we will use, for simplicity of notation we give the proof only for that case. Consider n_{1} ≥ ld. Observe that (l + 1)(l + 2)/2 is the dimension of the space of homogeneous polynomials P in three variables of degree l. Let γ(t) be a local parametrisation of C at p_{1}. The composition P (γ(t)) can be written as P (γ(t)) = ∑_{h=1}^{∞} L_{h}(P)t^{h}. Therefore the conditions, imposed to P, that the local intersection multiplicity i_{p_{1}}(C, {P = 0}) is > n_{1}, is given by the equations L_{h}(P) = 0 for 0 ≤ h ≤ n_{1}. On the other hand, as the semigroup Γ_{1} is the collection of intersection multiplicities of C with other curves at p_{1}, the number of independent conditions is at most #(Γ_{1} ∩ [0, n_{1}]), i.e. L_{h}(P) = 0 for h ≤ n_{1} and h ∈ Γ_{1} implies the vanishing L_{h}(P) for all h ≤ n_{1}. Therefore, if #(Γ_{1} ∩ [0, n_{1}]) < (l + 1)(l + 2)/2, then there exists a non-zero polynomial P of degree l with i_{p_{1}}(C, {P = 0}) > n_{1}. But, by Bezout’s theorem any such polynomial must have a component in common with C, but, as C is irreducible, this is impossible.

Proof of Theorem 3. Our Conjecture 2 in the unicuspidal case implies that the polynomial

D(t) = ∑_{l=0}^{d−1}(#Γ_{1} ∩ ((l − 1)d, ld] − (l + 1))t^{l}

has the negative distribution property. We have proved in the previous proposition that D(t) has the positive distribution property. Hence conjecture implies that D(t) is identically 0.

3.5. Comparing with consequences of log Bogomolov-Miyaoka-Yau type inequalities.

We are rather surprised that the restrictions (CP_{l}), imposed already by the very first intervals I_{l}, are closely related with famous properties (conjectures) of rational plane curves. In order to exemplify this, we need some more notations.

Similarly as above, let [m_{1}, m_{2}, …] be the multiplicity sequence of (C, p), the unique singular point of C. Following Nagata [26], we define t to be the maximal positive integer such that m_{1} ≥ m_{2} + … + m_{t}.

We rewrite the elements of Γ_{(C,p)} as {0 = γ_{0} < γ_{1} < γ_{2} < …}. Recall that the minimal set of generators of Γ_{(C,p)} verifies β̄_{0} < … < β̄_{g}. In terms of the decorations of the Eisenbud-Neumann splice diagram β̄_{0} = p_{1} … p_{g} = m_{1}, β̄_{k} = a_{k}p_{k+1} … p_{g} for 1 ≤ k < g, and β̄_{g} = a_{g}, see 2.1. In fact, since a_{2} > a_{1}p_{1}p_{2} and p_{1} ≥ 2, one has β̄_{2} > 2β̄_{1}. In particular, the first (at least) three elements of Γ_{(C,p)} depend only on β̄_{0} and β̄_{1} (hence, all their properties can be verified essentially at the level of 12 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

germs with one Puiseux pair). Using these facts, one can verify that

\{ \begin{matrix}γ_{1} = m_{1}; γ_{2} = m_{1} + m_{2}; and \\ m_{1} + 2m_{t+1} ≤ γ_{3} ≤ 3m_{1}. \end{matrix}  (5)

3.5.1. Example. (CP_{1}). By the above notations, (CP_{1}) says that γ_{2} ≤ d < γ_{3}. The first part γ_{2} ≤ d can be verified easily. Indeed, if L_{p} denotes the tangent cone (line) of C at p, then i_{p}(C, L_{p}) ∈ Γ_{(C,p)} and it is strict larger than m_{1}, hence γ_{2} ≤ i_{p}(C, L_{p}) ≤ d (the second inequality by Bézout’s theorem).

For the second inequality, notice that if C satisfies the Nagata-Noether inequality d < m_{1} + 2m_{t+1}, then by (5) it satisfies d < γ_{3} too. Recall that any curve C which can be transformed by a Cremona transformation into a line satisfies the Nagata-Noether inequality [18, 23, 26]. On the other hand, it is conjectured that any cuspidal rational curve can be transformed into a line by a Cremona transformation (see [23]).

Notice also that via (5), (CP_{1}) (or d < γ_{3}) implies that d < 3m_{1}, an inequality valid for any cuspidal rational curve C, which was proved in [23] by the log Miyaoka inequality [20, 25]. In fact, d < γ_{3} also implies that i(C, L_{p}) = γ_{2} = m_{1} + m_{2}.

3.5.2. Finally, let us reconsider again the inequality m_{1}+m_{2} = γ_{2} ≤ d. Here, by a result of Yoshihara [47] (whose proof is based on the Abhyankar-Moh theorem), one has equality m_{1} + m_{2} = d if and only if C is an Abhyankar-Moh-Suzuki curve (cf.Definition 1). The Abhyankar-Moh theorem [1] says that such a curve can be transformed into a line, hence by the above discussion it satisfies (CP_{1}). In the next section we will show that, in fact, it satisfies all the restrictions (CP_{l}), l ≥ 1.

3.5.3. The interested reader is invited to analyze some other particular restrictions (CP_{l}), e.g. (CP_{l}) for l = 1, 2, 3, combined together; cf. also with the last statement of Remarks 1(1).

Remark. Notice also that the above distribution law (CP) is very different from Arnold’s prediction of the distribution of generic sub-semigroups of ℕ [3]. This shows that the geometric realization of (C, p) as the unique singular point of a degree d rational curve implies that the semigroup of (C, p) “is far to be generic” (a fact already suggested by the local Abhyankar-Azevedo theorem as well).

3.6. Comparison with Varchenko’s criterion.

The above negative distribution property has some analogies with the criterions provided by the semicontinuity of the spectrum [44, 45]. Namely, if {(C, p_{i})}_{i} are the local singular points of the degree d curve C, then the multisingularity ∑_{i}(C, p_{i}) is in the deformation of the universal plane germ (U, 0) := (x^{d}+y^{d}, 0). In particular, the collection of all spectral numbers Sp of the local plane curve singularities (C, p_{i})_{i} satisfies the semicontinuity property compared with the spectral numbers of (U, 0) for any interval (α, α + 1).

In order to exemplify the similarities, let us assume for simplicity that ν = 1. Since the spectral numbers of (U, 0) are of type l/d, the semicontinuity property for intervals (−1 + l/d, l/d) ( l = 2, 3, …, d − 1) reads as follows:

#{α ∈ Sp : α < l/d} ≤ (l − 2)(l − 1)/2.  (6)

On the other hand, the negative distribution property of D(t) is equivalent to

#{k ∈ Γ_{(C,p)} : k ≤ ld} ≤ (l + 1)(l + 2)/2  (7)

for any l = 1, 2, …, d − 3. Although the similarities are striking, it is not easy at all to compare the two set of inequalities: the transition from Sp(C, p) to Γ_{(C,p)} arithmetically is not very simple. (If we add to the discussion the semicontinuity intervals (−1+l/d, l/d) with l > d, then the comparison is even more difficult.) But even when Sp(C, p) and Γ_{(C,p)} can be easily compared (see e.g. below), still, the comparison of the inequalities is not obvious. Let us exemplify this in the case when (C, p) has only one Puiseux pair, say (a, b), a < b. In this case the semicontinuity transforms into

#{i ≥ 0, j ≥ 0 : ia + jb < abl/d} ≤ (l − 2)(l − 1)/2 + [al/d] + [bl/d] + 1.

(Recall that (a − 1)(b − 1) = (d − 1)(d − 2).) On the other hand, our criterion (7) is:

#{i ≥ 0, j ≥ 0 : ia + jb ≤ ld} ≤ (l + 2)(l + 1)/2.

In the next examples show that the two restrictions are (at least arithmetically) independent.

Example 1. The semicontinuity of the spectrum does not imply the negative distribution property. Indeed, (d, a, b) = (11, 4, 31) satisfies the semicontinuity but N(t) has some positive coefficients (Γ in I_{4} has six elements). A similar example is (19, 7, 52); here the first l when CP_{l} fails is l = 7, I_{l} has 9 semigroup elements. (The reader is invited to verify that these triples (d, a, b) cannot be realized geometrically, a fact compatible with our conjecture.)

In fact, one can even enter in the game another restriction, namely the sharp inequality of Orevkov [34]:

d < \frac{3 + \sqrt{5}}{2}(a + 1) + \frac{1}{\sqrt{5}},  (8)

and ask if the semicontinuity together with (8) would imply arithmetically the negative distribution property. The answer again is no, as (again) the above two examples show.

Nevertheless, for d large, ν = 1 and (C, p) with only one Puiseux pair, computer experiment shows that Orevkov inequality and the semicontinuity of the spectrum imply the vanishing of R(t).

Example 2. Notice that for ν = 1, Theorem 3 replaces the negative distribution property with the vanishing of N(t). This is a non-trivial additional restriction. E.g., (d, a, b) = (5, 3, 7) satisfies the semicontinuity property of the spectrum and also N(t) = −t. (Its geometric realization is excluded by Theorem 3.)

Example 3. The negative distribution property does not imply the semicontinuity of the spectrum. Consider the case (d, a, b) = (7, 3, 16). Then it satisfies the negative distribution property (with N(t) = −t^{2}) but it does not satisfy the semicontinuity of the spectrum.

Let us reconsider the triplet (d, a, b) = (7, 3, 16) from the point of view of Theorem 3: this is a candidate for a unicuspidal curve but with R(t) ≠ 0. According to Theorem 3 a geometric realization cannot exist, but the very existence of this 14 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

triplet shows that the proof of Theorem 3 cannot be replaced by a pure arithmetical argument (having starting point the non-positivity of the coefficients of R, and the fact that (C, p) has only one Puiseux pairs, and proving the vanishing of R).

Remark 1. For ν = 1, we do not know any example when R is zero but the situation geometrically is not realizable. (In particular, we do not know any example when R or N are vanishing, but the semicontinuity of the spectrum fails.) This motivates the following conjecture:

Conjecture 3. Conjecture 3.2 (or, equivalently, 3.3) covers an ‘if and only if’ property: the local topological type (C, p) ⊂ (ℙ^{2}, p) can be realized by a degree d unicuspidal rational curve if and only if the property (CP) is valid.

4. The validity of (CP) for Abhyankar-Moh-Suzuki curves.

The AMS type curves appeared naturally in the study of rational plane curves C meeting with a line L at only one point {p} = C∩L. By the Lin-Zaidenberg theorem [48] the curve C has at most another cusp as singularities. C is called a Abhyankar-Moh-Suzuki curve if C \ L is smooth, respectively Lin-Zaidenberg curve if C \ L is singular. K. Tono has given classifications, up to projective equivalence, of AMS curves in [37] and of LZ curves in [38]. Here we will not use these classifications, but we use the relation of these curves with automorphisms of the affine plane ℂ^{2} = ℙ^{2} \ L instead.

Definition 1. An irreducible plane curve C is said to be of Abhyankar-Moh-Suzuki type (AMS type for short) if there exists a line L ⊂ ℙ^{2} such that C \ L is isomorphic to ℂ. In our situation this means that ν = 1 and C∩L = Sing(C) = {p}.

Not any curve with ν = 1 is of AMS type, e.g. the examples (c)-(f) in Examples 1 are not (cf. also with 3.5.2). The simplest AMS curve is {zx^{d−1} + y^{d} = 0}. In this case Γ_{(C,p)} is generated by two elements, d − 1 and d, and (CP) can be easily verified. The goal of the present section is to prove the general case.

Theorem 4. (CP) is satisfied by any AMS curve; in other words R(t) ≡ 0.

Identify ℂ^{2} with ℙ^{2} \ L. In the next discussion we consider algebraic automorphisms ϕ : ℂ^{2} → ℂ^{2} with components (f, g). Recall, that by [1], a curve C is of AMS type if and only if it is the compactification in ℙ^{2} of the zero locus of a component of a certain automorphism ϕ. The embedding of ℂ^{2} into ℙ^{2} allows us to view any automorphism of ℂ^{2} as a birational transformation of ℙ^{2}. The point is that the combinatorics of the minimal embedded resolution of (C, p) is closely related to the combinatorics of the minimal resolution of the indeterminacy of ϕ as a birational transformation of ℙ^{2}, and this last one can be described precisely. In the sequel we make this statement more precise. For details see [10, 11].

Let π′ : X′ → ℙ^{2} the minimal resolution of the indeterminacy of ϕ. Then (π′)^{∗}L is a divisor with normal crossings and its dual graph A (weighted by the corresponding self intersections) has the form of Figure 1. Here the vertices of A are the “black” vertices. The strict transform of L is denoted

Figure 1.

Figure 2. An elementary graph of degree n

by a “white” vertex with decoration L. This graph can be obtained from elementary pieces.

An elementary graph of degree n is defined is Figure 2. The graph A is obtained by putting r elementary graphs of degrees n_{1}, …, n_{r} one after the other, identifying the last vertex of each graph with the first vertex of the next, and weighting the identified vertices with −3. We say that the graph A has r floors. The morphism π′ : X′ → ℙ^{2} is a composition of blowing ups, which are totally ordered by appearance. This gives a total order of the vertices of the graph (which can be recovered combinatorially by successively contracting (−1)-vertices).

Now, we concentrate on the unique singular point p of C. The minimal embedded resolution π : X → ℙ^{2} of (C, p) ⊂ (ℙ^{2}, p) is again a composition of blowing ups, and all these blowing ups appear in the minimal resolution of the indeterminacy of ϕ. In fact, there are two possibilities:

(a) π is the composition of all the blowing ups of π′ except the last n_{r} − 1 of them;

(b) π is the composition of the blowing ups of the first r − 1 floors of π′ except

the last n_{r−1} − 1 of them. If the second possibility holds then one can find a different automorphism ψ = (f, g) such that C is the closure of {f = 0} and the graph associated with ψ has r − 1 floors satisfying possibility (a). Therefore, one can always assume the validity of (a). This means that the embedded resolution graph of (C, p) can be obtained from the above graph A by deleting the last n_{r} − 1 (black) vertices, and changing the decoration of E_{n_{r}} into −1. The strict transform C̄ intersects E_{n_{r}}.

In particular, the singularity p of the curve C has r Newton pairs, and the decorations {(p_{k}, a_{k})}_{k=1}^{r} of its Eisenbud-Neumann splice diagram (cf. 2.1) can be computed by standard graph-determinant computations (cf. [7], section 21). One gets the following:

\{ \begin{matrix}d = i_{p}(z, C̄) = n_{1} … n_{r}; \\ p_{1} = n_{1} − 1; and p_{k} = n_{k} for k = 2, …, r; \\ a_{1} = n_{1}, and a_{k} = n_{1}^{2} … n_{k−1}^{2}n_{k} − 1 for k = 2, …, r. \end{matrix}  (9)

16 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

In particular, the generators β̄_{i} of Γ_{(C,p)} are β̄_{0} = (n_{1} − 1)n_{2} … n_{r}, β̄_{1} = n_{1} … n_{r}, and for 1 < k ≤ r one has β̄_{k} = (n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)n_{k+1} … n_{r}.

We will verify (CP) by induction over r. Assume that A = A_{r} has r floors of degrees n_{1}, …, n_{r}. Let Γ_{r} be the corresponding semigroup. If r = 1 then Γ_{1} is generated by n_{1} − 1 and n_{1}, d_{1} = n_{1} and (CP) can be easily verified. If r > 1, by (9) and 2.1, the degree d_{r} of of C is d_{r} = n_{1} … n_{r} = d_{r−1}n_{r}, and Γ_{r} is generated by n_{r}Γ_{r−1} and β̄_{r} := n_{1}^{2} … n_{r−1}^{2}n_{r} − 1. In fact, any element x ∈ Γ_{r} can be written in a unique way as x = n_{r}y + bβ̄_{r} with y ∈ Γ_{r−1} and 0 ≤ b < n_{r} (see e.g. [36]).

Now, the inductive step runs as follows:

∑_{x∈Γ_{r}} t^{⌈x/d_{r}⌉} = ∑_{b=0}^{n_{r}−1} ∑_{y∈Γ_{r−1}} t^{⌈(yn_{r}+bβ̄_{r})/d_{r}⌉} = ∑_{b=0}^{n_{r}−1} ∑_{y∈Γ_{r−1}} t^{bd_{r−1}+⌈y/d_{r−1}⌉}

= \frac{1 − t^{d_{r}}}{1 − t^{d_{r−1}}} · \frac{1 − t^{d_{r−1}}}{(1 − t)^{2}} .

5. The case ν = 1 and (C, p) with one Puiseux pair

Assume that the unique cusp of C has exactly one Puiseux pair (a, b) (where 1 < a < b). Then clearly (a − 1)(b − 1) = (d − 1)(d − 2), where d = deg(C) as above.

In the sequel we denote by {φ_{j}}_{j≥0} the Fibonacci numbers φ_{0} = 0, φ_{1} = 1, φ_{j+2} = φ_{j+1} +φ_{j}. The Fibonacci numbers have a remarkable amount of interesting properties, see e.g. Vajda’s book [43]. We also will use some of them which will be crucial in the next arguments and also in the proof of the conjecture for Kashiwara’s families (cf. section 6). E.g.:

(1) φ_{j}^{2} − φ_{j−1}φ_{j+1} = (−1)^{j+1};

(2) φ_{j}^{2} − φ_{j−2}φ_{j+2} = (−1)^{j}.

(3) gcd(φ_{j}, φ_{i}) = φ_{gcd(j,i)}.

Examples 1. We will consider the following pairs (a, b):

(a) (a, b) = (d − 1, d);

(b) (a, b) = (d/2, 2d − 1), where d = deg(C) is even;

(c) (a, b) = (φ_{j−2}^{2}, φ_{j}^{2}) and d = φ_{j−1}^{2} + 1 = φ_{j−2}φ_{j}, where j is odd and ≥ 5;

(d) (a, b) = (φ_{j−2}, φ_{j+2}) and d = φ_{j}, where j is odd and ≥ 5;

(e) (a, b) = (φ_{4}, φ_{8} + 1) = (3, 22) and d = φ_{6} = 8;

(f) (a, b) = (2φ_{4}, 2φ_{8} + 1) = (6, 43) and d = 2φ_{6} = 16.

All these cases are realizable: (a) e.g. by {zy^{d−1} = x^{d}}, (b) by {(zy − x^{2})^{d/2} = xy^{d−1}}, or by the parametrization [z(t) : x(t) : y(t)] = [1 + t^{d−1} : t^{d/2} : t^{d}]. Both cases (a) and (b) satisfy Yoshihara’s criterion m_{1}+m_{2} = d [47] (cf. here with 3.5.2), hence any curve with these data is AMS curve.

The existence of (c) and (d) is guaranteed by Kashiwara classification [19], Corollary 11.4. These two cases can be realized by a rational pencil of type (0, 1): the generic member of the pencil is (c), while the special member of the pencil is of type (d) (cf. also with the next section). Orevkov in [34] denoted the curves (d) by C_{j}, where a different construction is also given for them.

The cases (e) and (f) correspond to the sporadic cases C_{4} and C_{4}^{∗} of Orevkov

on rational cuspidal projective plane curves 17

Remarks 1. (1) The above list is not accidental. In [12] we prove that if C is a unicuspidal rational plane curve of degree d and if the singular point p of C has only one characteristic pair (a, b), then the triple (d, a, b) is one of the above cases. This classification is coordinated by the following integer. Let π : X → ℙ^{2} be the minimal embedded resolution of C ⊂ ℙ^{2}, and let C̄ be the strict transform of C. Then C̄^{2} = d^{2} − ab = 3d − a − b − 1, and in the above cases is as follows: it is positive for (a) and (b), it is zero for (c), equals −1 for (d), and = −2 for (e) and (f). [Notice that C̄^{2} < −1 if and only if a + b > 3d, i.e. if and only if the semigroup element a + b is not sitting in the first three intervals I_{l}. Notice also that C̄^{2} < −1 happens exactly for the cases (a)-(d), i.e. when κ̄(ℙ^{2} \ C) = −∞; cf. (3) below.]

(2) Since Γ_{(C,p)} is the semigroup generated by a and b, the verification of (CP) for the above triples (d, a, b) is purely combinatorial depending on these integers.

On the other hand, not any triple (d, a, b) (with (a − 1)(b − 1) = (d − 1)(d − 2)) satisfies (CP). E.g., (5, 3, 7) or (17, 6, 49) do not. (But curves with these data do not exist, cf. [12].)

(3) The log Kodaira dimensions κ̄(ℙ^{2} \ C) are the following (cf. [34]): −∞ for (a)-(d), and 2 for the last two sporadic cases.

(4) Let α = (3 + \sqrt{5})/2 be the root of α + \frac{1}{α} = 3. Notice that in family (d), d/a and b/d asymptotically equals α. In fact, for j odd, {φ_{j}/φ_{j−2}}_{j} are the increasing convergents of the continued fraction of α. Using this, another remarkable property of the family (d) can be described as follows (cf. [34], page 658). The convex hull of all the pairs (m, d) ∈ ℤ^{2} satisfying m + 1 ≤ d < αm (cf. with the sharp Orevkov inequality [34], or Example 1) coincides with the convex hull of all pairs (m, d) realizable by rational unicuspidal curves C (where d = deg(C) and m = mult(C, p)) with κ̄(ℙ^{2} \ C) = −∞; moreover, this convex hull is generated by curves with numerical data (a) and (d). (For curves with d > αm, see 9.)

Theorem 5. The identity (CP) (i.e. R(t) ≡ 0) is satisfied in the above cases (a)-(f).

The cases (a) and (b) are covered by Theorem 4. The cases (e) and (f) can be verified by hand: doing this the reader definitely will feel the mystery of this distribution pattern. In the sequel we will verify (c) and (d).

We start with (d). We fix j. The point is that a + b = 3d, hence a nice induction can be considered if we group the intervals I_{l} in blocks of three. Let us analyze the first block. I_{1} contains a and 2a; I_{2} contains 3a, 4a, and 5a; finally I_{3} has 6a, 7a, b and a + b. All this can be verified by the definition of the Fibonacci numbers. E.g. 3a > d iff 3φ_{j−2} > φ_{j−1} + φ_{j−2} iff 2φ_{j−2} > φ_{j−1} iff 2φ_{j−2} > φ_{j−2} + φ_{j−3} which is true.

It is clear that it is enough to analyze the intervals I_{l} for l < d. So fix such an l. The point is that an inequality of type kb > ld is true if and only if k/l > 1/α. One direction is easy, since d/b > 1/α. Assume, that k/l > 1/α, then d/b ≥ k/l > 1/α is not possible since d/b – being a convergent of α – is the best approximation of α among fractions with denominator ≤ b (see e.g. [33] (7.13)). Hence k/l > d/b. In particular, kb ∈ I_{l} if and only if ⌈kα⌉ = l.

There is a similar statement for ia, but its proof is slightly different. If ia ≤ ld then using d/a < α one gets i/l < α. Assume now that i/l < α then we wish to prove that d/a < i/l < α is not possible. For this notice that d/a < b/d < α (in fact 18 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

d/a and b/d are two consecutive convergents of α). Then i/l cannot be between b/d and α since b/d is one of the convergents and l < d; and also i/l cannot be between d/a and b/d since ab − d^{2} = 1 and l < d (cf. [33], p. 165, 5(a)). Hence ia ≤ ld if and only if i/l < α. This implies that ia ∈ I_{l} if and only if ⌈i/α⌉ = l.

Now, using the fact that a + b = 3d, we can move the first block to the second block (of three intervals ) by adding a + b = 3d. In the relevant intervals (l < d) the only terms not in the image of this translation have form ia or kb, and they are all distinct. An easy counting shows that the only fact we have to show is that in any interval there are exactly 3 terms of these types. Namely, we have to verify the following: Consider the numbers S_{1} of the form ⌈i/α⌉ and the numbers S_{2} of the form ⌈kα⌉. Then the claim is that in S_{1} ∪ S_{2} each positive integer appears exactly three times.

This can be proved as follows. Consider in the (x, y)-plane the semi-line ℓ_{1} : {y = αx} (in the first quadrant) and the semi-line ℓ_{2}: {y = −x/α} (in the forth quadrant). For any positive integer l consider the vertical segment connecting the two intersection points of the line {x = l} with ℓ_{1} and ℓ_{2}. Notice that the length of this segment is exactly 3l. Any horizontal line {y = i} (resp. {y = −k}) intersects the segment iff i/α ≤ l (resp. kα ≤ l). Since the segment can be intersected exactly by 3l horizontal lines of type y =integer (and all the numbers of type kα and i/α are distinct) the claim follows.

Next, we prove (c). We start as in (d). Fix j and notice that d = φ_{j−2}φ_{j}.

Step 1. Assume that l < φ_{j−2} + φ_{j}. If kb > ld then kφ_{j} > lφ_{j−2}, hence k/l > φ_{j−2}/φ_{j} > 1/α. Conversely, assume that k/l > 1/α. Consider the convergents φ_{j−2}/φ_{j} > φ_{j}/φ_{j+2} > 1/α. Since φ_{j+2} > l we conclude that k/l cannot be neither between φ_{j}/φ_{j+2} and 1/α nor between φ_{j−2}/φ_{j} and φ_{j}/φ_{j+2} by similar argument as in (d). Hence either k/l > φ_{j−2}/φ_{j} or l = φ_{j}. This shows that for l ∉ {φ_{j}, φ_{j} +1}, kb ∈ I_{l} if and only if ⌈kα⌉ = l.

Step 2. Assume again that l < φ_{j−2} + φ_{j}. If ia ≤ ld then i/l ≤ φ_{j}/φ_{j−2} < α. Conversely, assume that i/l < α and consider the three intervals φ_{j}/φ_{j−2} < φ_{j+2}/φ_{j} < φ_{j+4}/φ_{j+2} < α. Then i/l cannot be in the second and third interval by similar arguments as above. Moreover, it cannot be in the first one either, since the two end-points are two elements of the φ_{j}-Farey sequence, and the denominator of any rational number between them is at least φ_{j−2} + φ_{j} (cf. (6.4) [33]). Therefore, either l = φ_{j} or i/l ≤ φ_{j}/φ_{j−2}. Hence, if l ∉ {φ_{j}, φ_{j} + 1} then ia ∈ I_{l} if and only if ⌈i/α⌉ = l.

Step 3. The intervals I_{l} for l = φ_{j} and φ_{j} + 1 can be analyzed independently: kb ∈ I_{φ_{j}} if and only if k = φ_{j−2} (and then kb = φ_{j}d), and ia ∈ I_{φ_{j}} if and only if i = φ_{j+2} − 1 or i = φ_{j+2} − 2.

Similarly, I_{φ_{j}+1} contains no number of type kb, but contains three numbers of type ia for i = φ_{j+2}, φ_{j+2} + 1, φ_{j+2} + 2.

Combined the argument of (d) applied for Step 1 and Step 2 and the above two facts, we conclude that the distribution patern is true for any l < φ_{j} + φ_{j−2}.

Step 4. Notice that in the relevant intervals (i.e. for l < d), the semigroup element ia + kb determines uniquely the integers i and k.

Step 5. Using the identity aφ_{j} = φ_{j−2}d, we construct a well-defined injective map s_{l} : I_{l−φ_{j−2}} ∩Γ → I_{l} ∩Γ by x ↦ x+aφ_{j}. Denote by P_{l} the subset of semigroup elements of I_{l} ∩ Γ which have the form ia + kb with k ≥ 0 and a > i ≥ 0. Then (via step 4) I_{l} ∩ Γ is the disjoint union of the image of s_{l} and P_{l}. Therefore, it is enough to show that

#P_{l} = φ_{j−2}  (10)

for any φ_{j−2} ≤ l < d.

Step 6. Since the distribution property is true for any l < φ_{j−2} + φ_{j} (step 3), we conclude (by the arguments of step 5) that (10) is true for any φ_{j−2} − 1 ≤ l ≤ φ_{j−2} + φ_{j} − 1.

Step 7. We verify that there is a bijection P_{l} → P_{l+φ_{j}} given by (i, k) ↦ (i, k + φ_{j−2}) for any l ≥ φ_{j−2}. The facts that the map is well-defined and injective are clear, for the surjectivity one has to verify an (easy) inequality satisfied by the Fibonacci numbers. Hence (10) follows by step 6 and induction.

6. R(t) ≡ 0 for κ̄(ℙ^{2} \ C) = −∞ (i.e. for Kashiwara’s curves)

We start with the following lemma which helps us to compare distinct semigroups.

Lemma 1. Fix integers k ≥ 0 and m, d ≥ 1. Then

(a) in the set B := {⌈mk/d⌉ + j}_{j=0}^{m−1} the unique multiple of m is ⌈k/d⌉m.

(b) Fix a subset Γ ⊂ ℕ and define the series

η(t) := ∑_{k∈Γ} t^{⌈k/d⌉} and χ(t) := ∑_{k∈Γ} t^{⌈mk/d⌉}.

Then the series ψ(t) := χ(t)(1 − t^{m})/(1 − t) satisfies the identity

η(t^{m}) = \frac{1}{m} ∑_{ξ^{m}=1} ψ(ξt).

Proof. It is clear that there is only one multiple of m in B, say a. If k = qd the proof is also clear. Otherwise, write k = qd + r with r ∈ {1, …, d − 1}. Thus mq/d = mq + rm/d. Since 0 < r/d < 1 then 0 < rm/d < m and 0 < ⌈mr/d⌉ ≤ m. Finally mq < a ≤ mq + m + m − 1 < m(q + 2), and a = (q + 1).

6.1. The proof of the conjecture for Kashiwara’s curves.

As we have mentioned most of the previous curves appear as irreducible components of fibres of rational functions on ℙ^{2} of type (0, 1), that is, rational functions all whose fibres (once the indeterminacy point has been removed) are isomorphic to ℂ. Remark that any of these rational function ϕ has at most one indeterminacy point which will be the only possible singular point of any of its fibres. The classification of such rational functions is given in H. Kashiwara’s paper [19]. Moreover, a rational cuspidal curve C verifies κ̄(ℙ^{2} \ C) = −∞ if and only if C is an irreducible component of a fibre of a rational function of type (0, 1).

The number of multiple fibres that a rational function ϕ on ℙ^{2} of type (0, 1) can have is at most two. Kashiwara’s classification gives three strata inside the set of rational functions of type (0, 1): F_{0}, F_{I}, F_{II}, according to the number of multiple fibres will be 0, 1 or 2 respectively.

(1) The stratum F_{0} consists of all linear rational functions on ℙ^{2}, therefore there are no cuspidal rational curves as fibres.

(2) If ϕ ∈ F_{I} then the multiple fibre of ϕ turns out to be a line L, and ϕ is a component of an automorphism of ℂ^{2} = ℙ^{2} \ L, see Corollaire 8.1 in [19]. Moreover 20 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

every fibre of ϕ, but the multiple one, defines a rational cuspidal curve of AMS-type for which we have already checked (CP) in Theorem 4.

(3) A rational function ϕ ∈ F_{II} has two atypical values, say 0 and ∞. Its divisor will be denoted by (ϕ) = mS_{0} − nS_{∞} (for some integers that we may assume gcd(m, n) = 1 taking ϕ primitive). From the topological point of view the rational function ϕ has only three different fibres S_{0}, S_{∞} and the generic fibre ϕ_{ge}.

The minimal resolution graph of the indeterminacy of a rational function ϕ of type (0, 1) almost coincides with the minimal resolution graph of the generic fibre of ϕ. But this is not the case for special fibres. In her Théorème 6.1 the minimal resolution graphs of the indeterminacy point of the rational functions of type (0, 1) are given. The remaining case in which we are interested in is ϕ ∈ F_{II} where five different graphs appear:

Case 1. II(ℓ)^{∗} (with ℓ ≥ 0),

Case 2. II^{+}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N even ≥ 2),

Case 3. II^{+}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N odd ≥ 1),

Case 4. II^{−}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N even ≥ 2),

Case 5. II^{−}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N odd ≥ 2).

Here λ_{1}, …, λ_{N} are integers such that λ_{1}, …, λ_{N} ≥ 0 if ℓ ≥ 1 and λ_{1}, …, λ_{N} ≥ 1 if ℓ = 0.

In Case 1, for a rational function ϕ^{ℓ} whose graph belongs to II(ℓ)^{∗}, there are three distinct cuspidal rational curves appearing as fibres of ϕ^{ℓ}: S_{0}^{ℓ}, S_{∞}^{ℓ} and the generic fibre ϕ_{ge}^{ℓ}. The rational function ϕ^{ℓ} is constructed by induction based on a rational function ϕ^{ℓ−1} which implies that S_{∞}^{ℓ} is nothing but S_{0}^{ℓ−1}, see e.g. Corollaire 11.4. Thus for each graph in II(ℓ)^{∗} there are only two new unicuspidal rational plane curves: the generic member of the pencil, we will denote it by II(ℓ)_{ge}, and the new special member of the pencil, we will denote it by II(ℓ)_{sp}. We have already studied those curves (c) and (d) in Examples 1:

(c) II(ℓ)_{ge} is a unicuspidal rational curve of degree d = φ_{2ℓ+3}φ_{2ℓ+5} and only one characteristic pair (a, b) = (φ_{2ℓ+3}^{2}, φ_{2ℓ+5}^{2});

(d) II(ℓ)_{sp} is a unicuspidal rational curve of degree d = φ_{2ℓ+3} and only one characteristic pair (a, b) = (φ_{2ℓ+1}, φ_{2ℓ+5}).

Therefore in Theorem 5 we have proved the identity (CP) for II(ℓ)_{ge} and II(ℓ)_{sp}, ℓ ≥ 0.

The same fact happens for all rational functions whose graphs belongs to any of the remaining four cases, see e.g. Corollaire 11.6. The irreducible component of the ∞ fibre of the ℓ-rational function is nothing but the 0-fibre of the previous (ℓ − 1)-rational function. Therefore in each group we have just two new unicuspidal rational curves: the generic member of the pencil and one special member of the pencil.

The only difference between the generic rational curve in II^{+}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N even ≥ 2), (Case 2), and the generic rational curve in II^{+}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N odd ≥ 1), (Case 3), is the number of Puiseux pairs. In fact, it is possible to codify the invariants of the generic members of Cases 2 and 3 in one and the same sequence of Eisenbud-Neumann splice diagrams, which we denote by II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge}. The same can be done with the special members of the pencils, which we will denote by II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp}.

Finally the same can be done also with Cases 4 and 5, that is for rational functions with graphs in II^{−}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N even ≥ 2) or II^{−}(ℓ, N; λ_{1}, …, λ_{N})^{∗} (with N odd ≥ 2). The generic curve will be denoted by II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge} and the special curve by II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{sp}.

6.2. Degrees of curves and generators of semigroups.

Fix ℓ ≥ 0 and N ≥ 1 and non-negative integers λ_{1}, …, λ_{N}.

6.2.1. II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge}. For 1 ≤ i ≤ N define

\{ \begin{matrix}n_{i} := λ_{i}φ_{2ℓ+3}^{2} + φ_{2ℓ+3}φ_{2ℓ−1} − 1  for i odd; \\ n_{i} := λ_{i}φ_{2ℓ+3}^{2} + φ_{2ℓ+3}(φ_{2ℓ+3} − φ_{2ℓ−1}) − 1 for i even. \end{matrix}  (11)

The degree of the curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} is d = φ_{2ℓ+3}φ_{2ℓ+5}n_{1} … n_{N}, (see Proposition 7.2 in [19]). We remark that the integer m_{ℓ} in [19] is nothing but the Fibonacci number φ_{2ℓ+3}. This curve has only one singularity whose splice diagram can be easily deduced from the resolution graph of the corresponding rational function. This singularity has N + 1 Newton pairs and the decorations {(p_{k}, a_{k})}_{k=1}^{N+1} of the corresponding Eisenbud-Neumann splice diagram can be done computing the graph determinants (as it is explained in [7]). Thus

\{ \begin{matrix}p_{k} = n_{k} for 1 ≤ k ≤ N, and p_{N+1} = φ_{2ℓ+3}^{2}; \\ a_{k} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)/φ_{2ℓ+3}^{2}, for 1 ≤ k ≤ N and \\ a_{N+1} = φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2}. \end{matrix}  (12)

To check that the a_{k}’s are integers, let b_{k} denote the numerator φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k}− 1 of a_{k}, then b_{k} = b_{k−1}n_{k−1}n_{k} + n_{k−1}n_{k} − 1. By induction, to show that b_{k} = 0 mod φ_{2ℓ+3}^{2} it is enough to prove that n_{k−1}n_{k} − 1 = 0 mod φ_{2ℓ+3}^{2}. From the definition of n_{k}, see (11), n_{k−1}n_{k} − 1 = (φ_{2ℓ+3}φ_{2ℓ−1} − 1)(−φ_{2ℓ+3}φ_{2ℓ−1} − 1) − 1 = −(φ_{2ℓ+3}^{2}φ_{2ℓ−1}^{2} −1)−1 = 0 mod φ_{2ℓ+3}^{2}, after property (ii) of the Fibonacci numbers.

Thus the generators {β̄_{k}} of its semigroup Γ_{II^{+}(ℓ,N;λ)_{ge}} are β̄_{0} = φ_{2ℓ+3}^{2}n_{1} … n_{N}, β̄_{k} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)n_{k+1} … n_{N} for 1 ≤ k ≤ N, and β̄_{N+1} = φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2}.

6.2.2. II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp}. The curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp} is rational and unicuspidal and has degree d = φ_{2ℓ+5}n_{1} … n_{N}, (see Proposition 7.2 in [19]). This curve has also only one singularity which is a cusp with N + 1 Newton pairs. To compute the decorations {(p_{k}, a_{k})}_{k=1}^{N+1} of the corresponding splice diagram, observe that all the determinants, but the last two, are the same as in the previous case. Thus the invariants in this case are:

\{ \begin{matrix}p_{k} = n_{k} for 1 ≤ k ≤ N, and p_{N+1} = φ_{2ℓ+3}; \\ a_{k} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)/φ_{2ℓ+3}^{2}, for 1 ≤ k ≤ N and \\ a_{N+1} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2} + 1)/φ_{2ℓ+3}. \end{matrix}  (13)

Moreover, a_{N+1} is also integer because n_{i} = −1 mod φ_{2ℓ+3} and then φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2} + 1 = φ_{2ℓ+5}^{2} − 1 = (φ_{2ℓ+3}φ_{2ℓ+7} − 1) + 1 = 0 mod φ_{2ℓ+3}.

The generators {β̄_{k}} of its semigroup Γ_{II^{+}(ℓ,N;λ)_{sp}} are β̄_{0} = φ_{2ℓ+3}n_{1} … n_{N}, β̄_{k} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)n_{k+1} … n_{N}/φ_{2ℓ+3} for 1 ≤ k ≤ N, and β̄_{N+1} = (φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2} + 1)/φ_{2ℓ+3}.

22 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

6.2.3. II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge}. For 1 ≤ i ≤ N, define

\{ \begin{matrix}ñ_{i} := λ_{i}φ_{2ℓ+3}^{2} + φ_{2ℓ+3}φ_{2ℓ−1} − 1  for i even; \\ ñ_{i} := λ_{i}φ_{2ℓ+3}^{2} + φ_{2ℓ+3}(φ_{2ℓ+3} − φ_{2ℓ−1}) − 1 for i odd. \end{matrix}  (14)

Essentially the data of the curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge} can be obtained from the data of the curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} replacing φ_{2ℓ+5} by φ_{2ℓ+1}. The combinatorial reason for that is the identity φ_{2ℓ+3}^{2} = φ_{2ℓ+1}φ_{2ℓ+5} − 1. The degree of the rational curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge} is d = φ_{2ℓ+3}φ_{2ℓ+1}ñ_{1} … ñ_{N}, (see Proposition 7.2 in [19]). This curve has only one singular point which is a cusp with N +1 Newton pairs. The decorations {(p_{k}, a_{k})}_{k=1}^{N+1} of the corresponding Eisenbud-Neumann splice diagram obtained computing the graph determinants are

\{ \begin{matrix}p_{k} = ñ_{k} for 1 ≤ k ≤ N, and p_{N+1} = φ_{2ℓ+3}^{2}; \\ a_{k} = (φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{k−1}^{2}ñ_{k} − 1)/φ_{2ℓ+3}^{2}, for 1 ≤ k ≤ N and \\ a_{N+1} = φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{N}^{2}. \end{matrix}  (15)

In the same way as before one checks that the a_{k}’s are integers. The minimal set of generators of its semigroup Γ_{II^{−}(ℓ,N;λ)_{ge}} are β̄_{0} = φ_{2ℓ+3}^{2}ñ_{1} … ñ_{N}, β̄_{k} = (φ_{2ℓ+1}^{2}n_{1}^{2} … ñ_{k−1}^{2}ñ_{k} − 1)ñ_{k+1} … ñ_{N} for 1 ≤ k ≤ N, and β̄_{N+1} = φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{N}^{2}.

6.2.4. II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{sp}. The degree of the unicuspidal rational curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{sp} is d = φ_{2ℓ+1}n_{1} … n_{N}, (see Proposition 7.2 in [19]). Its singularity has N + 1 Newton pairs and the decorations {(p_{k}, a_{k})}_{k=1}^{N+1} of the corresponding Eisenbud-Neumann splice diagram are

\{ \begin{matrix}p_{k} = ñ_{k} for 1 ≤ k ≤ N,and p_{N+1} = φ_{2ℓ+3}; \\ a_{k} = (φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{k−1}^{2}ñ_{k} − 1)/φ_{2ℓ+3}^{2}, for 1 ≤ k ≤ N and \\ a_{N+1} = (φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{N}^{2} + 1)/φ_{2ℓ+3}. \end{matrix}  (16)

Again a_{N+1} is integer and the generators of its semigroup Γ_{II^{−}(ℓ,N;λ)_{sp}} are β̄_{0} = φ_{2ℓ+3}ñ_{1} … ñ_{N}, β̄_{k} = (φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{k−1}^{2}ñ_{k} − 1)ñ_{k+1} … ñ_{N}/φ_{2ℓ+3} for 1 ≤ k ≤ N, and β̄_{N+1} = (φ_{2ℓ+1}^{2}ñ_{1}^{2} … ñ_{N}^{2} + 1)/φ_{2ℓ+3}.

6.3. (CP) for generic members of the pencils.

The generic members of the pencils are the curves II^{ε}(ℓ, N; λ_{1}, …, λ_{N})_{ge}, ε = ±. To deal with the elements of their semigroups Γ_{II^{±}(ℓ,N;λ)_{ge}} it is better to multiply them by φ_{2ℓ+3}. Define

χ_{ge}^{ε}(t) := ∑_{k∈Γ_{II^{ε}(ℓ,N;λ)ge}} t^{⌈φ_{2ℓ+3}k/d⌉} and ψ_{ge}^{ε}(t) := \frac{χ_{ge}^{ε}(t)(1 − t^{φ_{2ℓ+3}})}{1 − t}, with ε = ±,

where either d = φ_{2ℓ+3}φ_{2ℓ+5}n_{1} … n_{N} if ε = + or d = φ_{2ℓ+3}φ_{2ℓ+1}ñ_{1} … ñ_{N} otherwise. If

CP_{ge}^{ε}(t) := ∑_{k∈Γ_{II^{ε}(ℓ,N;λ)ge}} t^{⌈k/d⌉} with ε = ±,

then Lemma 1 implies

CP_{ge}^{ε}(t^{φ_{2ℓ+3}}) = \frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} ψ_{ge}^{ε}(ξt) with ε = ±.  (17)

Proposition 3.

(a) For the curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} the following identity holds:

χ_{ge}^{+}(t) = \frac{(1 − t^{φ_{2ℓ+1}φ_{2ℓ+5}})(1 − t^{φ_{2ℓ+3}^{2}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})(1 − t^{φ_{2ℓ+1}})(1 − t^{φ_{2ℓ+5}})}.

(b) For the curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge} the following identity holds:

χ_{ge}^{−}(t) = \frac{(1 − t^{φ_{2ℓ+1}φ_{2ℓ+5}})(1 − t^{φ_{2ℓ+3}^{2}φ_{2ℓ+1}ñ_{1}...ñ_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})(1 − t^{φ_{2ℓ+1}})(1 − t^{φ_{2ℓ+5}})}.

Proof. We start with the curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} which has degree d = φ_{2ℓ+3}φ_{2ℓ+5}n_{1} … n_{N} and whose generators of its semigroup Γ_{II^{+}(ℓ,N;λ)_{ge}} has been described in (12). By 2.1, each element x in the semigroup can be written in a unique way as

x = x_{0}φ_{2ℓ+3}^{2} ∏_{i=1}^{N} n_{i} + ∑_{k=1}^{N} y_{k} a_{k} φ_{2ℓ+3}^{2}n_{k+1} … n_{N} + z_{0} φ_{2ℓ+5}^{2} n_{1}^{2} … n_{N}^{2},

with x_{0} ≥ 0, 0 ≤ y_{k} ≤ n_{k} − 1, for 1 ≤ k ≤ N, and 0 ≤ z_{0} ≤ φ_{2ℓ+3}^{2} − 1. Thus

\frac{φ_{2ℓ+3}x}{d} = x_{0} \frac{φ_{2ℓ+3}^{2}}{φ_{2ℓ+5}} + ∑_{k=1}^{N} y_{k} \frac{(φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)}{φ_{2ℓ+5}n_{1} … n_{k}} + z_{0}φ_{2ℓ+5}n_{1} … n_{N}.

Write x_{0} = q_{0}φ_{2ℓ+5} + r_{0} with 0 ≤ r_{0} ≤ φ_{2ℓ+5} − 1. Using φ_{2ℓ+3}^{2} = φ_{2ℓ+5}φ_{2ℓ+1} − 1, we get

⌈φ_{2ℓ+3}x/d⌉ = q_{0}φ_{2ℓ+3}^{2} + r_{0}φ_{2ℓ+1} + ∑_{k=1}^{N} y_{k}φ_{2ℓ+5}n_{1} … n_{k−1} + z_{0}φ_{2ℓ+5}n_{1} … n_{N},

because −1 < (−r_{0}n_{1} … n_{N} − ∑_{k=1}^{N} y_{k}n_{k+1} … n_{N})/φ_{2ℓ+5}n_{1} … n_{N} ≤ 0, for every 0 ≤ y_{k} ≤ n_{k} − 1, for 1 ≤ k ≤ N, and 0 ≤ r_{0} ≤ φ_{2ℓ+5} − 1. The result is proved because χ_{ge}^{+}(t) is equal to

∑_{q_{0}≥0} ∑_{r_{0}=0}^{φ_{2ℓ+5}−1} ∑_{k=1}^{N} ∑_{y_{k}=0}^{n_{k}−1} ∑_{z_{0}=0}^{φ_{2ℓ+3}^{2}−1} t^{q_{0}φ_{2ℓ+3}^{2}+r_{0}φ_{2ℓ+1}+∑_{k=1}^{N} y_{k}φ_{2ℓ+5}n_{1}...n_{k−1}+z_{0}φ_{2ℓ+5}n_{1}...n_{N}}

= \frac{(1 − t^{φ_{2ℓ+1}φ_{2ℓ+5}}) ∏_{k=1}^{N}(1 − t^{φ_{2ℓ+5}n_{1}...n_{k}})(1 − t^{φ_{2ℓ+3}^{2}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})(1 − t^{φ_{2ℓ+1}}) ∏_{k=1}^{N}(1 − t^{φ_{2ℓ+5}n_{1}...n_{k−1}})(1 − t^{φ_{2ℓ+5}n_{1}...n_{N}})}.

The proof for the curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge} is essentially the same replacing φ_{2ℓ+5} by φ_{2ℓ+1}.

Theorem 6. (CP) is true for the generic curves II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} and II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{ge}.

Proof. We do the case ε = + leaving the other one to the reader. Using Proposition 3 and (17) then

CP_{ge}^{+}(t^{φ_{2ℓ+3}}) = \frac{(1 − t^{φ_{2ℓ+3}})(1 − t^{φ_{2ℓ+3}^{2}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})} · \frac{1}{φ_{2ℓ+3}}

24 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

· ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{(1 − (ξt)^{φ_{2ℓ+1}φ_{2ℓ+5}})}{(1 − (ξt)^{φ_{2ℓ+1}})(1 − (ξt)^{φ_{2ℓ+5}})(1 − ξt)}.

Let Δ(t) be the characteristic polynomial of the rational unicuspidal curve of Example (d) in Examples 1. That is, it has degree φ_{2ℓ+3} and one characteristic pair (a, b) = (φ_{2ℓ+1}, φ_{2ℓ+5}). For such a curve we have already proved R(t) ≡ 0 in Theorem 5. Thus

\frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{(1 − (ξt)^{φ_{2ℓ+1}φ_{2ℓ+5}})}{(1 − (ξt)^{φ_{2ℓ+1}})(1 − (ξt)^{φ_{2ℓ+5}})(1 − ξt)}

= \frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{Δ(ξt)}{(1 − ξt)^{2}} = \frac{(1 − t^{φ_{2ℓ+3}^{2}})}{(1 − t^{φ_{2ℓ+3}^{2}})^{3}} .

Since the degree of II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{ge} is φ_{2ℓ+3}φ_{2ℓ+5}n_{1} … n_{N} then the result follows because

CP_{ge}^{+}(t^{φ_{2ℓ+3}}) = \frac{(1 − t^{φ_{2ℓ+3}^{2}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})^{2}}.

6.4. (CP) for special members of the pencils.

The special members of the pencils are the rational curves II^{ε}(ℓ, N; λ_{1}, …, λ_{N})_{sp}, with ε = ±. As in the previous case, to deal with the elements of their semigroups Γ_{II^{±}(ℓ,N;λ)_{sp}} we multiply them by φ_{2ℓ+3}. Define

χ_{sp}^{ε}(t) := ∑_{k∈Γ_{II^{ε}(ℓ,N;λ)sp}} t^{⌈φ_{2ℓ+3}k/d⌉} and ψ_{sp}^{ε}(t) := \frac{χ_{sp}^{ε}(t)(1 − t^{φ_{2ℓ+3}})}{1 − t}, with ε = ±,

where either d = φ_{2ℓ+5}n_{1} … n_{N} if ε = + or d = φ_{2ℓ+1}ñ_{1} … ñ_{N} otherwise. If

CP_{sp}^{ε}(t) := ∑_{k∈Γ_{II^{ε}(ℓ,N;λ)sp}} t^{⌈k/d⌉} with ε = ±,

then Lemma 1 implies

CP_{sp}^{ε}(t^{φ_{2ℓ+3}}) = \frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} ψ_{sp}^{ε}(ξt) with ε = ±.  (18)

Theorem 7. (CP) is true for the special curves II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp} and II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{sp}.

Proof. We do the proof for the curve II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp} and the proof for the curve II^{−}(ℓ, N; λ_{1}, …, λ_{N})_{sp} is essentially the same replacing n_{i} by ñ_{i} and φ_{2ℓ+5} by φ_{2ℓ+1}.

The degree of II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{g} is d = φ_{2ℓ+5}n_{1} … n_{N} and the generators of the semigroup Γ_{II^{+}(ℓ,N;λ_{1},...,λ_{N})_{sp}} have been described in (13). Each element x in Γ_{II^{+}(ℓ,N;λ)_{sp}} can be written in a unique way as

x = x_{0}φ_{2ℓ+3} ∏_{i=1}^{N} n_{i} + ∑_{k=1}^{N} y_{k} a_{k} φ_{2ℓ+3} n_{k+1} … n_{N} + z_{0} \frac{φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2} + 1}{φ_{2ℓ+3}},

with x_{0} ≥ 0, 0 ≤ y_{k} ≤ n_{k} − 1, for 1 ≤ k ≤ N, and 0 ≤ z_{0} ≤ φ_{2ℓ+3} − 1. Thus

\frac{φ_{2ℓ+3}x}{d} = x_{0} \frac{φ_{2ℓ+3}^{2}}{φ_{2ℓ+5}} + ∑_{k=1}^{N} y_{k} \frac{(φ_{2ℓ+5}^{2}n_{1}^{2} … n_{k−1}^{2}n_{k} − 1)}{φ_{2ℓ+5}n_{1} … n_{k}} + z_{0} \frac{φ_{2ℓ+5}^{2}n_{1}^{2} … n_{N}^{2} + 1}{φ_{2ℓ+5}n_{1} … n_{N}}.

Write x_{0} = q_{0}φ_{2ℓ+5} + r_{0} with 0 ≤ r_{0} ≤ φ_{2ℓ+5} − 1. Since φ_{2ℓ+3}^{2} = φ_{2ℓ+5}φ_{2ℓ+1} − 1 then

⌈φ_{2ℓ+3}x/d⌉ = q_{0}φ_{2ℓ+3}^{2} + α(r_{0}, y, z_{0}) + ⌈β(r_{0}, y, z_{0})⌉.

where β(r_{0}, y, z_{0}) := (z_{0} − r_{0}n_{1} … n_{N} − ∑_{k=1}^{N} y_{k}n_{k+1} … n_{N})/φ_{2ℓ+5}n_{1} … n_{N} and

α(r_{0}, y, z_{0}) := r_{0}φ_{2ℓ+1} + ∑_{k=1}^{N} y_{k}φ_{2ℓ+5}n_{1} … n_{k−1} + z_{0}φ_{2ℓ+5}n_{1} … n_{N}.

Set A := {(r_{0}, y, z_{0}) : 0 ≤ r_{0} ≤ φ_{2ℓ+5} − 1, 0 ≤ z_{0} ≤ φ_{2ℓ+3} − 1, 0 ≤ y_{k} ≤ n_{k} − 1, for k = 1, …, N}. Since for (r_{0}, y_{1}, …, y_{N}, z_{0}) ∈ A, one gets −1 < β(r_{0}, y, z_{0}) < 1 then ⌈ β(r_{0}, y, z_{0})⌉ is either 0 or 1. Consider the subset P of A defined by {z_{0} − r_{0}n_{1} … n_{N} − ∑_{k=1}^{N} y_{k}n_{k+1} … n_{N} > 0} and Q its complement in A. Thus ⌈ β(r_{0}, y, z_{0})⌉ is 1 if and only if (r_{0}, y, z_{0}) ∈ P and it is zero otherwise. This implies

χ_{sp}^{+}(t) = \frac{∑_{(r_{0},y,z_{0})∈P} t^{α(r_{0},y,z_{0})+1} + ∑_{(r_{0},y,z_{0})∈Q} t^{α(r_{0},y,z_{0})}}{(1 − t^{φ_{2ℓ+3}^{2}})} .

Remark 2. If (r_{0}, y, z_{0}) ∈ P then 0 < z_{0} −r_{0}n_{1} … n_{N} −∑_{k=1}^{N} y_{k}n_{k+1} … n_{N} < φ_{2ℓ+3}.

By (18), CP_{sp}^{+}(t^{φ_{2ℓ+3}}) is equal to

\frac{(1 − t^{φ_{2ℓ+3}})}{(1 − t^{φ_{2ℓ+3}^{2}})}·\frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{∑_{(r_{0},y,z_{0})∈P} (ξt)^{α(r_{0},y,z_{0})+1} + ∑_{(r_{0},y,z_{0})∈Q}(ξt)^{α(r_{0},y,z_{0})}}{(1 − ξt)} .

Claim: CP_{sp}^{+}(t^{φ_{2ℓ+3}}) is also equal to:

\frac{(1 − t^{φ_{2ℓ+3}})}{(1 − t^{φ_{2ℓ+3}^{2}})} · \frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{∑_{(r_{0},y,z_{0})∈P} (ξt)^{α(r_{0},y,z_{0})} + ∑_{(r_{0},y,z_{0})∈Q}(ξt)^{α(r_{0},y,z_{0})}}{(1 − ξt)} .

For each (r_{0}, y, z_{0}) ∈ P consider the series t^{α(r_{0},y,z_{0})+1}/(1 − t) = t^{α(r_{0},y,z_{0})+1}(1 + t + t^{2} + ..). Then

\frac{1}{φ_{2ℓ+3}} ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{(ξt)^{α(r_{0},y,z_{0})+1}}{(1 − ξt)}

is nothing but the coefficients whose power is a multiple of φ_{2ℓ+3} in t^{α(r_{0},y,z_{0})+1}(1+ t + t^{2} + ..). Therefore to prove the claim it is enough to check that for every (r_{0}, y, z_{0}) ∈ P then α(r_{0}, y, z_{0}) ≠ 0 mod φ_{2ℓ+3}. Write α(r_{0}, y, z_{0}) as

r_{0}(φ_{2ℓ+3} − φ_{2ℓ+2}) + (∑_{k=1}^{N} y_{k}n_{1} … n_{k−1} + z_{0}n_{1} … n_{N}) (φ_{2ℓ+2} + 2φ_{2ℓ+3}).

Since gcd(φ_{2ℓ+2}, φ_{2ℓ+3}) = 1 (see property (iii) of the Fibonacci numbers) then 26 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

α(r_{0}, y, z_{0}) = 0 mod φ_{2ℓ+3} if and only if

−r_{0} + ∑_{k=1}^{N} y_{k}n_{1} … n_{k−1} + z_{0}n_{1} … n_{N} = 0 mod φ_{2ℓ+3}.

Moreover, since for all k, n_{k} = −1 mod φ_{2ℓ+3} then

−r_{0} + ∑_{k=1}^{N} y_{k}(−1)^{k−1} + z_{0}(−1)^{N} = 0 mod φ_{2ℓ+3}

⟺ −(−1)^{N}r_{0} − ∑_{k=1}^{N} y_{k}(−1)^{N−k} + z_{0} = 0 mod φ_{2ℓ+3}.

But this last identity implies z_{0} − r_{0}n_{1} … n_{N} − ∑_{k=1}^{N} y_{k}n_{k+1} … n_{N} = sφ_{2ℓ+3} for s ∈ ℤ which contradicts the definition of the set P, see also Remark 2.

Since

\frac{∑_{(r_{0},y,z_{0})∈P} t^{α(r_{0},y,z_{0})} + ∑_{(r_{0},y,z_{0})∈Q} t^{α(r_{0},y,z_{0})}}{(1 − t^{φ_{2ℓ+3}^{2}})}

= \frac{(1 − t^{φ_{2ℓ+1}φ_{2ℓ+5}})(1 − t^{φ_{2ℓ+3}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}})(1 − t^{φ_{2ℓ+1}})(1 − t^{φ_{2ℓ+5}})},

the claim implies that CP_{sp}^{+}(t^{φ_{2ℓ+3}}) can be written as:

\frac{(1 − t^{φ_{2ℓ+3}})(1 − t^{φ_{2ℓ+3}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}^{2}}) · φ_{2ℓ+3}} · ∑_{ξ^{φ_{2ℓ+3}} =1} \frac{Δ(ξt)}{(1 − ξt)^{2}} = \frac{(1 − t^{φ_{2ℓ+3}φ_{2ℓ+5}n_{1}...n_{N}})}{(1 − t^{φ_{2ℓ+3}})^{2}}

where again Δ(t) is the characteristic polynomial of the rational unicuspidal curve of degree φ_{2ℓ+3} which has one characteristic pair (a, b) = (φ_{2ℓ+1}, φ_{2ℓ+5}), see (d) in Examples 1. Since the degree of II^{+}(ℓ, N; λ_{1}, …, λ_{N})_{sp} is φ_{2ℓ+5}n_{1} … n_{N} then Theorem 7 is proved.

7. The case κ̄(ℙ^{2} \ C) = 1 and ν = 1.

7.1. Tono’s classification theorem.

We recall the following classification result of K. Tono [39]. Let [x : y : z] be a system of homogeneous coordinates in ℙ^{2}. Assume that C is a unicuspidal rational plane curve with κ̄ = 1. Then C is projectively equivalent to one of the following curves C′:

Type I. C′ is given by

((f_{1}^{s}y + ∑_{i=2}^{s+1} a_{i}f_{1}^{s+1−i}x^{ia−a+1})^{a} − f_{1}^{as+1})/x^{a−1} = 0,

where f_{1} = x^{a−1}z + y^{a}, a ≥ 3, s ≥ 1, a_{2}, …, a_{s+1} ∈ ℂ with a_{s+1} ≠ 0.

In this case, d = a^{2}s + 1, and the multiplicity sequence of (C′, p) is [(a^{2} − a)s, (sa)_{2a−1}, a_{2s}].

Type IIa. C′ is given by

((yf_{2}^{n} + x^{2n+1})^{4n+1} − f_{3}^{2n+1})/f_{2}^{n} = 0,

where f_{2} = xz − y^{2}, f_{3} = f_{2}^{2n}z + 2x^{2n}yf_{2}^{n} + x^{4n+1} and n ≥ 2.

In this case d = 8n^{2} + 4n + 1 and the multiplicity sequence of (C′, p) is [(n(4n + 1))_{4}, (4n + 1)_{2n}, 3n + 1, n_{3}]. Type IIb. C′ is given by

((f_{3}^{2s−1}(f_{2}^{n}y + x^{2n+1}) + ∑_{i=1}^{s} a_{i}f_{3}^{2(s−i)}f_{2}^{i(4n+1)−n})^{4n+1} − f_{3}^{2((4n+1)s−n)})/f_{2}^{n} = 0,

where n ≥ 2, s ≥ 1, a_{1}, …, a_{s} ∈ ℂ with a_{s} ≠ 0.

The degree of C′ is d = 2(4n+1)^{2}s−4n(2n+1). Set a^{∗} := 4n+1 and s^{∗} := 4s−1. The multiplicity sequence for s = 1 is [(3na^{∗})_{4}, (3a^{∗})_{2n}, (a^{∗})_{3}, 3n + 1, n_{3}], otherwise it is

[(s^{∗}a^{∗}n)_{4}, (s^{∗}a^{∗})_{2n}, (sa^{∗})_{3}, (s − 1)a^{∗}, (a^{∗})_{2(s−1)}, 3n + 1, n_{3}].

7.2. (C, p) in the case Type I

After a computation, one has the following facts:

If s = 1, then (C′, p) has two characteristic pairs. The decorations of the splice diagram (for notation see 3.5) are the following:

p_{1} = a − 1, p_{2} = a, a_{1} = a, and a_{2} = a(d + 1) + 1 = a^{3} + 2a + 1.

In particular, the semigroup Γ is generated by the elements a^{2}−a, a^{2} and a^{3}+2a+1.

If s > 1 the (C′, p) has three pairs: p_{1} = a − 1, p_{2} = s, p_{3} = a, a_{1} = a, a_{2} = d and a_{3} = as(d + 1) + 1. The semigroup is generated by

β̄_{0} = (a^{2} − a)s, β̄_{1} = a^{2}s, β̄_{2} = a(a^{2}s + 1) = ad, β̄_{3} = 1 + as(d + 1) = a_{3}.

Notice that if in this second case we consider s = 1, we get a non-minimal splice diagram (with p_{2} = 1) of the first case. In particular, any argument for general s > 1 can be adopted to the s = 1 situation by a simple substitution s = 1 and by elimination of the semigroup generator β̄_{2} = ad. Therefore, in the sequel we will consider the general s ≥ 1 situation (which can be specialized to s = 1 as described above).

Theorem 8. Type I satisfies the distribution property (CP): R(t) ≡ 0.

Proof. We will use the notations of 3.3. We will prove the inequality

#Γ ∩ I_{l+1} ≤ 1 + #Γ ∩ I_{l}, (in_{l})

for any l. Notice that this implies the negative distribution property, hence the vanishing of R(t) would follow from Theorem 3.

Since Γ ∩ I_{0} = {0} and Γ ∩ I_{1} = {(a^{2} − a)s, a^{2}s}, the inequality (in_{0}) follows. Since d(d − 3) + 1 is the largest element in ℕ \ Γ, (in_{l}) for l ≥ d − 2 also follows. Next, we fix an l with 1 ≤ l ≤ d − 3. Denote by i_{0} := (l − 1)d + 1 and consider the map

ϕ_{l} : Γ ∩ I_{l} \ {i_{0}} → Γ ∩ I_{l+1}

defined by ϕ_{l}(x) = x + β̄_{1} = x + a^{2}s. Clearly, ϕ_{l} is injective.

28 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

The following facts can be verified by elementary computations (using the short hints):

(a) Using 2.1 (especially the formula for Δ) one gets that any γ ∈ Γ can be written in a unique way in the form

γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} + nβ̄_{1},  (19)

where 0 ≤ k < a, 0 ≤ m < a, 0 ≤ t < s and n ≥ 0. Here k is the remainder of γ modulo a.

(b) If in (19) n > 0 then γ = γ′+a^{2}s for some γ′ ∈ Γ. In particular, Γ∩I_{l+1}\im(ϕ_{l}) has two types of elements:

Type A. γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} with 0 ≤ k < a, 0 ≤ m < a, 0 ≤ t < s; or

Type B. γ = (l + 1)d.

Here we do not exclude the situation when some semigroup element has both types. Let S_{A} (resp. S_{B}) be the set of elements of type A (resp. B).

(c) #S_{A} ≤ 2.

Indeed, assume that γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} and γ′ = k′β̄_{3} + m′β̄_{0} + t′β̄_{2} are both elements of S_{A}. If k′ < k then (a − 1)(a^{2} − a)s + (s − 1)ad ≥ m′β̄_{0} + t′β̄_{2} > ld − k′β̄_{3} ≥ −d + (k − k′)β̄_{3} ≥ −d + β̄_{3}, a contradiction. Hence k = k′. Next, consider the difference γ − γ′ = (m − m′)β̄_{0} + (t − t′)da. Since |γ − γ′| < d and |m − m′|β̄_{0} < (a − 1)d one gets |t − t′|ad < ad, hence t = t′. Therefore, if γ′ > γ then γ′ = γ + (a^{2} − a)s (since 2(a^{2} − a)s ≥ d).

(d) One has the following identity for any γ of type A:

kβ̄_{3} + mβ̄_{0} + tβ̄_{2} = (kas + m + at)d + (k − m)(1 + sa),  (20)

with |k − m| ≤ a − 1.

If S_{A} = ∅, or S_{A} = {γ} and (l + 1)d ∉ Γ, or S_{A} = {(l + 1)d} then obviously (in_{l}) follows. Hence we only have to analyze the following two cases:

Case 1. #S_{A} = 2.

Write S_{A} = {γ, γ′} where γ′ = γ + (a^{2} − a)s (see the proof of (c)). Set γ = ld + r for some r > 0. Since ld + r + (a^{2} − a)s = γ′ ≤ ld + d, one gets r ≤ as + 1.

Case 1a. First we verify that the case 0 < r ≤ as cannot occur. Indeed, assume that r ≤ as and write γ in the form (20). Here there are two possibilities: either kas + m + at = l and k − m > 0, or kas + m + at = l + 1 and k − m < 0. The first possibility is eliminated by r = (k−m)(as+1) > as ≥ r. Hence kas+m+ta = l+1, and d−r = (m−k)(1+as). If m−k ≤ a−2 then a^{2}s+1−as ≤ d−r ≤ (a−2)(1+sa), a contradiction. Finally, if m − k = a − 1 then necessarily k = 0 and m = a − 1, hence γ′ = tβ̄_{2} + (a − 1)β̄_{1} is not of type A, again a contradiction.

Case 1b. Therefore, r = as+1. In other words, γ = ld+as+1 and γ′ = (l+1)d. In particular, γ′ is of also of type B. This implies that #Γ ∩ I_{l+1} = 2 + #im(ϕ_{l}). Next, write γ as in (20). Then an elementary computation shows that kas+m+at = l+1 is not possible. Therefore kas + m + at = l and k − m = 1. In particular,

i_{0} = mβ̄_{3} + (m + a − 1)β̄_{0} + tβ̄_{2} + (a(s − 1) + 1)β̄_{1},

hence i_{0} ∈ Γ ∩ I_{l}. This implies #Γ ∩ I_{l} = 1 + #im(ϕ_{l}), hence (in_{l}) follows with equality.

on rational cuspidal projective plane curves 29

Case 2. S_{A} = {γ}, (l + 1)d ∈ Γ and γ ≠ (l + 1)d.

Our goal is to show that i_{0} ∈ Γ. Since (l + 1)d is not of type A, it follows that (l + 1)d = γ̃ + a^{2}s for some γ̃ ∈ Γ – in fact, for γ̃ = ld + 1. Hence we have to verify the following fact: If γ̃ = ld + 1 ∈ Γ, and there exists γ ≠ (l + 1)d of type A in Γ ∩ I_{l+1}, then i_{0} ∈ Γ.

Write γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} with 0 ≤ k < a, 0 ≤ m < a, 0 ≤ t < s. Similarly, set some representation γ̃ = k′β̄_{3} + m′β̄_{0} + t′β̄_{2} + n′β̄_{1}. Here we do not impose any inequality for the coefficients k′, t′, m′, n′; hence k′ is uniquely determined by γ̃, but m′ and n′ not (because of the relation a(a^{2} − a) = a^{2}(a − 1)). Notice also that i_{0} = γ̃ − d. Therefore, the identity

d = β̄_{3} + 2β̄_{0} − (1 + sa)β̄_{1}

shows that if there is a representation of γ̃ with k′ ≥ 1 and some m′ ≥ 2 then i_{0} ∈ Γ.

First we verify that (in any representation of γ̃ as above) k′ ≥ 1. Indeed, assume that k′ = 0. Then (taking γ̃ = ld + 1 modulo a) one gets that l = −1 + La for some 1 ≤ L < as. Since ld + 1 ≤ γ ≤ (l + 1)d − 1, one has

La(a^{2}s + 1) − a^{2}s ≤ kβ̄_{3} + mβ̄_{0} + tβ̄_{2} ≤ La(a^{2}s + 1) − 1.  (21)

This implies that

⌈\frac{kβ̄_{3} + mβ̄_{0} + tβ̄_{2}}{a(a^{2}s + 1)} ⌉ = L.

But one also has

(t + ks)a(a^{2}s + 1) < kβ̄_{3} + mβ̄_{0} + tβ̄_{2} ≤ (t + ks + 1)a(a^{2}s + 1).

Hence L = t + ks + 1. But for this L the left inequality of (21) fails.

Next, we show that there exists a representation with m′ ≥ 2. E.g., if in some “bad” representation of γ̃ one has m′ = 0, then taking k′β̄_{3} + t′β̄_{2} + n′β̄_{1} modulo d one gets n′ ≡ −1 + k′(as + 1) modulo d. Since 1 ≤ k′ ≤ a − 1, this implies that n′ ≥ as, hence n′a^{2} can be rewritten as (n′ − a + 1)a^{2} + a(a^{2} − a). Similar argument works for m′ = 1 as well.

7.3. Addendum to the proof of Theorem 8.

In the verification of the conjecture for Type II curves we will use the results valid for the Type I curves. We will need the following additional facts.

Assume that (C′, p) is of Type I as above, and we keep the notations of 7.2 and 8. 8 shows that (∗) #Γ ∩ [0, ld] = (l + 1)(l + 2)/2 for any 0 ≤ l < d. We first assume that d ≡ 0, a ≡ 1 and s ≡ −1 mod 4.

Recall that any γ ∈ Γ can be written in a unique way as γ = kβ̄_{3}+mβ̄_{0}+tβ̄_{2}+nβ̄_{1}. It is easy to see that γ ≡ −n mod 4. Set #_{l,i} := #{γ ∈ Γ, γ ≤ ld, n ≡ i mod 4} for i = 0, 1, 2, 3 and 0 ≤ l < d. Then, using the function x ↦ x + β̄_{1} = x + d − 1, one gets

#_{l,1} = #_{l−1,2}, #_{l,2} = #_{l−1,3}, #_{l,3} = #_{l−1,0}, #_{l,0} = #_{l−1,1} + A

for some A ∈ ℕ. Here one uses that 4|ld. Taking the sum over i and using (∗), one gets that A = l + 1. Hence #_{l,i} can be computed inductively. In particular, for any l < d, #_{l,0} = (l + 1) + #_{l−4,0} = (l + 1) + (l − 3) + #_{l−8,0} = …. More precisely, if 30 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

l = 4k + r (with 0 ≤ r ≤ 3), then

#_{l,0} = (k + 1)(2k + r + 1) for any l < d.

E.g., this can be rewritten into

#_{2l,0} = (l + 1)(l + 2)/2, whenever 2l < d. (#_{2l})

The very last identity is true even if s = 1, a ≡ 1 and d ≡ 2 mod 4. Indeed, in this case β̄_{2} is eliminated, β̄_{0} ≡ β̄_{3} ≡ β̄_{1} − 1 ≡ 0 mod 4, hence γ ≡ n mod 4. Therefore, #_{l,1} = #_{l−1,0} and #_{l,3} = #_{l−1,2} similarly as above, since ld is even. Although #_{l,2} = #_{l−1,1} may not be true in general, but for l even it is true. Writing #_{l,0} = #_{l−1,3}+A, the formula for (#_{2l}) works again by a similar argument as above.

7.4. (C′, p) in the case Type II

One can verify the following facts:

In case IIa one has two characteristic pairs, the splice decorations are: p_{1} = n, p_{2} = 4n + 1, a_{1} = 4n + 1 and a_{2} = (2n + 1)d − n(4n + 1).

In case IIb one has three characteristic pairs, and the splice decorations can be uniformly described for any s ≥ 1. They are: p_{1} = n, p_{2} = s^{∗}, p_{3} = a^{∗}, a_{1} = a^{∗}, a_{2} = d/2 and a_{3} = a^{∗}s^{∗}d/2 + (s − 1)a^{∗} + 3n + 1.

In fact, the IIa case also can be considered as a ‘specialization’: if in the formulas of IIb (including the formula of d as well) we substitute s = 1/2 then we get the non-minimal splice diagram (with p_{2} = s^{∗} = 1) of the IIa case. Hence in the sequel we will handle the case II uniformly (using the formulas of IIb, where s is any positive integer or 1/2).

It is easy to verify that for Type II the degree d satisfies 2d = (a^{∗})^{2}s^{∗} + 1, and the generators of Γ are

β̄_{0} = \frac{(a^{∗})^{2} − a^{∗}}{4} s^{∗}; β̄_{1} = 2d − 1 = (a^{∗})^{2}s^{∗}; β̄_{2} = a^{∗}d/2; β̄_{3} = \frac{a^{∗}s^{∗}}{4} (2d + 1) + \frac{1}{4}.

If s = 1/2, or s^{∗} = 1, then one should eliminate β̄_{2}.

Theorem 9. The Conjecture is true for Type II curves.

Proof. One can also think about the integers a^{∗} and s^{∗} as parameters of a Type I curve (a^{∗}, s^{∗} corresponding to a and s) with semigroup Γ^{∗}. Their other Type I invariants are (cf. 7.2):

d^{∗} = (a^{∗})^{2}s^{∗} + 1; β̄_{0}^{∗} = ((a^{∗})^{2} − a^{∗})s^{∗}; β̄_{1}^{∗} = d^{∗} − 1 = (a^{∗})^{2}s^{∗};
β̄_{2}^{∗} = a^{∗}d^{∗}; β̄_{3}^{∗} = a^{∗}s^{∗}(d^{∗} + 1) + 1.

Notice that

d^{∗} = 2d, β̄_{0}^{∗} = 4β̄_{0}, β̄_{1}^{∗} = β̄_{1}; β̄_{2}^{∗} = 4β̄_{2}; β̄_{3}^{∗} = 4β̄_{3}.

We fix an integer l < d and we wish to determine

#{γ ∈ Γ : γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} + nβ̄_{1} ≤ ld}.

Multiplying by 4, one gets that this equals

#{γ^{∗} ∈ Γ^{∗} : γ^{∗} = kβ̄_{3}^{∗} + mβ̄_{0}^{∗} + tβ̄_{2}^{∗} + 4nβ̄_{1}^{∗} ≤ 2ld^{∗}}.

on rational cuspidal projective plane curves 31

If s ≥ 1, then d^{∗} ≡ 0; a^{∗} ≡ 1; s^{∗} ≡ −1 mod 4. If s = 1/2, then d^{∗} ≡ 2 and a^{∗} ≡ 1 mod 4, and s^{∗} = 1. Hence this wanted number is #_{2l,0} computed in 7.3 (notice that 2l < 2d = d^{∗}), and equals (l + 1)(l + 2)/2. Hence

#{γ ∈ Γ : γ = kβ̄_{3} + mβ̄_{0} + tβ̄_{2} + nβ̄_{1} ≤ ld} = (l + 1)(l + 2)/2

for any s. In particular, conjecture follows for Type II as well.

8. Lin-Zaidenberg curves: κ̄ = 1 and ν = 2.

In this section we use an equivalent definition of Lin-Zaidenberg type curves given in terms of automorphisms of the affine plane, (see section 4 for another definition). Consider a coordinate system x, y in ℂ^{2}. Take a pair (p, q) of relatively prime numbers with p > q. Let C_{p,q} be the curve of ℂ^{2} defined by the equation y^{p} + x^{q} = 0. Consider ℙ^{2} together with a projective reference (X, Y, Z). We embed ℂ^{2} into ℙ^{2} declaring that the image of the embedding is the open subset U_{Z} defined by Z ≠ 0 and that (x, y) = (X/Z, Y/Z). We will denote by the same symbol the curve C_{p,q} and its compactification to ℙ^{2}. Let ϕ : ℂ^{2} → ℂ^{2} be an algebraic automorphism. The embedding of ℂ^{2} into ℙ^{2} allows us to view any automorphism of ℂ^{2} as a birational transformation of ℙ^{2}.

Definition 2. A curve C is said to be of Lin-Zaidenberg type, (LZ type for short) if it is equal to ϕ^{−1}(C_{p,q}) for a certain automorphism ϕ viewed as birational transformation of ℙ^{2}. (Taking the inverse ϕ^{−1} instead of ϕ is just for notational convenience).

For a rational bicuspidal plane curve, K. Tono [38] has proved that C is of LZ type if and only if κ̄(ℙ^{2} \ C) = 1.

Theorem 10. (CP) is satisfied by any LZ type curve; in other words R(t) ≡ 0.

Let C be a LZ type curve. The curve C has two singularities, one at the origin of ℂ^{2} and the other at infinity. Let L_{1}(t) and L_{2}(t) denote respectively the Poincaré series of their corresponding semigroups, see 2.1. From the very definition (4) of R(t), the vanishing R(t) ≡ 0 is equivalent to

\frac{1}{d} ∑_{ξ^{d}=1} L_{1}(ξt)L_{2}(ξt) = \frac{1 − t^{d^{2}}}{(1 − t^{d})^{3}} .  (22)

The singularity at the origin is isomorphic to y^{p} + x^{q} = 0, therefore

L_{1}(t) = \frac{(1 − t^{pq})}{(1 − t^{p})(1 − t^{q})}.  (23)

The combinatorics of the minimal embedded resolution of the singularity at infinity of C are closely related to the combinatorics of the minimal resolution of the indeterminacy of ϕ as a birational transformation of ℙ^{2}, and to the combinatorics of the minimal embedded resolution of the singularity that C_{p,q} has at infinity.

We will need the facts on algebraic automorphisms of ℂ^{2} from section 4; we refer to [10], [11] for details and more precise statements. Let π : X → ℙ^{2} the minimal resolution of the indeterminacy of ϕ. Let L := ℙ^{2} \ ℂ^{2} denote the line at 32 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

infinity and consider A the dual graph of the normal crossing divisor π^{∗}L. Recall that we have a total order of the vertices of the graph (this order is recovered combinatorially by successively contracting (−1) vertices and adjusting weights at each step). Denote by A^{∗} the graph A minus its first vertex (the one corresponding to the line at infinity). The automorphism ϕ admits a unique decomposition as ψ ∘ H where H is an affine transformation of ℂ^{2} and ψ is the automorphism of ℂ^{2} which is composition of π^{−1} and the successive contraction of all the irreducible components of π^{∗}L with self intersection (−1). Moreover all the components except the one corresponding to the last vertex of A are contracted. For our purposes it is sufficient to consider automorphisms for which H is the identity; we will assume this in the sequel. The automorphism ϕ admits a decomposition as ϕ = ϕ_{r} ∘ ... ∘ ϕ_{1}, where ϕ_{i} is the result considering first the blowing ups whose exceptional divisors belong to the i-th floor and then contracting all possible components with selfintersection (−1). The graph associated to the resolution of indeterminacy of ϕ_{i} is the elementary graph of length n_{i}.

The singularity at infinity of C_{p,q} is at the point (1 : 0 : 0) of ℙ^{2} and is defined by the equation y^{p} + z^{p−q} = 0. Let σ : Y → ℙ^{2} the composition of blowing ups giving its minimal embedded resolution of this singularity. By the irreductibility of C_{p,q} at (1 : 0 : 0) the blowing ups whose composition gives rise to σ are totally ordered. Let E = σ^{−1}(1 : 0 : 0) be the exceptional divisor of σ; denote by G its dual graph. Each vertex of G corresponds to an irreducible component of E, and hence to one of the blowing-ups whose composition is σ, and therefore they are totally ordered. It is known that G is a linear graph whose first vertex is one of the extremes. Although the ordering given to the vertices is not the same that the order given by the linearity of the graph we will denote by v_{1} and v_{2} the two extremal vertices of the graph (the vertex v_{1} is the first vertex in the given order, but v_{2} need not be the second). We decorate the graph adding an arrow to the vertex corresponding with the irreducible component of E.

We must distinguish two cases:

Case 1: the first indetermination point of ϕ^{−1} is different to the point at infinity of C_{p,q}. Let ψ : Z → ℙ^{2} be the blowing up process giving the minimal embedded resolution of the singularity that C has at infinity. It is easy to check that ψ is the sequence of blowing ups needed to resolve the indeterminacy of ϕ, followed by the sequence of blowing ups providing the embedded resolution of the singularity of C_{p,q} at infinity. The divisor ψ^{∗}C can be decomposed as

ψ^{∗}C = C̃ + ∑_{i=1}^{k} m_{k}E_{k},

where C̃ is the strict transform of C and E_{1},..., E_{k} the irreducible components of the exceptional divisor ordered by appearance.

In this case it is easy to see that the decorated dual graph B of the exceptional divisor is obtained by joining with an edge the last vertex of A^{∗} with the first vertex v_{1} of G. We give a total order to the vertices of the resulting graph are totally ordered by considering first the vertices of A^{∗} and after the vertices of G with the previously given order. This order coincides with the natural order obtained defined identifying the vertices with the divisors E_{1},...,E_{k}.

With this information it is easy to compute the multiplicity sequence of the singularity, and to deduce from it the degree of C and the coefficients m_{1},...,m_{k}.

on rational cuspidal projective plane curves 33

The degree is d := deg(C) = pn_{1} … n_{r}. For computing the Poincaré series L_{2} we need to know the coefficients m_{i} of the univalent and trivalent vertices of B. We observe that the univalent vertices are the first vertex of each floor of A^{∗} (which in the picture are the univalent central vertices of the floors), the second vertex of the first floor, and the vertex v_{2} of G. We have exactly one trivalent vertex for each floor in addition to the last vertex of G, where the arrow decorating the graph is attached. The Poincaré series of the semigroup of the singularity at infinity obtained is:

L_{2}(t) = \frac{(1 − t^{n_{1}^{2}...n_{r}^{2}p^{2}−pq}) ∏_{i=1}^{r}(1 − t^{(n_{1}^{2}...n_{i}^{2}−1)n_{i}...n_{r}p})}{(1 − t^{n_{1}^{2}...n_{r}^{2}p−q})(1 − t^{n_{1}...n_{r}p}) ∏_{i=1}^{r}(1 − t^{(n_{1}^{2}...n_{i}^{2}−1)n_{i+1}...n_{r}p})}.  (24)

Case 2: the first indetermination point of ϕ^{−1} is equal to the point at infinity of C_{p,q}. We can further assume that the singularities at infinity of ϕ_{r}^{−1}(C_{p,q}) and C do not have the same resolution graph: if this were the case we could work with ϕ_{r}^{−1}(C_{p,q}) instead of C_{p,q} and with ϕ_{r−1} ∘ ... ∘ ϕ_{1} ∘ H instead of ϕ, and we would be in Case 1. Let k be the only integer such that p − kq < q < p − (k − 1)q; then the condition that ϕ_{r}^{−1}(C_{p,q}) and C do not have the same resolution graph is equivalent to n_{r} ≥ k + 1.

Let π′ : X → ℙ^{2} be the minimal blowing-up process resolving the indeterminacy of ϕ^{−1}. Viewing ϕ as a composition of a sequence of blowing ups followed by a sequence of contractions it is clear that the dual graph of (π′)^{∗}L can be naturally identified with the graph A, but with a different ordering of the vertices (in particular the first vertices are those of the r-th floor of A). Noticing that n_{r} ≥ k + 1, a comparison between the blowing up sequences π′ and σ shows that their first k + 1 blowing up processes are the same in both sequences. Call σ′ the composition of these blowing-ups and let W:={w′_{1}, ..., w′_{k+1}} be the ordered set of vertices of G corresponding to the exceptional divisors of these blowing-ups. We can decompose σ as σ = σ_{2} ∘ σ_{1}, where σ_{1} is the composition of blowing-ups whose associated vertex does not belong to W and σ_{2} is the composition of the remaining blowing-ups. Suppose that w′_{i} is the first of the vertices of W such that its associated exceptional divisor meets the strict transform of C_{p,q} by σ_{2}. Clearly w′_{i} can be regarded also as a vertex of A. Denote by π_{1} the composition of those blowing ups of π such that the vertex associated to their exceptional divisor is smaller or equal than w′_{i} with the ordering in A induced by π.

Let ψ : Z → ℙ^{2} be the blowing up process giving the minimal embedded resolution of the singularity that C has at infinity. Then it is easy to check that ψ = π_{1}∘σ_{1}. Due to this decomposition the decorated dual graph B of the exceptional divisor of ψ can be obtained as follows:

Let A′ be the graph obtained from A^{∗} by deleting the last k vertices. If k > 1 let G′ be the graph obtained from G by deleting the first k − 1 vertices; if k = 1 let G′ be the graph obtained by deletion of the first vertex of G. The graph G′ is linear. If k = 1 then its first vertex is a extreme of it, and is denoted by w. If k > 0 both its first and second vertices are extremes of it; in this case we alter the ordering in the vertices of G′ by interchanging the order of the first two vertices, and denote by w the first vertex of G′ with the altered order. The graph B is is obtained by identifying (not joining with an edge) the last vertex of A′ with the vertex w of G′. We give a total order to the vertices of the resulting graph by considering first the vertices of A′ and after the vertices of G′ with the altered order. This order coincides with the natural order obtained defined identifying the vertices with the irreducible components of the exceptional divisor of ψ. The univalent vertices of B 34 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

are the first vertex of each floor of A (in the picture these are the univalent central vertices of the floors), the second vertex of the first floor, and the extreme of G′ different from w. We have exactly one trivalent vertex for each floor in addition to the last vertex of G′, where the arrow decorating the graph is attached. The degree of C is d = n_{1} … n_{r}q and the Poincaré series of the singularity at infinity is:

L_{2}(t) = \frac{(1 − t^{n_{1}^{2}...n_{r}^{2}q^{2}−pq}) ∏_{i=1}^{r}(1 − t^{(n_{1}^{2}...n_{i}^{2}−1)n_{i}...n_{r}q})}{(1 − t^{n_{1}^{2}...n_{r}^{2}q−p})(1 − t^{n_{1}...n_{r}q}) ∏_{i=1}^{r}(1 − t^{(n_{1}^{2}...n_{i}^{2}−1)n_{i+1}...n_{r}q})}.  (25)

Remark 3. The formulas for the degree and L_{2}(t) obtained in Case 2 are the result of interchanging the role of p and q in the formulas obtained in Case 1.

Due to (22), (23), (24), (25) and 3, we need to prove the following fact. For any two coprime positive integers p and q (without imposing p > q), for n_{1}, …, n_{r} positive integers and d := n_{1} … n_{r}p the following identity holds:

\frac{1}{d} ∑_{ξ^{d}=1} \frac{(1 − (ξt)^{pq})(1 − (ξt)^{n_{1}^{2}...n_{r}^{2}p^{2}−pq})}{(1 − (ξt)^{p})(1 − t^{q})(1 − (ξt)^{n_{1}^{2}...n_{r}^{2}p−q})(1 − (ξt)^{n_{1}...n_{r}p})}

▪ ∏_{i=1}^{r} \frac{(1 − (ξt)^{(n_{1}^{2}...n_{i}^{2}−1)n_{i}...n_{r}p})}{(1 − (ξt)^{(n_{1}^{2}...n_{i}^{2}−1)n_{i+1}...n_{r}p})} = \frac{1 − t^{d^{2}}}{(1 − t^{d})^{3}} .

Multiplying in both sides by d(1−t^{d})^{3} and extracting common factors of numerator and denominator we see that the last equality is equivalent to:

(1 − t^{d}) ∑_{ξ^{d}=1}[(∑_{j=0}^{p−1}(ξt)^{jq})(∑_{j=0}^{p−1}(ξt)^{j(n_{1}^{2}...n_{r}^{2}p−q)})( ∑_{j=0}^{n_{1}...n_{r}−1}(ξt)^{jp})▪  (26)

▪ ∏_{i=1}^{r}( ∑_{j=0}^{n_{i}−1}(ξt)^{j(n_{1}^{2}...n_{i−1}^{2}n_{i}−1)n_{i+1}...n_{r}p})] = d(1 − t^{d^{2}}).

The left hand side of (26) is equal to (1 − t^{d}) times

∑_{ξ^{d}=1}[ ∑_{i,j=0}^{p−1} ∑_{l=0}^{n_{1}...n_{r}−1} ∑_{α=1}^{r} ∑_{k_{α}=0}^{n_{α}−1}(ξt)^{iq+j(n_{1}^{2}...n_{r}^{2}p−q)+lp+∑_{α=1}^{r} k_{α}(n_{1}^{2}...n_{α−1}^{2}n_{α}−1)n_{α+1}...n_{r}p}].

(27)

The exponent of (ξt) in the last expression can be written as A(i, j, l, k) + dB(i, j, l, k), for

A(i, j, l, k) := (i − j)q + lp − ∑_{α=1}^{r} k_{α}n_{α+1} … n_{r}p,

B(i, j, l, k) := jn_{1} … n_{r} + ∑_{α+1}^{r} k_{α}n_{1} … n_{α−1}.

But ∑_{ξ^{m}=1} ξ^{m} = 0 for any integer m non-divisible by d. In the present case m = A(i, j, l, k).

Lemma 2. If A(i, j, l, k) is divisible by d then i = j and l = ∑_{α=1}^{r} n_{α+1} … n_{r}.

As d = n_{1} … n_{r}p we deduce that p|A(i, j, l, k). It immediately follows that i = j and that

n_{1} … n_{r}|l − ∑_{α=1}^{r} k_{α}n_{α+1} … n_{r}.

We claim that then 0 = l − ∑_{α=1}^{r} k_{α}n_{α+1} … n_{r}. The claim is proved by induction on r. For r = 1 it is obvious since |l − k_{1}| < n_{1}. For the induction step we write

l − ∑_{α=1}^{r} k_{α}n_{α+1} … n_{r} = l − k_{r} + n_{r} ∑_{α=1}^{r−1} k_{α}n_{α+1} … n_{r−1};

as n_{1} … n_{r} divides the last quantity we have n_{r}|l − k_{r}. Therefore we can express l as l = k_{r} + n_{r}l′. Then

n_{1} … n_{r−1}|l′ − ∑_{α=1}^{r−1} k_{α}n_{α+1} … n_{r−1},

and we conclude by induction.

Therefore, by Lemma 2 and a computation, the left hand side of (26) becomes

(1 − t^{d}) · d ∑_{j=0}^{p−1} ∑_{α=1}^{r} ∑_{k_{α}=0}^{n_{α}−1} t^{jn_{1}^{2}...n_{r}^{2}p+∑_{α=1}^{r} n_{1}^{2}...n_{α−1}^{2}n_{α}...n_{r}p}.

A computation shows that the result equals d(1 − t^{d^{2}}), as desired.

9. (CP) for Orevkov’s curves

In this section we will consider the families {C_{4k}} and {C_{4k}^{∗}} (k ≥ 2) of Orevkov [34]. There is a special interest in these curves: Orevkov proved that they satisfy d > αm (where α = (3 + \sqrt{5})/2 as above); moreover he conjectured that these are the only rational cuspidal curves (together with the curves C_{4} and C_{4}^{∗} from Examples 1(e)-(f)) satisfying this inequality. Also, for these curves one has κ̄(ℙ^{2} \ C) = 2, in contrast with the previous sections 3-7. In particular, Theorem 11 shows that (CP) is not a speciality of curves with κ̄ < 2 (and of some finitely many sporadic curves of general type).

In this case the number of characteristic pairs of (C, p) is again two, which makes the structure of the semigroups rather interesting. The numerical invariants of the curves are the following (cf. [34]). In both cases j ≡ 0 mod 4, and j ≥ 8.

9.1. C_{j} has degree d = φ_{j+2}.

The germ (C_{j}, p) has two characteristic pairs with numerical invariants:

p_{1} = φ_{j}/3, p_{2} = 3; a_{1} = φ_{j+4}/3, a_{2} = 1 + φ_{j}φ_{j+4}/3;
β̄_{0} = φ_{j}, β̄_{1} = φ_{j+4}, β̄_{2} = a_{2}.

9.2. C_{j}^{∗} has degree d^{∗} = 2φ_{j+2}.

The germ (C_{j}, p) has two characteristic pairs with numerical invariants:

p_{1} = φ_{j}/3, p_{2}^{∗} = 6; a_{1} = φ_{j+4}/3, a_{2}^{∗} = 1 + 2φ_{j}φ_{j+4}/3;

36 j. f. de bobadilla, i. luengo, a. melle-hernández, a. némethi

β̄_{0}^{∗} = 2φ_{j}, β̄_{1}^{∗} = 2φ_{j+4}, β̄_{2}^{∗} = a_{2}^{∗}.

Theorem 11. C_{4k} and C_{4k}^{∗} satisfy (CP).

Proof. We start with the case C_{j}.

Step 1. Assume that l < d. Notice that gcd(β̄_{1}, d) = 1, hence kβ̄_{1} = ld implies l = k = 0. In all the other cases, by a similar argument as in the proof of 5, kβ̄_{1} < ld is equivalent to k/l < 1/α. Hence kβ̄_{1} ∈ I_{l} if and only if ⌈kα⌉ = l.

Step 2. Assume l < d. If iβ̄_{0} > ld then i/l > φ_{j+2}/φ_{j} > α. Conversely, if i/l > α then consider φ_{j+2}/φ_{j} > φ_{j+4}/φ_{j+2} > α. Similarly as in the proof of 5, we get that i/l ≥ φ_{j+2}/φ_{j}. Hence, if l ∉ {φ_{j}, φ_{j} + 1} then iβ̄_{0} ∈ I_{l} if and only if ⌈i/α⌉ = l.

Step 3. Notice that x := φ_{j}φ_{j+4}/3 = (φ_{j+2}^{2} − 1)/3 is not a multiple of d, hence x and a_{2} = x + 1 are in the same interval I_{l_{0}}. Let Γ_{0} be the semigroup generated by β̄_{0} and β̄_{1}. Hence for l < l_{0} one has Γ_{0} ∩ I_{l} = Γ ∩ I_{l}.

Moreover, l_{0} − 1 < φ_{j}. Indeed, (l_{0} − 1)φ_{j+2} < x = (φ_{j+2}^{2} − 1)/3 < φ_{j+2}φ_{j} (because φ_{j+2} < 3φ_{j}).

Notice also that the very first semigroup element iφ_{j} + kφ_{j+4} of Γ_{0} which can be written in two different ways is x = (φ_{j+4}/3)φ_{j} = (φ_{j}/3)φ_{j+4}.

Hence, for any l ≤ l_{0} − 1, by similar argument as in the proof of 5(d), and using Step 1 and Step 2, we get that the distribution patern is true.

The proof now bifurcates in two cases.

First assume that d = φ_{j+2} = 3t − 1 for some t. Then x = 3t^{2} − 2t and l_{0} = t. Step 4. One can verify that the following numbers are in increasing order: x−2φ_{j},

(l_{0} − 1)d, x − φ_{j}, x, l_{0}d, x + φ_{j}, x + 3φ_{j}, (l_{0} + 1)d, x + 4φ_{j}, x + 6φ_{j}, (l_{0} + 2)d, x + φ_{j+4}, x + 7φ_{j}. E.g., x + φ_{j} > l_{0}d reduces to φ_{j} > t, or to 3φ_{j} > φ_{j+2} + 1 which is true. The other verifications are similar.

Step 5. Since φ_{j+4} + φ_{j} = 3φ_{j+2}, the map s_{l} : I_{l−3} ∩ Γ_{0} → I_{l} ∩ Γ_{0}, given by y ↦ y + φ_{j+4} + φ_{j} is well-defined. Clearly, it is injective. For l = l_{0} the complement of its image has two elements, namely x and x − φ_{j} (cf. Steps 3 and 4).

For l > l_{0} the map s_{l} is surjective. This follows from Step 4 and the identities iφ_{j} = (i − φ_{j+4}/3)φ_{j} + (φ_{j}/3)φ_{j+4} (for i > φ_{j+4}/3) and kφ_{j+4} = (k − φ_{j}/3)φ_{j+4} + (φ_{j+4}/3)φ_{j} (for k > φ_{j}/3). Therefore

∑_{k∈Γ_{0}} t^{⌈k/d⌉} = 1 + 2t + … + l_{0}t^{l_{0}−1} + (l_{0} + (l_{0} − 1)t + l_{0}t^{2}) · (t^{l_{0}} + t^{l_{0}+3} + t^{l_{0}+6} + …).

Step 6. Consider now the intervals l_{0} ≤ l ≤ 2l_{0} − 1. Since (cf. Step 4) 2x + d > 2x + 2φ_{j} > 2l_{0}d, we get that 2a_{2} is not situating in these intervals.By a mod 3 argument, for any such l, Γ ∩ I_{l} is the disjoint union of I_{l} ∩ Γ_{0} and I_{l} ∩ (a_{2} + Γ_{0}). Moreover, in all these intervals any element of a_{2} + Γ_{0} has a unique representation in the form a_{2} + iφ_{j} + kφ_{j+4}.

In particular, we have to understand the distribution of a_{2} + Γ_{0}. By a similar computation as in Step 4, we get that I_{l_{0}} contains only a_{2}, I_{l_{0}+1} contains a_{2} + iφ_{j} for i = 1, 2, 3; I_{l_{0}+2} contains a_{2} + iφ_{j} for i = 4, 5, 6. In particular, for intervals l = l_{0}, l_{0} + 1, l_{0} + 2 the distribution of Γ follows.

Step 7. Assume again l_{0} ≤ l ≤ 2l_{0} −1. Consider the injective map s′_{l} : I_{l−3} ∩(a_{2} + Γ_{0}) → I_{l}∩(a_{2}+Γ_{0}) given by y ↦ y+φ_{j} +φ_{j+4}. We claim that for l_{0}−3 ≤ l ≤ 2l_{0}−1 the complement of the image of s′_{l} has exactly three elements. There are two types of elements in the complement. The first type is a_{2} +kφ_{j+4} = 1+(k +φ_{j}/3)φ_{j+4} =

on rational cuspidal projective plane curves 37

1 + k′φ_{j+4}. Since k′φ_{j+4} is never a multiple of d (in the relevant intervals) we get (via Step 1) that 1 + k′φ_{j+4} ∈ I_{l} if and only if k′φ_{j+4} ∈ I_{l} if and only if ⌈k′α⌉ = l.

The other type is a_{2} + iφ_{j} = 1 + (i + φ_{j+4}/3)φ_{j} = 1 + i′φ_{j}. Notice that i′φ_{j} can be a multiple of d in these intervals, namely for i′ = d. This fact combined with Step 2 we get that 1 + i′φ_{j} ∈ I_{l} if and only if ⌈i′/α⌉ = l. Now, by the end of the proof of 5(d) the above claim follows.

In particular, the distribution is true for any l < 2l_{0}.

Step 8. Since (CP_{l}) is true if and only if (CP_{d−2−l}) is true, one gets (CP_{l}) for all the remaining cases.

The proof in the other case d = φ_{j+2} = 3t + 1 is similar. The only differences are the following. First one has the following increasing numbers:

x − φ_{j}, (l_{0} − 1)d, x, x + φ_{j}, l_{0}d, x + 2φ_{j}, x + 4φ_{j}, (l_{0} + 1)d, x + 5φ_{j}, x + 6φ_{j}, (l_{0} + 2)d, x + 7φ_{j}. Also x + φ_{j+4} < (l_{0} + 2)d. In particular,

∑_{k∈Γ_{0}} t^{⌈k/d⌉} = 1+2t+…+l_{0}t^{l_{0}−1}+((l_{0}−1)+(l_{0}−1)t+l_{0}t^{2})·(t^{l_{0}} +t^{l_{0}+3}+t^{l_{0}+6}+…).

Moreover, a_{2} + Γ_{0} has 2, 3, resp. 3 elements in I_{l} for l = l_{0}, l_{0} + 1, resp. l_{0} + 2, namely a_{2} + iφ_{j} for i = 0, 1 (l = l_{0}); a_{2} + iφ_{j} for i = 2, 3, 4 (l = l_{0} + 1); and finally a_{2} + iφ_{j} for i = 5, 6 and a_{2} + φ_{j+4} in the last case. Otherwise all the arguments are similar.

In the next paragraph we show that the case C_{j}^{∗} can be reduced to the case C_{j}. Assume first that l < φ_{j+2}. In these intervals I_{l}(C_{j}), for any element of the

semigroup of C_{j}, the coefficient c_{2} of a_{2} is ≤ 2. Indeed, 3a_{2} = 3 + φ_{j}φ_{j+4} = 2+φ_{j+2}^{2}. Moreover, by a computation one can verify that there no elements of type 2a_{2} + iφ_{j} + kφ_{j+4} (i.e. with c_{2} = 2) which equals some lφ_{j+2} + 1 (the first entry of some interval). Using this, one gets a bijection Γ(C_{j})∩I_{l}(C_{j}) → Γ(C_{j}^{∗})∩I_{l}(C_{j}^{∗}) given by x ↦ 2x − c_{2} (which sends β̄_{i} into β̄_{i}^{∗}). Therefore, (CP_{l}) is true for C_{j}^{∗} for any l ≤ d^{∗}/2 − 1. By symmetry, (CP_{l}) is true for any l ≥ d^{∗} − 2 − (d^{∗}/2 − 1) = d^{∗}/2 − 1, hence for any l.

Acknowledgements. We thank J. Stevens for drawing our attention to an error in the original version of 3.6, and K. Tono and M.G. Zaidenberg for helpful information and comments.

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J. Fernández de Bobadilla

Current address:

Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, C/ Senda del Rey 9, E-28040, Madrid,

Spain

Present long-term address

Department of Mathematics, University of Utrecht, Postbus 80010, 3508TA Utrecht,

The Netherlands

bobadilla@math.uu.nl I. Luengo-Velasco and A. Melle-Hernández Facultad de Matemáticas,

Universidad Complutense, Plaza de Ciencias, E-28040, Madrid,

Spain

iluengo@mat.ucm.es amelle@mat.ucm.es

A. Némethi

Current address:

Rényi Institute of Mathematics,

Budapest, Hungary. Present long-term address Department of Mathematics, Ohio

State University, Columbus, OH 43210,

USA

nemethi@math.ohio-state.edu nemethi@renyi.hu

ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

J. FERNÁNDEZ DE BOBADILLA, I. LUENGO-VELASCO, A. MELLE-HERNÁNDEZ and A. NÉMETHI

Abstract for On-Line Publication

In 2002 L. Nicolaescu and the forth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg-Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg-Witten conjecture led us to a very interesting and deep set of ‘compatibility properties’ of these curves (generalising the Seiberg-Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (e.g. the Noether-Nagata or the log Bogomolov-Miyaoka-Yau inequalities). Namely, we provide a set of ‘compatibility conditions’ which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer d if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree d. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than two.