Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 309-329.

Rational 1- and 2-cuspidal Plane Curves

Torsten Fenske

Fakultät für Mathematik, Ruhr-Universität Bochum Geb. NA 2/31, Universitätsstr. 150, D-44780 Bochum e-mail: Torsten.Fenske@ruhr-uni-bochum.de

Abstract. We prove the existence of series of plane rational 1- and 2-cuspidal curves by constructing suitable Cremona transformations which transform the curves V (xy^{d−1} −z^{d}) resp. V (xy^{d−1} −z^{d} −z^{d−1}y) into the desired ones. Further, we obtain a complete list of existing types of rational cuspidal plane curves having one or two cusps with maximal multiplicity m_{max} = deg C − 2 resp. m_{max} = deg C − 3. MSC 1991: 14H20, 14H10, 14D15, 14N05

1. Introduction

In this paper we assume the ground field to be the field of complex numbers. Let C ⊂ ℙ^{2} be a rational curve with only cusps as singularities. By a cusp we mean an analytic irreducible singular point. Let (C, p) ⊂ (ℂ^{2}, 0) an irreducible analytic plane curve germ and let

V_{n} ⟶^{σ_{n}} V_{n−1} ⟶ ⋯ ⟶ V_{1} ⟶^{σ_{1}} V_{0} = ℂ^{2}  (1)

be the minimal embedded resolution of the singularity of C at p. The complete preimage of C is a simple normal crossing (SNC) divisor D = E + C_{n} where C_{n} is the proper preimage of C and E is the exceptional divisor. Let C_{i} ⊂ V_{i} resp. E_{i} ⊂ V_{i} denote the proper transform of C resp. the exceptional divisor under the blowing up σ_{i}. Then the multiplicities of C_{i−1} at p_{i−1} = σ_{i}(E_{i}) form a sequence m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{n} = 1 of decreasing integers called the multiplicity sequence of C at p (see [3]). Since the number of times the multiplicity 1 occurs in the sequence equals the last multiplicity greater than 1 in the sequence (see Proposition 3.1) we omit the multiplicity 1 and denote such a sequence by \overline{m}_{p} = [m_{0}, …, m_{l}] where

m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{l} > m_{l+1} = 1

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

for a suitable l ≥ 0. We sometimes write [m̂_{0_{r_{0}}}, …, m̂_{k_{r_{k}}}] for a multiplicity sequence which means that the multiplicity m̂_{i} occurs r_{i} times for i = 0, …, k. For example, [4_{2}, 2_{3}] means [4, 4, 2, 2, 2].

Then we ask whether there exists a rational cuspidal curve of given degree and given multiplicity sequences. In his classification of plane curves up to degree 5, Namba has given explicit equations for all possible rational cuspidal curves of degree 3, 4 and 5 ([10]). Only a few examples are known for the case deg C ≥ 6.

Recently, Flenner and Zaidenberg classified rational cuspidal plane curves with at least three cusps and maximal multiplicity deg C −2 [3] and deg C −3 [4]. They showed that those curves cannot have more than three cusps. They further showed that the only existing curves with maximal multiplicity deg C − 2 are of type (i): [a + b], [2_{a}], [2_{b}] with a + b = deg C − 2 and of type (ii): [3a, 2_{a}], [3_{a}], [2_{1}] if the maximal multiplicity is deg C − 3, a and b being positive integers. They proved the existence of these curves by two different methods. One method was to compute the rational parameterization and the other was to perform suitable Cremona transformations of ℙ^{2} transforming the rational plane quadric C_{2} = V (xy −z^{2}) into curves of type (i) and the rational plane cubic C_{3} = V (xy^{2} − z^{3}) into curves of type (ii).

Following their methods, we obtain rational 1- and 2-cuspidal plane curves by suitable Cremona transformations of ℙ^{2} transforming the rational plane curve C_{d,t} = V (xy^{d−1} − z^{d} − t z^{d−1}y) which are 1-cuspidal rational curves with a cusp of type [d − 1] and a d-fold point if t = 0 and a (d − 1)-fold point if t ≠ 0.

We obtain the following

Theorem 1.1. Let d ≥ 2 and 0 ≤ b < a be integers. The following types of rational cuspidal plane curves exist.

deg C_{a,d} type of cusps  χ(Θ_{V} ⟨D⟩)
1  da + d  [da, d_{a+b}, d − 1] , [d_{a−b}]  1 − a
1a  da + d  [da, d_{2a}, d − 1]  \begin{matrix}3 − d − a & if & d > 2 \\ 1 − a & if & d = 2 \end{matrix}
2  da + d  [da, d_{a+b}] , [d_{a−b}, d − 1]  a − 1
2a  da + d  [da, d_{2a}] , [d − 1]  \begin{matrix}−a & if & d > 2 \\ 1 − a & if & d = 2 \end{matrix}
3 da + d + 1 [da + 1, d_{a}] , [d_{a+1}]  1 − a
4 da + d + 1 [da, d_{a+1}] , [(d + 1)_{a}]  0
5 da + d + 1 [da, d_{a}] , [(d + 1)_{a}, d]  0
6 da + d + 2 [da + 1, d_{a}] , [(d + 1)_{a+1}]  0
7 da + 2 d − 1 [da + d − 1, d_{a}, d − 1] , [d_{a+1}, d − 1]  1 − a
8  a + 2  [a] , [2_{a}]  0

Here, χ(Θ_{V} ⟨D⟩) denotes the Euler-characteristic of the sheaf Θ_{V} ⟨D⟩ of germs of vector fields on V tangent along D where the pair (V, D) is an SNC-pair consisting of a complete surface V and an SNC-divisor D obtained by a minimal embedded resolution ϕ : V ⟶ ℙ^{2} of the singularities of C with D = ϕ^{−1}(C)_{red} such that D has only simple normal crossings as singularities.

It turns out that all existing types of rational 1- and 2-cuspidal plane curves with maximal multiplicity m_{max} = d − 2 and m_{max} = d − 3 are already contained in the table of Theorem 1.1. We complete the results of Flenner and Zaidenberg and give the complete list of existing types of rational cuspidal plane curves with at most two cusps and maximal multiplicity deg C − 2 resp. deg C − 3.

Theorem 1.2. Let a ≥ 1 and 0 ≤ b < a be integers.

(1) The only existing type of a rational cuspidal plane curve with one cusp and maximal multiplicity deg C − 2 is of type [2a, 2_{2a}] of degree 2a + 2 with χ(Θ_{V} ⟨D⟩) = 1 − a.

(2) The only existing types of rational cuspidal plane curves with two cusps and maximal multiplicity deg C −2 are (1) [2a+1, 2_{a}], [2_{a+1}] of degree 2a+3 with χ(Θ_{V} ⟨D⟩) = 1−a,

(2) [a],[2_{a}] of degree a + 2 with χ(Θ_{V} ⟨D⟩) = 0 and (3) [2a, 2_{a+b}], [2_{a−b}] of degree 2a + 2 with χ(Θ_{V} ⟨D⟩) = 1 − a.

For m_{max} = deg C − 3 we obtain

Theorem 1.3. Let a ≥ 1 and 0 ≤ b < a be integers.

(1) The only existing types of rational cuspidal plane curves with one cusp and maximal multiplicity deg C − 3 are (1) [3a, 3_{2a}, 2] of degree 3a + 3 with χ(Θ_{V} ⟨D⟩) = −a and (2) [2_{6}] of degree 5 with χ(Θ_{V} ⟨D⟩) = 0.

(2) The only existing types of rational cuspidal plane curves with two cusps and maximal

multiplicity deg C − 3 are the following

deg C_{i} type of cusps  χ(Θ_{V} ⟨D⟩)
C_{1}  7  [4] , [3_{3}]  0
C_{2}  6  [3] , [3_{2}, 2]  0
C_{3}  5  [2_{4}] , [2_{2}]  0
C_{4} 2a + 3 [2a, 2_{a}] , [3_{a}, 2]  0
C_{5} 2a + 4 [2a + 1, 2_{a}] , [3_{a+1}]  0
C_{6} 2a + 3 [2a, 2_{a+1}] , [3_{a}]  0
C_{7} 3a + 3 [3a, 3_{2a}] , [2]  −a
C_{8} 3a + 4 [3a + 1, 3_{a}] , [3_{a+1}]  0
C_{9} 3a + 3 [3a, 3_{a+b}, 2] , [3_{a−b}]  1 − a
C_{10} 3a + 3 [3a, 3_{a+b}] , [3_{a−b}, 2]  1 − a
C_{11} 3a + 5 [3a + 2, 3_{a}, 2] , [3_{a+1}, 2]  1 − a

Remark 1.4. The case m_{max} = deg C − 2 was independently done by F. Sakai and K. Tono In [7], Matsuoka and Sakai proved that for the maximal multiplicity m_{max} of a rational cuspidal curve of degree d the inequality d < 3 m_{max} must hold. Thus, by Theorems 1.2 and 1.3 we obtain a complete list of rational cuspidal plane curves of degree 6.

Corollary 1.5. ([1]) The only rational cuspidal plane curves of degree 6 with s singular points are the ones in the following list.

(1) s = 1  type of cusp χ(Θ_{V} ⟨D⟩) C_{1} [5]  −2 C_{2} [4, 2_{4}]  −1 C_{3} [3_{3}, 2]  −1

(2) s = 2  type of cusps χ(Θ_{V} ⟨D⟩) C_{4} [3_{3}] , [2]  −1 C_{5} [3_{2}, 2] , [3]  0 C_{6} [3_{2}] , [3, 2]  0 C_{7} [4, 2_{3}] , [2]  −1 C_{8} [4, 2_{2}] , [2_{2}]  −1 C_{9} [4] , [2_{4}]  0

(3) s = 3  type of cusps χ(Θ_{V} ⟨D⟩) C_{10} [4] , [2_{3}] , [2]  0 C_{11} [4] , [2_{2}] , [2_{2}]  0

In particular, there is no rational cuspidal plane sextic with more than three cusps.

In Section 2, we prove Theorem 1.1. We do this by constructing Cremona transformations transforming the curves C_{d,t} into curves of desired type. In Section 3, we sum up some results which impose strong restrictions on the multiplicity sequences for a curve to exist. Section 3 contains the proofs of Theorem 1.2 and 1.3. We apply the results of Section 3 to all possible types of rational plane curves having one or two cusps and maximal multiplicity m_{max} = deg C − 2 resp. m_{max} = deg C − 3. It turns out that the remaining types of curves appear in the table of Theorem 1.1. In Section 5, we give concrete parameterizations and defining equations for the rational cuspidal plane curves of degree 6 which we obtained by explicit computations of the parameterization with the computer algebra system Maple and by elimination of variables with the program Singular [5].

The author thanks Hubert Flenner for many valuable conversations concerning this material.

2. Proof of Theorem 1.1

Let X be a surface being isomorphic to a Hirzebruch surface Σ_{n} = Proj (O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(n)) for some n > 0, F a fibre of X and E the horizontal section of X with E^{2} = −n. We define an elementary transformation of F at p to be a blowing up of X at p followed by a blowing down of the proper transform of the fibre F. Then we obtain a new Hirzebruch surface X′ where the fibre F is replaced by a new fibre F′ being the exceptional divisor of the blowing up. Let E′ be the proper transform of the horizontal section E. If p ∈ E then E′^{2} = E^{2} − 1 otherwise we have E′^{2} = E^{2} + 1.

Let σ_{0} : X_{1} ⟶ ℙ^{2} be the blowing up of ℙ^{2} at r_{0}. Then X_{1} ≅ Σ_{1} is a Hirzebruch surface with a ruling π_{1} : X_{1} ⟶ ℙ^{1} given by the lines through r_{0} in ℙ^{2}. We start with the curve C_{d,i} = V (xy^{d−1} − z^{d} − i · z^{d−1}y) ⊂ ℙ^{2} with i = 0, 1. It is easily checked that the curve C_{d,0} has a cusp of type [d − 1] at p_{0} = (1 : 0 : 0) ∈ ℙ^{2}, a d-fold point at q_{0} = (0 : 1 : 0) ∈ ℙ^{2} if i = 0 and a (d − 1)-fold point at q_{0} if i = 1. The strategy is to construct a sequence

C′_{d,i} ⊂ ℙ^{2} ⟵^{σ_{n}} X_{n} ⟶^{σ_{n−1}} X_{n−1} ⟶^{σ_{n−2}} ⋯ ⟶^{σ_{1}} X_{1} ⟶^{σ_{0}} X_{0} = ℙ^{2} ⊃ C_{d,i}  (2)

of birational transformations of Hirzebruch surfaces beginning with the blowing up σ_{0} followed by (n − 1) elementary transformations σ_{i} for i = 1, …, n − 1 of a fibre F_{i} at a point r_{i} ∈ F_{i} such that E_{n}^{2} = −1 followed by a blowing down τ of E_{n}. The composition of these maps defines a Cremona transformation.

Before constructing the Cremona transformations we fix some notations. F_{i+1} denotes the new fibre of the ruling π_{i+1} : X_{i+1} ⟶ ℙ^{1} replacing the fibre F_{i} under σ_{i} for i = 1, …, n − 1. Let σ_{i} : X_{i+1} ⟶ X_{i} be a birational transformation of Hirzebruch surfaces. Let E_{i} be the horizontal section with negative self-intersection in X_{i}, C_{i} be any curve on X_{i} and p_{i} ∈ C_{i}. Then E_{i+1} resp. C_{i+1} always denotes the proper transforms of E_{i} resp. C_{i} under σ_{i} and p_{i+1} the point of C_{i+1} being infinitesimally near to p_{i} for i = 0, …, n.

Cases 1, 1a. Let C_{0} = C_{d,0}. We begin with the blowing-up σ_{0} at T_{p_{0}}C ∩ T_{q_{0}}C = {r_{0}}. Let F_{1} resp. F′_{1} be the fibre being the proper transform of the tangent line T_{p_{0}}C resp. T_{q_{0}}C. Then by Bézout’s theorem, we have C_{1}.F_{1} = d · p_{1} and C_{1}.F′_{1} = d · q_{1}. For i = 1, …, b we perform elementary transformations σ_{i} at r_{i} ∈ E_{i} ∩ F_{i} followed by elementary transformations σ_{i} at r′_{i} ∈ E_{i}∩F′_{i} for i = b+1, …, a. Since we performed the elementary transformations at a point lying on E_{i} we arrive at a Hirzebruch surface X_{a+1} with E_{a+1}^{2} = −a − 1, C_{a+1}.F_{a+1} = d · p_{a+1}, C_{a+1}.F′_{a+1} = d · q_{a+1}. p_{a+1} resp. q_{a+1} is a cusp of type [d_{b}, d − 1] resp. [d_{a−b}]. Now, for i = a + 1, …, 2a we perform elementary transformations at s_{i} ∈ F_{i}\E_{i}. Thus, X_{2a+1} is a Hirzebruch surface with E_{2a+1}^{2} = −1. So, we can blow down E_{2a+1} and arrive at ℙ^{2}. Let C′ be the proper transform under 2. Then, C′ is a rational cuspidal plane curve of type 1 resp. 1a of the table.

Cases 2, 2a. This sequence is constructed as the sequence in Case 1 with the following difference: For i = 1, …, b the elementary transformations are performed at r′_{i} ∈ E_{i} ∩ F′_{i}, for i = b + 1, …, a at r_{i} ∈ E_{i} ∩ F_{i} and for i = a + 1, …, 2a at any point s′_{i} ∈ F′_{i}\E_{i}.

Case 3. We begin with C_{0} = C_{d+1,0}. Then we perform the blowing up σ_{0} at the (d + 1)-fold point q_{0}. Let F_{1} resp. F′_{1} be the proper transform of L = \overline{p_{0}q_{0}} resp. T_{q_{0}}C. Then C_{1}.F_{1} = d·p_{1}, C_{1}.F′_{1} = d·q_{1} and C_{1}.E_{1} = 1·q_{1}. Now, for i = 1, …, a we perform elementary transformations σ_{i} at r_{i} ∈ E_{i}∩F_{i} followed by elementary transformations σ_{i} at r′_{i} ∈ F′_{i}\E_{i} for i = a+1, …, 2a. Hence, X_{2a+1} is a Hirzebruch surface with E_{2a+1}^{2} = −1 and C_{2a+1}.E_{2a+1} = (da + 1) · q_{2a+1}. Thus, after blowing down E_{2a+1} we arrive at ℙ^{2} and the proper transform C′ of C_{d} is a curve of type 3 in the table.

Case 4. Let C_{0} = C_{d+1,0}. Let L be any line through p_{0} which is neither the tangent line nor the line through q_{0}. Then by Bezout’s theorem, L intersects C transversally in exactly one further point, say r_{0}. Let s ∈ T_{q_{0}}C ∩L. We perform the first blowing up σ_{0} at s. Let F_{1} resp. F′_{1} be the proper transform of L resp. T_{q_{0}}C. Then C_{1}.F_{1} = d · p_{1} + 1 · r_{1}, C_{1}.F′_{1} = (d + 1) · q_{1} and C_{1} ∩ E_{1} = ∅. Now, for i = 1, …, a perform elementary transformations σ_{i} at r_{i} followed by elementary transformations σ_{i} at r′_{i} ∈ E_{i} ∩ F′_{i} for i = a + 1, …, 2a. Hence, X_{2a+1} is a Hirzebruch surface E_{2a+1}^{2} = −1 with C_{2a+1}.E_{2a+1} = da · p_{2a+1}. Thus, after blowing down E_{2a+1} we arrive at ℙ^{2} and the proper transform C′ of C is a curve of type 4 in the table.

Case 5. We start with C_{0} = C_{d+1,1}. By Bezout’s theorem T_{q_{0}}C intersects C transversally in a further point r′_{0}. Now, we perform the blowing up σ_{0} at T_{p_{0}}C ∩ T_{q_{0}}C = {r_{0}}. Let F_{1} resp. F′_{1} be the proper transforms of T_{p_{0}}C resp. T_{q_{0}}C. Then C_{1}.F_{1} = (d+1)·p_{1}, C_{1}.F′_{1} = d·q_{1}+1·r′_{1} and C_{1} ∩ E_{1} = ∅. Now, for i = 1, …, a we perform elementary transformations σ_{i} at r_{i} ∈ E_{i} ∩ F_{i} followed by elementary transformations σ_{i} at r′_{i} for i = a + 1, …, 2a. Then E_{2a+1}^{2} = −1 and C_{2a+1}.E_{2a+1} = da · q_{2a+1}. Thus, after blowing down E_{2a+1} we arrive at ℙ^{2} and the proper transform C′ of C is a curve of type 5 in the table.

Case 6. Now, let C = C_{d+2,1}. By Bezout’s theorem, T_{q_{0}}C intersects C in one further point r′_{0}. We perform the blowing up σ_{0} at q_{0}. Let F_{1} resp. F′_{1} be the proper transforms of \overline{p_{0}q_{0}} resp. T_{q_{0}}C. Then C_{1}.F_{1} = (d + 1) · p_{1}, C_{1}.F′_{1} = d · q_{1} + 1 · r′_{1} and C_{1}.E_{1} = 1 · q_{1}. For i = 1, …, a we perform elementary transformations σ_{i} at r_{i} ∈ E_{i}∩F_{i} followed by elementary transformations at r′_{i} for i = a + 1, …, 2a. We have E_{2a+1}^{2} = −1 and C_{2a+1}.E_{2a+1} = (da + 1) · q_{2a+1}. After blowing down E_{2a+1} we arrive at ℙ^{2} and the proper transform C′ of C is a rational curve of type 6 in the table.

Case 7. We start with C = C_{d,0} and perform the blowing up σ_{0} at T_{p_{0}}C ∩T_{q_{0}}C = {r_{0}}. Let F_{1} resp. F′_{1} be the proper transform of T_{p_{0}}C resp. T_{q_{0}}C. Then C_{1}.F_{1} = d · p_{1}, C_{1}.F′_{1} = d · q_{1} and C_{1} ∩ E_{1} = ∅. We perform an elementary transformation σ_{1} at q_{1} followed by an elementary transformation σ_{2} at r_{2} ∈ E_{2} ∩ F_{2}. We obtain a Hirzebruch surface X_{3} with E_{3}^{2} = −1 and C_{3}.F_{3} = d · p_{3}, C_{3}.F′_{3} = d · q_{3} and C_{3}.E_{3} = (d − 1) · q_{3}. The point p_{3} is a cusp of type [d, d − 1] and the point q_{3} a cusp of type [d − 1]. For i = 3, …, a + 2 we perform elementary transformations σ_{i} at r_{i} ∈ E_{i} ∩ F_{i} followed by elementary transformations σ_{i} at r′_{i} ∈ F′_{i}\E_{i} for i = a + 3, …, 2a + 2. Hence E_{2a+3}^{2} = −1 and C_{2a+3}.E_{2a+3} = (da + d − 1) · q_{2a+3}. After blowing down E_{2a+3} we arrive at ℙ^{2} and the proper transform C′ is a curve of type 7 in the table.

Case 8. We start with C = C_{3,0}. Let q_{0} ∈ C be any point which is not a 3-fold point. Let L = \overline{p_{0}q_{0}}. By Bezout’s theorem, T_{q_{0}}C intersects C in a further point, say r′_{0}. Then we perform σ_{0} at q_{0}. Let F_{1} resp. F′_{1} be the proper transform of L resp. T_{q_{0}}C. For i = 1, …, a − 1, we perform elementary transformations σ_{i} at r′_{i} followed by elementary transformations σ_{i} at r_{i} ∈ E_{i}∩F_{i} for i = a, …, 2a−2. Then C_{2a−1}.F_{2a−1} = 2·p_{2a−1}, C_{2a−1}.F′_{2a−1} = 1·r′_{2a−1}+1·q_{2a−1} and C_{2a−1}.E_{2a−1} = a·q_{2a−1}. C_{2a−1} has a cusp of type [2_{a}] at p_{2a−1}. After blowing down E_{2a−1} we arrive at ℙ^{2} and the proper transform C′ is a curve of type 8 in the table.

Let (V, D) be an (SNC)-pair constructed from a minimal embedded resolution of a rational cuspidal curve C ⊂ ℙ^{2}. In order to compute the Euler-characteristic of Θ_{V} ⟨D⟩ we use the following formula ([3], Lemma 2.3):

χ(Θ_{V} ⟨D⟩) = K_{V} (K_{V} + D) = −3(deg C − 3) + ∑_{p∈Sing C} χ_{p}

where K_{V} denotes the canonical divisor of V, χ_{p} = M_{p} + ω_{p} − 1 and

M_{p} = ∑_{i=0}^{r}(m_{i} − 1) ω_{p} = ∑_{i=1}^{r}(⌈\frac{m_{i−1}}{m_{i}} ⌉ − 1)

for a singular point p ∈ Sing C with multiplicity sequence [m_{0}, …, m_{r}]. By ⌈a⌉, we denote the smallest integer ≥ a.

ω_{p_{i}} denotes the number of inner blowing up in the resolution of p_{i}. The blowing up σ_{i+1} of V_{i} at p_{i} in Case 1 is called inner, if p_{i} ∈ E_{i} ∩ E_{i−k}^{(k)} for some k > 0 where E_{i−k}^{(k)} is the proper transform of the exceptional divisor E_{i−k} of σ_{i−k} in V_{i}. σ_{i+1} is called outer in the opposite. (cf. [3] or [7])

Now, for the case 1 we get

∑_{p∈Sing C} M_{p} = da − 1 + (d − 1)(a + b) + d − 2 + (d − 1)(a − b) = 3da + d − 2a − 3

and

∑_{p∈Sing C} ω_{p} = a − 1 + 1 + d − 2 + d − 1 = 2d + a − 3

and therefore

χ(Θ_{V} ⟨D⟩) = −3(da + d) + 9 + 3da + d − 2a − 3 + 2d + a − 3 − 2 = −a + 1.

By analogous computations we obtain the Euler-characteristic of the cases 1a to 7.

3. Restrictions on rational cuspidal plane curves

We begin with the characterization of multiplicity sequences as given in [3], Prop. 1.2. Let (C, p) ⊂ (ℂ^{2}, 0) be a rational cuspidal curve germ with multiplicity sequence m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{n} = 1 coming from a minimal embedded (SNC) resolution.

Proposition 3.1. The multiplicity sequence \overline{m}_{p} has the following two properties:

(1) For each i = 1, …, n there exists a k ≥ 0 such that

m_{i−1} = m_{i} + ⋯ + m_{i+k}

where m_{i} = m_{i+1} = ⋯ = m_{i+k−1}.

(2) If m_{n−r} > m_{n−r+1} = ⋯ = m_{n} = 1 then m_{n−r} = r − 1.

From the genus formula for plane projective curves we obtain a formula relating the multiplicity sequences to the degree of the curve. Let C ⊂ ℙ^{2} be a rational cuspidal curve of degree d and let p_{1}, …, p_{s} be the singular points and \overline{m}_{p_{i}} = [m_{i,0}, …, m_{i,r_{i}}] the multiplicity sequences. The arithmetical genus of C is

p_{a}(C) = \binom{d − 1}{2}

and also p_{a}(C) = gC̃ + ∑_{i=1}^{s} δ_{p_{i}} where gC̃ is the genus of the normalization C̃ of C and δ_{p_{i}} the number of virtual double points at p_{i} ([8]). In our case, we can compute δ_{p_{i}} by the formula

δ_{p_{i}} = ∑_{j=0}^{r_{i}} \frac{m_{i,j}(m_{i,j} − 1)}{2} .

Hence, we get

\binom{d − 1}{2} = ∑_{i=1}^{s} ∑_{j=0}^{r_{i}} \frac{m_{i,j}(m_{i,j} − 1)}{2} .  (3)

Now by Proposition 3.1 and (3), we are able to write down a finite list of possible types of rational cuspidal plane curves of given degree.

If we intersect C with a line L which is either the tangent line T_{p}C of a singular point p or a line through two different singular points, Bézout’s theorem yields

Proposition 3.2. Let C ⊂ ℙ^{2} be a rational cuspidal curve and p_{1}, …, p_{s} its cusps with multiplicity sequences \overline{m}_{p_{i}} = [m_{i,0}, …, m_{i,r_{i}}]. Then

(1) m_{i,0} + m_{i,1} ≤ d

(2) m_{i,0} + m_{j,0} ≤ d for i ≠ j.

We obtain another useful restriction if we apply the general Plücker formula

2 − 2g = 2d^{∗} − d − ∑_{p∈Sing C}(m_{p} − r_{p})  (4)

where d^{∗} is the degree of the dual curve C^{∗}, g the genus of C, m_{p} the multiplicity of C at p and r_{p} the number of branches at p ∈ C.

Proposition 3.3. Let C, p_{i}, \overline{m}_{p_{i}} be as in Proposition 3.2. Then

3d − 6 ≥ 2 ∑_{i=1}^{s}(m_{i,0} − 1) + ∑_{i=1}^{s}(m_{i,1} − 1) .

Proof. First, we apply the general Plücker formula to the dual curve.

2 − 2g = 2d − d^{∗} − ∑_{p∈Sing C^{∗}}(m_{p}^{∗} − r_{p}^{∗}) .  (5)

Now, if we put (4) and (5) together, we get

3d − 6(1 − g) = 2 ∑_{p∈Sing C}(m_{p} − r_{p}) + ∑_{p∈Sing C^{∗}}(m_{p}^{∗} − r_{p}^{∗}) .

We are left to prove the inequality m_{p_{i}}^{∗} ≥ m_{i,1} for i = 1, …, s. Set p = p_{i} and \overline{m}_{p_{i}} = \overline{m} = [m_{0}, …, m_{r}] for some i.

Let Φ(t) = (1 + ⋯: t^{1+α_{1}} + ⋯: t^{2+α_{1}+α_{2}} + ⋯) be the parameterization of C near the singular point p with 1 + α_{1} = m_{0} and 2 + α_{1} + α_{2} = (T_{p}C.C)_{p} = m_{0} + ⋯ + m_{k} with m_{0} = ⋯ = m_{k−1}. Moreover,

1 + α_{2} = m_{0} + ⋯ + m_{k} ≥ m_{1} .

The Plücker embedding for C near p yields the parameterization

Φ^{∗}(t) = ((1 + α_{2})t^{2+α_{1}+α_{2}} + ⋯: −(2 + α_{1} + α_{2})t^{1+α_{2}} + ⋯: (1 + α_{2}) + ⋯)

of the dual curve. Hence m_{0}^{∗} ≥ m_{1}. □

In [2], Flenner and Zaidenberg proved the following: If an SNC-pair (V, D) admits a generic ℙ^{1}-fibration over some smooth curve C such that D.F ≤ 3 for a generic fibre F, then the dimension of H^{2}(V; Θ_{V} ⟨D⟩) is zero.

If C ⊂ ℙ^{2} is a cuspidal curve with maximal multiplicity m_{max} ≥ d − 2 at p ∈ C then a blowing up of ℙ^{2} at p yields a ℙ^{1}-fibration over ℙ^{1} and by Bezout’s theorem, the proper transform intersects the generic fibre in d − m_{max} ≤ 2 points. Hence, by resolving all singularities we obtain an SNC-pair (V, D) satisfying the assumptions of the proposition.

Proposition 3.4. Let C ⊂ ℙ^{2} be a rational cuspidal curve with maximal multiplicity m_{max} ≥ d − 2. Then

H^{2}(V; Θ_{V} ⟨D⟩) = 0  □

In [4], they extended this result as follows:

Proposition 3.5. Let π : S ⟶ ℙ^{1} be a Hirzebruch surface with a section ℙ^{1} ≅ E ⊂ S of π and let C ⊂ S be a rational cuspidal curve such that π_{|C∪E} : C ∪ E ⟶ ℙ^{1} has degree 4. Assume that q ∈ C ∪ E is a branch point of π_{|C∪E} and that F_{π(q)} meets C ∪ E in three different points. Then

(1) π_{∗}(Θ_{S}⟨C ∪ E⟩) = 0

(2) If (V, D) is the (SNC)-pair obtained by a minimal embedded resolution V ⟶ S of

C ∪ E then

H^{2}(V; Θ_{V} ⟨D⟩) = 0

The proof is based on the following Lemma ([4] Lemma 2.3).

Lemma 3.6. Let Δ ⊆ ℂ be an open connected subset, and let B ⊆ S := Δ × ℙ^{1} be a closed curve such that the projection π : B ⟶ Δ is finite with d ≥ 3 sheets. Assume that there is a vector field ξ ≠ 0 on S which is tangent to B. Then either B ⟶ Δ is unramified over a point a ∈ Δ, or the fibre F_{a} := {a} × ℙ^{1} intersects B in at most two different points. □

Proof of Proposition 3.5. (2) follows from (1). Indeed, by Serre duality, we have

H^{2}(V; Θ_{V} ⟨D⟩) ≅ H^{0}(V; Ω^{1}⟨D⟩ ⊗ ω_{V})

The perfect pairing

Ω_{V}^{1} ⟨D⟩ ⊗ Ω_{V}^{1} ⟨D⟩ ⟶ ω_{V} (D)

induces an isomorphism Ω_{V}^{1} ⟨D⟩ ≅ Θ_{V} ⟨D⟩ ⊗ ω_{V} (D). Therefore

H^{2}(Θ_{V} ⟨D⟩) ≅ H^{0}(Θ_{V} ⟨D⟩ ⊗ ω_{V}^{2} (D)) ⊆ H^{0}(S; Θ_{S}⟨E ∪ C⟩ ⊗ ω_{S}^{2}(E ∪ C))  (∗)

Since E ∪ C ⟶ ℙ^{1} has degree 4, there is a Zariski open subset U in ℙ^{1} such that ω_{S}^{2}(E ∪ C)_{|π^{−1}(U)} ≅ O_{|π^{−1}(U)}. Therefore, (1) implies π_{∗}(Θ_{S}⟨E ∪ C⟩ ⊗ ω_{S}^{2}(E ∪ C)) = 0 saying that the last cohomology group in (∗) vanishes.

In order to prove (1) we assume that π_{∗}(Θ_{S}⟨E ∪ C⟩) ≠ 0 and apply Lemma 3.6 to B := E ∪ C, to the fibre F_{π(q)} and to a small neighbourhood of π(q) giving a contradiction which proves (1). □

Corollary 3.7. Let C ⊂ ℙ^{2} be a rational 1- or 2-cuspidal curve of degree d having a cusp p ∈ C with multiplicity sequence \overline{m}_{p} = [d − 3, m_{p,1}, …, m_{p,r_{p}}]. Assume that C satisfies at least one of the following conditions:

(1) C has a further cusp q with multiplicity sequence (2_{b}) and b > 1,

(2) m_{p,1} = 1,

(3) m_{p,1} = 2 and 0 ≡ d − 3 mod 2. Then

H^{2}(V; Θ_{V} ⟨D⟩) = 0

where (V, D) is the (SNC)-pair obtained by a minimal embedded resolution V ⟶ ℙ^{2} of the singularities of C.

Proof. We perform a blowing up σ_{0} : X_{1} ⟶ ℙ^{2} at p. Then X_{1} is isomorphic to the Hirzebruch surface Σ_{1} with a ruling π_{1} : X_{1} ⟶ ℙ^{1} given by the lines of ℙ^{2} through p. Let C_{1} be the proper transform of C under σ_{0} and let E_{1} be the exceptional divisor and F_{1} the fibre corresponding to the tangent line of C at p.

First, we assume that (1) holds. Since b > 1 it follows by Bezout’s theorem that F1^{′}.(C_{1} ∪ E_{1}) = 2 · q_{1} + q1^{′} + q1^{′′} where F1^{′} is the fibre corresponding to the line \overline{pq} and where q_{1} = σ^{−1}(q) is a branch point of the degree 4 mapping π_{1|C_{1}∪E_{1}} : C_{1} ∪ E_{1} ⟶ ℙ^{1}. Now, the result follows directly from Proposition 3.4.

Let us assume that (2) holds. By Bezout’s theorem, C_{1}.E_{1} = (d − 3) · p_{1} and either (a) F_{1}.(C_{1} ∪ E_{1}) = 2 · p_{1} + p1^{′} + p1^{′′} or (b) F_{1}.(C_{1} ∪ E_{1}) = 2 · p_{1} + 2 · p1^{′}. Hence, p_{1} is a branch point of π_{|C_{1}∪E_{1}} and in case (b) p1^{′} is a branch point π_{|C_{1}∪E_{1}}, too. Now, for (a) the result follows directly from Proposition 3.4.

In order to prove the statement for case (b) we perform elementary transformations of the fibre F_{1}. We use the same notation as in Section 2. For i = 1, …, d − 3 we perform elementary transformations of F_{i} at p_{i} ∈ E_{i} ∩ F_{i}. Then for i < d − 3, E_{i}.C_{i} = (d − 3 − i) · p_{i} and F_{i}.(C_{i} ∪ E_{i}) = 2 · p_{i} + 2 · pi^{′}. If i = d − 3 then we arrive at a Hirzebruch surface X_{d−2} with a ruling π_{d−2} : X_{d−2} ⟶ ℙ^{1} such that C_{d−2}.E_{d−2} = 0 and F_{d−2}.(C_{d−2} ∪ E_{d−2}) = p_{d−2} + pd−2^{′′} + 2 · pd−2^{′}. Hence pd−2^{′} is a branch point of π_{d−2}_{|C_{d−2}∪E_{d−2}}. From Proposition 3.4 it follows π_{∗}(Θ_{X_{d−2}}⟨C_{d−2} ∪ E_{d−2}⟩) = 0. Since π_{∗}(Θ_{X_{1}}⟨C_{1} ∪ E_{1}⟩) is torsion free this sheaf is also zero. Now, the same argument as in the proof of Proposition 3.4 (1) proves the statement for case (b).

Now, we assume that (3) holds. Let n := \frac{d−3}{2}. We have (C_{1} ∪ E_{1}).F_{1} = 3 · p_{1} + p1^{′} and E_{1}.C_{1} = (d − 3) · p_{1}. Now, for i = 1, …, n we perform elementary transformations σ_{i} at

p_{i} ∈ E_{i} ∩ F_{i}. Then, for i < n, E_{i}.C_{i} = (d − 3 − 2i) · p_{1} and (C_{i} ∪ E_{i}).F_{i} = 3 · p_{i} + pi^{′}.

Since we always performed the elementary transformation at a point lying on the horizontal section we arrive at a Hirzebruch surface X_{n+1} with E_{n+1}^{2} = −(n + 1), C_{n+1}.E_{n+1} = 0 and F_{n+1}.(C_{n+1} ∪ E_{n+1}) = 2 · p_{n+1} + pn+1^{′} + pn+1^{′′} where p_{n+1} is a branch point of the ruling π_{n+1} : X_{n+1} ⟶ ℙ^{1}. By Proposition 3.4 it follows π_{∗}(Θ_{X_{n+1}}⟨C_{n+1} ∪ E_{n+1}⟩) = 0. Hence π_{∗}(Θ_{X_{1}}⟨C_{1} ∪ E_{1}⟩) = 0. □

Remark 3.8. (1) If C has at least three cusps and maximal multiplicity deg C − 3 then it follows from Proposition 3.5 that H^{2}(V; Θ_{V} ⟨D⟩) = 0 [4]. By [12] the complement ℙ^{2}\C has logarithmic Kodaira-dimension 2 and therefore by [2], H^{0}(V; Θ_{V} ⟨D⟩) = 0. Hence, we have χ(Θ_{V} ⟨D⟩) ≤ 0.

(2) In order to prove a similar result for curves satisfying the conditions of Corollary 3.7 we need to check that H^{0}(V; Θ_{V} ⟨D⟩) = 0. It is sufficient to show that H^{0}(ℙ^{2}; Θ_{ℙ^{2}}⟨C⟩) = 0, i.e. that there are no infinitesimal automorphisms. Therefore, C must not be an integral curve of a vector field of ℙ^{2}. In [2], Flenner and Zaidenberg computed all vector fields and integral curves of ℙ^{2}. It turned out that the only rational cuspidal curves which occur as integral curves of vector fields of ℙ^{2} are those with affine defining equation u^{r} − cv^{s} = 0, c ∈ ℂ and (r, s) = 1 for suitable affine charts.

Matsuoka and Sakai [7] proved a lower bound for the maximal multiplicity of a rational cuspidal plane curve. The main ingredient in their proof was the well-known log-Miyaoka-Yau inequality [9].

Proposition 3.9. Let C ⊂ ℙ^{2} be a rational cuspidal curve of degree d. Let m_{max} denote the maximum of the multiplicities of its cusps. Then the inequality d < 3m_{max} holds.

4. Proof of Theorem 1.2 and 1.3

Now, we apply the restrictions and formulas for the existence of rational cuspidal plane curves to the case where C has maximal multiplicity m_{max} = deg C − 2 resp. m_{max} = deg C − 3. The pair (V, D) always denotes the SNC-pair coming from the minimal embedded resolution of singularities of C ⊂ ℙ^{2} as described in Section 1. Remember that the Euler characteristic of Θ_{V} ⟨D⟩ is given by the formula

χ(Θ_{V} ⟨D⟩) = −3(deg C − 3) + ∑_{p∈Sing C} χ_{p}

where χ_{p} = M_{p} + ω_{p} − 1 (see Section 2).

Proof of Theorem 1.2. Let C ⊂ ℙ^{2} be a rational cuspidal curve of degree d and let [d − 2, m_{1}, …, m_{r}] be the multiplicity sequence of a cusp p of C. Then by Bezout’s theorem, m_{i} ≤ 2 for i = 2, …, r. Hence, the only possible multiplicity sequences for a cusp with multiplicity d − 2 are [d − 2] and [d − 2, 2_{a}]. If p ∈ Sing C has multiplicity sequence [d − 2] then

M_{p} = d − 3, ω_{p} = d − 3 and χ_{p} = 2d − 7

If p has multiplicity sequence [d − 2, 2_{a}] then by Proposition 3.1

2|d − 2 and d − 2 ≤ 2a or d − 2 = 2a + 1  (†)

and

M_{p} = d − 3 + a, ω_{p} = ⌈\frac{d − 2}{2}⌉ and χ_{p} = d − 4 + a + ⌈\frac{d − 2}{2}⌉ .

If q ∈ Sing C with multiplicity sequence [n_{0}, …, n_{s}] is another cusp then again by Bezout’s theorem, n_{j} ≤ 2 for j = 0, …, s. Hence, the only possible multiplicity sequence is [2_{b}] where

M_{q} = b, ω_{q} = 1 and χ_{q} = b .

By the genus formula, we obtain

d − 2 = \binom{d − 1}{2} − \binom{d − 2}{2} = ∑_{i=1}^{r} \binom{m_{i}}{2} + ∑_{i=0}^{s} \binom{n_{i}}{2} = a + b .

The only possible types of 1- and 2-cuspidal curves with mmax = d − 2 are

{[d − 2, 2_{a}]} where a = d − 2  (1)
{[d − 2], [2_{b}]} where b = d − 2  (2)
{[d − 2, 2_{a}], [2_{b}]} where a + b = d − 2 .  (3)

Case 1. If 1 ≡ d − 2 mod 2 then by (†) d − 2 = 2a + 1. On the other hand, a = d − 2 which is a contradiction. If 0 ≡ d − 2 mod 2 then C must be of type

[2a, 2_{2a}] with 2a = d − 2 .

By Theorem 1.1 (1a), such a curve exists.

Case 2. A curve of type

[a], [2_{a}] with a = d − 2

always exists by Theorem 1.1 (8).

Case 3. If 1 ≡ d − 2 mod 2 then (†) yields d − 3 = 2a. It follows d − 1 = 2b. Hence, this is a curve of type

[2a + 1, 2_{a}], [2_{a+1}] with 2a + 1 = d − 2

which exists by Theorem 1.1 (3). If 0 ≡ d − 2 mod 2 then by (†) 2a ≥ d − 2. Hence, this is a curve of type

[2a′, 2_{a′+b′}], [2_{a′−b′}] with 2a′ = d − 2 and a′ + b′ = a

which exists by Theorem 1.1 (1). This completes the proof. □

Proof of Theorem 1.3. First we determine those types of rational cuspidal curves which are integral curves of ℙ^{2}. By Remark 3.8, their affine defining equation must be of the form u^{d−3} − cv^{d} = 0, c ∈ ℂ for suitable affine charts. It defines a rational cuspidal curve of type

{[3a + 1, 3_{a}], [3_{a+1}]} where 3a + 4 = d

or

{[3a + 2, 3_{a}, 2], [3_{a+1}, 2]} where 3a + 5 = d .

For all other rational cuspidal curves with mmax = d − 3 we have H^{0}(V; Θ_{V} ⟨D⟩) = 0.

By Proposition 3.1, the only possible multiplicity sequences for a cusp with m_{max} = d − 3 are [d − 3], [d − 3, 2_{a}], [d − 3, 3_{a}] or [d − 3, 3_{a}, 2]. If p ∈ Sing C has multiplicity sequence \overline{m}_{p} = [d − 3] then

M_{p} = d − 4, ω_{p} = d − 4 and χ_{p} = 2d − 9 .

If \overline{m}_{p} = [d − 3, 2_{a}] then again by the characteristic property of 3.1

2|d − 3 and d − 3 ≤ 2a or d − 3 = 2a + 1  (∗)

and

M_{p} = d − 4 + a, ω_{p} = ⌈\frac{d − 3}{2}⌉ and χ_{p} = d − 5 + a + ⌈\frac{d − 3}{2}⌉ .

If \overline{m}_{p} = [d − 3, 3_{a}] then

3|d − 3 and d − 3 ≤ 3a or d − 3 = 3a + 1  (∗∗)

and

M_{p} = d − 4 + 2a, ω_{p} = ⌈\frac{d − 3}{3}⌉ + 1 and χ_{p} = d − 4 + 2a + ⌈\frac{d − 3}{3}⌉ .

If \overline{m}_{p} = [d − 3, 3_{a}, 2] then

3|d − 3 and d − 3 ≤ 3a or d − 3 = 3a + 2  (∗ ∗ ∗)

and

M_{p} = d − 3 + 2a, ω_{p} = ⌈\frac{d − 3}{3}⌉ + 1 and χ_{p} = d − 3 + 2a + ⌈\frac{d − 3}{3}⌉ .

Let C ⊂ ℙ^{2} be a rational cuspidal curve of degree d with one cusp of type [d−3, m_{1}, …, m_{r}]. By the genus formula

2d − 5 = \binom{d − 1}{2} − \binom{d − 3}{2} = ∑_{i=1}^{r} \binom{m_{i}}{2}

and by Bézout’s theorem, it follows m_{i} ≤ 3. Thus, the following multiplicity sequences are possible.

[d − 3, 2_{a}] where a = 2d − 5  (1)
[d − 3, 3_{a}] where 3a = 2d − 5  (2)
[d − 3, 3_{a}, 2] where 3a = 2d − 6 .  (3)

Case 1. A curve with a multiplicity sequence as in (1) satisfies condition (1) of Corollary 3.7. Therefore, χ(Θ_{V} ⟨D⟩) ≤ 0. Since a = 2d − 5 and 0 ≡ d − 3 mod 2 we have

χ(Θ_{V} ⟨D⟩) = −3d + 9 + d − 5 + 2d − 5 + \frac{d − 3}{2} = \frac{d − 3}{2} − 1 ≤ 0

Hence, only d = 5 is possible. By Namba [10], a 1-cuspidal rational curve of degree 5 of type [2_{6}] exists.

Case 2. If 0 ≡ d − 3 mod 3 then 3|d. Hence 3 |̸ 2d − 5. Thus, such a curve cannot exist. If 1 ≡ d − 3 mod 3 then a = 2d − 5 and by (∗∗) 2d − 5 = d − 4. But this is impossible.

Case 3. If 0 ≡ d − 3 mod 3 then this is exactly Theorem 1.1 (1a). Hence, such a curve exists. If 2 ≡ d − 3 mod 3 then 3a = 2d − 6 and because of (∗ ∗ ∗) d − 3 = 2d − 4 which is impossible for d ≠ 1. This completes the proof for 1-cuspidal curves.

In order to prove (2) we list all possible types of 2-cuspidal curves of degree d with maximal multiplicity d − 3. If q ∈ Sing C with multiplicity sequence \overline{n}_{q} = [n_{0}, …, n_{s}] is another cusp then again by Bezout’s theorem, n_{i} ≤ 3 for i = 0, …, s. Hence, by Proposition 3.1, the only multiplicity sequences being possible are either \overline{n}_{q} = [2_{b}] where

M_{q} = b, ω_{q} = 1 and χ_{q} = b

or \overline{n}_{q} = [3_{b}] where

M_{q} = 2b, ω_{q} = 2 and χ_{q} = 2b + 1

or \overline{n}_{q} = [3_{b}, 2] where

M_{q} = 2b + 1, ω_{q} = 2 and χ_{q} = 2b + 2 .

We obtain the following list of possible types of 1- and 2-cuspidal curves with maximal multiplicity m_{max} = d − 3.

{[d − 3], [2_{b}]} where b = 2d − 5  (1)
{[d − 3], [3_{b}]} where 3b = 2d − 5  (2)
{[d − 3], [3_{b}, 2]} where 3b = 2d − 6  (3)
{[d − 3, 2_{a}], [2_{b}]} where a + b = 2d − 5  (4)
{[d − 3, 2_{a}], [3_{b}, 2]} where a + 3b = 2d − 6  (5)
{[d − 3, 2_{a}], [3_{b}]} where a + 3b = 2d − 5  (6)

{[d − 3, 3_{a}], [2_{b}]} where 3a + b = 2d − 5  (7)
{[d − 3, 3_{a}, 2], [2_{b}]} where 3a + b = 2d − 6  (8)
{[d − 3, 3_{a}], [3_{b}]} where 3a + 3b = 2d − 5  (9)
{[d − 3, 3_{a}, 2], [3_{b}]} where 3a + 3b = 2d − 6  (10)
{[d − 3, 3_{a}], [3_{b}, 2]} where 3a + 3b = 2d − 6  (11)
{[d − 3, 3_{a}, 2], [3_{b}, 2]} where 3a + 3b = 2d − 7 .  (12)

Curves with multiplicity sequences of type (1), (2) or (3) satisfy condition (2) of Corollary 3.7. In these cases, χ(Θ_{V} ⟨D⟩) ≤ 0.

Case 1. We have

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (2d − 9) + (2d − 5) = d − 5 ≤ 0

It follows that for d > 5 such a curve cannot exist. But for d = 5, such a curve does not exist either ([10]).

Case 2. Since 3a = 2d − 5 we have 3|2d − 5, i.e. 1 ≡ d mod 3. The Euler-characteristic is

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (2d − 9) + (2\frac{2d − 5}{3} + 1) = \frac{d − 7}{3} ≤ 0

For d > 7 such a curve cannot exist. For d = 7 it is a curve with two cusps of type

[4] , [3_{3}] .

This is a curve of type [da, d_{a+1}], [(d + 1)_{a}] with d = 3 and a = 1. By Theorem 1.1 (4) such a curve exists.

Case 3. We have 3a = 2d − 6. Hence 3|2d − 6, i.e. 0 ≡ d mod 3. The Euler-characteristic is

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (2d − 9) + (2\frac{2d − 6}{3} + 2) = −d + \frac{4d − 12}{3} + 2 = \frac{d − 6}{3} ≤ 0

For d > 6 such a curve cannot exist. For d = 6 such a curve exists by Theorem 1.1. Indeed, their multiplicity sequences are

[3] , [3_{2}, 2]

which is a curve of type [da, d_{2a−1}, d − 1], [d] for d = 3 and a = 1 which exists by Theorem 1.1 (1).

Case 4. From the genus formula we have a + b = 2d − 5. From Corollary 3.7 (1) it follows χ(Θ_{V} ⟨D⟩) ≤ 0 if b > 1. If b = 1 then a = 2d − 6 and hence 0 ≡ d − 3 mod 2. Therefore χ(Θ_{V} ⟨D⟩) ≤ 0 by Corollary 3.7 (3). We compute the Euler-characteristic:

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (d − 5 + a + ⌈\frac{d − 3}{2}⌉) + (b) = ⌈\frac{d − 3}{2}⌉ − 1 ≤ 0 .

Let 0 ≡ d − 3 mod 2. Then χ(Θ_{V} ⟨D⟩) = \frac{d−3}{2} − 1 > 0 for d ≥ 7.

If 1 ≡ d − 3 mod 2 then χ(Θ_{V} ⟨D⟩) = \frac{d−2}{2} − 1 > 0 for d ≥ 6. By [10], for d = 5 there is a curve with two cusps of type

[2_{4}] , [2_{2}] .

For curves with multiplicity sequences of type (5) or (6) we have χ(Θ_{V} ⟨D⟩) ≤ 0 if 0 ≡ d − 3 mod 2 (Corollary 3.7 (3)).

Case 5. First, let 1 ≡ d − 3 mod 2. Then, since (∗) and a + 3b = 2d − 6 holds we have

2a = d − 4 and 6b = 3d − 8

But 3 |̸ 3d − 8. A contradiction. Hence, such a curve does not exist.

Now, let 0 ≡ d − 3 mod 2. Taking into account that a + 3b = 2d − 6 we obtain

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (d − 5 + \frac{d − 3}{2} + a) + (2b + 2) = \frac{d − 3}{2} − b ≤ 0 .

On the other hand, (∗) holds. We obtain the inequality

3b = 2d − 6 − a ≤ \frac{4d − 12}{2} − \frac{d − 3}{2} = \frac{3d − 9}{2} .

Putting these inequalities together we get

3d − 9 ≤ 6b ≤ 3d − 9 .

Hence, only 2b = d − 3 is possible and therefore 2a = d − 3. This is a rational cuspidal curve with two cusps of type

[2a, 2_{a}] , [3_{a}, 2] for a ≥ 1

By Theorem 1.1 (5), such a curve exists.

Case 6. Let 1 ≡ d − 3 mod 2. Then 2a = d − 4 and 2b = d − 2. A curve with cusps of type

[2a + 1, 2_{a}] , [3_{a+1}] for a ≥ 1

exists by Theorem 1.1 (6).

Now let 0 ≡ d − 3 mod 2. Since 2a ≥ d − 3 and a + 3b = 2d − 5 we have

3b = 2d − 5 − a ≤ \frac{4d − 10}{2} − \frac{d − 3}{2} = \frac{3d − 7}{2} .

The Euler-characteristic is

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (d − 5 + 2d − 5 − 3b + \frac{d − 3}{2}) + (2b + 1) = \frac{d − 3}{2} − b ≤ 0 .

Therefore

3d − 9 ≤ 6b ≤ 3d − 7 .

Only 2b = d − 3 is possible. Then 2a = d − 1. Hence, the multiplicity sequences are

[2a, 2_{a+1}] , [3_{a}] for a ≥ 1 .

By Theorem 1.1 (4) such a curve exists.

Case 7. First, let 1 ≡ d − 3 mod 3. Then 3a = d − 4 and b = d − 1. We have

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (d − 4 + 2\frac{d − 4}{3} + \frac{d − 1}{3}) + (d − 1) = 1 .

Since b > 1 it follows from Corollary 3.7 (1) that χ(Θ_{V} ⟨D⟩) ≤ 0. A contradiction. Hence, such a curve does not exist. The case 0 ≡ d − 3 mod 3 is treated in the next Lemma.

Case 8. If 2 ≡ d − 3 mod 3 then 3a = d − 5 and b = d − 1. Hence

χ(Θ_{V} ⟨D⟩) = −3d + 9 + (d − 3 + 2\frac{d − 5}{3} + \frac{d − 2}{3}) + (d − 1) = 1 .

As in Case 7 b > 1 implies χ(Θ_{V} ⟨D⟩) ≤ 0 contradicting the upper computation. The case 0 ≡ d − 3 mod 3 is treated in the next Lemma.

Case 9. Let 0 ≡ d − 3 mod 3. From the genus formula it follows 3(a + b) = 2d − 5, i.e. 3|2d − 5 which implies 1 ≡ d mod 3. A contradiction.

If 1 ≡ d − 3 mod 3 then 3a = d − 4 and 3b = d − 1. We obtain a curve with cusps of type

[3a + 1, 3_{a}] , [3_{a+1}] for a ≥ 1

of degree 3a + 4 which clearly exists. By Theorem 1.1 (3) such a curve exists.

Case 10. If 2 ≡ d − 3 mod 3 then 3a = d − 5, i.e. 3|d − 5 and from the genus formula, it follows 3b = 2d − 6 − d + 5 = d − 1, i.e. 3|d − 1 which is a contradiction.

If 0 ≡ d − 3 mod 3 then because of (∗∗) 3a ≥ d − 3. These are curves of type

[3a′, 3_{a′+b′}, 2] , [3_{a′−b′}] with 3a′ = d − 3 and a′ + b′ = a .

By Theorem 1.1 (1) those curves exist.

Case 11. If 1 ≡ d − 3 mod 3 then 3a = d − 4, i.e. 3|d − 4 and from the genus formula, it follows 3b = 2d − 6 − d + 4 = d − 2, i.e. 3|d − 2 which is a contradiction.

If 0 ≡ d − 3 mod 3 then because of (∗∗) 3a ≥ d − 3. These are curves of type

[3a′, 3_{a′+b′}] , [3_{a′−b′}, 2] with 3a′ = d − 3 and a′ + b′ = a .

By Theorem 1.1 (2) those curves exist.

Case 12. Let 0 ≡ d − 3 mod 3. We have 3a + 3b = 2d − 7 by the genus formula. Hence, 3|2d − 7 which contradicts the assumption.

Otherwise, if 2 ≡ d − 3 mod 3 then 3a = d − 5 and 3b = d − 2: a curve with two cusps of type

[3a + 2, 3_{a}, 2] , [3_{a+1}, 2] .

By Theorem 1.1 (7) those curves exist.

We complete the proof with the following lemma.

Lemma 4.1. Let d ≡ 0 mod 3 and 0 < b ≤ a be positive integers.

(1) A curve of degree d and of type [d − 3, 3_{a}], [2_{b}] exists if and only if b = 1.

(2) A curve of degree d of type [d − 3, 3_{a}, 2], [2_{b}] does not exist.

Proof. As in the proof of Theorem 3.1 we transform the curves by a suitable Cremona transformation. We use the notation of Section 2.

(1) Suppose, C is a curve of type [d − 3, 3_{a}], [2_{b}]. Let p_{0} ∈ C be the cusp of type [d − 3, 3_{a}] and q_{0} ∈ C be the cusp of type [2_{b}].

(a) If a = \frac{d−3}{3}, then b = d − 2. Perform the blowing up σ_{0} at p_{0}. Let F_{1} resp. F′_{1} be the proper transform of T_{p}C resp. L = \overline{pq}. Then by Bezout’s theorem, C_{1}.F_{1} = 3 · p_{1}, C_{1}.F′_{1} = 2 · q_{1} + q′_{1} and C_{1}.E_{1} = (d − 3) · p_{1}. C_{1} has two cusps, one at p_{1} of type [3_{a}] and one at q_{1} of type [2_{b}]. For i = 1, …, a + 2, we perform elementary transformations σ_{i} at p_{i} ∈ E_{i} ∩ F_{i} followed by elementary transformations σ_{i} at q_{i} ∈ C_{i} for i = a + 3, …, 2a + 4. We arrive at X_{2a+5} with E_{2a+5}^{2} = −1. Then C_{2a+5}.F_{2a+5} = 3 · p_{2a+5}, C_{2a+5}.F′_{2a+5} = 2 · q_{2a+5} + q′_{2a+5} and C_{2a+5}.E_{2a+5} = (a+2)·q′_{2a+5}. C_{2a+5} has a cusp of type [3_{2}] at p_{2a+5} and a cusp of type [2_{b−a−2}] at q_{2a+5}. Note that b − a − 2 > 0 if and only if d ≥ 5. Hence, after blowing down E_{2a+5} we arrive at ℙ^{2} and the proper transform C′ is a rational cuspidal curve of type [a + 2], [3_{2}] and [2_{d−a−4}] of degree a + 5. By [4], such a curve cannot exist.

(b) Let a > \frac{d−3}{3} =: ω. We start as in the case before. But then for i = 1, …, ω, σ_{i} is an elementary transformation of F_{i} at p_{i} ∈ E_{i} ∩ F_{i} followed by an elementary transformation σ_{i} of F′_{i} at q′_{i} for i = ω + 1, …, 2ω. Then, we arrive at X_{2ω+1} and C_{2ω+1}.F_{2ω+1} = 3p_{2ω+1}, C_{2ω+1}.F′_{2ω+1} = 2q_{2ω+1} + q′_{2ω+1} and C_{2ω+1}.E_{2ω+1} = 2ωq_{2ω+1}. C_{2ω+1} is a curve with a cusp of type [3_{a−ω}] at p_{ω+1} and a cusp of type [2_{b+ω}]. After blowing down E_{ω+1} we arrive at ℙ^{2} and the proper transform C′ is a rational cuspidal curve of type [2ω, 2_{b+ω}] and [3_{a−ω}] and degree 2ω + 3. But this is Case 6. Hence

a − ω + 1 = b + ω .

Moreover, the equality 3a + b = 2d − 5 holds. b = 1 and 3a = 2d − 6 is the unique solution. Therefore such a curve must be of type

[3a, 3_{2a}] , [2] with a ≥ 1  (13)

which exists by Theorem 1.1 (1a).

(2) Suppose such a curve exists. Let ω := \frac{d−3}{3}. We perform the same transformations as described in (1) (b). We obtain the following rational cuspidal plane curves of degree 2ω + 3.

[2] , [2ω, 2_{b+ω}] if a = ω  (14)

and

[3_{a−ω}, 2] , [2ω, 2_{b+ω}] if a > ω .  (15)

(14) is Case 4 of the proof. Hence such a curve cannot exist. (15) is Case 5. Such a curve exists if and only if

b + ω = a − ω .

The equality 3a + b = 2d − 6 also holds. The unique solution is b = 0 and 3a = 2d − 6 which is a contradiction to b > 0. Hence, such a curve cannot exist. □

5. Parameterization and defining equations of the rational cuspidal plane curves of degree 6

Finally, we will give the parameterizations and defining equations of the curves of Corollary 1.5 up to projective equivalence.

1-cuspidal curves:

1. C_{1} is projectively equivalent to one of the curves

(a) Ψ(s, t) = (st^{5} : t^{6} : s^{6}) resp.

f = zy^{5} − x^{6}

(b) Ψ(s, t) = (st^{5} : t^{6} : s^{6} + s^{5}t + as^{4}t^{2} + bs^{3}t^{3}) resp.

f = x^{6} + x^{5}y + ax^{4}y^{2} + bx^{3}y^{3} − y^{5}z

(c) Ψ(s, t) = (st^{5} : t^{6} : s^{6} + as^{5}t + s^{4}t^{2} + bs^{3}t^{3}) resp.

f = x^{6} + ax^{5}y + x^{4}y^{2} + bx^{3}y^{3} − y^{5}z

(d) Ψ(s, t) = (st^{5} : t^{6} : s^{6} + as^{5}t + bs^{4}t^{2} + s^{3}t^{3}) resp.

f = x^{6} + ax^{5}y + bx^{4}y^{2} + x^{3}y^{3} − y^{5}z

with a, b ∈ ℂ.

2. C_{2} is projectively equivalent to one of the curves

(a) Ψ(s, t) = (s^{2}t^{4} + st^{5} : t^{6} : s^{6} + 3s^{5}t + as^{4}t^{2} + (2a − 5)s^{3}t^{3} + bst^{5}) resp.

f = x^{6} + (2a − 6)x^{5}y + (a^{2} − 8a + 15)x^{4}y^{2} + (−2a^{2} + 11a − b − 15)x^{3}y^{3}
+ (−ab + 3b)x^{2}y^{4} + (−ab − b^{2} + 3b)xy^{5} − 2x^{3}y^{2}z + (6 − 2a)x^{2}y^{3}z
+ (2a − 6)xy^{4}z + (a + b − 3)y^{5}z + y^{4}z^{2}

(b) Ψ(s, t) = (s^{2}t^{4} : t^{6} : s^{6} + as^{4}t^{2} + st^{5}) resp.

f = −x^{6} − 2ax^{5}y − a^{2}x^{4}y^{2} + xy^{5} + 2x^{3}y^{2}z + 2ax^{2}y^{3}z − y^{4}z^{2}

with a, b ∈ ℂ.

3. C_{3} is projectively equivalent to one of the curves

(a) Ψ(s, t) = (s^{3}t^{3} : t^{6} : s^{6} + bst^{5}) resp.

f = −x^{6} − b^{3}xy^{5} + 3x^{4}yz − 3x^{2}y^{2}z^{2} + y^{3}z^{3}

(b) Ψ(s, t) = (s^{3}t^{3} + st^{5} : t^{6} : s^{6} + 2s^{4}t^{2} + s^{2}t^{4} + bst^{5}) resp.

f = x^{6} + b^{2}x^{2}y^{4} + b^{3}xy^{5} − 3x^{4}yz − b^{2}y^{5}z + 3x^{2}y^{2}z^{2} − y^{3}z^{3}

(c) Ψ(s, t) = (s^{3}t^{3} + s^{2}t^{4} + ast^{5} : t^{6} : s^{6} + (1 + 2a)s^{4}t^{2} + (a^{2} − 2a)s^{2}t^{4} + bst^{5}) resp.

f = − x^{6} + 6ax^{5}y + (b − 10a^{2})x^{4}y^{2} − 4abx^{3}y^{3}

+ (8a^{4} − 4a^{5} − 4a^{3}b + 4a^{2}b − ab^{2})x^{2}y^{4} − (4a^{4}b + 4a^{2}b^{2} + b^{3})xy^{5} + 3x^{4}yz − 12ax^{3}y^{2}z + (8a^{2} − 2b)x^{2}y^{3}z + (8a^{3} + 4ab)xy^{4}z + (4a^{5} + 4a^{3}b + ab^{2})y^{5}z − 3x^{2}y^{2}z^{2} + 6axy^{3}z^{2} + (2a^{2} + b)y^{4}z^{2} + y^{3}z^{3}

with a, b ∈ ℂ.

2-cuspidal curves:

1. C_{4} is projectively equivalent to one of the curves

(a) Ψ(s, t) = (t^{6} : s^{3}t^{3} + s^{2}t^{4} : s^{6} + 2s^{5}t + s^{4}t^{2} + as^{3}t^{3}) with a ≠ 0 resp.

\begin{aligned}f = & − a^{3}x^{4}y^{2} − 2a^{2}x^{3}y^{3} − ax^{2}y^{4} − y^{6} + 2a^{2}x^{4}yz + 2ax^{3}y^{2}z \\ & + 3xy^{4}z − ax^{4}z^{2} − 3x^{2}y^{2}z^{2} + x^{3}z^{3} \end{aligned}

(b) Ψ(s, t) = (t^{6} : s^{3}t^{3} : s^{6} + s^{2}t^{4}) resp.

f = x^{4}y^{2} + y^{6} − 3xy^{4}z + 3x^{2}y^{2}z^{2} − x^{3}z^{3}

2. C_{5} is projectively equivalent to Ψ(s, t) = (t^{6} : s^{3}t^{3} : s^{6} + s^{4}t^{2}) resp.

f = −y^{6} + 3xy^{4}z − 3x^{2}y^{2}z^{2} + x^{3}z^{3} − x^{2}y^{4}

3. C_{6} is projectively equivalent to Ψ(s, t) = (t^{6} : s^{3}t^{3} : s^{6} + s^{5}t) resp.

f = x^{3}z^{3} − xy^{5} − y^{6} + 3xy^{4}z − 3x^{2}y^{2}z^{2}

4. C_{7} is projectively equivalent to Ψ(s, t) = (t^{6} : s^{2}t^{4} : s^{6} + as^{4}t^{2} + s^{3}t^{3}) with a ≠ 0 resp.

f = x^{3}y^{3} − a^{2}x^{2}y^{4} − 2axy^{5} − y^{6} + 2ax^{3}y^{2}z + 2x^{2}y^{3}z − x^{4}z^{2}

5. C_{8} is projectively equivalent to Ψ(s, t) = (t^{6} : s^{2}t^{4} : s^{6} + s^{5}t + as^{4}t^{2}) with a ≠ \frac{1}{4} resp.

f = (1 − 2a)xy^{5} − y^{6} + 2ax^{3}y^{2}z + 2x^{2}y^{3}z − x^{4}z^{2} − a^{2}x^{2}y^{4}

6. C_{9} is projectively equivalent to Ψ(s, t) = (st^{5} + t^{6} : s^{2}t^{4} : s^{6} − 2s^{5}t + 2s^{4}t^{2}) resp.

f = 4xy^{5} − 4x^{2}y^{4} − 5y^{6} + 4x^{3}y^{2}z − 2x^{2}y^{3}z + 2xy^{4}z + y^{5}z − x^{4}z^{2}

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Received February 11, 1998