ON RATIONAL CUSPIDAL CURVES I. SHARP

ESTIMATE FOR DEGREE VIA MULTIPLICITIES

S.Yu. Orevkov

Abstract. Let C ⊂ P^{2} be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that d < 3m. We show that d < αm + const where α = 2.61... is the square of the ”golden section”. We also construct examples which show that this estimate is asymptotically sharp. When κ̄(P^{2} − C) = −∞, we show that d > αm and this estimate is sharp.

The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types.

Let C be an algebraic curve in P^{2} (over C) which is homeomorphic to CP^{1}.

This means that C is rational and cuspidal, i.e. all its singularities are cusps where cusp means a singularity at which the curve has only one analytic branch. Such curves were studied, for instance, in [Y1-Y3], [Ka], [MS], [FZ2], [OZ3].

Denote by m_{p} the multiplicity of C at a point p (i.e. m_{p} = (C · L)_{p} for a generic line L). Let d be the degree of C and m = max_{p∈C} m_{p}.

Put α = (3 + \sqrt{5})/2 = 2.6180... (α is a root of α + \frac{1}{α} = 3).

Theorem A. d < α(m + 1) + 1/\sqrt{5} = αm + 3.0652...

For m > 10 this estimate is stronger than the estimate d < 3m obtained by T. Matsuoka and F. Sakai [MS] though the method of the proof is very similar. Denote by κ̄(Y) the logarithmic Kodaira dimension of a non-complete surface Y (see [F]).

Theorem B.

(a). If κ̄(P^{2} \ C) = −∞ then d < αm.

(b). If κ̄(P^{2} \ C) = 2 then d < α(m + 1) − 1/\sqrt{5} = αm + 2.1708...

(c). κ̄(P^{2} \ C) ≠ 0.

Denote by φ_{0}, φ_{1}, φ_{2}, … the Fibonacci numbers: φ_{0} = 0, φ_{1} = 1, φ_{k+2} = φ_{k} + φ_{k+1}.

Theorem C. For any j > 0, j ≢ 2 mod 4 there exists a rational cuspidal curve C_{j} of degree d_{j} = φ_{j+2} which has a single cusp of multiplicity m_{j} = φ_{j}. Thus, lim d_{j}/m_{j} = α.

(a). ([Ka]; see Remark 3) If j is odd then κ̄(P^{2} \ C_{j}) = −∞ and the cusp of C_{j} has one characteristic pair (m_{j}, n_{j}) where n_{j} = φ_{j+4}.

(b). If j is even (and hence, divisible by 4) then κ̄(P^{2} \ C_{j}) = 2. The cusp of C_{j} for j = 8, 12, ... has two characteristic pairs^{1} (φ_{j}, φ_{j+4}) and (3, 1); the cusp of C_{4} has one characteristic pair (φ_{4}, φ_{8} + 1) = (3, 22).

Typeset by AMS-TEX

Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502

1See the definition in §3.

(c). For any j > 0, j ≡ 0 mod 4 there exist a rational cuspidal curve C_{j}^{∗} of degree d_{j}^{∗} = 2φ_{j+2} which has a single cusp of multiplicity m_{j}^{∗} = 2φ_{j}. κ̄(P^{2} \ C_{j}) = 2. The cusp of C_{j}^{∗} for j = 8, 12, ... has two characteristic pairs (2φ_{j}, 2φ_{j+4}) and (6, 1); the cusp of C_{4}^{∗} has one characteristic pair (2φ_{4}, 2φ_{4} + 1) = (6, 43).

Since {d_{j}/m_{j}} with odd j is the sequence of the upper convergents of the continuous fraction of α, Theorem C.a shows that the estimate in Theorem B.a is ”sharp” in the following sense: the convex hull of all pairs (m, d) ∈ Z^{2} satisfying m + 1 ≤ d < αm coincides with the convex hull of all pairs (m, d) realizable by rational cuspidal curves C with κ̄(P^{2} \C) = −∞ (obviously, the opposite bound m+1 ≤ d is sharp due to the curves y = x^{d}).

In contrary, the curves C_{4k} and C_{4k}^{∗} provide examples where d > αm.

Conjecture. C_{4k} and C_{4k}^{∗} are the only rational cuspidal curves with d > αm.

It is possible to prove that this conjecture follows from Conjecture 2.3. The first six pairs (m, d) with d > αm which are not realized by Theorem C and which are neither forbidden by Theorem A nor by the theorem of Matsuoka—Sakai [MS] are (4, 11), (5, 14), (6, 17), (7, 19), (7, 20), and (φ_{6}, φ_{8}) = (8, 21). Using successive birational quadratic transformations the author checked that the cases (4, 11) and (8, 21) are not realizable. For example, in the case (8, 21) the only candidate is the curve of degree 21 with two cusps of types (8, 55) and (2, 3). After 8 quadratic transformations such a curve would be transformed into the cuspidal cubic with two inflection points.

Remark 1. The curve C_{1} is a conic. C_{3} is a quintic with the cusp A_{12} (see [DG], [Y1]).

Remark 2. The idea how to pass from the construction of C_{4k} to the construction of C_{4k}^{∗} belongs to E. Artal.

Remark 3. A curve C is rational cuspidal and satisfies the condition κ̄(P^{2} \ C) = −∞ if and only if C is a fiber of a rational function on P^{2} all whose fibers (with the indeterminacy point removed) are isomorphic to C^{1}. H. Kashiwara [Ka] has classified all such functions. Our curve C_{2l+3} (l ≥ 0) is one of the multiple fibers^{2} of a function of the type II(l)^{∗}. The degree is computed in [Ka; 7.2]; It is possible to compute by induction the multiplicity of the cusp using the recurrence relations (written in [Ka; 11.1]) for the defining polynomial.

Remark 4. It is shown in [MS] that each of the rational cuspidal curves known up to that moment, can be mapped onto a line by a birational transformation of P^{2}. It is clear from the construction that the curves C_{j} from Theorem C satisfy this property. The only known to the author new series of examples appeared in literature after the paper [MS], is the series of rational tricuspidal curves constructed by H. Flenner and M. Zaidenberg [FZ2]. It is shown in [A] that they also can be mapped onto a line by Cremona transformations.

The main tool used, is the logarithmic version of Bogomolov — Miyaoka — Yau inequality. In the case of rational cuspidal curves we give (see the end of §2) an interpretation of this inequality in the form #equ ≤ 5 + #var where #var and #equ are respectively the number of variables and the number of the equations in the system of simultaneous equations which appears if one wants to construct a rational curve of a degree d with the given list of types of singular branches using a parameterization by polynomials with indeterminate coefficients.

I am grateful to E. Artal and M.G. Zaidenberg for useful discussions. I am grateful to university of Zaragoza and Max-Planck-Institut für Mathematik for hospitality.

2The left one on the picture in [Ka; 6.1] where the graphs of resolution are described.

§1. M-number of a singularity of a plane curve

Let (C, 0) be a germ of an analytically irreducible curve in a neighborhood U of 0 ∈ C^{2} and let σ : (X, D, E) → (U, C, 0) be the minimal resolution of singularity of C at 0, i.e. σ is birational, D = σ^{−1}(C) is a curve with simple normal crossings (SNC-curve) and (X, D) is minimal possible under these conditions.

Let V_{E} be the vector space over Q formally generated by the irreducible components E_{1}, ..., E_{n} of E and V_{E}^{+} = {Σx_{i}E_{i} | x_{i} ≥ 0}. Define D_{E} and K_{E} as the elements of V_{E} such that E_{i}D_{E} = E_{i}D and E_{i}(E_{i} + K_{E}) = −2 for any i = 1, ..., n. Since the intersection form on E is non-degenerate, these elements exist and are unique.

Following [F], we define a twig of D as T = E_{i_{1}} + ... + E_{i_{m}} such that E_{i_{j}}E_{i_{k}} = 1 for |j − k| = 1, E_{i_{j}}E_{i_{k}} = 0 for |j − k| > 1, E_{i_{1}}(D − T) = 1 and E_{i_{j}}(D − T) = 0 for j > 1. By the local Zariski–Fujita decomposition near E we mean K_{E} + D_{E} = H_{E} + N_{E} where H_{E} ∈ V_{E} and N_{E} ∈ V_{E}^{+} are such that

(i). supp N_{E} is contained in the union of all twigs of D;

(ii). H_{E} · E_{i} ≥ 0 for all i = 1, ..., n; (iii). H_{E} · E_{i} = 0 for E_{i} ⊂ supp N_{E}. Let μ = μ_{(C,0)} be the Milnor number of (C, 0) and put

M = M_{(C,0)} = μ + H_{E}^{2}; \overline{M} = \overline{M}_{(C,0)} = μ + (K_{E} + D_{E})^{2}.

Since the intersection form is negatively definite, we have μ > \overline{M} > M. In §4 we prove some other inequalities for μ and M.

We shall call the number \overline{M} (resp. M) the rough (resp. fine) M-number of the singularity (C, 0). When the singularity is analytically irreducible, the rough M-number \overline{M} will be called also the parametric codimension of (C, 0) (see Proposition 3.4 for the motivation of this term).

In this paper we are going to use the language of Puiseux characteristic pairs but the authors of [MS] studying similar problems use an alternative language of multiplicity sequences. To compare these approaches, we give an expression of \overline{M} in terms of the multiplicity sequence. It will not be used in the sequel. Let (m_{1}, m_{2}, …, m_{n}) be the multiplicity sequence of (C, 0), i.e. the sequence of multiplicities of C in all the points which were blown up during the resolution process. Following [MS], we say that a blowup is subdivisional if it is performed at an intersection point of two exceptional curves. Otherwise, it is called sprouting. By convention, the first blow-up is subdivisional. Let ω = ω_{(C,0)} denote the number of subdivisional blow-ups. Then

μ = ∑ m_{i}(m_{i} − 1) (see [Mi]); \overline{M} = ω − 2 + ∑(m_{i} − 1) (see [OZ2; (37)]).

§2. BMY-inequalities

Let X be a smooth projective surface and D ⊂ X a reduced (maybe, reducible) SNCcurve. As usual, by κ̄ we denote the logarithmic Kodaira dimension. Let K = K_{X} be the canonical class of X. If κ̄(X \ D) ≥ 0, i.e. |m(K + D)| ≠ 0 for infinitely many m, then (see [F]) there exists the Zariski decomposition K + D = H + N where H, N ∈ Pic X ⊗ Q are such that (i). N is effective and the intersection form is negatively definite on the subspace V_{N} ⊂ Pic X ⊗ Q generated by the irreducible components of N; (ii). HC ≥ 0 for any effective C ∈ Pic X; (iii). H is orthogonal to V_{N}.

By (iii) we have (K + D)^{2} = H^{2} + N^{2} where N^{2} ≤ 0. Thus, H^{2} ≥ (K + D)^{2}. We shall use the following logarithmic form of the Bogomolov – Miyaoka – Yau (log-BMY) inequality.

Theorem 2.1. (a). (Miyaoka [My]) If κ̄(X \ D) ≥ 0 then (K + D)^{2} ≤ 3e(X \ D); (b). (Kobayashi—Nakamura—Sakai [KNS]) If κ̄(X \ D) = 2 then H^{2} ≤ 3e(X \ D); where e() denotes the topological Euler characteristic.

Let C ⊂ P^{2} be an algebraic curve of degree d and σ : (X, D) → (P^{2}, C) be the minimal resolution of the singular points p_{1}, ..., p_{s} of C. Let E_{i} = σ^{−1}(p_{i}) and denote by V_{i}, i = 1, ..., s the subspace of V = Pic X ⊗ Q generated by the irreducible components of E_{i}. Let V_{0} = σ^{∗}((Pic P^{2}) ⊗ Q), i.e. V_{0} is generated by the transform of a generic line. Then V = V_{0} ⊕ ... ⊕ V_{s} is a direct sum of pairwise orthogonal subspaces.

Suppose that κ̄(X\D) ≥ 0 and let K+D = H+N be the Zariski decomposition. Denote by K_{i}, D_{i}, H_{i} the orthogonal projections of K, D, H onto V_{i} (i = 0, ..., s). Obviously, that K_{i} and D_{i} (i > 0) coincide with K_{E_{i}}, D_{E_{i}} introduced in §1 and it follows from [F, Theorem (6.20)] that H_{i} = H_{E_{i}} under some additional conditions which are satisfied in out case when κ̄(P^{2} \ C) = 2. Clearly, that (K_{0} + D_{0})^{2} = H_{0}^{2} = (d − 3)^{2}, hence,

(K + D)^{2} = (d − 3)^{2} + ∑_{i=1}^{s}(K_{i} + D_{i})^{2};  H^{2} = (d − 3)^{2} + ∑_{i=1}^{s} H_{i}^{2};  (1)

If C is rational and cuspidal (see Introduction) then by genus formula we have

∑ μ_{i} = (d − 1)(d − 2)  (2)

and e(X \ D) = 1. Combining (1) and (2), we rewrite the log-BMY inequalities as follows.

Corollary 2.2. Let C ⊂ P^{2} be a rational cuspidal curve of degree d and \overline{M}_{1}, ..., \overline{M}_{s} (resp. M_{1}, ..., M_{s}) be the rough (resp. fine) M-numbers of its singularities (see §1). Then

\overline{M}_{1} + ⋯ + \overline{M}_{s} ≤ 3d − 4,  if κ̄(P^{2} − C) ≥ 0;  (3)
M_{1} + ⋯ + M_{s} ≤ 3d − 4,  if κ̄(P^{2} − C) = 2.  (4)

The factor of the family of rational curves in P^{2} by the action of the group P GL(3) is (3d − 9)-dimensional. Thus, according to Proposition 3.4 below, the inequality (3) has a very natural interpretation as follows.

Given a list of singularity types satisfying (2), let us try to realize this list by a rational curve of a given degree. To this end let us write down its polynomial parameterization (f_{1}(t) : f_{2}(t) : f_{3}(t)). Fix some coefficients to cancel the action of P GL(3) and P GL(2) and consider the others as indeterminates. Add also indeterminates t_{1}, ..., t_{s} which are the values of t where we are going to put the singularities. Then each singularity type imposes some number of equations for the indeterminates. Thus, (3) means:

If (the total number of the equations) > 5 + (the number of the indeterminates) then the system of simultaneous equations has no solution.

This speculation leads us to the following

Conjecture 2.3. If C ∈ P^{2} is a rational (not necessarily cuspidal) curve then ∑ \overline{M}_{i} ≤ 3d − 9 where the sum is taken over all irreducible analytical branches of C.

The question ”What singularities may have a plane rational curve of a given degree?” is discussed in [SS] for affine curves parametrized by two polynomials in one variable. In this case, 2.3 can be formulated as a generalization of a Davenport theorem which estimates deg(p(t)^{a} − q(t)^{b}) via deg p and deg q.

§3. Puiseux characteristic sequence

Let (C, 0) be a germ of an analytically irreducible curve in C^{2}. of multiplicity m and let x = t^{m}, y = a_{m}t^{m} + a_{m+1}t^{m+1} + … be its analytic parameterization. Put k_{0} = d_{1} = m and define recursively

k_{i} = min{k | a_{k} ≠ 0 & k ≢ 0 mod d_{i}},  d_{i+1} = gcd(d_{i}, k_{i}),  i = 1, 2, ...

Then m = k_{0} < k_{1} < k_{2} < … and m = d_{1} > d_{2} > . . .. Let h is defined by the conditions d_{h} > 1, d_{h+1} = 1 and put q_{1} = k_{1}, q_{i} = k_{i} − k_{i−1}, i = 2, ..., h. The sequence Ch(C, 0) = (m; q_{1}, q_{2}, ..., q_{h}) will be called the characteristic sequence of the singularity (C, 0). and the pairs (d_{i}, q_{i}) will be called the characteristic pairs^{3}. When C is smooth at p we put h = 0, Ch(C, p) = (1; ) . The following statement is well-known.

Proposition 3.1. Let σ : X → C^{2} be the blowing up of the origin, E = σ^{−1}(0), and p a point on E. Let (C, p) be the germ at p of an analytically irreducible curve on X. Let k = (C · E)_{p} and Ch(C, p) = (m; q_{1}, q_{2}, ..., q_{h}). Then

Ch(σ(C), 0) = \{ \begin{matrix}(k; k + m, q_{2}, …, q_{h}) & if k = q_{1}, \\ (k; k + m, q_{1} − k, q_{2}, …, q_{h}) & if k < q_{1}. \end{matrix}

If a curve C_{1} ⊂ X is transversal to E and (C_{1} · C)_{p} = k_{1} then (σ(C_{1}) · σ(C))_{0} = k + k_{1}.

For a complete SNC-curve E on a smooth surface X let us denote by d(E) the discriminant of its dual weighted graph which is defined as d(E) = det A_{E} where A_{E} = || − E_{i} · E_{j}||_{i,j=1}^{n} is the intersection matrix of the irreducible components of E. Put by definition d(∅) = 1.

Proposition 3.2. (see, for instance, [EN]). Let (X, D, E) → (C^{2}, C, 0) be the minimal resolution of the singularity of C (like in §1). Then the dual graph of E ∪ C looks like

− −^{\frac{q_{1}}{d_{2}}} − ∘_{|}_{|} −_{\frac{d_{1}}{d_{2}}} −^{\frac{q_{2}}{d_{3}}} − ∘_{|}_{|} −_{\frac{d_{2}}{d_{3}}} −^{\frac{q_{3}}{d_{4}}} − ∘_{|}_{|} −_{\frac{d_{3}}{d_{4}}} … − ∘_{|}_{|} − −^{q_{h}} − ∘_{|}_{|}−_{d_{h}}−∘^{C′}

where the dashed lines mean linear chains of vertices, the numbers written near them are their discriminants and C′ denotes the proper transform of C.

Denote by ⌈a⌉ the minimal integer which is ≥ a.

Proposition 3.3. Let μ be the Milnor number and \overline{M}, M the rough and fine M-numbers (see §1) of (C, 0). Then one has

μ = 1 − d_{1} + ∑_{i=1}^{h} q_{i}(d_{i} − 1);  (5)

\overline{M} = d_{1} − 2 + ∑_{i=1}^{h} (q_{i} − ⌈\frac{q_{i}}{d_{i}} ⌉);  (6)

3This definition differs from the standard one.

M = d_{1}(1 − q_{1}^{−1}) − 1 + ∑_{i=1}^{h} q_{i}(1 − d_{i}^{−1}).  (7)

The formula (5) is classical (see [Mi, p.93]); a proof of (6) (and of (5) as well) is written in [OZ1; (11), (13)]; (7) can be proved the same way as [OZ1; 5.2(ii)] (see also [OZ2; p.169]);

The next statement (as well as its proof) is not quite rigorous and we shall not use it in the sequel. We formulate it only as a motivation of the Conjecture 2.3.

Proposition 3.4. \overline{M} coincides with the codimension of the stratum of topological equisingularity of a generic family of mappings Δ → C^{2} where Δ ⊂ C^{1} is a disk (parametrical analogue of the μ = const stratum).

Proof. Let F = {f_{u}}_{u∈U}, f_{u}(τ) = (x_{u}(τ), y_{u}(τ)), τ ∈ C^{1}, u = (u_{1}, u_{2}, ...) be a given family and m the multiplicity of f_{0}. Let F_{e} be the equisingularity stratum of f_{0} and F_{0} = {f_{u} | ∃τ_{0} such that x′_{u}(τ_{0}) = ... = x_{u}^{(m−1)}(τ_{0}) = 0}. Then F_{e} ⊂ F_{0} ⊂ F and codim F_{0} = m − 1. Let v = (v_{1}, v_{2}, ...) be coordinates on F_{0}. For f_{v} ∈ F_{0} there exists a unique (up to multiplication by ^{m}\sqrt{1}) reparametrization τ = τ(t) such that x_{v}(τ(t)) − x_{v}(τ_{0}) = t^{m}. Put y_{v}(τ(t)) = ∑ a_{k}t^{k}. Then a_{k} are analytic functions of v. The fact that the characteristic sequence of f_{v} coincides with that of f_{0} is equivalent to vanishing of certain finite number of coefficients a_{k}. The number of them is easy to calculate from the definition of the characteristic sequence. It remains to compare the answer with (6).

§4. Some estimates for M-numbers

Let the notation be the same as in §3.

Lemma 4.1.

\overline{M} − \frac{μ}{m} > m − 3.  (8)

Proof. By (6), using that d_{1} = m and ⌈q_{i}/d_{i}⌉ ≤ (q_{i} + d_{i} − 1)/d_{i}, we obtain

\overline{M} ≥ m − 2 + ∑_{i=1}^{h}(q_{i} − 1)(1 − d_{i}^{−1}).  (9)

Dividing (5) by m, we get

\frac{μ}{m} = \frac{1}{m} − 1 + ∑_{i=1}^{h} \frac{q_{i}d_{i}}{m} (1 − \frac{1}{d_{i}} ) ,  hence,  (10)

\overline{M} − \frac{μ}{m} = m − 2 + ∑_{i≥2} (q_{i}(1 − \frac{d_{i}}{m}) − 1)(1 − \frac{1}{d_{i}} )  subtract (10) from (9)
≥ m − 2 − ∑_{i≥2} d_{i}m^{−1}(1 − d_{i}^{−1})  use q_{i} ≥ 1

≥ m − 2 − ∑_{i≥2} d_{i}m^{−1}.

It remains to note that d_{i+1} ≤ d_{i}/2, hence, d_{i} ≤ m/2^{i−1}, thus, ∑_{i≥2} d_{i}/m < 1. □

Corollary 4.2. [OZ1; 6.2].

\overline{M} ≥ \frac{μ}{m}  (11)

Proof. For m > 2 apply (8). If m = 2 then \overline{M} = μ/m. □

Lemma 4.3.

M ≥ \frac{μ − 1}{m} + m − \frac{m(m − 1)}{μ + m − 1}  (12)

and one has ”=” in (12) if and only if h = 1.

Proof. Denote μ + m − 1 by A. Since d_{1} = m, we have A = ∑_{i=1}^{h} q_{i}(d_{i} − 1). Then by (5),

(7) we have

M − \frac{μ}{m} − m + \frac{1}{m} = −\frac{d_{1}}{q_{1}} + ∑_{i≥1} \frac{q_{i}}{d_{i}} (d_{i} − 1) − ∑_{i≥1} \frac{q_{i}}{d_{1}} (d_{i} − 1) = B − \frac{d_{1}}{q_{1}} ,

where B = ∑_{i≥2} q_{i}(d_{i} − 1)(d_{i}^{−1} − d_{1}^{−1}). In this notation (12) is equivalent to B ≥ d_{1}/q_{1} − m(m − 1)/A, and transforming the right hand side as

\frac{d_{1}}{q_{1}} − \frac{m(m − 1)}{A} = \frac{d_{1}(A − q_{1}(d_{1} − 1))}{q_{1}A} = \frac{d_{1}}{q_{1}A} ∑_{i≥2} q_{i}(d_{i} − 1),

we rewrite (12) in the form

∑_{i≥2} q_{i}(d_{i} − 1)(d_{i}^{−1} − d_{1}^{−1}) ≥ \frac{d_{1}}{q_{1}A} ∑_{i≥2} q_{i}(d_{i} − 1)

To prove this inequality, it is sufficient to verify that for each i ≥ 2 one has d_{i}^{−1} − d_{1}^{−1} > d_{1}/(q_{1}A). Indeed, d_{1} ≥ 2d_{2} ≥ 2d_{i}, hence d_{1} − d_{i} ≥ d_{i}. We have also q_{1} > d_{1} and A ≥ q_{1}(d_{1} − 1) ≥ q_{1} > d_{1}. The product of the last three inequalities yields (d_{1} − d_{i})q_{1}A > d_{i}d_{1}^{2}. It remains to divide the both sides by d_{1}d_{i}q_{1}A. □

Corollary 4.4.

M ≥ \frac{μ}{m} + m − 1 − \frac{1}{m(m + 1)}  (13)

Proof. Apply to the μ in the denominator in (12) the estimate (see (7)). μ ≥ 1 − d_{1} + q_{1}(d_{1} − 1) = (q_{1} − 1)(d_{1} − 1) ≥ m(m − 1). □

§5. Proofs of Theorems A and B.

Let C ⊂ P^{2} be a rational cuspidal curve such that κ̄(P^{2} \C) = −∞. According to [W], in this case C has only one cusp (denote it by p). Let (X, D) → (P^{2}, C) be the minimal resolution of the singularity of C at p. Denote by C′ the proper transform of C.

Lemma 5.1. (C′)^{2} ≥ −1.

Proof. X is a smooth surface and D is an SNC-curve on it such that X \ D is Q-acyclic and κ̄(X \ D) = −∞. All such pairs (X, D) are described more or less explicitely in [FZ1; 4.9, 4.10]. One can see from this description that if such a pair is minimal (D contains no (−1)-curve of valence 2) then D contains a twig (see §1) whose intersection matrix is not negatively definite.

In our case the dual graph of D is shown in Proposition 3.2 and all the linear chains denoted by dashed lines are negatively definite. Thus, (C′)^{2} can not be ≤ −2 because this would imply that the graph is minimal and all linear chains are negatively definite. □

Let d = deg(C) and (d_{i}, q_{i}), i = 1, ..., h, d_{1} = m be the characteristic pairs of C at p (see §3). By (2) and (5) (recall that C has only one cusp) we have

d^{2} − 3d + 2 = 1 − m + ∑ q_{i}(d_{i} − 1).  (15)

It is not difficult to see that (C′)^{2} = d^{2} − ∑ q_{i}d_{i} (see, for instance, the last formula on p.109 in [OZ1]). Hence, by Lemma 5.1, we have d^{2} − ∑ q_{i}d_{i} ≥ −1. Subtracting (15) from this formula, we obtain

3d − m ≥ ∑ q_{i}.  (16)

Thus, (d^{2} − 3d + 2) + (m − 1) =^{(15)} ∑ q_{i}(d_{i} − 1) ≤ ∑ q_{i}(m − 1) ≤^{(16)} (3d − m)(m − 1). Hence,

d^{2} − 3dm + m^{2} ≤ −1 and the part (a) of Theorem B is proven.

Remark. The curves from Theorem C.a provide ”=” in all the above inequalities.

Proof of the part (c) of Theorem B. The classification of pairs (X, D) where X is a smooth surface and D a minimal SNC-curve on X such that κ̄(X \ D) = 0 under some additional conditions is obtained by Fujita [F, (8.64)] and those of them which are Q-acyclic are listed in [FZ1, 5.14 and Fig. 16]. The dual graphs of four of them are shown on Fig. 1(a)

– (d) and all the others are obtained from the graph on Fig. 1(e) by the operation which is called in [FZ1] a comb-attachment. It is a sequence of blow-ups such that the center of the next one lies on the exceptional curve of the previous one and the last exceptional curve is not included into D. Thus, if C ⊂ P^{2} is a rational cuspidal curve with κ̄(P^{2} \ C) = −∞ and (X, D) → (P^{2}, C) is its minimal resolution then either D coincides with one of the graphs listed above (case 1), or D is obtained from one of them by a sequence of blow-ups and C′ (the proper transform of C) is the is the last exceptional curve (case 2).

Fig. 1.

Case 1. Easy to see that none of the vertices on the graphs (a) – (d) can be chosen as the curve C′, because each vertex has a connected component Γ of the complement such that d(Γ) ≠ 1. Let us show that the graph (e) is also impossible. Denote by p the point where the comb-attachment was done and by D_{0} the central component. If C′ ≠ D_{0} and p ∉ D_{0} then the complement of C′ contains a vertex with weight 0. Otherwise, for each choice of C′, one of the connected components of its complement contains a vertex incident to two twigs, each of them is −∘^{−2}. But the Proposition 3.2 shows that two twigs with equal discriminants at the same vertex can not exist.

Case 2. Since D \ C′ is obtained as a minimal resolution of singularities, each vertex connected to C′ with an edge is a (−1)-curve of valence 3 (in D). This is possible only for the graph (e) when C′ is connected with the central vertex and we can apply the same arguments as in the case 1. □

Proof of Theorem A. If κ̄(P^{2} \ C) = −∞ then the part (a) of Theorem B provides even a stronger estimate, so, we suppose that κ̄(P^{2} \ C) ≥ 0 and we may apply (3). Let p_{1}, ..., p_{s} be the singular points of C. Denote by m_{i}, μ_{i}, \overline{M}_{i} respectively the multiplicity, the Milnor number and the rough M-number (see §1) of p_{i}. Without loss of generality we may suppose that m = m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{s}. By (2) we have μ_{1} = (d − 1)(d − 2) − (μ_{2} + μ_{3} + …), hence, by (3) and (8),

3d − 4 ≥ ∑ \overline{M}_{i} > \frac{μ_{1}}{m} + m − 3 + ∑_{i≥2} \overline{M}_{i} = \frac{d^{2} − 3d + 2}{m} + m − 3 + ∑_{i≥2} (\overline{M}_{i} − \frac{μ_{i}}{m} ).

By (11) the last sum is non-negative, hence, f(d) < 0 where f(x) = (x^{2} − 3x + 2)m^{−1} − 3x+m+1. But f(α(m+1)+5^{−1/2}) = 11/(5m) > 0 and f′(x) > 0 for x ≥ 3(m+1)/2. □

Proof of the part (b) of Theorem B. Repeating word by word the proof of Theorem A (replacing \overline{M}_{i} with M_{i} and using (13) instead of (8), we arrive to f(d)+2−(m^{2} + m)^{−1} ≤ 0. Evaluating this expression for d = α(m + 1) − 5^{−1/2} we obtain (1/5)(m − 4)(m^{2} + m)^{−1}. This quantity is positive for m > 4. It remains to note that for m ≤ 4 the required estimate is weaker than the estimate d < 3m proven in [MS]. □

§6. Examples (proof of Theorem C)

Let, as in Introduction, φ_{0} = 0, φ_{1} = 1, φ_{i+2} = φ_{i} + φ_{i+1} be the Fibonacci numbers. Put also φ_{−a} = (−1)^{a+1}φ_{a}. We shall need the following identities:

a). φ_{a−2} + φ_{a+2} = 3φ_{a};  c). φ_{a}^{2} − φ_{a−2}φ_{a+2} = (−1)^{a};
b). φ_{a}^{2} − φ_{a−1}φ_{a+1} = (−1)^{a+1}; d). φ_{a−3}φ_{a+3} − φ_{a−1}φ_{a+1} = 3 · (−1)^{a}.  (17)

Proof. (a) is just the sum of the identities φ_{a−2} + φ_{a−1} = φ_{a}, φ_{a+1} − φ_{a−1} = φ_{a} and φ_{a+2} − φ_{a+1} = φ_{a}; (b) can be easily proven by induction [V; (1.8)].

(c). (−1)^{a} =^{by (b)} φ_{a−1}φ_{a+1} − φ_{a}^{2} = (φ_{a} − φ_{a−2})(φ_{a+2} − φ_{a}) − φ_{a}^{2} = (φ_{a−2} + φ_{a+2})φ_{a} −

2φ_{a}^{2} − φ_{a−2}φ_{a+2} =^{by (a)} φ_{a}^{2} − φ_{a−2}φ_{a+2}.

(d). By (a) we have φ_{a±3} = 3φ_{a±1} − φ_{a∓1}. Putting this into (d), we transform (d) into (b) multiplied by 3. □

Fix a nodal cubic N ⊂ P^{2}. Let p_{0} be its node and P_{1}, P_{2} the analytic branches of N at p_{0}. Define a birational transformation f : P^{2} − − → P^{2} as follows. Blow up 7 infinitely close points of P_{1} at p_{0}. Denote by E_{1}, …, E_{7} the exceptional curves and by E_{0} the proper transform of N. Then E_{0}^{2} = E_{7}^{2} = −1, E_{1}^{2} = ⋯ = E_{6}^{2} = −2, E_{0} · E_{1} = E_{1} · E_{2} = ⋯ = E_{6} · E_{7} = E_{7} · E_{0} = 1 and the other intersections are zeros. E_{0} and E_{7} are symmetric to each other in the obtained configuration. Hence, we can blow down E_{0}, …, E_{6} and the image of E_{7} will be again a nodal cubic. Since all nodal cubics are projectively equivalent, we may assume that E_{7} is mapped onto N and E_{0}, …, E_{6} onto infinitely close points of P_{2}.

Denote by C_{−3} the tangent to P_{1}, by C_{−1} the tangent to P_{2}, and by C_{0} the tangent to N at a flex point. Let C_{0}^{∗} be a conic^{4} which intersects N only in one point which is a smooth point of N. To find such a conic, note that v = u(1 − u)^{2} and v = u − 2u^{2} have the contact of order 3 at (0, 0), hence v^{2} = u(1 − u)^{2} and v^{2} = u − 2u^{2} have the contact of order 6.

Now, let us define recurrently:

C_{j} = f(C_{j−4}), j > 0, j ≢ 2 mod 4; C_{4k}^{∗} = f(C_{4k−4}^{∗}), k = 1, 2, …

Since f is biregular on P^{2} \ N, these curves are rational and unicuspidal. The fact that their characteristic sequences at p_{0} are like it was described in Theorem C, can be easily proven by the simultaneous induction together with the assertion that

k_{j,1} = φ_{j},  k_{j,2} = φ_{j+4},  j = −3, −1, 0, 1, 3, …
k_{j,1}^{∗} = 2φ_{j},  k_{j,2}^{∗} = 2φ_{j+4},  j = 4, 8, 12, …

where k_{j,ν} = C_{j} · P_{ν} and k_{j,ν}^{∗} = C_{j}^{∗} · P_{ν}. Indeed, using Proposition 3.1, easy to check that m_{j+4} = k_{j+4,1} = k_{j,2} and k_{j+4,2} = 7k_{j,2} − m_{j}. It remains to apply the evident identity φ_{j−4} − 7φ_{j} + φ_{j+4} = 0 which is the sum of the identities (17.a) for a = j ± 2

and three times for a = j. To find d_{j} = deg(C_{j}), note that (d_{j} − 1)(d_{j} − 2) =^{(2),(5)}

(φ_{j} −1)(φ_{j+4} −1)+1+(−1)^{j} =^{(17a,c)} (φ_{j+2} −1)(φ_{j+2} −2). Hence, d_{j} = φ_{j+2}. Analogously, deg(C_{j}^{∗}) = 2φ_{j+2}.

The fact that κ̄(P^{2} \ C_{2k+1}) = −∞ follows from [Ka] (see Remark 3 in Introduction). Let us show that κ̄(P^{2}\C_{4k}) = 2. The fact that κ̄ ≠ 0 and κ̄ ≠ −∞, follows from Theorem B.c and Lemma 5.1 respectively. To show that κ̄ ≠ 1, we shall use the explicit classification of Q-acyclic surfaces with κ̄ = 1 obtained in [GM] and exposed in [FZ1; 5.7–5.11] in a form convenient for our purposes. We shall use the terminology and notation of [FZ1]. The pair (X̄, D̄) is called pre-Q-acyclic if D̄ is an SNC-divisor on a smooth surface X̄ and maybe after some comb-attachments (see [FZ1] or the proof of B.c in §5) one can obtain a pair (X, D) such that X \ D is Q-acyclic. All the pre-Q-acyclic pairs with κ̄ = 1 belong to four classes which are denoted in [FZ1] by (A1), (A2), (B1) and (B2).

A computation of the discriminant of the dual graphs (it is done, for instance, in [O]) easily shows that the surfaces in the cases (A2) and (B2) are already Q-acyclic and the surfaces (A1) and (B1) become Q-acyclic after a single comb-attachment.

Hence, the surfaces from (A2) and (B2) are impossible for our curve C_{4k} because their graphs do not coincide with the graph of the resolution (In the case (B2) the number of the broken chains must be 2, but then the sum of the weights of the vertices of valence 3 is −2).

The case (A1) is impossible because any choice of the comb-attachment preserves a vertex incident to two twigs of the form −∘^{−2}. It remains to consider the case (B1).

4The idea to use this conic belongs to E. Artal

By [FZ1; 5.11] the number of the broken chains must be two. Hence, the graph of the resolution is obtained by a single comb-attachment from a graph >∘^{−n}−∘^{0}− ∘^{n−2} < . The resulting graph has only two vertices of valence ≥ 3 and they are connected by an edge. It is possible only if a single blow-up was performed and if it was done at the central vertex. The result of this operation is > ∘^{1−n} − ∘^{n−1} < . Such a graph is impossible because the weights of the vertices of valence 3 must be negative. The case of C_{4k}^{∗} is treated the same way.

Now let us write the formulas for the constructed curves. To this end we decompose f into a product of quadratic transformations. Let f = π_{2} ∘ π_{1}^{−1}, where π_{i} : X → P^{2} are the birational morphisms. Denote by L and C the proper transform on X (with respect to π_{1}) of C_{−3} and f^{−1}(C_{−1}) — the line and the conic which have the maximal possible contact with P_{1}. Then f can be presented as

[E_{0}, L, C] → [E_{0}, E_{3}, C] → [E_{0}, E_{3}, E_{6}] → [E_{7}, E_{4}, E_{1}] → [E_{7}, E_{4}, L] → [E_{7}, C, L].

This means that the result of each quadratic transformation is the variety obtained from X by blowing down all the curves L, C, E_{0}, …, E_{7} except the three curves listed in the brackets. Each of these quadratic transformation can be easily written by formulas:

x_{1} = xy,  x_{2} = x_{1}z_{1} − y^{2}, x′_{2} = y_{2}z_{2}, x′_{1} = x′_{2}z′_{2},  x′ = x′_{1}^{2};
y_{1} = y^{2},  y_{2} = y_{1}z_{1},  y′_{2} = x_{2}z_{2}, y′_{1} = y′_{2}z′_{2} + x′_{2}^{2}, y′ = x′_{1}y′_{1};
z_{1} = yz − x^{2}, z_{2} = z_{1}^{2},  z′_{2} = x_{2}y_{2}, z′_{1} = z′_{2}^{2},  z′ = x′_{1}z′_{1} + y′_{1}^{2}.

The last two transformations are chosen to be symmetric to the first two, to guarantee that N → N and P_{1} → P_{2}. Performing all the substitutions we obtain f(x : y : 1) = (x′ : y′ : z′) where

x′ = p_{1}q^{2}, y′ = qp_{3}, z′ = p_{1}^{3} − xy(4y − x^{2})p_{1}^{2} + y^{4}(2y + x^{2})p_{1} − xy^{7},

and p_{1} := y − x^{2}, q := xy − x^{3} − y^{3}, and p_{3} := p_{1}^{2} − 2xy^{2}p_{1} + y^{5}. The equations p_{1} = 0, p_{3} = 0, q = 0 define C_{1}, C_{3}, and N. Put p_{0} := 3x + 3y + 1. Then p_{0} = 0 can be chosen as C_{0} and substituting u = −4(x + y)/p_{0}, v = 4(x − y)/p_{0} into the above equation of C_{0}^{∗} we get 21x^{2} − 22xy + 21y^{2} − 6x − 6y + 1 = 0.

The multiplicity sequences of C_{j} at p_{0} is (φ_{j}, S_{j}, S_{j−4}, …, S_{ν}) and where j = 4k + ν, ν = 3, 4, 5, k = 0, 1, … and S_{i} denotes the subsequence (φ_{i}, φ_{i}, φ_{i}, φ_{i}, φ_{i}, φ_{i} − φ_{i−4}). In the case of C_{j}^{∗} all the multiplicities should be multiplied by 2.

§7. Open fans and linear chains of rational curves

To deal with linear chains of rational curves on a smooth surface X, it is convenient to use the following language. We shall call an open fan a sequence of vectors c = (v_{0}, …, v_{n+1}), v_{i} ∈ Z^{2} such that v_{i} ∧ v_{i+1} = 1 for i = 0, …, n. We use here the natural identification ∧^{2}Z^{2} = Z assuming (a, b) ∧ (c, d) = ad − bc. Given any sequence of integers (a_{1}, …, a_{n}), one can construct an open fan c as above such that v_{i} ∧ v_{i+1} = 1 for i = 0, …, n and v_{i−1} ∧ v_{i+1} = −a_{i} for i = 1, …, n. Clearly that c is uniquely defined up to the action of SL_{2}(Z) and we shall write c = c(a_{1}, …, a_{n}).

Let X be a smooth surface and D = D_{1} + ⋯ + D_{n} a linear chain of rational curves on X (i.e. D_{i}D_{j} = 0 for |i − j| > 1, D_{i}D_{j} = 1 for |i − j| = 1). Put c(D) = c(D_{1}^{2}, …, D_{n}^{2}). It is convenient to denote by D_{0} (resp. D_{n+1}) the germ of a generic analytic curve which meets D_{1} (resp. D_{n+1}) transversally.

Given an open fan c = (v_{0}, …, v_{n+1}), we define its rotation number as rot(c) = ∑_{i=0}^{n} a_{i} where a_{i} is the oriented angle from v_{i} to v_{i+1} (all a_{i} are positive by the definition of open fan). This definition depends on the choice of a base in Z^{2}.

Given a vector v ∈ Z^{2}, let A_{v} be the triangular automorphism of Z^{2} defined by A_{v}u = u + (v ∧ u)v. Recall that d(D) denotes the discriminant of D (see §3).

Proposition 7.1. (a). If X̃ is obtained by blowing up a point D_{i} ∩ D_{i+1} (i = 0, ..., n) and D̃ is the total transform of D then c(D̃) = (v_{0}, …, v_{i}, v_{i} + v_{i+1}, v_{i+1}, …, v_{n+1}).

(b). If X′ is obtained by blowing up a smooth point of D_{i} (i = 1, ..., n) and D′ is the strict transform of D then c(D′) = (v_{0}, …, v_{i}, A_{v_{i}}v_{i+1}, …, A_{v_{i}}v_{n+1}).

(c). d(D) = v_{0} ∧ v_{n+1}.

(d). Let n_{+} be the number of positive squares in a diagonalization over Q of the intersection matrix A_{D} = ||D_{j} · D_{j}||. Then n_{+} = ⌈rot(c)/π⌉ − 1.

(e). D can be blown down to a smooth point iff v_{0} ∧ v_{n+1} = 1 and rot(c) < π.

Proof. (a,b) evident; (c) induction by n; (d) follows from (c) and the Sylvester formula.

(e). D can be blown down iff A_{D} is negatively definite and det A_{D} = ±1 (see [Mu]). Thus, (e) follows from (c) and (d). □

Let Σ be a primitive fan in Z^{2} and X = X_{Σ} the corresponding toric variety (see [D]). Let v_{0}, …, v_{n+1} be a sequence of generators of one-dimensional cones of Σ such that all v_{0}, …, v_{n} are distinct but maybe v_{n+1} = v_{0}. Suppose that each pair (v_{i}, v_{i+1}) forms the positively oriented base of some two-dimensional cone of Σ. Let D_{i} be the closure of the one-dimensional orbit of X corresponding to v_{i}. It is not difficult to show that v_{i−1} ∧ v_{i+1} = −D_{i}^{2}, thus, c(D_{1} + ⋯ + D_{n}) = (v_{0}, …, v_{n+1}).

§8. Another construction of the curves C_{j}

Fix an integer j ≥ 3, j ≢ 2 mod 4.

Case (a): j is odd. Consider three vectors v_{0} = −(φ_{j}^{2}, φ_{j+2}^{2}), v_{1} = (φ_{j−2}, φ_{j+2}), v_{2} = (φ_{j}, φ_{j+4}) in Z^{2}. Since the determinants

v_{0} ∧ v_{1} =^{(17.c)} φ_{j+2},  v_{1} ∧ v_{2} =^{(17.d)} 3,  v_{2} ∧ v_{0} =^{(17.c)} φ_{j},  (18)

are positive, we can consider the complete fan Σ spanned by v_{0}, v_{1}, v_{2}.

Let X_{Σ} be the smooth two-dimensional toric variety associated with the minimal primitive subdivision of Σ. Denote by D_{i} the closure of the one-dimensional orbit corresponding to v_{i} (i = 0, 1, 2) and by D the closure of X_{Σ} − (X_{0} ∪ D_{0}) where X_{0} is the open orbit of X_{Σ}. Let us blow up two generic points p_{1} ∈ D_{1} and p_{2} ∈ D_{2}. Denote by E_{1} and E_{2} the exceptional curves and by D′ (resp. D′_{i}) the strict preimage of D (resp. D_{i}).

As we pointed out above, c(D) = (v_{0}, …, v_{1}, …, v_{2}, …, v_{0}). Hence, by 7.1(b), c(D′) =

(v_{0} − (v_{1} ∧ v_{0})v_{1}, …, v_{1}, …, v_{2}, …, v_{0} + (v_{2} ∧ v_{0})v_{2} =^{(18)} (e_{1}, …, v_{1}, …, v_{2}, …, e_{2}) where

v_{i} corresponds to D′_{i} and e_{1} = (1, 0), e_{2} = (0, 1) is the base of Z^{2}.

Hence, by 7.1(e), D′ can be blown down to a smooth point, denote it by p. Counting blow-ups and blow-downs, we see that the resulting surface is P^{2}. By 3.2, 7.1(c), E_{2} is mapped onto a rational unicuspidal curve C_{2} with one characteristic pair (φ_{j}, φ_{j+4})).

By construction P^{2} \ C contains the affine two-dimensional toric variety isomorphic to C × (C \ 0) which corresponds to the vector v_{0}. Hence, κ̄(P^{2} \ C) = −∞.

Case (b): j is even. Recall that j ≢ 2 mod 4, hence, j is divisible by 4. Clearly that the complement of a rational cuspidal curve is a Q-acyclic surface. We shall apply a general method due to T. tom Dieck and T. Petri [tDP] to construct a Q-acyclic surface starting with a line arrangement.

Let us consider the arrangement of six lines L_{1}, …, L_{6} on P^{1} ×P^{1} where L_{1}, L_{3}, L_{5} are horizontal and L_{2}, L_{4}, L_{6} are vertical. Let p_{ij} = L_{i}∩L_{j} and X \D be the Q-acyclic surface obtained from this line arrangement by cutting cycles at p_{16}, p_{25}, p_{14} and p_{36} (see [tDP]) according to Fig. 2. This means that X is the result of blowing up at these four points and at some of their infinitely close points and D is the total preimage of L_{1} ∪⋯∪L_{6} with the last (−1)-curves excluded. The strict transforms of L_{i} are denoted on Fig. 2by D_{i}. We cut the cycles at p_{16}, p_{25} and p_{36} as it is shown on Fig. 2 where the strict transforms of L_{i} are denoted by D_{i}. We cut the cycle at p_{14} with the multiplicities (\frac{1}{3}φ_{j+4} − φ_{j}, \frac{1}{3}φ_{j}).

Fig. 2.

We are going to show that D − C (see Fig. 2) can be blown down to a smooth point p and the image of C is the required curve.

Let σ′ : X′ → P^{1} × P^{1} cuts the cycle at p_{14} and σ : X → X′ cuts the other three cycles. Denote by D′_{i} the strict transform of D_{i} on X′ . Consider P^{1} × P^{1} as the toric variety associated with the complete fan spanned on the vectors v_{4} = (1, 0), v_{3} = (0, 1), v_{2} = (−1, 0), v_{1} = (0, −1) whose orbits are L_{1}, …, L_{4}.

Put v_{0} = (\frac{1}{3}φ_{j+4} −φ_{j}, −\frac{1}{3}φ_{j}). Then X′ is the toric variety associated with the minimal primitive subdivision of the fan spanned on the vectors v_{0}, …, v_{4} and D′_{0} correspond to v_{0}. Put D′_{14} = σ′^{−1}(D_{1}+⋯+D_{4})−D′_{0}. Then c(D′_{14}) = (v_{0}, …, v_{4}, v_{3}, v_{2}, v_{1}, …, v_{0}). Let D_{14} be the strict transform of D′_{14} on X. Then by 7.1(b) we have c(D_{14}) = (v_{0}, …, v_{4}, …, Av_{0}) where A = A_{v_{3}}^{2}A_{v_{2}}^{2}A_{v_{1}}.

Blowing down successively D_{5}, D_{6} and D_{7} we map D − C onto a linear chain D̄_{14} and by 7.1(e) it suffices to prove that d(D̄_{14}) = 1. Since D̄_{14} is the result of the inverse of the operation described in Proposition 7.1(b) applied three times to D_{14} at D_{4}, we have c(D̄_{14}) = (v_{0}, …, v_{4}, …, Bv_{0}) where B = A_{v_{4}}^{−3}A. Clearly that

A_{v_{1}} = A_{v_{3}} = ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) and A_{v_{2}} = A_{v_{4}} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), hence, A = ( \begin{matrix}−1 & 2 \\ 1 & −3 \end{matrix} ) and B = ( \begin{matrix}−4 & 11 \\ 1 & −3 \end{matrix} ),

hence, Bv_{0} = \frac{1}{3}(−4φ_{j+4} + φ_{j}, φ_{j+4}). Thus, by 7.1(c) we have

d(D̄_{14}) = v_{0} ∧ Bv_{0} = \frac{1}{9}(φ_{j+4} + φ_{j})^{2} − φ_{j}φ_{j+4} =^{by (17.a)} φ_{j+2}^{2} − φ_{j}φ_{j+4} =^{by (17.c)} 1.

Using 3.2 and 7.1(c) we find the characteristic pairs of the image of C on P^{2}. For j ≥ 8 we have (q_{1}, d_{1}) = d_{2} · (v_{4} ∧ Bv_{0} , v_{0} ∧ v_{4}) = (φ_{j+4}, φ_{j}).

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ON RATIONAL CUSPIDAL CURVES 15

[Y3] H. Yoshihara, A note on the existance of some curves, in: Algebraic geometry and commutative algebra in honor of M. Nagata, Kinokuniya, Tokyo, 1987. Steklov Mathematical Institute, Gubkina 8, Moscow, Russia Max-Planck-Institute für Mathematik, Gottfried-Claren-Str. 26, Bonn, Germany