ON RATIONAL CUSPIDAL CURVES, OPEN SURFACES AND LOCAL SINGULARITIES

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}. The classification of such curves, up to the action of the automorphism group P GL(3, ℂ) on ℙ^{2}, is a very difficult open problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when C is a rational curve.

The problem remains very difficult even if we aim much less, e.g. the determination of the maximal number of cusps among all the rational cuspidal plane curves (a problem proposed by F. Sakai in [13]), — this number is expected to be small. (In [47] K. Tono recently proved that it is less than nine; the maximal number of cusps known by the authors is four; and, in fact, it is expected to be four.)

This remarkable problem of classification 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, and in the classical algebraic geometry.

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, regarding these surfaces, Flenner and Zaidenberg in [7] formulated the rigidity conjecture. This says that every ℚ-acyclic affine surfaces Y with logarithmic Kodaira dimension κ̄(Y) = 2 must be rigid. 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. (Notice that if C has at least three cusps then κ̄(ℙ^{2} \ C) = 2 by [53]; and, in fact, all the curves with κ̄(ℙ^{2} \ C) < 2 are classified, see below.) Many known examples support the rigidity conjecture, see [6, 7, 8, 9].

Another related, very old, famous open problem has its roots in early algebraic geometry, and wears the name of Coolidge and Nagata, see [2, 26]. It predicts that every rational cuspidal curve can be transformed by a Cremona transformation into a line.

The aim of the present article is to present some of these conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors.

Key words and phrases. Cuspidal rational plane curves, logarithmic Kodaira dimension, Nagata-Coolidge problem, Flenner-Zaidenberg rigidity conjecture, surface singularities, ℚ-homology spheres, Seiberg-Witten invariant, graded roots, Heegaard Floer homology, Ozsváth-Szabó invariants.

The first three author is partially supported by Ramón y Cajal contact. The first three authors thank Marie Curie Fellowship for the Transfer of Knowledge supporting their visit at the Rényi Institute, Budapest, Hungary; the first three authors are partially supported by MTM2004-08080-C02-01; the last author is also supported by the Marie Curie Fellowship and OTKA Grant.

2 J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi

In section 3 we present the Nagata-Coolidge problem. The main theme of section 4 is Orevkov’s conjecture [37], which formulates an inequality involving the degree d and numerical invariants of local singularities. In a different formulation, this is equivalent with the positivity of the virtual dimension of the space of curves with fixed degree and certain local type of singularities which can be geometrically realized. (This was used, as a ‘first test’, by the second author to check that some singularities might be realized or not.) The equivalence of the two inequalities is proved via some properties of the numerical, local and global deformation invariants; this material is presented in sections 2 and 3.

In section 5 starts with some classification results: the classification of projective plane curves with κ̄(ℙ^{2} \C) < 2 by the work of Kashiwara, Lin, Miyanishi, Sugie, Tsunoda, Tono, Wakabayashi, Yoshihara, Zaidenberg (among others). Also, we recall the classification (of the authors) of rational unicuspidal curves whose cusp has one Puiseux pair. The end of this section deals with the rigidity conjecture of Flenner and Zaidenberg.

In section 6 we present the author’s ‘compatibility property’ (a sequence of inequalities), conjecturally satisfied by the degree and local invariants of the singularities of a rational cuspidal curve. It turns out that in the unicuspidal case, the inequalities are true if and only if they are (in fact) equalities. One of the reformulations (valid in the unicuspidal case) of this conjectural series of identities is the semigroup distribution property, which is a very precise compatibility property connecting the semigroup of the local singularity and the degree of the curve. We also present one of its equivalent statements suggested to us by Campillo. Section 7 explains the relation of the semigroup distribution property (for the unicuspidal case) with the ‘Seiberg-Witten invariant conjecture’ (of the forth author and Nicolaescu [32]), which basically lead us to this compatibility property. The last section present another connection with the Seiberg-Witten theory (based on the articles [28, 29, 30]), but now exploiting the relation with the Heegaard-Floer homology (introduced and studied by Ozsváth and Szabó [39]). Here, a crucial intermediate object is the ‘graded root’ (introduced by the forth author [28]), which provides a completely different interpretation of the semigroup distribution property.

The authors thank A. Campillo, K. Tono and M.G. Zaidenberg for helpful information, comments and interesting discussions.

The first three authors are grateful for the warm hospitality of the Alfréd Rényi Institute of Mathematics (Budapest, Hungary) where they spent a fruitful period in an ideal mathematical environment.

2. Local data

One of our main goals is to characterise the local embedded topological types of local singular germs (C, p_{i}) ⊂ (ℙ^{2}, p_{i}) which can be realized by a projective plane curve C of degree d. (Here, the points {p_{i}} are not fixed.) In this section we collect some results about the (deformation of) local invariants (C, p_{i}).

2.1. Let (C, p) be the germ at p of a reduced curve C ⊂ ℙ^{2} and let f ∈ O_{ℙ^{2},p} be a function defining the singularity (C, p) in some local coordinates x and y.

2.1.1. Let ϕ : C_{(C,p)} → S_{(C,p)} be a semi-universal deformation of (C, p). The base space S_{(C,p)} is smooth and its tangent space is isomorphic to the vector space O_{C,p}/(f_{x}, f_{y}). In particular its dimension is equal to the Tjurina number τ(C, p) := dim_{ℂ}(O_{ℙ^{2},p}/I^{ea}(C, p)), where I^{ea}(C, p) is the ideal (f, f_{x}, f_{y}).

Rational cuspidal curves, open surfaces and singularities 3

2.1.2. Let S_{(C,p)}^{es} ⊂ S_{(C,p)} be the smooth subgerm of a semi-universal equisingular deformation of (C, p), see [55], Theorem 7.4. Let T_{ε} := Spec(ℂ[ε]/(ε^{2})) be the base space of first order infinitesimal deformations. The equisingularity ideal of (C, p) is the ideal

I^{es}(C, p) = {g ∈ O_{ℙ^{2},p} | f + εg is equisingular over T_{ε}}.

It contains I^{ea}(C, p). Moreover the tangent space of S_{(C,p)}^{es} ⊂ S_{(C,p)} is isomorphic to the vector space I^{es}(C, p)/I^{ea}(C, p). In particular the codimension of S_{(C,p)}^{es} in S_{(C,p)} is the topological invariant

τ^{es}(C, p) := dim_{ℂ}(O_{ℙ^{2},p}/I^{es}(C, p)).  (1)

The equisingular stratum S_{(C,p)}^{es} coincides with the μ-constant stratum of S_{(C,p)}, where μ = μ(C, p) is the Milnor number of (C, p).

2.1.3. Let n : O_{C,p} ↪ Õ_{C,p} := ∏_{i=1}^{r} ℂ{t} be the normalisation map, where r is the number of local irreducible components of (C, p). Let δ(C, p) be the dimension of the cokernel Õ_{C,p}/n(O_{C,p}). Let cond(O_{C,p}) be the conductor ideal of O_{C,p}, that is the annihilator of the O_{C,p}-module Õ_{C,p}/n(O_{C,p}).

The δ-constant stratum S_{(C,p)}^{δ} ⊂ S_{(C,p)} is the locus of points where δ is constant, see [3, 42]. In general, S_{(C,p)}^{δ} is not longer smooth at C, but it is locally irreducible. Even more, the tangent cone of S_{(C,p)}^{δ} at C is always a linear space which is identified with the vector space cond(O_{C,p})/I^{ea}(C, p), [3], Theorem 4.15. In particular, the codimension of S_{(C,p)}^{δ} in S_{(C,p)} is equal to

codim (S_{(C,p)}^{δ} ⊂ S_{(C,p)}) = δ(C, p).  (2)

2.1.4. S. Diaz and J.Harris [3] showed the inclusion of ideals

I^{ea}(C, p) ⊂ I^{es}(C, p) ⊂ cond(O_{C,p}) ⊂ O_{C,p}.

Since the equisingular stratum S_{(C,p)}^{es} is contained in the δ-constant stratum S_{(C,p)}^{δ} then

M̄(C, p) := codim (S_{(C,p)}^{es} ⊂ S_{(C,p)}^{δ}) = τ^{es}(C, p) − δ(C, p) ≥ 0.  (3)

In fact, M̄(C, p) is a topological invariant of the singularity (see next paragraph). S. Orevkov and M. Zaidenberg used M̄(C, p) in a different situation, cf. [37, 38].

2.2. The minimal embedded resolution of (C, p) is obtained via a sequence of blowing-ups at infinitely near points to p. The topological numerical invariants δ(C, p), μ(C, p), τ^{es}(C, p) and M̄(C, p) can be written in terms of the multiplicity sequence [m_{1}^{p}, …, m_{k_{p}}^{p}] of (C, p) associated with this minimal resolution. This is the set of multiplicities of the strict transform of C at all the infinitely near points along all the branches. (We do not omit the 1’s; for this notation we follow [24] and [7].)

Using Milnor’s formula μ(C, p) = 2δ(C, p) − r + 1 one gets

μ(C, p) + r − 1 = 2δ(C, p) = ∑_{i=1}^{k_{p}} m_{i}^{p}(m_{i}^{p} − 1).  (4)

A blow-up in the minimal good resolution is called inner (or subdivisional) if its center is at the intersection point of two exceptional curves of the resolution process (such a center is called a satellite point in [54]). If the center is situated on exactly one exceptional divisor, then the blow-up is called outer (or sprouting). Notice that the first blow-up is neither inner nor outer. A center is called a free infinitely near point if it is either outer or it is p, the center of the very first blow up. It is convenient to count this first infinitely near point by two. Let ω_{p}, resp. ρ_{p}, denote the number of inner, respectively of outer, blow-ups. Then k_{p} − 1 = ω_{p} + ρ_{p} and the number of free infinitely near points is L_{p} := 2 + ρ_{p} = k_{p} − ω_{p} + 1.

C.T.C. Wall in Theorem 8.1 of [54] proved the following formula for τ^{es}(C, p)

τ^{es}(C, p) = ∑_{i=1}^{k_{p}} \frac{(m_{i}^{p} − 1)(m_{i}^{p} + 2)}{2} + ω_{p} − 1 = ∑_{i=1}^{k_{p}} \frac{m_{i}^{p}(m_{i}^{p} + 1)}{2} − L_{p}.  (5)

Different proofs of this formula have been given by Mattei, Theo de Jong and Shustin. The parametric codimension is equal to

M̄(C, p) = ∑_{i=1}^{k_{p}}(m_{i}^{p} − 1) + ω_{p} − 1 = −L_{p} + ∑_{i=1}^{k_{p}} m_{i}^{p}.  (7)

For other equivalent formulae of M̄(C, p) see [38].

3. Global data

3.1. Let ℙ^{N}, where N = d(d + 3)/2, be the Hilbert scheme parametrising algebraic projective plane curves of degree d. The locus V_{d,g} in ℙ^{N} of reduced and irreducible curves of degree d and genus g is irreducible (see e.g. [15]). The locus of reduced and irreducible curves of degree d and genus g having only nodes as singularities is smooth of dimension 3d − 1 + g and is dense in V_{d,g}.

Recall that our goals is to characterise the local embedded topological types of local singular germs (C, p_{i}) ⊂ (ℙ^{2}, p_{i}) which can be realized by a projective curve C of degree d. Assume that a projective reduced plane curve C of degree d exists with fixed topological types S_{1}, …, S_{ν}. Let V (S_{1}, …, S_{ν}) denote the locally closed subscheme of reduced curves on ℙ^{2} having exactly ν-singularities with topological types S_{1}, …, S_{ν}.

Greuel and Lossen in Theorem 3.6 of [12] proved that the dimension of V (S_{1}, …, S_{ν}) at C satisfies

dim(V (S_{1}, …, S_{ν}), C) ≥ C^{2} + 1 − p_{a}(C) − τ^{es}(C) = \frac{d(d + 3)}{2} − τ^{es}(C),  (8)

where τ^{es}(C) := ∑_{i=1}^{ν} τ^{es}(C, p_{i}). The right hand side

expdim(V (S_{1}, …, S_{ν}), C) := \frac{d(d + 3)}{2} − τ^{es}(C)

is called expected dimension of V (S_{1}, …, S_{ν}) at C. The study of the locally closed subscheme V (S_{1}, …, S_{ν}) of ℙ^{N} and its properties have been studied intensively in the literature (see for instance works by Artal Bartolo, Diaz, Greuel, Harris, Karras, Lossen, Shustin, Tannenbaum, Wahl, Zariski among many others).

3.2. The action of P GL(3, ℂ). Consider a reduced curve C of degree d in ℙ^{2} and the action of the group P GL(3) := P GL(3, ℂ) on the space ℙ^{N}, which parametrises plane curves of degree d. The orbit of a curve C is a quasi-projective variety of dimension dim P GL(3)− dim Stab_{P GL(3)}(C). For a general curve, the dimension of the orbit is 8, that is, its stabiliser is 0-dimensional.

Since the topological types S_{1}, …, S_{ν} of singularities of C and the degree d of C remain constant under the P GL(3)-action, we consider the virtual dimension of V (S_{1}, …, S_{ν}) at C defined by

virtdim(V (S_{1}, …, S_{ν}), C) := expdim(V (S_{1}, …, S_{ν}), C) − (8 − dim Stab_{P GL(3)}(C)). (9)

Curves with small orbits have been studied and classified by P. Allufi and C. Faber in [1]. According to this classification, C is always a configuration of rational curves. Moreover, C consists of irreducible components of the form below, with arbitrary multiplicities. We reproduce here their list together with the dimension of the stabiliser Stab_{P GL(3)}(C).

(1) C consists of a single line; dim Stab_{P GL(3)}(C) = 6.

(2) C consists of 2 (distinct) lines; dim Stab_{P GL(3)}(C) = 4.

(3) C consists of 3 or more concurrent lines; dim Stab_{P GL(3)}(C) = 3.

(4) C is a triangle (consisting of 3 lines in general position); dim Stab_{P GL(3)}(C) = 2.

(5) C consists of 3 or more concurrent lines, together with 1 other (non-concurrent) line; dim Stab_{P GL(3)}(C) = 1.

(6) C consists of a single conic; dim Stab_{P GL(3)}(C) = 3.

(7) C consists of a conic and a tangent line; dim Stab_{P GL(3)}(C) = 2.

(8) C consists of a conic and 2 (distinct) tangent lines; dim Stab_{P GL(3)}(C) = 1.

(9) C consists of a conic and a transversal line and may contain either one of the tangent lines at the 2 points of intersection or both of them; dim Stab_{P GL(3)}(C) = 1.

(10) C consists of 2 or more bitangent conics (conics in the pencil y^{2}+λxz) and may contain the line y through the two points of intersection as well as the lines x and/or z, tangent lines to the conics at the points of intersection; again, dim Stab_{P GL(3)}(C) = 1.

(11) C consists of 1 or more (irreducible) curves from the pencil y^{b}+λz^{a}x^{b−a}, with b ≥ 3, and may contain the lines x and/or y and/or z; dim Stab_{P GL(3)}(C) = 1.

(12) C contains 2 or more conics from a pencil through a conic and a double tangent line; it may also contain that tangent line. In this case, dim Stab_{P GL(3)}(C) = 1.

3.3. The Coolidge-Nagata conjecture. Let C_{1} and C_{2} be two curves in ℙ^{2}. The pairs (ℙ^{2}, C_{1}) and (ℙ^{2}, C_{2}) are birationally equivalent if there exist a birational map σ : ℙ^{2} −−→ ℙ^{2} such that the proper image of C_{1} by f coincides with C_{2}. Traditionally, the map σ is called a Cremona transformation.

Let π : V → ℙ^{2} be an embedded resolution of singularities of C, let C̄ be the strict transform of C by π and let K_{V} be the canonical divisor of V. One can show that h^{0}(m(K_{V} + C̄)) and the Kodaira dimension κ(K_{V} + C̄, V) are birational invariants of C as plane curves (see [18], [20]). Thus, one defines κ[C] := κ(K_{V} + C̄, V).

Let D be the reduced total preimage of C by the embedded resolution π. One can also show that h^{0}(m(K_{V} + D)) and the Kodaira dimension κ(K_{V} + D, V) are birational invariants of the open surface Y := ℙ^{2} \ C (see [18]). Thus one defines the logarithmic Kodaira dimension κ̄(Y) := κ(K_{V} + D, V) of Y.

One of the open problems regarding projective plane curves is the following famous Coolidge-Nagata problem/conjecture, cf. [2] and [26]: every rational cuspidal curve can be transformed by a Cremona transformation into a straight line.

In [2], J.I. Coolidge proved:

3.4. Theorem. A rational curve can be transformed into a straight line by a Cremona transformation if and only if all the conditions for special adjoints of any index are incompatible.

A special adjoint of index m is an effective divisor in the complete linear system |mK_{V} + C̄|. In fact one can check that κ[C] = −∞ if and only if C has no special adjoints. One has the following criterium due to N.M. Kumar and M.P. Murthy (cf. Corollary 2.4 in [20]), cf. also with Iitaka (Proposition 12 in [18]):

3.5. Theorem. Let C be an irreducible rational plane curve. The following conditions are equivalent:

a) the curve C can be transformed into a straight line by a Cremona transformation,

b) κ[C] = −∞,

c) |2K_{V} + C̄| = ∅,

d) |2(K_{V} + C̄)| = ∅. In [20] N.M. Kumar and M.P. Murthy also showed that a sufficient condition is C̄^{2} ≥ −3.

The Nagata-Coolidge problem has been solved for cuspidal rational plane curves with logarithmic Kodaira dimension of the complement κ̄(ℙ^{2} \ C) < 2 (using the classification listed in 5.2) and for all known curves with κ̄ = 2 (see below, cf. also with 5.8).

4. On rational curves

From now on we are interested in rational projective plane curves. Let C be a reduced rational projective plane curve of degree d in the complex projective plane with singular points {p_{j}}_{j=1}^{ν}. Let S_{j} be the topological type of the singularity (C, p_{j}).

The main theme of this section is Orevkov’s conjecture [37]. Since this conjecture is not true for a small number of curves (with positive dimensional stabiliser Stab_{P GL(3)}(C), all of them with κ̄(Y) < 2), we correct the original version by adding a contribution provided by this stabiliser; and we also present some equivalent reformulations.

4.1. Virtual dimension conjecture. The virtual dimension virtdim(V (S_{1}, …, S_{ν}), C) of reduced rational projective plane curve is non-negative:

expdim(V (S_{1}, …, S_{ν}), C) − 8 + dim Stab_{P GL(3)}(C)) ≥ 0.  (10)

This version was also conjectured independently by I. Luengo.

Since C is rational, if the multiplicity sequence of the singularity (C, p) is [m_{1}^{p}, …, m_{k}^{p}], then (the genus-formula reads as):

(d − 1)(d − 2) = ∑_{p∈Sing(C)} ∑_{i=1}^{k} m_{i}^{p}(m_{i}^{p} − 1) = ∑_{p∈Sing(C)} (∑_{i=1}^{k}(m_{i}^{p})^{2} − ∑_{i=1}^{k} m_{i}^{p}) .  (11)

Eliminating d^{2} from (9) and (5), one gets

virtual dim = 3d − 9 − ∑_{p∈Sing(C)} ∑_{i=1}^{k} m_{i}^{p} + ∑_{P ∈Sing(C)} L_{p} + dim Stab_{P GL(3)}(C).

Substituting (7) in this equality, one gets

virtual dim = 3d − 9 − ∑_{p∈Sing(C)} M̄(C, p) + dim Stab_{P GL(3)}(C).  (12)

In particular, (10) is equivalent to

3d − 9 − ∑_{p∈Sing(C)} M̄(C, p) + dim Stab_{P GL(3)}(C) ≥ 0.  (13)

4.2. Orevkov’s conjecture. S. Orevkov in [37] conjectured the following inequality.

For a rational cuspidal curve

∑_{p∈Sing(C)} M̄(C, p) ≤ 3d − 9.  (14)

Orevkov’s conjecture is stated for any rational curve (without restrictions). In such a case, the sum is over all irreducible branches at each singular point. Note that (14) is not true for the curve C defined by x^{2}y + z^{3} = 0 since M̄(C, p) = 1 at its singular point. Nevertheless, dim Stab_{P GL(3)}(C) = 1 and (10) and (13) hold. We believe that the correct statement of the conjecture is (10), or equivalently (13). Nevertheless, one can prove (cf. Lemma 5.3) that for irreducible, cuspidal, rational plane curve with with κ̄(Y) = 2 the statements of (10) and (14) are equivalent.

4.3. The C̄^{2}–Conjecture. Let π : V → ℙ^{2} be the minimal good embedded resolution of C ⊂ ℙ^{2}, and let C̄ be the strict transform of C and D = π^{−1}(C) be the reduced preimage of C as above. One of the integers which plays a special role in the classification problem is the self-intersection of C̄ in V. It equals

C̄^{2} = d^{2} − ∑_{p∈Sing(C)} ∑_{i=1}^{k}(m_{i}^{p})^{2}.  (15)

From (10) and (15) one gets

3d = C̄^{2} + 2 + ∑_{p∈Sing(C)} ∑_{i=1}^{k} m_{i}^{p}.

Thus, via (12):

virtual dim = C̄^{2} − 7 + ∑_{p∈Sing(C)} L_{p} + dim Stab_{P GL(3)}(C).  (16)

Therefore, inequality (10) is equivalent to:

C̄^{2} − 7 + ∑_{p∈Sing(C)} L_{p} + dim Stab_{P GL(3)}(C) ≥ 0.  (17)

5. Cuspidal rational curves and the rigidity conjecture.

Let C be an irreducible curve of degree d in the complex projective plane. One of the main invariants of such curves is the logarithmic Kodaira dimension κ̄ = κ̄(Y), where Y := ℙ^{2}\C. The following result of Wakabayashi [53] is crucial in the classification procedure.

5.1. Theorem. [53] Let C be an irreducible curve of degree d in ℙ^{2}.

(1) If g(C) ≥ 1 and d ≥ 4 then κ̄(ℙ^{2} \ C) = 2.

(2) If g(C) = 0 and C has at least 3 cuspidal points then κ̄(ℙ^{2} \ C) = 2.

(3) If g(C) = 0 and C has at least 2 singular points and one of them is locally reducible then κ̄(ℙ^{2} \ C) = 2.

(4) If g(C) = 0 and C has 2 cuspidal points then κ̄(ℙ^{2} \ C) ≥ 0.

5.2. Recall that the open surface Y := ℙ^{2} \ C is ℚ-acyclic if and only if C is a rational cuspidal curve. From now on we assume that C is a rational cuspidal plane curve of degree d, in particular C is irreducible. Let (C, p_{i})_{i=1}^{ν} be the collection of local plane curve singularities, all of them locally irreducible.

(a) If κ̄ = −∞ then ν = 1 by [53]. Moreover, all these curves are classified by M. Miyanishi and T. Sugie [25] (see also H. Kashiwara [19]). The family contains as an important subfamily the Abhyankar-Moh-Suzuki curves.

(b) The case κ̄ = 0 cannot occur by a result of Sh. Tsunoda [48], see also Orevkov’s article [37].

(c) If κ̄ = 1 then by the above result of Wakabayashi [53] one has ν ≤ 2. In the case ν = 1, K. Tono provides the possible equations of the curves [46]. (Notice that Tsunoda’s classification in [49] is incomplete.). On the other hand, by another result of Tono [45], the case ν = 2 corresponds exactly to the Lin-Zaidenberg bicuspidal rational plane curves.

5.3. Lemma. If C is an irreducible, cuspidal, rational plane curve with with κ̄(Y) = 2 then dim Stab_{P GL(3)}(C) = 0.

Proof. According with the above classification of Aluffi and Faber, C is a rational cuspidal plane curve with dim Stab_{P GL(3)}(C) > 0 if and only if C is a generic member of the pencil y^{d} + λz^{a}x^{d−a} with d ≥ 3 and (d, a) = 1. If a = d − 1 then C is of Abhyankar-Moh-Suzuki type (with κ̄ = −∞). Otherwise a ≠ d−1 and C is of Lin-Zaidenberg type (with κ̄ = 1). □

5.4. From a different point of view, one can classify triples (d, a, b) such that there exists a unicuspidal rational plane curve C of degree d whose singularity has only one Puiseux pair of type (a, b), where 1 < a < b. Let {φ_{j}}_{j≥0} denote the Fibonacci numbers φ_{0} = 0, φ_{1} = 1, φ_{j+2} = φ_{j+1} + φ_{j}.

5.5. Theorem. [11] The Puiseux pair (a, b) can be realized by a unicuspidal rational curve of degree d if and only if (d, a, b) appears in the following list.

(a) (a, b) = (d − 1, d);

(b) (a, b) = (d/2, 2d − 1), where d 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.

In the first four cases κ̄ = −∞ and they can be realized by some particular curves which appear in Kashiwara’s classification [19]. The last two sporadic cases have κ̄ = 2 and were found by Orevkov and Artal-Bartolo, cf. [37].

5.6. The rigidity conjecture of Flenner and Zaidenberg. Let Y be an ℚ-acyclic affine surface, and fix one of its ‘minimal logarithmic compactifications’ (V, D). This means that V is a smooth projective surface with a normal crossing divisor D, such that Y = V \D, and (V, D) is minimal with these properties.

The sheaf of the logarithmic tangent vectors Θ_{V} ⟨D⟩ controls the deformation theory of the pair (V, D), cf. [7]. E.g., H^{0}(V, Θ_{V} ⟨D⟩) is the set of infinitesimal automorphisms, H^{1}(V, Θ_{V} ⟨D⟩) is the space of infinitesimal deformations, and H^{2}(V, Θ_{V} ⟨D⟩) is the space of obstructions. Iitaka showed in [16] that if κ̄(Y) = 2 then the automorphism group of the surface Y is finite (this also provides a different proof of 5.3). Therefore h^{0}(Θ_{V} ⟨D⟩) = 0. In [7, 8, 57] Flenner and Zaidenberg conjectured the following

Rigidity conjecture: Every ℚ-acyclic affine surfaces Y with logarithmic Kodaira dimension κ̄(Y) = 2 is rigid. That is,

h^{1}(Θ_{V} ⟨D⟩) = 0  and  h^{2}(Θ_{V} ⟨D⟩) = 0.  (18)

In particular, the Euler characteristic χ(Θ_{V} ⟨D⟩) = h^{2}(Θ_{V} ⟨D⟩) − h^{1}(Θ_{V} ⟨D⟩) must vanish.

This can be applied in our situation as follows. Consider a projective curves C, and write Y := ℙ^{2} \ C. The ℚ-acyclicity of Y is equivalent to the fact that C is rational and cuspidal. For V one can take the minimal embedded resolution of the pair (ℙ^{2}, C).

The conjecture for Y = ℙ^{2} \ C implies the projective rigidity of the curve C. This means that every equisingular deformation of C in ℙ^{2} would be projectively equivalent to C. Thus V (S_{1}, …, S_{ν}) has expected dimension 8.

In Corollary 2.5 of [8], Flenner and Zaidenberg show that for any cuspidal rational plane curve

χ(Θ_{V} ⟨D⟩) = K_{V} (K_{V} + D) = −3(d − 3) + ∑_{p∈Sing(C)} M̄(C, p).  (19)

By (12) and Lemma 5.3 then

virtual dim = −χ(Θ_{V} ⟨D⟩).  (20)

The vanishing of χ(Θ_{V} ⟨D⟩) implies any of the equivalent equalities (10), (13) or (17). On the other hand, if (10), (13) or (17) hold, then

χ(Θ_{V} ⟨D⟩) ≤ 0.  (21)

5.7. Proposition. [Tono] For cuspidal rational plane curves with κ̄ = 2 the following inequality holds

χ(Θ_{V} ⟨D⟩) ≥ 0.  (22)

Proof. (22) follows from the article [47] of K. Tono in the following way. F. Sakai in [41] introduce the invariant γ_{2} := h^{0}(2K_{V} + D). Lemma 4.1 in [47] states that if the pair (V, D) satisfies the following three conditions (for details see [loc. cit.])

(A1) κ̄(V \ D) = 2,

(A2) (V, D) is almost minimal, and

(A3) D contains neither a rod consisting of (−2)-curves nor a fork consisting of (−2)- curves, then

γ_{2} = K_{V} (K_{V} + D) + \frac{D(D + K_{V})}{2} + χ(O_{V}).

(The main point here is that by a vanishing theorem h^{1}(2K_{V} +D) = 0, by an easy argument h^{2}(2K_{V} + D) = 0 too, hence γ_{2} = χ(2K_{V} + D) can be computed by Riemann-Roch.)

One can check that in our case the minimal embedded resolution satisfies these conditions. Moreover, χ(O_{V}) = 1 and (since D is a rational tree, the adjunction formula implies) K_{V} D + D^{2} = −2. Thus γ_{2} = K_{V} (K_{V} + D). Therefore, via (19), one has:

χ(Θ_{V} ⟨D⟩) = h^{0}(2K_{V} + D) ≥ 0.
□

5.8. Corollary. Let C be an irreducible, cuspidal, rational plane curve with κ̄(ℙ^{2}−C) = 2. The following conditions are equivalent:

(i) χ(Θ_{V} ⟨D⟩) = 0, (ii) virtdim(V (S_{1}, …, S_{ν}), C) ≥ 0, i.e. (10) holds, where S_{j} is the topological type of

the corresponding uni-branch singularity (C, p_{j}). (iii) χ(Θ_{V} ⟨D⟩) ≤ 0.

In such a case, the curve C can be transformed by a Cremona transformation of ℙ^{2} into a straight line (i.e., the Coolidge-Nagata problem has a positive answer).

Proof. (i) ⇒ (ii) ⇒ (iii) follows from (20) and (21). (iii) ⇒ (i) follows from 5.7 and (21). Finally, the characterisation 3.5 shows that C can be transform into a straight line by a Cremona transformation. Indeed, h^{0}(2K_{V} + D) = χ(Θ_{V} ⟨D⟩) = 0, but O_{V} (2K_{V} + C̄) is a subsheaf of O_{V} (2K_{V} + D), hence h^{0}(2K_{V} + C̄) = 0 as well. □

6. The semigroup distribution property

6.1. The characterisation problem of 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. (For a — non-complete — list of some of these restrictions, see e.g. [10, 11].)

In [10] we proposed a new compatibility property — valid for rational cuspidal curves C. 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 polynomial 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.

6.2. Conjecture A. [10] 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 then

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

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. We also mention that examples with strict inequality may occur, cf. [10] (in all these examples known by the authors ν > 1).

The main result of [10] is :

6.3. Theorem. [10] If the logarithmic Kodaira dimension κ̄ := κ̄(ℙ^{2} \ C) is ≤ 1, then the above conjecture A is true (in fact with n_{l} = 0).

There is an additional surprising phenomenon in the above conjecture. Namely, in the unicuspidal case one can show the following.

6.4. Proposition. [10] If ν = 1 then c_{l} ≥ (l + 1)(l + 2)/2 for 0 ≤ l ≤ d − 3.

Therefore, conjecture 6.2 in this case can be reformulated as follows:

6.5. Conjecture B1. With the notations of 6.2, if ν = 1, then

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

In fact, if ν = 1, we can do more. 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. In the next discussion we will replace Δ by Γ_{(C,p)}. Recall that the semigroup Γ_{(C,p)} ⊂ ℕ consists of all the possible intersection multiplicities I_{p}(C, h) at the point p for all h ∈ O_{(ℂ^{2},p)}.

Hence, one can reformulate conjecture B1 in the language of the semigroup of the germ (C, p) and the degree d. It turns out that the collection of vanishings of all the coefficients n_{l} is replaced by a very precise and mysterious distribution of the elements of the semigroup with respect to the intervals I_{l} := ( (l − 1)d, ld]:

6.6. Conjecture B2. Assume that ν = 1. Then for any l > 0, the interval I_{l} contains exactly min{l + 1, d} elements from the semigroup.

In other words, for every rational unicuspidal plane curve C of degree d, the above conjecture is equivalent with the conjectural identity

D(t) ≡ 0,  (DP)

where:

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

For an explanation of the equivalences of conjectures B1 and B2, see 7.5. Here we only mention a key connection between the coefficients c_{l} and the semigroup Γ_{(C,p)}.

First, consider the identity (cf. [14]) Δ(t) = (1 − t) · L(t), where L(t) = ∑_{k∈Γ_{(C,p)}} t^{k} is the Poincaré series of Γ_{(C,p)}. Write Δ(t) = 1 − P (t)(1 − t) for some polynomial P (t), then L(t) + P (t) = 1/(1 − t) = ∑_{k≥0} t^{k}. In particular, P (t) = ∑_{k∈ℕ\Γ_{(C,p)}} t^{k}. Then

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

Hence c_{l} = #{k ∉ Γ_{(C,p)} : k > (d−3−l)d}. Since k ∈ Γ_{(C,0)} if and only if μ−1−k ∉ Γ_{(C,p)} for any 0 ≤ k ≤ μ − 1, one gets c_{l} = #{k ∈ Γ_{(C,p)} ; k ≤ ld}.

6.7. The following equivalent formulation was suggested to us by A. Campillo.

6.8. Theorem. Let C be a unicuspidal rational plane curve of degree d. The curve C satisfies the semigroup compatibility property (DP) (i.e. conjectures B1 and/or B2) if and only if the elements of the semigroup Γ_{(C,p)} in [0, ld] are realized by projective (possibly non-reduced) curves of degree l for l ≤ d − 3.

Proof. The proof of the ‘if’ part is easy. For the ‘only if’ part fix a projective coordinate system [X : Y : Z] such that the affine chart Z ≠ 0 contains the singular point p. Let V be the vector space of polynomials of degree l in variables (X/Z, Y/Z). Its dimension is N := (l + 1)(l + 2)/2, which, in fact, equals the number of elements of the semigroup in the interval [0, ld]. Let these semigroup elements be denoted by 0 = s_{1}, ..., s_{N}, ordered in an increasing way.

Consider the decreasing filtration V_{1} ⊃ V_{2} ⊃ ⋯ ⊃ V_{N}, defined by

V_{i} := {f ∈ V : I_{p}(C, f) ≥ s_{i}}.

First, we verify that dim(V_{i}/V_{i+1}) is at most 1. Indeed, assume that I_{p}(C, f_{i}) = I_{p}(C, f_{2}) = I. Let n : (ℂ, 0) → (C, p) be the normalisation of (C, p), and write f_{i} ∘ n(t) = a_{i}t^{I} + ⋯ with a_{i} ≠ 0, for i = 1 and 2. Then I_{p}(C, a_{2}f_{1} − a_{1}f_{2}) > I. Since there is no semigroup element between s_{i} and s_{i+1}, the inequality dim(V_{i}/V_{i+1}) ≤ 1 follows.

Next, notice that to prove the theorem it is enough to show that each dimension dim(V_{i}/V_{i+1}) is exactly 1.

But, if dim(V_{i}/V_{i+1}) = 0 for some i then dim(V_{N}) is at least 2. Since for any f ∈ V_{N} one has I_{p}(C, f) ≥ s_{N}, dim(V_{N}) ≥ 2 would imply (by similar argument as above) the existence of an f ∈ V_{N} with I_{p}(C, f) > s_{N}. Since I_{p}(C, f) is an element of the semigroup and the last element of the semigroup in the interval [0, ld] is s_{N}, we get that I_{p}(C, f) > ld, which contradicts the irreducibility of C by Bézout Theorem. □

6.9. A counterexample to an ‘extended’ version. In [10] we formulated the following conjecture, as an extension of the conjecture 6.6 to an ‘if and only if’ statement.

6.10. ‘Conjecture’ C. 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 (DP) is valid.

In the sequel we present a counterexample to the ‘if’ part (i.e. to the ‘extension’).

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} ≥ 1 (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 [4]:

[figure: C̄; a_{1}; a_{2}; a_{g−1}; a_{g}; p_{1}; p_{2}; p_{g−1}; p_{g}; ⋯]

Consider now the local singularity whose Eisenbud-Neumann splice diagram is decorated by two pairs (p_{1}, a_{1}) = (2, 7) and (p_{2}, a_{2}) = (4, 73). A local equation for such singularity can be (x^{2} − y^{7})^{4} + x^{33}y = 0. Its multiplicity sequence is [8_{3}, 4_{6}, 1_{4}]. A minimal set of generators of its semigroup Γ_{(C,p)} is given by ⟨8, 28, 73⟩. Its Milnor number is 16 · 15, hence a possible unicuspidal plane curve C of degree 17 might exist with such local singularity. Moreover the distribution property (DP) of the semigroup is also satisfied. Nevertheless, such a curve C does not exist. To prove this, one can either use Cremona transformations to transform C into another curve for which one sees that it does not exist, or one uses Varchenko’s semi-continuity criterium for the spectrum of the singularity [51, 52]. Here we will follow the second argument.

The spectrum of the irreducible singularity (C, 0) can be computed from the Newton pairs of the singularity. The forth author provided such a formula in [27]. It is convenient to consider the spectrum Sp(C, 0) = ∑_{r} n_{r}(r) as an element of ℤ[ℚ ∩ (0, 2)]. We write Sp_{(0,1)}(C, 0) for the collection of spectral elements situated in the interval (0, 1).

6.10.1. Theorem. If the irreducible germ (C, 0) has g Newton pairs {(p_{k}, q_{k})}_{k=1}^{g} then

Sp_{(0,1)}(C, 0) = ∑_{k=1}^{g} S_{k} where S_{k} = ∑ (\frac{i/a_{k} + j/p_{k} + t}{p_{k+1}p_{k+2} ⋯ p_{g}}) ,

where the second sum is over 0 < i < a_{k}, 0 < j < p_{k}, i/a_{k} + j/p_{k} < 1 and 0 ≤ t ≤ p_{k+1}p_{k+2} ⋯ p_{g} − 1 (if k = g then S_{g} = ∑(l/a_{g} + k/p_{g}) where the sum is over 0 < l < a_{g}, 0 < k < p_{g}, l/a_{g} + k/p_{g} < 1).

If the local singular type {(C, p)} can be realized by a degree d plane curve C, then (C, p) 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(C, p) of the local plane curve singularity (C, p) satisfies the semi-continuity property compared with the spectral numbers of (U, 0) for any interval (α, α+1). Since the spectral numbers of (U, 0) are of type l/d, the semi-continuity property for intervals (−1 + l/d, l/d) (l = 2, 3, …, d − 1) reads as follows:

#{α ∈ Sp(C, p) : α < l/d} ≤ (l − 2)(l − 1)/2.  (23)

In our case, for d = 17 and l = 12, using Theorem 6.10.1 we get

#{α ∈ Sp(C, p) : α < 12/17} − (12 − 2)(12 − 1)/2 = 1,

which contradicts to (23). Thus the rational unicuspidal plane curve C of degree 17 with such singularity cannot exist.

Although, in the realization problem of the above case (p_{1}, a_{1}; p_{2}, a_{2}; d) cannot be eliminated by the semigroup distribution property (DP), but can be by the semi-continuity of the spectrum, it is not true that the semi-continuity implies (DP). For a more precise discussion see [10].

7. The semigroup compatibility property and surface singularities

7.1. Superisolated singularities. The theory of normal surface singularities (in fact, of isolated hypersurface surface singularities) ‘contains’ in a canonical way the theory of complex projective plane curves via the family of superisolated singularities. These singularities were introduced by the second author in [22]. A hypersurface singularity f : (ℂ^{3}, 0) → (ℂ, 0), f = f_{d} + l^{d+1} (where f_{d} is homogeneous of degree d and l is linear) is superisolated if the projective plane curve C := {f_{d} = 0} ⊂ ℙ^{2} is reduced, and its singularities {p_{i}}_{i=1}^{ν} are not situated on {l = 0}. The equisingular type of f depends only on f_{d}, i.e. only on the projective curve C ⊂ ℙ^{2}. In particular, all the invariants (of the equisingular type) of f can be determined from the invariants of the pair (ℙ^{2}, C). In order to exemplify this, we will use the above notation for C, namely: let μ_{i} and Δ_{i} be the Milnor number and the characteristic polynomial of the local plane curve singularities (C, p_{i}). Set 2δ := ∑_{i} μ_{i}, Δ := ∏_{i} Δ_{i}, and Δ̄(t) := t^{−δ}Δ(t). We assume that C is irreducible.

In the next discussion we follow [23, 10]. There is a standard procedure which provides the plumbing graph of the link M of f from the embedded resolution graphs of (C, p_{i})’s and the integer d. The point is that the link M is rational homology sphere if and only if C is rational and cuspidal. In this section, we will assume that these conditions are satisfied. Let (V, D) be the minimal embedded resolution of the pair (ℙ^{2}, C) as above. The minimal plumbing graph of M (or, equivalently, the minimal good resolution graph of {f = 0}) can be obtained from the dual graph of D by decreasing the decoration (self-intersection) of C̄ by d(d + 1). In the language of topologists, if C is unicuspidal (ν = 1), then M = S_{−d}^{3}(K) (i.e. M is obtained via surgery of the 3-sphere S^{3} along K with surgery coefficient −d), where K ⊂ S^{3} is the local knot of (C, p). One can also verify that H_{1}(M, ℤ) = ℤ_{d}.

Another topological invariant of f is the following. Let Z → ({f = 0}, 0) be the minimal good resolution, K_{Z} be the canonical divisor of Z and # the number of irreducible exceptional divisor (which equals the number of irreducible divisors of D). Then K_{Z}^{2} + # is a well-defined invariant of f, which, in fact, can be computed from the link M (or, from its graph) as well. In our case, surprisingly, in this invariant of the link M all the information about the local types (C, p_{i}) are lost: K_{Z}^{2} + # = 1 − d(d − 2)^{2}, it depends only on d.

The same is true about the Euler characteristic χ(F), or the signature σ(F) of the Milnor fiber F of f, or about the geometric genus p_{g} of f. In fact, it is well-known that for any hypersurface singularity, any of p_{g}, σ(F) and χ(F) determines the remaining two modulo K_{Z}^{2} + #. E.g., one has the relation:

8p_{g} + σ(F) + K_{Z}^{2} + # = 0.  (24)

In our case, for the superisolated singularity f, one has p_{g} = d(d − 1)(d − 2)/6, hence the smoothing invariants χ(F) and σ(F) depend only on the degree d.

7.2. For a normal surface singularity with rational homology sphere link (and with some additional analytic restriction, e.g. complete intersection or Gorenstein property) there is a subtle connection between the Seiberg-Witten invariants of its link M and some analytic/smoothing invariants. The hope is that the geometric genus (or, equivalently, χ(F) or σ(F), see (24) and the discussion nearby), can be recovered from the link. The starting point is an earlier conjecture of Neumann and Wahl [35]:

7.2.1. For any isolated complete intersection whose link M is integral homology sphere, σ(F) is exactly eight times the Casson invariant λ(M) of the link.

Notice that the link of a hypersurface superisolated singularity is never an integral homology sphere. The generalised conjecture, valid for rational homology spheres (Conjecture SW C below) was proposed by the forth author in a joint work with L. Nicolaescu in [32] involving the Seiberg-Witten invariant of the link. It was verified for rather large number of non-trivial special families (rational and elliptic singularities, suspension hypersurface singularities f(x, y)+z^{n} with f irreducible, singularities with good ℂ^{∗} action) [32, 33, 34, 28, 31]. But the last three authors of the present article in [23] have shown that the conjecture fails at that high generality how it was originally stated. The counterexamples were provided exactly by superisolated singularities and/or their universal abelian covers. Nevertheless, in the next paragraph we will recall this conjecture (in its original form), since this have guided us to the semigroup compatibility property, and we believe that it hides a deep mathematical substance (even if at this moment it is not clear for what family we should expect its validity).

Let sw_{M}(can) be the Seiberg-Witten invariant of the link M associated with the canonical spin^{c} structure (this is induced by the complex structure of {f = 0} \ {0}, and it can be identified combinatorially from the graph of M; in this article we will not discuss the invariants associated with the other spin^{c} structures).

7.3. ‘Conjecture’ SWC. [32] For a ℚ-Gorenstein surface singularity whose link M is a rational homology sphere one has

sw_{M}(can) − (K_{Z}^{2} + #)/8 = p_{g}.

In particular, if the singularity is Gorenstein and admits a smoothing, then −sw_{M}(can) = σ(F)/8 (cf. (24)).

If M is an integral homology sphere then sw_{M}(can) = −λ(M). If M is a rational homology sphere then by a result of Nicolaescu [36], sw_{M}(can) = 𝒯_{M} − λ(M)/|H_{1}(M, ℤ)|, where λ(M) is the Casson-Walker invariant of M (normalised as in [21]), and 𝒯_{M} denotes the sign refined Reidemeister-Turaev torsion (associated with the canonical spin^{c} structure) [50].

7.4. In our present situation, when M is the link of the superisolated singularity f, one shows, cf. [23] (using the notations of 7.1), that

𝒯_{M} = \frac{1}{d} ∑_{ξ^{d}=1≠ξ} \frac{Δ(ξ)}{(ξ − 1)^{2}} and λ(M) = −\frac{Δ̄(t)′′(1)}{2} + \frac{(d − 1)(d − 2)}{24}. (25)

Therefore, since p_{g} and K_{Z}^{2} + # depend only on d, the SW C imposes serious restriction on the local invariant Δ. This condition, for some cases when the number of singular points of C is ≥ 2, is not satisfied (hence SW C fails, cf. [23]); nevertheless, as we will see, the SW C identity in the unicuspidal case is equivalent with Conjecture B2 of section 6 about the distribution property of the semigroup. In order to explain this, let us assume that C is unicuspidal, and consider (motivated by (25))

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

Similarly,

N(t) := ∑_{l=0}^{d−3} (c_{l} − \frac{(l + 1)(l + 2)}{2})t^{d−3−l}; and D(t) := ∑_{k∈Γ_{(C,p)}} t^{⌈k/d⌉} − \frac{1 − t^{d}}{(1 − t)^{2}} .

Notice that this D(t) agrees with the one defined in (DP), section 6. In [10] the following facts are verified:

R(t) = D(t^{d})/(1 − t^{d}) = N(t^{d}).  (26)

N(t) (hence R(t) too) has non-negative coefficients.  (27)
R(1) = sw_{M}(can) − \frac{K^{2} + #}{8} − p_{g}.  (28)

Therefore, in this case, we have the equivalence of the ‘Seiberg-Witten invariant conjecture’ with the ‘semigroup distribution property’:

7.5. Theorem. Assume that ν = 1. Then the following facts are equivalent: (a) R(1) = 0, i.e. Conjecture SW C (7.3) is true;

(b) R(t) ≡ 0;

(c) N(t) ≡ 0, i.e. Conjecture B1 (6.5) is true;

(d) D(t) ≡ 0, i.e. Conjecture B2 (6.6) is true.

8. The semigroup distribution property and Heegaard Floer homology

8.1. The presentation of this section is based on some recent results of the forth author in [28, 29, 30]. In the sequel we assume that C is unicupidal, and we keep the notations of the previous section.

There is another way to compute the Seiberg-Witten invariant of the link M via its Heegaard-Floer homology. For any oriented rational homology 3-sphere M the Heegaard Floer homology HF^{+}(M) was introduced by Ozsváth and Szabó in [39] (cf. also with their long list of articles). HF^{+}(M) is a ℤ[U]-module with compatible ℚ-grading. Moreover, HF^{+}(M) has a natural direct sum decomposition (compatible with the ℚ-grading) corresponding to the spin^{c}-structures of M: In this article we write HF^{+}(M, can) for the Heegaard-Floer homology associated with the canonical spin^{c} structure.

For some (negative definite) plumbed rational homology 3-spheres M, one can compute the Heegaard Floer homology of HF^{+}(M, can) of M (equivalently, of −M) in a purely combinatorial way from the plumbing graph G. This will be true for all the 3-manifolds discussed in this section. This is done via some intermediate objects, the graded root associated with G (in fact, one has a graded root corresponding to each spin^{c}-structure of M, but here we will discuss only the ‘canonical’ one). The theory of graded roots, from the point of view of singularity theory, is rather interesting by itself, and we plan to exploit this connection in the future more.

Next, we provide a short presentation of abstract graded roots (cf. [28]).

8.2. Definition of the ‘abstract graded root’ (R, χ). Let R be an infinite tree with vertices V and edges E. We denote by [u, v] the edge with end-points u and v. We say that R is a graded root with grading χ : V → ℤ if

(a) χ(u) − χ(v) = ±1 for any [u, v] ∈ E;

(b) χ(u) > min{χ(v), χ(w)} for any [u, v], [u, w] ∈ E;

(c) χ is bounded below, χ^{−1}(n) is finite for any n ∈ ℤ, and #χ^{−1}(n) = 1 if n ≫ 0.

8.3. Examples. (1) For any integer n ∈ ℤ, let R_{n} be the tree with V = {v^{k}}_{k≥n} and E = {[v^{k}, v^{k+1}]}_{k≥n}. The grading is χ(v^{k}) = k.

(2) Let I be a finite index set. For each i ∈ I fix an integer n_{i} ∈ ℤ; and for each pair i, j ∈ I fix n_{ij} = n_{ji} ∈ ℤ with the next properties: (i) n_{ii} = n_{i}; (ii) n_{ij} ≥ max{n_{i}, n_{j}}; and (iii) n_{jk} ≤ max{n_{ij}, n_{ik}} for any i, j, k ∈ I. For any i ∈ I consider R_{n_{i}} with vertices {v_{i}^{k}} and edges {[v_{i}^{k}, v_{i}^{k+1}]}, (k ≥ n_{i}). In the disjoint union ∐_{i} R_{n_{i}}, for any pair (i, j), identify v_{i}^{k} and v_{j}^{k}, resp. [v_{i}^{k}, v_{i}^{k+1}] and [v_{j}^{k}, v_{j}^{k+1}], whenever k ≥ n_{ij}, and take the induced χ.

(3) Any map τ : {0, 1, …, r} → ℤ produces a starting data for construction (2). Indeed, set I = {0, …, r}, n_{i} := τ(i) (i ∈ I), and n_{ij} := max{n_{k} : i ≤ k ≤ j} for i ≤ j. Then the root constructed in (2) using this data will be denoted by (R_{τ} , χ_{τ}).

8.3.1. Here are two (typical) graded roots (cf. with 8.11):

[figure: 0; −1; −2; −3; −4; −5; Σ(5, 5, 6); S_{−5}^{3}(T_{3,7}); S_{−5}^{3}(T_{2,13}), S_{−5}^{3}(T_{4,5}); χ]

8.4. The canonical graded root (R, χ) of M. [28] Next, we define for any (negative definite, plumbed) rational homology sphere M a graded root.

We fix a plumbing graph G and denote by L the corresponding lattice: the free ℤ-module of rank # with fixed basis {A_{j}}_{j}, and bilinear form (A_{i}, A_{j})_{i,j}. (In our case, a possible choice is the dual resolution graph and the corresponding intersection form associated with the minimal good resolution Z → ({f = 0}, 0).) Set L′ = Hom_{ℤ}(L, ℤ) ⊂ L ⊗ ℚ. Let K_{Z} ∈ L′ be the canonical cycle defined by K_{Z}(A_{j}) + A_{j}^{2} + 2 = 0 for any j. Then define χ : L → ℤ by (the Riemann-Roch formula) χ(x) := −(K_{Z}(x) + x^{2})/2.

The definition of the graded root captures the position of the lattice points in the different ellipsoids χ^{−1}(n). For any n ∈ ℤ, one constructs a finite 1-dimensional simplicial complex L̄_{≤n} as follows. Its 0-skeleton is L_{≤n} := {x ∈ L : χ(x) ≤ n}. For each x and j, with both x and x + A_{j} ∈ L_{≤n}, we consider a unique 1-simplex with endpoints at x and x + A_{j} (e.g., the segment [x, x + A_{j}] in L ⊗ ℝ). We denote the set of connected components of L̄_{≤n} by π_{0}(L̄_{≤n}). For any v ∈ π_{0}(L̄_{≤n}), let C_{v} be the corresponding connected component of L̄_{≤n}.

Next, we define (R, χ) as follows. The vertices V(R) are ∪_{n∈ℤ}π_{0}(L̄_{≤n}). The grading V(R) → ℤ, still denoted by χ, is χ|π_{0}(L̄_{≤n}) = n. If v_{n} ∈ π_{0}(L̄_{≤n}), and v_{n+1} ∈ π_{0}(L̄_{≤n+1}), and C_{v_{n}} ⊂ C_{v_{n+1}}, then [v_{n}, v_{n+1}] is an edge of R. All the edges are obtained in this way.

8.5. Example. [29] Recall that the link of the superisolated singularity f (where C is rational and unicuspidal of degree d) is the surgery manifold S_{−d}^{3}(K), where K ⊂ S^{3} is the local knot of (C, p). The graded root of M can be represented by a function τ as in 8.3(3) associated with the Alexander polynomial Δ of K ⊂ S^{3}. Similarly as in section 6, set μ = 2δ for the degree of Δ (which equals (d−1)(d−2)), and write Δ(t) as 1+δ(t−1)+(t−1)^{2}Q(t) for some polynomial Q(t) = ∑_{i=0}^{μ−2} α_{i}t^{i}. Set c_{l} := α_{(d−3−l)d} (cf. with 6.1). Then define τ : {0, 1, …, 2d − 4} → ℤ by

τ(2l) = \frac{l(l−1)}{2}d − l(δ − 1), τ(2l + 1) = τ(2l + 2) + c_{d−3−l}.

Then (R, χ) = (R_{τ} , χ_{τ}).

8.6. Example. Let Σ(d, d, d + 1) be the Seifert 3-manifold (d, d, d + 1); equivalently, the link of the Brieskorn singularity x^{d} + y^{d} + z^{d+1} = 0. Its graded root also can be represented by the ‘τ-construction’ (for the more general situation of Seifert manifolds, see [28]).

For any 0 ≤ l ≤ d − 3 define c_{l}^{u} := (l + 1)(l + 2)/2, and 2δ := (d − 1)(d − 2). Then define τ^{u} : {0, 1, …, 2d − 4} → ℤ by

τ^{u}(2l) = \frac{l(l−1)}{2}d − l(δ − 1), τ^{u}(2l + 1) = τ^{u}(2l + 2) + c_{d−3−l}^{u}.

Then (R, χ) = (R_{τ^{u}}, χ_{τ^{u}}).

Notice the shocking similarities of 8.5 and 8.6: the graded roots associated with S_{−d}^{3}(K) and Σ(d, d, d + 1) coincide exactly when c_{l} = c_{l}^{u} for all l.

To any graded root, one can associate a natural graded ℤ[U]-module.

8.7. Definition. The ℤ[U]-module associated with a graded root. Consider the ℤ[U]-module ℤ[U, U^{−1}], and (following [40]) denote by T_{0}^{+} its quotient by the submodule U · ℤ[U]. It is a ℤ[U]-module with grading deg(U^{−h}) = 2h.

Now, fix a graded root (R, χ). Let ℍ(R, χ) be the set of functions ϕ : V → T_{0}^{+} with the property that whenever [v, w] ∈ E with χ(v) < χ(w) one has U ·ϕ(v) = ϕ(w). Then ℍ(R, χ) is a ℤ[U]-module via (Uϕ)(v) = U · ϕ(v). Moreover, ℍ(R, χ) has a grading: ϕ ∈ ℍ(R, χ) is homogeneous of degree h ∈ ℤ if for each v ∈ V with ϕ(v) ≠ 0, ϕ(v) ∈ T_{0}^{+} is homogeneous of degree h − 2χ(v).

In the sequel, the following notation is useful: If P is a ℚ-graded ℤ[U]-module with hhomogeneous elements P_{h}, then for any r ∈ ℚ we denote by P [r] the same module graded in such a way that P [r]_{h+r} = P_{h}.

8.8. Theorem. [28, 40] Assume that M is either S_{−d}^{3}(K) or Σ(d, d, d + 1). Then

HF^{+}(−M, can) = ℍ(R, χ)[−(K_{Z}^{2} + #)/4].

In other words, for these 3-manifolds, the Heegaard-Floer homology can be recovered from the graded root via ℍ(R, χ) modulo a shift in grading by −(K_{Z}^{2} + #)/4. (The shift in the above two examples are different; in the case of S_{−d}^{3}(K) one has K_{Z}^{2} +# = 1−d(d−2)^{2}, while for Σ(d, d, d + 1) one has K_{Z}^{2} + # = −d(d − 1)(d − 3).)

Now, Conjecture B1 (6.5) and the above discussion/examples read as follows:

8.9. Theorem. Assume that ν = 1. Then the following facts are equivalent: (a) Conjecture B1 (6.5) is true,

(b) The canonical graded roots of S_{−d}^{3}(K) and Σ(d, d, d + 1) are the same.

(c) The canonical Heegaard-Floer homologies of −S_{−d}^{3}(K) and −Σ(d, d, d + 1) are the same modulo a shift in the grading, namely:

HF^{+}(−S_{−d}^{3}(K), can)[1 − d(d − 2)^{2}] = HF^{+}(−Σ(d, d, d + 1), can)[−d(d − 1)(d − 3)].

Proof. The equivalence (a) ⇔ (b) is clear from the above discussion, while (b) ⇔ (c) can be deduced by a direct computation, or from an easy formula which provides ℍ(R_{τ} , χ_{τ}) from τ, cf. [28]. (Nevertheless, see another argument below.) □

8.10. Remark. Regarding the Seiberg-Witten invariant of M = S_{−d}^{3}(K), one has

sw_{M}(can) − \frac{K_{Z}^{2} + #}{8} = ∑_{l≥0} τ(2l + 1) − τ(2l + 2) = ∑_{l≥0} c_{l},  (29)

and there is a similar formula for M = Σ(d, d, d + 1) with the obvious replacements. Therefore, to the equivalences (8.9) one can add:

(d) sw_{M}(can) − \frac{K_{Z}^{2} + #}{8} |_{M=S_{−d}^{3}(K)} = sw_{M}(can) − \frac{K_{Z}^{2} + #}{8} |_{M=Σ(d,d,d+1)}.

Since the Conjecture SW C (7.3) is true for the Brieskorn singularity f_{BR} := x^{d} + y^{d} + z^{d+1} (cf. [33]), and the geometric genus of the superisolated singularity f equals the geometric genus of f_{BR} (both equal d(d − 1)(d − 2)/6), this last identity (d) is also equivalent with the validity of the SW C for f — a fact already proved in 7.5.

(Notice also that the expression sw_{M}(can) − (K_{Z}^{2} + #)/8 can be deduced from ℍ, a fact which implies (c) ⇒ (d), while (d) ⇒ (a) follows from (29).)

8.11. Example. Assume that d = 5 and C is unicuspidal whose singular point has only one Puiseux pair (a, b) with a < b. Then by the genus formula the possible values of (a, b) are (4, 5), (3, 7) and (2, 13). It turns out that the first and the third cases can be realized, while the second not. The corresponding graded roots (together with the root of Σ(5, 5, 6)) are drawn in the above figure (8.3.1).

References

[1] Aluffi, P. and Faber, C: Plane curves with small linear orbits II, International Journal of Mathematics, 11 (2000), 591–608.

[2] Coolidge, J.L.: A treatise of algebraic plane curves, Oxford Univ. Press. Oxford, 1928.

[3] S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468.

[4] D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110, (Princeton University Press, Princeton, 1985).

[5] Fenske, T.: Rational 1- and 2-cuspidal plane curves, Beiträge Algebra Geom. 40 (1999), no. 2, 309–329.

[6] Fenske, T.: Rational cuspidal plane curves of type (d, d − 4) with χ(Θ_{V} ⟨D⟩) ≤ 0. Manuscripta Math. 98 (1999), no. 4, 511–527,

[7] Flenner, H. and Zaidenberg, M: ℚ-acyclic surfaces and their deformations, Contemporary Math. 162 (1994), 143–208.

[8] Flenner, H. and Zaidenberg, M.: On a class of rational cuspidal plane curves. Manuscripta Math. 89 (1996), no. 4, 439–459.

[9] Flenner, H. and Zaidenberg, M.: Rational cuspidal plane curves of type (d, d−3). Math. Nachr. 210 (2000), 93–110.

[10] J. Fernández de Bobadilla, I Luengo-Velasco, A. Melle-Hernández and A. Némethi: On rational cuspidal projective plane curves, to appear in Proc. London Math. Soc.

[11] J. Fernández de Bobadilla, I Luengo-Velasco, A. Melle-Hernández and A. Némethi: Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair, submitted.

[12] G-M. Greuel and C. Lossen: Equianalytic and equisingular families of curves on surfaces, manuscripta math. 91 (1996) 323–342.

[13] Gurjar R.V., Kaliman S., Kumar N.M., Miyanishi M., Russell R., Sakai F., Wright D., Zaidenberg M.: Open problems on open algebraic varieties. arXiv:math.AG/9506006;

[14] Gusein-Zade, S.M., Delgado, F. and Campillo, A.: On the monodromy of a plane curve singularity and the Poincaré series of the ring of functions on the curve, Functional Analysis and its Applications, 33(1) (1999), 56-67.

[15] Harris, J.; Morrison, I.: Moduli of curves, Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.

[16] Iitaka, S.: On logarithmic Kodaira dimension of algebraic varities, Complex Analysis and Algebraic Geometry , Cambridge Univ. Press, Cambridge, 1977, 175–190.

[17] Iitaka, S.: On irreducible plane curves, Saitama Math. J. 1 (1983) 47–63.

[18] Iitaka, S.: Birational geometry of plane curves, Tokyo J. Math. 22 (1999) 289–321.

[19] Kashiwara, H.: Fonctions rationelles de type (0,1) sur le plan projectif complexe, Osaka J. Math. 24 (1987) 521–577.

[20] Kumar, N.M. and Murthy, M.P.: Curves with negative self intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1983) 767–777.

[21] Lescop, C.: Global Surgery Formula for the Casson-Walker Invariant, Annals of Math. Studies, vol. 140, Princeton University Press, 1996.

[22] Luengo, I.: The μ-constant stratum is not smooth, Invent. Math., 90 (1), 139-152, 1987.

[23] Luengo, I., Melle-Hernández, A. and Némethi, A.: Links and analytic invariants of superisolated singularities, Journal of Algebraic Geometry, 14 (2005), 543-565.

[24] Matsuoka, T. and Sakai, F.: The degree of rational cuspidal curves, Math. Ann., 285 (1989), 233–247.

[25] Miyanishi, M. and Sugie, T.: On a projective plane curve whose complement has logarithmic Kodaira dimension ∞, Osaka J. Math. , 18 (1981), 1–11.

[26] Nagata, M.: On rational surfaces I. Irreducible curves of arithmetic genus 0 and 1, Memoirs of College of Science, Univ. of Kyoto, Series A, Vol. XXXII (3) (1960), 351-370.

[27] Némethi, A.: On the spectrum of curve singularities, Proceedings of the Singularity Conference, Oberwolfach, July 1996; Progress in Mathematics, Vol. 162, 93-102, Birkhäuser 1998.

[28] Némethi, A.: On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds (math.AG/0310083), Geometry and Topology 9 (2005), 991-1042.

[29] Némethi, A.: On the Heegaard Floer homology of S_{−d}^{3}(K) and unicuspidal rational plane curves, to appear in Fields Institute Communications, Vol. 47, 2005.

[30] Némethi, A.: On the Heegaard Floer homology of S_{−p/q}^{3}(K), submitted (math.GT/0410570).

[31] Némethi, A.: Line bundles associated with normal surface singularities, submitted (math.AG/0310084).

[32] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities, Geometry and Topology, Volume 6 (2002), 269-328.

[33] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities II (singularities with good C^{∗}-action), J. London Math. Soc. (2) 69 (2004), no. 3, 593–607.

[34] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities III (splicings and cyclic covers), math.AG/0207018. to appear in Selecta Math.

[35] Neumann, W. and Wahl, J.: Casson invariant of links of singularities, Comment. Math. Helv. 65, 58-78, 1991.

[36] Nicolaescu, L.I.: Seiberg-Witten invariants of rational homology 3-spheres, Comm. in Contemp. Math., 6 (2004), 833-866.

[37] Orevkov, S.Yu.: On rational cuspidal curves, I. Sharp estimate for degree via multiplicity, Math. Ann. 324 (2002), 657-673.

[38] Orevkov, S.Yu., Zaidenberg, M.G.: On the number of singular points of plane curves, In: Algebraic Geometry. Proc. Conf., Saintama Univ., March 15–17, 1995, alg-geom/9507005

[39] Ozsváth, P.S. and Szabó, Z.: Holomorphic disks and topological invariants for closed threemanifolds. Ann. of Math. (2) 159 (2004), no. 3, 1027–1158.

[40] Ozsváth, P.S. and Szabó, Z.: On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185-224.

[41] Sakai, F.: Kodaira dimensions of complements of divisors, Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 239–257.

[42] Teissier, B.: Résolution simultanée—I, II, Séminaire sur les Singularités des Surfaces. Edited by Michel Demazure, Henry Charles Pinkham and Bernard Teissier. Lecture Notes in Mathematics,

777. Springer, Berlin, 1980. 71–146

[43] Teissier, B. and Zariski, O. Le problème des modules pour les branches planes, (Hermann, Paris, 1986, Appendice).

[44] Tono, K.: Defining equations of certain rational cuspidal plane curves, Manuscripta Math. 103 (2000) 47–62.

[45] Tono, K.: Defining equations of certain rational cuspidal plane curves, doctoral thesis, Saitama University, 2000.

[46] Tono, K.: On rational unicuspidal plane curves with κ̄ = 1, RIMS-Kôkyûroku 1233 (2001) 82–89.

[47] Tono, K.: On the number of cusps of cuspidal plane curves, Math. Nachr. 278 (2005) 216–221.

[48] Tsunoda, Sh.: The complements of projective plane curves, RIMS-Kôkyûroku 446 (1981) 48–56.

[49] Tsunoda, Sh.: The Structure of Open Algebraic Surfaces and Its Application to Plane Curves, Proc. Japan Acad. Ser. A 57 (1981) 230–232.

[50] Turaev, V.G.: Torsion invariants of Spin^{c}-structures on 3-manifolds, Math. Res. Letters, 4 (1997), 679-695.

[51] Varchenko, A.N.: On the change of discrete characteristics of critical points of functions under deformations, Uspekhi Mat. Nauk, 38:5 (1983), 126-127.

[52] Varchenko, A.N.: Asymtotics of integrals and Hodge structures. Science rewievs: current problems in mathematics 1983, 22, 130-166; J. Sov. Math. 1984, Vol. 27.

[53] Wakabayashi, I.: On the Logarithmic Kodaira Dimension of the Complement of a Curve in ℙ^{2}, Proc. Japan Acad., 54, Ser. A, (1978), 167162.

[54] Wall, C. T. C.: Notes on the classification of singularities, Proc. London Math. Soc. 48(3) (1984), no. 3, 461–513.

[55] Wahl, J. M.: Equisingular deformations of plane algebroid curves, Trans. Amer. Math. Soc. 193 (1974), 143–170.

[56] Zaidenberg, M.G. and Lin, V. Yu: An irreducible simply connected algebraic curve in ℂ^{2} is equivalent to a quasihomogeneous curve, Soviet Math. Dokl. 28 (1983) 200–204.

[57] Zaidenberg, M.G.: Selected problems, arkiv:AG/0501457.

Departamento de Matemáticas Fundamentales,, Facultad de Ciencias U.N.E.D., c/ Senda del Rey 9, 28040 Madrid, Spain.

Facultad de Matemáticas, Universidad Complutense, Plaza de Ciencias, E-28040, Madrid, Spain

Department of Mathematics, Ohio State University, Columbus, OH 43210,USA; and Rényi Institute of Mathematics, Budapest, Hungary.

  E-mail address: javier@mat.uned.es
  E-mail address: iluengo@mat.ucm.es
  E-mail address: amelle@mat.ucm.es
  E-mail address: nemethi@math.ohio-state.edu; nemethi@renyi.hu