On dicritical foliations and Halphen pencils

Luís Gustavo Mendes and Paulo Sad

Abstract

The aim of this article is to provide information on the number and on the geometrical position of singularities of holomorphic foliations of the projective plane. As an application it is shown that certain foliations are in fact Halphen pencils of elliptic curves. The results follow from Miyaoka’s semipositivity theorem, combined with recent developments on the birational geometry of foliations.

1 Introduction

We consider in this paper holomorphic foliations of the complex projective plane and study the problem of giving bounds to the number of singularities (counted with multiplicities) in terms of the degree of the foliation. There are different possibilities for the concept of multiplicity of a singularity. For example, if we take as multiplicity the Milnor number of the singularity, the classical Darboux’s Theorem states that the sum of Milnor numbers of all the singularities of a foliation of degree d is equal to d^{2} + d + 1. When we consider a different notion of multiplicity (cf. Definition 1), we obtain in Theorem 1 (under certain conditions) sharper estimations for the number of singularities and information on their geometrical position in the plane, as a consequence of Miyaoka’s Theorem ([Miy], [McQ1]).

The paper deals also with the question of finding global rational first integrals for dicritical foliations, i.e. foliations having dicritical singularities (in other words, the question of deciding when a dicritical foliation is a pencil of algebraic curves). We remark that meromorphic first integrals may not exist even locally in the case of dicritical singularities (cf. [Su], [K]). Also we recall that the presence of dicritical singularities makes it impossible in general to relate data of a dicritical foliation to data of invariant algebraic curves (for example, the so called Poincaré problem has in general a negative answer, see e.g. [LN]). Nevertheless, using Theorem 1 and a theorem of [McQ2] (see also the exposition [Br2]) we show in Theorem 2 that for special dicritical foliations the data of the foliation and of the invariant curves do agree, yielding then the existence of global first integrals.

Let us start with a definition analogous to the definition of algebraic multiplicity of a curve at a point.

Definition 1. Let G be a foliation on the surface M induced locally by η = 0 at the singularity p, where η is a holomorphic 1-form with an isolated zero. Let σ be the blow up of M at p. Then we define l(p) = l(p, G) as the vanishing order of σ^{∗}(η) along the exceptional divisor σ^{−1}(p).

This number can be easily computated from the vanishing order of η at the point p ∈ M. We remark that l(p) = 1 for reduced singularities (in Seidenberg’s sense [Se]), although the Milnor number may be arbitrarily high.

Let us consider a foliation F on CP^{2} and perform a sequence of blowup’s σ_{1}, …, σ_{k} in order to have only reduced singularities for σ^{∗}F , where σ = σ_{1} ∘ ⋯ ∘ σ_{k}. Starting with some singularity p_{0} of F_{0} = F, let p_{j} be the singularity of F_{j} = (σ_{1} ∘ … ∘ σ_{j})^{∗}F blown up by σ_{j+1} and l(p_{j}) = l(p_{j}, F_{j}). We define

L(F) := ∑_{j=0}^{k−1}(l(p_{j}) − 1)^{2}.

Given a reduced algebraic curve C ⊂ CP^{2}, let us define also

L(F, C) := ∑_{j=0}^{k−1} ν_{p_{j}} · (l(p_{j}) − 1),

where ν_{p_{j}} is the multiplicity of (the strict transform of) C at p_{j}. These integers will be used to obtain information on the number and on the position of the singularities of F.

Definition 2. A curve S ⊂ M is a special invariant curve for a foliation G when S is smooth, isomorphic to CP^{1}, G-invariant, Sing(G) ∩ S = {q} and there are local coordinates for which q = (0, 0) and f(x, y) = x^{m} · y (m ∈ N) is a local holomorphic first integral for G.

Special invariant curves necessarily appear in the study of certain singularities, e.g. in the reduction of the cusp d(y^{2} − x^{3}) = 0.

Theorem 1. Let F be a foliation of CP^{2} of degree d(F) ≥ 1 and C ⊂ CP^{2} be an algebraic curve of degree d(C). Suppose that F has a reduction of singularities free of special invariant curves. Then

L(F) ≤ (d(F) − 1)^{2}.

and, for d(F) ≥ 1,

L(F, C) ≤ d(C) · (d(F) − 1).

Theorem 2 below shows that both inequalities cannot be improved.

We remark that the simplest dicritical singularity is the radial one, locally induced by η(x, y) = xdy − ydx + h.o.t. = 0; we easily see that l(p) = 2 at a radial point. More generally, we will consider the following singularities:

Definition 3. A dicritical singularity is removable by one blow up if the exceptional divisor introduced by blowing it up is not invariant for the transformed foliation and contains only singularities which admit local holomorphic first integral.

We recall that a Halphen pencil of index l − 1 (l ∈ N, l ≥ 2) is a pencil of elliptic curves of degree 3(l − 1) with 9 base points (possibly infinitely near). It is well-known that Halphen pencils of index l − 1 ≥ 2, contain a unique (reduced) cubic curve, which is smooth at the base points and is the support of a non-reduced curve of the pencil with multiplicity l − 1 (cf. [C-D]).

Theorem 2. Let l ≥ 2. Consider a foliation F of CP^{2} which has a reduction of singularities free of special invariant curves. Suppose also that the singular set of F is composed by singularities p_{1}, …, p_{k} which are removable by one blow up and singularities which have local holomorphic first integral. If d(F) = 3l − 2 and l(p_{i}) = l for all 1 ≤ i ≤ k, then F is a Halphen pencil of index l − 1.

Along the proof of Theorem 2, it will become clear that the hypotheses lead to ”generic” Halphen pencils, in the sense that there are 9 distinct base points and the elliptic fibration associated to the pencil has singular fibers which are cycles of rational curves.

One should be aware of examples where special invariant curves are present and both Theorems 1 and 2 do not hold. One of them is the pencil of rational curves generated by two smooth conics; the degree of the associated foliation is two, there are 4 radial points and 3 reduced singularities. In this example, there are two straight lines in the pencil whose strict transforms under blow up’s at the base points are special invariant curves. Contrasting with this example, when applied to a foliation whose singular set is composed by a) k radial singularities and b) reduced singularities, Theorem 1 yields k ≤ (d(F) − 1)^{2}.

Another example can be found in [LN]: foliations of degree four, whose singular set is composed by 12 radial points and 9 reduced singularities with local holomorphic first integral. Contrasting with the second assertion of Theorem 1, in such examples 4 radial points belong to an invariant straight line, 7 radial points belong to a (degenerate) invariant conic, 10 radial points belong to a (degenerate) invariant cubic. The strict transforms of such curves are special curves. Among these examples one finds pencils of elliptic curves with arbitrarily high degree generic curves and foliations without global rational first integrals, although the singularities have local holomorphic or meromorphic first integral.

The first author thanks I. Pan for useful conversations. He also thanks the support of the “Instituto de Matemática Pura e Aplicada - IMPA” and the “Pontifícia Universidade Católica do Rio de Janeiro- PUC”, where he started working in the present paper.

2 Preliminaries

We will collect some facts involving line bundles associated to foliations (see [Br1] and [Br2] for more information).

Let S be an irreducible compact curve and G a foliation of a compact complex surface M.

If S is not invariant by G, then

T_{G}^{∗} · S = tang(G, S) − S · S,

where T_{G}^{∗} is the cotangent bundle of G, tang(G, S) ≥ 0 is the sum of the orders of tangencies between the foliation and the curve and S · S is the self-intersection number of S.

When S is G-invariant, we get

T_{G}^{∗} · S = GSV (G, S) + 2p_{a}(S) − 2,

where GSV (G, S) is the sum of the local indices defined in [G-S-V] and p_{a}(C) is the arithmetical genus.

Next comes the formula for the variation of the cotangent bundle T_{G}^{∗} under a blow up π : N → M at a point p:

T_{π^{∗}G}^{∗} = π^{∗}(T_{G}^{∗}) ⊗ O_{N}(−(l(p) − 1)E_{p}),

where E_{p} is the exceptional divisor. We will need also to deal later with the normal bundle N_{G} of the foliation G; its variation under blow up’s is given by

N_{π^{∗}G} = π^{∗}(N_{G}) ⊗ O_{N}(−l(p)E_{p}).

Recall that a line bundle (divisor) D over a surface is nef (numerically effective) if D · S ≥ 0 for any irreducible curve S of the surface.

We say that a line bundle (divisor) L is pseudo-effective if L · D ≥ 0 for any nef line bundle D. A fundamental fact in the study of holomorphic foliations is Miyaoka’s Theorem ([Miy], [McQ1]), which can be stated for algebraic surfaces as : if G is not a foliation by rational curves then T_{G}^{∗} is pseudo-effective.

We will need the notions of Kodaira dimension k(G) and numerical Kodaira dimension ν(G) of a reduced foliation G. The reader may consult [Men],

[McQ2] and [Br2] for main properties.

We assign to G a non negative Kodaira dimension if h^{0}(M, T_{G}^{∗⊗m}) ≥ 1 for some n ∈ N; we define k(G) = k ≥ 1 if h^{0}(M, T_{G}^{∗⊗n}) = k+1, for some n ∈ N, and there are global sections s_{0}, ⋯ s_{k} of T_{G}^{∗⊗n} giving rise to k algebraically independent rational functions of M:

f_{1} = \frac{s_{1}}{s_{0}} , ⋯, f_{k} = \frac{s_{k}}{s_{0}} .

For completeness, k(G) := −∞ if h^{0}(M, T_{G}^{∗⊗n}) = 0 ∀n ≥ 1.

We recall the Zariski decomposition of a pseudo-effective divisor. Supposing that T_{G}^{∗} is pseudo-effective, there exists a numerical decomposition

T_{G} = P + N,

where i) some integral multiple of P is a nef divisor; ii) N is a uniquely defined Q^{+}-divisor, whose support is contractible to a normal singularity of a surface and iii) P · N = 0. This enable us to define ν(G) (without ambiguities) as

1) ν(G) = −∞ if G is a rational fibration.

2) ν(G) = 0 if P is numerically trivial.

3) ν(G) = 1 if P is not numerically trivial but P · P = 0.

4) ν(F) = 2 if P · P > 0.

Both dimensions are invariants of reduced foliations under birational transformations between smooth surfaces (cf. [Men]) or normal surfaces (cf. [McQ2]). It follows from a Theorem of [McQ2] that, when k(G) ≥ 0 one has

k(G) = ν(G).

Finally, given a foliation F of the projective plane CP^{2}, there are isomorphisms

T_{F}^{∗} = O_{CP^{2}}(d(F) − 1) and N_{F} = O_{CP^{2}}(d(F) + 2).

3 Proof of Theorem 1

Let us start giving an alternative description for the special invariant curves.

Lemma 1. Let G be a foliation with reduced singularities on a projective surface, whose cotangent bundle T_{G}^{∗} is pseudo-effective. Then, an irreducible curve S is a special invariant curve if and only T_{G}^{∗} · S < 0. Proof. The only part is immediate from the intersection formula given in the Preliminaries.

Let us suppose that T_{G}^{∗} · S < 0 and prove that S is a special invariant curve for G. Since T_{G}^{∗} is pseudo-effective, then S · S < 0.

We claim that S is G-invariant. Otherwise we would have

T_{G}^{∗} · S = tang(G, S) − S · S

and T_{G}^{∗} · S > 0. Now we assert that S is smooth and isomorphic to CP^{1}. One has

T_{G}^{∗} · S = GSV (G, S) + 2p_{a}(S) − 2,

and GSV (G, S) ≥ 0 because G has reduced singularities; thus p_{a}(S) ≤ 0. Since S is an irreducible curve, p_{a}(S) ≥ 0, that is, p_{a}(S) = 0 and this implies that S is in fact a smooth curve, isomorphic to CP^{1} (cf. [B-P-V], pg. 68). It follows from T_{G}^{∗} · S < 0 and p_{a}(S) = 0 that GSV (G, S) ∈ {0, 1}. Let us remark that GSV (G, S) = 0 occurs only when there are no singularities of the foliation along the smooth curve S, which is impossible in the present situation because S · S < 0. We consider finally the case GSV (G, S) = 1, that is, Sing(G) ∩ S = {q} and the vanishing order of G along C is one. Since the leaf S \ {q} is the complex line C, the group of holonomy of this leaf is trivial. Therefore q is not a saddle-node, q is linearizable and admits a local holomorphic first integral f(x, y) = x^{m} · y (m ∈ N) where (x, y) are the coordinates after linearization (cf. [M-M]). Consequently, S is a special invariant curve for G.

Let us then prove Theorem 1. Consider a foliation F of the projective plane which has a reduction of singularities σ^{∗}F free of special invariant curves.

We claim that Miyaoka’s Theorem implies that T_{σ^{∗}F}^{∗} is a pseudoeffective line bundle. The reason is that σ^{∗}F is not a rational fibration: in fact, except for the case d(F) = 0 , the reduction of singularities of any pencil of plane rational curves is a rational fibration having at least one singular fiber (see e.g. [Br1]). Now, a singular fiber of a rational fibration is a tree of rational curves (see [B-P-V], pg. 142), so in this fiber there exists necessarily a special invariant curve. But we can suppose d(F) ≥ 1 (for d(F) = 0 the first assertion of Theorem 1 is trivial) and σ^{∗}F has no special invariant curves.

We may therefore apply Lemma 1 and conclude that T_{σ^{∗}F}^{∗} is a nef line bundle. Since T_{σ^{∗}F}^{∗} is simultaneously nef and pseudo-effective divisor, we have

T_{σ^{∗}F}^{∗} · T_{σ^{∗}F}^{∗} ≥ 0.

In order to compute this number from data of the foliation, we use the isomorphism

T_{F}^{∗} = O_{CP^{2}}(d(F) − 1)

and the formula for the variation of the cotangent bundle under a blow up (cf Preliminaries). Applying repeatedly this formula and using the definition of L(F), one gets

T_{σ^{∗}F}^{∗} · T_{σ^{∗}F}^{∗} = (d(F) − 1)^{2} − L(F),

so that the inequality L(F) ≤ (d(F − 1)^{2} follows.

Let now \overline{C} := σ^{∗}(C) − ∑_{j} ν_{p_{j}}E_{j} be the strict transform of a plane curve C by the reduction of singularities ; E_{j} stands for the exceptional divisor introduced at the (j-1)-th step of the desingularization. We have then

T_{σ^{∗}F}^{∗} · \overline{C} = d(C) · (d(F) − 1) − L(F, C).

Since T_{σ^{∗}F}^{∗} is a nef divisor, we conclude that L(F, C) ≤ d(C) · (d(F) − 1), as desired.

4 Relation to Halphen pencils

In order to prove Theorem 2, we will need some Lemmata. Let us take once more a plane projective foliation F whose resolution of singularities σ^{∗}F is free of special invariant curves; we assume also that its singularities are a) dicritical singularities p_{1}, …, p_{k} removable by one blow up and b) singular points with local holomorphic first integral.

Lemma 2. Let l ≥ 2. The following conditions are equivalent.

i) d(F) = 3l − 2 and l(p_{i}) = l for 1 ≤ i ≤ k.

ii) k = 9, ∑_{i=1}^{9}(l(p_{i}) − 1)^{2} = (d(F) − 1)^{2} and ∑_{i=1}^{9}(l(p_{i}) − 1) = 3(d(F) − 1).

Proof. Suppose that i) holds. The resolution of singularities σ : N → CP^{2} of F can be regarded as a composition σ = σ_{2} ∘ σ_{1}, where σ_{1} : N′ → CP^{2} denotes the composition of k blow ups at p_{1}, ⋯, p_{k}.

According to Baum-Bott’s formula (cf. [B-B] or [Br1]):

N_{σ_{1}^{∗}F} · N_{σ_{1}^{∗}F} = ∑_{q∈N′} BB(σ_{1}^{∗}F, q),

where BB(σ_{1}^{∗}F, q) ∈ C is the Baum-Bott index at a singularity q of σ_{1}^{∗}F. By the hypothesis on the singular set of F, all the singularities of σ_{1}^{∗}F have local holomorphic first integrals. It is well-known that in this case BB(σ_{1}^{∗}F, q) is a non-positive integer; we conclude that

N_{σ_{1}^{∗}F} · N_{σ_{1}^{∗}F} ≤ 0.

In order to compute this number, we proceed in the same way as in the proof of Theorem 1. One has the isomorphism

N_{F} = O_{CP^{2}}(d(F) + 2)

and the formula for the variation of the normal bundle of a foliation under a blow up (cf. Preliminaries). Applying repeatedly this formula one gets

N_{σ^{∗}F} · N_{σ^{∗}F} = (d(F) + 2)^{2} − ∑_{i=1}^{k} l(p_{i})^{2},

so that

(d(F) + 2)^{2} − ∑_{i=1}^{k} l(p_{i})^{2} ≤ 0.

Since l(p_{i}) = l (for i = 1, ⋯, k), we get^{1}:

(d(F) + 2)^{2} ≤ kl^{2},

that is, 9 ≤ k, since d(F) = 3l −2. By another side, according to Theorem 1:

∑_{i=1}^{k}(l(p_{i}) − 1)^{2} ≤ (d(F) − 1)^{2},

1This inequality is already found in [Po] in the particular case when F is a pencil of plane curves.

that is, k(l − 1)^{2} ≤ (3l − 3)^{2} and then k ≤ 9, so that k = 9 as desired. The other two equalities of ii) follow trivially.

Suppose now that ii) holds. Let us show that the solutions of the equations

∑_{i=1}^{9}(l(p_{i}) − 1)^{2} = (d(F) − 1)^{2} and ∑_{i=1}^{9}(l(p_{i}) − 1) = 3(d(F) − 1)

are given by d(F) = 3l − 2 and l(p_{i}) = l (l ≥ 2).

Since d(F) ≥ 2 , we may define

r_{i} := \frac{l(p_{i}) − 1}{d(F) − 1}

and consider the equations

∑_{s=1}^{9} r_{i}^{2} = 1, and ∑_{s=1}^{9} r_{i} = 3.

The unique real solution to this pair of equations is

(r_{1}, ⋯, r_{9}) = (\frac{1}{3}, ⋯, \frac{1}{3}).

Then i) is proved.

Let us remark at this point that ∑_{i=1}^{9}(l(p_{i}) − 1)^{2} = (d(F) − 1)^{2} implies that l(q) = 1 for all the singularities of σ_{1}^{∗}F. Since q has a holomorphic first integral, its reduction is the same as the reduction of its separatrix; we conclude that the only possibilities are (modulo local change of coordinates) d(xy) = 0, d(y^{2} − x^{2m}) = 0 and d(y^{2} − x^{2m+1}) = 0, where m ∈ N. The last two cases give rise to special invariant curves, so that we are left only with the first case, that is, all the singularities of σ_{1}^{∗}F are of Morse type. In particular, σ = σ_{1} (up to biholomorphisms).

Lemma 3. The Kodaira dimension of σ^{∗}F is equal to 1.

Proof. First of all let us prove that T_{σ^{∗}F}^{∗} admits a global holomorphic section. As remarked above, l(q) = 1 for all the singularities having local holomorphic first integral that appear along the resolution of F; thus, there exists a line bundle isomorphism

T_{σ^{∗}F}^{∗} = σ^{∗}(O_{CP^{2}}(3(l − 1))) ⊗ O_{N}(− ∑_{i=1}^{9}(l − 1)E_{i}),

where E_{i} is the exceptional line introduced by the blow up of p_{i}, 1 ≤ i ≤ 9. Now we observe that a curve (l − 1)C, where C is a plane cubic curve (not necessarily smooth) passing through the points p_{1}, ⋯, p_{9}, corresponds to a global holomorphic section of T_{σ^{∗}F}^{∗} (more precisely, consider a section s of T_{σ^{∗}F}^{∗} which vanishes along the transform of a straight line with multiplicity 3(l − 1) and has poles of order (l − 1) along each E_{i}, 1 ≤ i ≤ 9; the global holomorphic section we look for is s̄_{1} = (P^{l−1}∘σ)·s, where P = 0 is a reduced affine equation for C). Hence, by definition,

k(σ^{∗}F) ≥ 0.

Notice at this point that thanks to the inequality

∑_{i=1}^{9} ν_{p_{i}}(C) · (l − 1) ≤ 3(l − 1) · d(C),

stated in Theorem 1, any cubic curve C passing by p_{1} ⋯ p_{9} is reduced (that is, C_{red} is not one or two straight lines) and C must be smooth at all the points p_{1}, ⋯, p_{9}.

Since T_{σ^{∗}F}^{∗} is a pseudo-effective divisor which is also a non-trivial nef divisor with T_{σ^{∗}F}^{∗} · T_{σ^{∗}F}^{∗} = 0, by definition the numerical Kodaira dimension is

ν(σ^{∗}F) = 1

and we conclude (cf. Preliminaries) that k(σ^{∗}F) = 1.

Remark 1. As a consequence of Lemma 3 we have that h^{0}(N, T_{σ^{∗}F}^{∗⊗n}) ≥ 2 for some n ≥ 1. Taking independent global sections s_{1} = s̄_{1}^{n}, s_{2} of T_{σ^{∗}F}^{∗⊗n}, we define the rational mapping

H′ : N − − > CP^{1},
H′(x) := (s_{1}(x) : s_{2}(x)).

This map is free of indetermination points, that is, it defines a fibration H′ : N → CP^{1}; the reason is that the divisors (s_{1}) and (s_{2}) (which are linearly equivalent to T_{σ^{∗}F}^{∗⊗n})) do not intersect:

(s_{1}) · (s_{2}) = n^{2} T_{σ^{∗}F}^{∗} · T_{σ^{∗}F}^{∗} = 0.

By construction, (H′^{−1}(∞))_{red} is the strict transform C̄ of C by σ (the fact that the curve C is smooth at the points p_{1}, ⋯, p_{9} is crucial here), so that it is connected and the exceptional divisors E_{1}, ⋯, E_{9} are multisections to H′. Also, H′^{−1}(∞) has multiplicity n(l − 1) for the map H′. Using Stein Factorization ([B-P-V], p. 25), we may change H′ by H in order to have connected fibers (notice that (H′^{−1}(∞))_{red} = (H^{−1}(∞))_{red} although the multiplicity may change); a fiber H′ of H′ becomes linearly equivalent to mH, where m ∈ N and H is a fiber of H (in particular we have T_{σ^{∗}F}^{∗} ·H = 0). After contracting the exceptional divisors, we conclude for the moment that σ_{∗}(H) is a Halphen pencil of index l′ − 1, l′ ≥ 2, whose base points are p_{1}, ⋯, p_{9}.

Lemma 4. H is a relatively minimal elliptic fibration.

Proof. In fact, mH is linearly equivalent to (s_{1}); using the wel-known formula for the variation of canonical line bundles:

K_{N} = σ^{∗}(K_{CP^{2}}) ⊗ O_{N}(∑_{i}^{9} E_{i}),

it follows that the Euler characteristic χ(mH) may be computed as:

χ(mH) = χ((s_{1})) = −T_{σ^{∗}F}^{∗⊗n} · (T_{σ^{∗}F}^{∗⊗n} ⊗ K_{N})
= −T_{σ^{∗}F}^{∗⊗n} · K_{N}
= −[T_{F}^{∗⊗n} · K_{CP^{2}} + ∑_{i=1}^{9}(l(p_{i}) − 1)n]
= −[−9n(l − 1) + 9n(l − 1)] = 0.

so that χ(H) = 0. One has also that it is relatively minimal (see [C-D], pg. 348).

It is worth noticing that H has at least one singular fiber, since the Euler characteristic of a rational surface is positive.

Lemma 5. Any irreducible component of a singular fiber of H is σ^{∗}Finvariant.

Proof. Let us consider an irreducible component H_{j} of a singular fiber H_{s} of H. Kodaira’s classification of singular fibers of elliptic fibrations (see [B-P-V], pg. 150) tell us that H_{j} · H_{j} = −2 when H_{j} is smooth. In any case, if H_{j} is not σ^{∗}F-invariant, we get

0 < tang(σ^{∗}F, H_{j}) − H_{j}^{2} = T_{σ^{∗}F}^{∗} · H_{j}.

Since T_{σ^{∗}F}^{∗} is a nef divisor,

T_{σ^{∗}F}^{∗} · H_{j} ≤ T_{σ^{∗}F}^{∗} · H_{s}.

In order to finish the proof, we use that T_{σ^{∗}F} · H_{s} = 0 (see Remark 1), contradiction.

Lemma 6. H = σ^{∗}F (as foliations).

Proof. Suppose by absurd that H ≠ σ^{∗}F. First of all let us notice that, in this case, σ^{∗}F is transverse to H at all points of the (generic) fiber H of H. In fact, T_{σ^{∗}F}^{∗} · H = 0 and H · H = 0 imply that

0 = T_{σ^{∗}F}^{∗} · H = tang(σ^{∗}F, H).

Consequently the analytic type of the generic fibers of H is constant: H is an isotrivial elliptic fibration. It is well known (cf. [B-P-V], pg. 159) that any singular fiber H_{s} of an isotrivial (relatively minimal) elliptic fibration has one of the following types (in Kodaira’s notation): II, III, IV, I_{b}^{∗}, II^{∗}, III^{∗}, IV^{∗}. We use Lemma 5 to guarantee that H_{s} is σ^{∗}F-invariant.

If H_{s} is of type II, III, or IV, we conclude that σ^{∗}F has a non-reduced singularity, a contradiction.

If H_{s} is of type I_{b}^{∗}, II^{∗}, III^{∗} or IV^{∗}, there exists a smooth component H_{j_{0}} of H_{s} which intersects three other components; in particular, there are at least three singular points of σ^{∗}F along H_{j}. It follows that GSV (σ^{∗}F, H_{j}) ≥ 3 and hence

T_{σ^{∗}F}^{∗} · H_{j} = GSV (σ^{∗}F, H_{j}) − 2 > 0.

Since T_{σ^{∗}F}^{∗} is a nef divisor, then:

T_{σ^{∗}F}^{∗} · H_{s} ≥ T_{σ^{∗}F}^{∗} · H_{j} > 0,

contradicting T_{σ^{∗}F}^{∗} · H_{s} = 0.

We may finish now the proof of Theorem 2.

We have shown before that σ_{∗}H a Halphen pencil of index l′ − 1 having 9 distinct base-points in the plane. We assert that, as a foliation, its degree is 3l′ − 2. In fact, the generic curve of the pencil has degree 3(l′ − 1) and has an ordinary singularity of multiplicity l′ − 1 at each base-point; we conclude easily that the vanishing order of the foliation at these points is l′ − 1, and so l(p_{i}) = l′ for 1 ≤ i ≤ k. Now

TH^{∗} = σ^{∗}(O_{CP^{2}}(d(σ_{∗}H) − 1)) ⊗ O_{N}(− ∑_{i=1}^{9}(l′ − 1)E_{i}),

so that

TH^{∗} · H = 3(l′ − 1)(d(σ_{∗}H) − 1) − 9(l′ − 1)^{2}.

From TH^{∗} · H = 0 we obtain d(σ_{∗}H) = 3l′ − 2; since d(F) = 3l − 2, we conclude that l = l′, and F is a Halphen pencil of index l − 1.

Remark 2. Using similar arguments as in Lemma 6, we conclude that the singular fibers of H = σ^{∗}F are of type I_{b} (nodal rational curve or a cycle of rational curves) and that the support of the multiple fiber is smooth or it is of type I_{b}, b = 1, 2 or 3. All the singularities of H are of Morse-type, and there are 12 of them.

It is not difficult to find examples of elliptic fibrations with the properties we have just described. The pencil given in homogeneous coordinates by:

X_{0}^{3} + X_{1}^{3} + X_{2}^{3} + λX_{0}X_{1}X_{2} = 0, λ ∈ CP^{1},

produces after reduction of singularities an elliptic fibration with 4 singular fibers of type I_{3}; it has no multiple fibers. In order to create a multiple fiber, we may for example select a smooth fiber of the fibration (type I_{0}) and apply to it a logarithmic transformation (cf. [B-P-V], pg.164) to become a multiple fiber with smooth reduction (type _{m}I_{0}). Although logarithmic transformations are quite different from birational transformations, it is known that starting with a rational elliptic surface the resulting surface is still a rational elliptic surface; in our case it has Euler number equal to 12 and we conclude that the fibration comes from a Halphen pencil of the plane after blow-ups at 9 (distinct) points.

5 Final Remarks

Let us discuss a more direct approach for finding the elliptic fibration H of the last section (cf. Lemmata 3, 4).

We take any cubic curve C that passes through the dicritical singularities p_{1}, ⋯, p_{9}; we know from Theorem 1 that C is reduced, so that the possibilities are (modulo change of coordinates and leaving aside for the moment the smooth case):

1) a cusp y^{2} − x^{3} = 0.

2) a conic tangent to a straight line y(y − x^{2}) = 0.

3) three concurrent lines xy(y − x) = 0.

4) a rational curve with a node y^{2} = x^{2}(x − 1).

5) a conic and a line meeting at two points (y − 1)(y − x^{2}) = 0.

6) three lines forming a triangle xy(x + y − 1) = 0.

Theorem 1 implies that C is smooth at the points p_{1}, ⋯, p_{9}. Also, for cases 2), 3), 5), 6) above, Theorem 1 implies that there are at most 3 dicritical singularities in components of C having degree one and at most 6 dicritical singularities in components of C having degree two. Since the cubic C passes by all 9 dicritical points, then there are exactly 3 dicritical points along the components of degree 1 and exactly 6 dicritical singularities along the components of degree 2. Let us denote a component of C by C′ and let C̄′ be the strict transform by σ. It follows that C̄′ · C̄′ = −2 and

T_{σ^{∗}F}^{∗} · \overline{C′} = 3(l − 1) − 3(l − 1) = 0,

if C′ has degree one and also

T_{σ^{∗}F}^{∗} · \overline{C′} = 6(l − 1) − 6(l − 1) = 0,

if C′ has degree two. If C′ were not F-invariant, we would find

tang(σ^{∗}F, \overline{C′}) = −2,

which is impossible. Therefore all the possibilities 2), 3), 5), 6) correspond to F-invariant cubics.

As for the cases 1) and 4), one has that T_{σ^{∗}F}^{∗} · C̄ = 0 and C̄ · C̄ = 0, so that tang(σ^{∗}F, C̄) = 0 if C is not invariant, which again is impossible due to the presence of a cusp or a node of \overline{C}. Therefore we may then assume in all cases above that (each component of) the cubic curve C is F-invariant; let us allow the possibility of C being smooth and F-invariant. Using that the non dicritical singularities of F are reduced , we exclude cases 1), 2) and 3).

Let us look now to a global rational vector field ZF inducing F, having isolated zeros at the singularities of F and pole of order d(F) − 1 = 3 · (l − 1) along the line at infinity L_{∞} (L_{∞} can be supposed transverse to C). Let ZP be a rational vector field wich extends (with a pole of order one along L_{∞}) a polynomial vector field inducing the foliation dP = 0, where P = 0 is a reduced affine equation of C along CP^{2} − L_{∞}. The singularities p_{1}, ⋯, p_{9} of F being removable by one blow up, then the restriction (ZF)_{| C} is a rational vector field with zeroes of multiplicity l − 1 at these points ; also, the zeroes of (ZF)_{| C} at each branch of the singular points of C (when they exist) have multiplicity one. By another side, the vector field (ZP)_{| C} has zeroes with multiplicity one at the same branches. There is a meromorphic function f over C verifying:

(ZF)_{| C} = f · (ZP)_{| C}

and we observe that f has zeroes of order l − 1 at the points p_{1}, ⋯, p_{9} and poles of order 3(l − 1) at the points q_{1}, q_{2} and q_{3} of C ∩ L_{∞}. Therefore the divisor D = ∑_{k=1}^{9} p_{k} − ∑_{j=1}^{3} q_{j} of C has torsion: (l − 1)D = (f) is principal.

The same is true, although by a different reason, when C is smooth but not F-invariant. We consider the meromorphic function f = (dP · ZF)_{| C} along C, where P = 0 is a reduced affine equation of C; we show easily that p_{1}, ⋯, p_{9} are zeroes of order l−1 (this is the vanishing order of ZF at each of these singularities). Again if we consider the divisor D = ∑_{k=1}^{9} p_{k} − ∑_{j=1}^{3} q_{j}, then (l − 1)D = (f) is principal. It is important to notice that D is special in the sense that its support is disjoint from the singular points of C.

Starting from the meromorphic function f and the torsion divisor D above, there exists a well known way to move C inside a pencil of elliptic curves as to have that C appears as the support of a (l − 1)-multiple curve. The reader may consult [S] (Theorem 7), for a smooth support; it extends naturally to our situation, because of the property supp(D) ∩ sing(C) = ∅.

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Instituto de Matemática Pura e Aplicada 110 Estrada D. Castorina 22460-320 – Jardim Botânico – Rio de Janeiro – Brasil e-mail: sad@impa.br