Symplectic bifurcations of plane curves and isotropic liftings

G. Ishikawa and S. Janeczko

The isotropic bifurcation problem is reduced to the classification of varieties by symplectomorphisms in the reduced space. The complete symplectic classification of Bruce-Gaffney’s plane curve singularites is provided and is applied to obtain naturally the Lagrangian opennings.

2000 Mathematics Subject Classification: Primary 58K40; Secondary 14H50, 70H15.

1 Introduction.

We start with a coisotropic fibration H : M^{2n} → R^{n−k}, 0 ≤ k ≤ n − 1, of the symplectic space (M^{2n}, ω) of dimension 2n, n ≥ 2. By the Jacobi-Liouville theorem, locally, there exist relative Darboux coordinates p_{1}, … p_{n}, q_{1}, …, q_{n} of M such that ω = ∑_{i=1}^{n} dp_{i} ∧ dq_{i}, and

H(p, q) = (q̄) = (q_{k+1}, …, q_{n}).

(cf. [1], p. 301). Then we consider the family of canonical reductions (Marsden-Weinstein symplectic reduction)

q̄ ↦ πq̄ : H^{−1}(q̄) → H^{−1}(q̄)/∼q̄ ≅ T^{∗}R^{k},
πq̄ : (p_{1}, …, p_{k}, q_{1}, …, q_{k}, p_{k+1}, …, p_{n}) ↦ (p_{1}, …, p_{k}, q_{1}, …, q_{k}).

Then we have the total projection

π : M^{2n} → N^{n+k} = T^{∗}R^{k} × R^{n−k}
π(p, q) = (p_{1}, …, p_{k}, q_{1}, …, q_{k}; q_{k+1}, …, q_{n}).

Let L^{m} ⊂ M^{2n} be an isotropic variety (ω|_{L} = 0, m ≤ n). Then the projection π(L) ⊂ N = T^{∗}R^{k} × R^{n−k} provides a bifurcation of (m − n + k)- dimensional isotropic varieties πq̄(H^{−1}(q̄) ∩ L) in T^{∗}R^{k} with the parameter space R^{n−k}. In this paper we consider the bifurcation of curves, namely, the case m = n − k + 1.

Any map-germ F : (R^{n−k+1}, 0) → N = T^{∗}R^{k} × R^{n−k} is called a symplectic bifurcation problem of curves. F is called transverse if F is transverse to T^{∗}R^{k} × {0} at 0. Moreover any symplectic bifurcation problem of curves is called an isotropic bifurcation of curves if there exists an isotropic map-germ F̃ : (R^{n−k+1}, 0) → ((M^{2n}, ω), 0) such that π ∘ F̃ = F, and F̃^{∗}ω = 0. (cf. [17], p. 29). Then, in this paper we show that any transverse symplectic bifurcation problem of curves is isotropic (Proposition 2.3).

In the paper [22], V. M. Zakalyukin classified the simple stable Lagrangian submanifold-germs (m = n) by symplectomorphisms which preserve a given coisotropic fibration. Contrary to it, admitting Lagrangian or isotropic varieties, we study the liftability and the classification problem of varieties in the reduced space, namely about the ”bottom-up” construction. The idea appeared before in PhD by M. Mikosz and part of it is published in [19].

Moreover we see that there exist, even in the simplest case n = 2, k = 1, many examples of non-transverse bifurcation problems of curves F : (R^{2}, 0) → N = R^{2} × R which are not liftable to isotropic mappings into M (Examples 2.6 and 2.7).

To make clear the problem we are going to study, let us consider the bifurcation problem of “cross caps” or “Whitney umbrellas” in the three space R^{2} × R = T^{∗}R × R with the symplectic foliation. Then we observe the four typical examples of bifurcations F : (R^{2}, 0) → (R^{2} × R), namely,

(1) transverse immersions or no bifurcation,

(2) transverse Whitney umbrellas or cusp bifurcations,

(3) hyperbolic Whitney umbrellas or X-pinch bifurcations, and

(4) elliptic Whitney umbrellas or figure 8 bifurcations See Fig. 1. See also §5. The classification is similar to the classification of functions on cross-caps. See [7]. However we remark that the non-transverse immersions are never isotropically liftable (Proposition 2.5), so they do not enter into our list, contrary to the ordinary singularity theory. The first two are transverse and the last two are not. Remark that there occur the changes of irreducible components through the bifurcations of types (3) and (4), while the transverse bifurcations of curves provide just irrreducible curves, like the cases (1) and (2).

Figure 1: Typical isotropic bifurcations

In this paper we study the transverse bifurcation problem of curves in detail. In §2, we show the unique isotropic liftability of transverse bifurcations. In §3, we study the general classification problem of parametrized varieties in the reduction space under a natural equivalence relation, liftable equivalence, that corresponds to the classification of isotropic varieties by symplectomorphisms preserving π-fibers for the given coisotropic fibration H : R^{2n} → R^{n−k}. We show a liftable diffeomorphism is actually a diffeomorphism preserving the symplectic foliation of N = T^{∗}R^{k} × R^{n−k}. Thus the classification under the liftable equivalence turns out to be reduced completely to the classification and deformation problem of curves on the symplectic space T^{∗}R^{k} = R^{2k}, k ≥ 1.

In §4, we give the classification of symplectic bifurcation problems of curves under the liftable equivalence.

In this paper, in particular, we study the simplest case k = 1, namely, the bifurcation problem of curves on the symplectic plane, in detail.

Now we try to clarify the fundamental problem we are facing: Suppose two plane curve-germs are transformed to each other by an orientation preserving diffeomorphism on the plane. Then, it is natural to ask: Are they transformed by a symplectomorphism on the plane? Does the singularity theory by using symplectomorphisms have any difference to the ordinary singularity theory even for the plane curves? Then we encounter “symplectic ghosts” in the sense of Arnold [3]. After several preliminaries in §6 and §7, answering to the question, we describe, in §8, for a plane curve-germ f : (R, 0) → (R^{2}, 0), the difference sp-codim(f) − codim(f) between the codimension sp-codim(f) of f under the “symplectic equivalence” and the right-left (i.e. A_{e}) codimension codim(f) of f in an explicit way. We call this difference the symplectic defect or the symplectic ghost number of f. Remark that our ghosts have moduli, while the ghosts appeard in [3] are discrete.

Obviously the symplectic defect is an invariant under the symplectic equivalence, namely a symplectic invariant. However we show that the symplectic defect is, in fact, an A-invariant of f, not just a symplectic invariant: If the symplectic defect is positive, then the classification of plane curves by symplectomorphisms differs from the classification by diffeomorphisms, while the difference depends only on the A-equivqlence class of the plane curves. Therefore also the symplectic codimension itself is an A-invariant: If two plane curve germs f and f′ are A-equivalent, then sp-codim(f) = spcodim(f′). In fact we show that sp-codim(f) is equal to the δ-invariant (the number of complex double points after a perturbation) of f. Remark that, a priori, the δ-invariant has nothing to do with the symplectic equivalence. Also we remark that, in the case k ≥ 2, the symplectic codimension of f : (R, 0) → T^{∗}R^{k} is, in fact, not necessarily an A-invariant (Remark 8.5).

Moreover, we show in §10, if the symplectic defect equals zero, that the symplectic classification of plane curves coincides with their isotopy classification. As an application, we have normal forms under the liftable equivalence for certain map-germs.

Also, we calculate symplectic defects for the A-simple plane curves

A_{2ℓ}, E_{6ℓ}, E_{6ℓ+2}, W_{12}, W_{18}, W_{1,2q−1}^{#}

classified by Bruce and Gaffney [5] and give the complete symplectic classification of the A-simple plane curves (Theorems 9.2 and 9.6). Moreover we obtain the symplectically mini-versal unfoldings of them (Proposition 9.9). As a byproduct of our approach, we give the classification (A_{2ℓ}, E_{6}, E_{8}^{±}) of “symplectically simple” plane curves ( Corollary 9.8).

As typical well-known examples of Lagrangian varieties, there are open Whitney umbrellas and open swallowtails. In this paper, in a systematic way, we characterize the opening processes of Whitney umbrellas and swallowtails. See §7, §9, §11 and §12. We show that the Lagrangian liftings are obtained from open Whitney umbrellas by the reduction process, in many cases (Corollary 7.4). Also we study the polynomial construction of open swallowtails from the viewpoint of the present paper (§11). In §12, we treat the construction of open swallowtails (Figure 2 (1)) and also “open folded umbrellas” (Figure 2 (2)), via the notion of “frontal-symplectic” versality.

Figure 2: Swallowtail and Folded Umbrella

2 Lagrangian liftability.

Let F : (R^{n−k+1}, 0) → (N, 0) = (T^{∗}R^{k} × R^{n−k}, 0) be a smooth map-germ.

Definition 2.1 By an isotropic lifting of F, we call the smooth map-germ F̃ : (R^{n−k+1}, 0) → (M, 0) which is isotropic, i.e. F̃^{∗}ω = 0, and π ∘ F̃ = F:

(R^{n−k+1}, 0) \underrightarrow{F̃} ((M, 0), ω)
∥  ↓^{π}
(R^{n−k+1}, 0) ⟶^{F} (N, 0).

When k = 1, we call an isotropic lifting also a Lagrangian lifting.

Now we have the following sufficient condition for the isotropic liftability of f.

Definition 2.2 A smooth map-germ F : (R^{n−k+1}, 0) → (N, 0) = (T^{∗}R^{k} × R^{n−k}, 0) is called transverse if F is tansverse to T^{∗}R^{k} × {0}.

The result mentioned in Introduction is proved in the following exact form:

Proposition 2.3 Let F : (R^{n−k+1}, 0) → (N, 0) be a transverse smooth mapgerm. Then there exists a smooth isotropic lifting F̃ : (R^{n−k+1}, 0) → M = R^{2n} for F. In fact, for some coordinates x = (t, λ_{1}, …, λ_{n−k}) with F (t, λ) = (f_{λ}(t), λ), the remaining components of F̃ are given by

p_{j}(x) = ∫_{0}^{t} (∑_{i=1}^{k} \frac{∂p_{i}}{∂λ_{j}} \frac{∂q_{i}}{∂t} − \frac{∂p_{i}}{∂t} \frac{∂q_{i}}{∂λ_{j}} ) dt, (k + 1 ≤ j ≤ n).

Moreover the Lagrangian liftings of F are equivalent to each other by symplectomorphisms on M preserving fibers of H.

Proof : By the equation F̃^{∗}ω = 0 we get the necessary condition for the germ to be a solution of the first-order partial differential equation. Actually we discuss as follows.

By the assumption, there exist coordinates

x = (x_{1}, x_{2} …, x_{n}) = (t, λ_{1}, …, λ_{n−k})

such that

F (t, λ_{1}, …, λ_{n−k}) = (q_{1}(t, λ), p_{1}(t, λ), …, q_{k}(t, λ), p_{k}(t, λ), λ_{1}, …, λ_{n−k}),

F̃(t, λ_{1}, …, λ_{n−k}) = (F (t, λ); p_{k+1}(t, λ), …, p_{n}(t, λ)).

Set θ = ∑_{i=1}^{n} p_{i}dq_{i}, the Liouville form on M = T^{∗}R^{n}. Then ω = dθ. Since the condition F̃^{∗}ω = 0 is equivalent to that F̃^{∗}θ is closed, we set, locally, F̃^{∗}θ = de, for some function e on (R^{n−k+1}, 0). Then we get

\frac{∂e}{∂t} = ∑_{i=1}^{k} p_{i}(x)\frac{∂q_{i}}{∂t} , \frac{∂e}{∂λ_{j}} = p_{k+j}(t, λ) + ∑_{i=1}^{k} p_{i}(t, λ)\frac{∂q_{i}}{∂λ_{j}} ,

(j = 1, …, n − k). If we take a function e(x) satisfying the first equation, then, we obtain p_{k+j}(x), (j = 1, …, n − k) by the second equality. Now we set

e = ∫_{0}^{t} ∑_{i=1}^{k} p_{i}(x)\frac{∂q_{i}}{∂t} dt + φ(λ), λ = (λ_{1}, …, λ_{n−k}),

for some function φ which is independent of t. Then we have

\frac{∂e}{∂λ_{j}} = ∫_{0}^{t} (∑_{i=1}^{k} \frac{∂p_{i}}{∂λ_{j}} \frac{∂q_{i}}{∂t} + p_{i} \frac{∂^{2}q_{i}}{∂t∂λ_{j}} ) dt + \frac{∂φ}{∂λ_{j}}

= ∫_{0}^{t} (∑_{i=1}^{k} \frac{∂p_{i}}{∂λ_{j}} \frac{∂q_{i}}{∂t} − \frac{∂p_{i}}{∂t} \frac{∂q_{i}}{∂λ_{j}} ) dt + ∑_{i=1}^{k} p_{i} \frac{∂q_{i}}{∂λ_{j}} + \frac{∂φ}{∂λ_{j}} .

Thus we have

p_{k+j}(t, λ) = ∫_{0}^{t} (∑_{i=1}^{k} \frac{∂p_{i}}{∂λ_{j}} \frac{∂q_{i}}{∂t} − \frac{∂p_{i}}{∂t} \frac{∂q_{i}}{∂λ_{j}} ) dt + \frac{∂φ}{∂λ_{j}} , (1 ≤ j ≤ n − k).

Remark that λ = (λ_{1}, …, λ_{n−k}) is equal to H ∘ F̃. Then the local symplectomorphism

(p, q) ↦ (p_{1}, …, p_{k}, p_{k+1} − \frac{∂φ(q̄)}{∂q_{k+1}}, …, p_{n} − \frac{∂φ(q̄)}{∂q_{n}}, q),

q̄ = (q_{k+1}, …, q_{n}), eliminates the terms \frac{∂φ}{∂λ_{j}}. □

Remark 2.4 (The coincidence of singular loci for the lifting and the original.) The singular locus of F̃ coincides with the singular locus of F:

Sing(F̃) = Sing(F) = \{(t, λ) ∈ (R^{n−k+1}, 0) |\frac{∂q_{i}}{∂t} = \frac{∂p_{i}}{∂t} = 0, 1 ≤ i ≤ k\} .

In the case k = 1, moreover we have the following:

Proposition 2.5 If F : (R^{n}, 0) → N = T^{∗}R × R^{n−1} is an immersion germ, then the transversality condition is the necessary and sufficient condition for the existence of Lagrangian liftings for F.

Proof : Assume that F is an immersion and not transverse to T^{∗}R × {0}. Then we see that the image F_{∗}(T_{0}R^{n}) contains T_{0}(T^{∗}R × {0}) by the simple argument of dimension. This means that, for any lifting F̃ of F, F̃_{∗}(T_{0}R^{n}) contains the p_{1}-q_{1} plane in T_{0}M. Then it is impossible that F̃ is isotropic. □

Example 2.6 (Non-transversal and liftable germs.) There exist non-tansverse and liftable map-germs. For example, if a Lagrangian immersion L : (R^{n}, 0) → M^{2n} is not transverse to the H-level H^{−1}(H(L(0))) for H : M^{2n} → R^{n−1}, then the tangent space L_{∗}(T_{0}R^{n}) contains a characteristic direction and F = π ∘ L is not an immersion, while F is liftable to L. In particular, a Lagrange surface in R^{4} projects to a Whitney umbrella. This procedure is related to the study on smooth perturbations of singular surfaces in R^{4} ([10]).

Example 2.7 (Non-transverse and non-liftable germs.) Let F : (R^{2}, 0) → (N, 0) = (R^{2} × R, 0) be a non-transverse map-germ of the form

F (x_{1}, x_{2}) = (q_{1}, p_{1}, q_{2}) = (q_{1}(x), p_{1}(x), \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})).

Consider the second order differential operator Δ = \frac{∂^{2}}{∂x_{1}^{2}} ± \frac{∂^{2}}{∂x_{2}^{2}} associated

to \frac{1}{2}(x_{1}^{2} ± x_{2}^{2}). If F has a Lagrangian lifting, then dp_{1} ∧ dq_{1} + dφ ∧ d(\frac{1}{2}(x_{1}^{2} ± x_{2}^{2})) = 0, for some function φ = φ(x_{1}, x_{2}). Then we see, by simple formal calculations, that the Jacobian J(p_{1}, q_{1}) satisfies the infinite condition (Δ^{ℓ}J(p_{1}, q_{1}))(0, 0) = 0, (ℓ = 0, 1, 2, …). So, if (Δ^{ℓ}J(p_{1}, q_{1}))(0, 0) ≠ 0, for some ℓ, then F is never liftable. For example,

F (x_{1}, x_{2}) = (q_{1}, p_{1}, q_{2}) = (x_{1}, x_{2}^{3}, \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})),

is not liftable. On the other hand, (x_{1}, x_{1}x_{2}, \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})) is liftable.

3 Liftable equivalence.

Let ϕ : (N, 0) → (N, 0) be a diffeomorphism-germ. We call ϕ a symplectically liftable diffeomorphism if there exists a symplectomorphism-germ Φ : ((M, 0), ω) → ((M, 0), ω) such that the following diagram commutes

((M, 0), ω) ⟶^{Φ} ((M, 0), ω)
π↓  ↓^{π}
(N, 0) ⟶^{ϕ} (N, 0).

Definition 3.1 We say that the two map-germs F_{1}, F_{2} : (R^{n−k+1}, 0) → (N, 0) are liftably equivalent if the following diagram commutes

(R^{n−k+1}, 0) ⟶^{F_{1}} (N, 0) ⟵^{π} ((M, 0), ω)
ψ↓  ^{ϕ}↓  ↓^{Φ}
(R^{n−k+1}, 0) ⟶^{F_{2}} (N, 0) ⟵^{π} ((M, 0), ω),

where ψ, ϕ are diffeomorphism-germs and Φ is a symplectomorphism-germ, i.e. ϕ is a symplectically liftable diffeomorphism.

The symplectically liftable diffeomorphisms of (N, 0) form the subgroup G_{symp} of the group G of diffeomorphism-germs. Classification of singularities of F according to the equivalence group G_{symp} is similar to the standard rightleft classification of singularities of map-germs (R^{n}, 0) → (R^{n+1}, 0), however this is specified to the space of map-germs which have the Lagrangian liftings.

Now we describe symplectically liftable diffeomorphisms in an explicit way.

Proposition 3.2 For a diffeomorphism-germ ϕ : (N, 0) = (T^{∗}R^{k}×R^{n−k}, 0) → (T^{∗}R^{k} × R^{n−k}, 0), the following conditions are equivalent:

(1) ϕ is a symplectically liftable diffeomorphism.

(2) ϕ is a Poisson diffeomorphism (for the Poisson structure on N induced from M by π).

(3) ϕ is a family of symplectic diffeomorphisms on T^{∗}R^{k} with parameter q̄ = (q_{k+1}, …, q_{n}). Namely, if we set

ϕ(q_{1}, p_{1}, …, q_{k}, p_{k}, q̄) = (Q_{1}, P_{1}, …, Q_{k}, P_{k}, Q̄),

then \overline{Q} = (Q_{k+1}, …, Q_{n}) depends only on q̄, and

(q_{1}, p_{1}, …, q_{k}, p_{k}) ↦ (Q_{1}, P_{1}, …, Q_{k}, P_{k})

is a symplectomorphism on (T^{∗}R^{k}, 0) for each fixed q̄ = (q_{k+1}, …, q_{n}).

(4) ϕ has a symplectic lifting Φ : (M, 0) → (M, 0) preserving fibers of H, namely, there exists a diffeomorphism-germ σ : (R^{n−k}, 0) → (R^{n−k}, 0) such that the following diagram commutes:

(N, 0) ⟵^{π} ((M, 0), ω) ⟶^{H} (R^{n−k}, 0)
ϕ↓  ^{Φ}↓  ↓^{σ}
(N, 0) ⟵^{π} ((M, 0), ω) ⟶^{H} (R^{n−k}, 0).

Proof :

(1) ⇒ (2): Let Φ be a symplectic lifting of ϕ on (M, 0). Then, for any function h, k on (N, 0), we have {h ∘ ϕ, k ∘ ϕ}_{N} = {h ∘ ϕ ∘ π, k ∘ ϕ ∘ π}_{M} = {h ∘ π ∘ Φ, k ∘ π ∘ Φ}_{M} = {h ∘ π, k ∘ π}_{M} = {h, k}_{N}. Thus ϕ is a Poisson diffeomorphism.

(2) ⇒ (3): The Poisson structure on N induces the foliation by symplectic leaves, T^{∗}R^{k} × {q̄}, intrinsically. So naturally ϕ induces a family of symplectomorphisms on T^{∗}R^{k}. Also we can argue in more direct way as follows: The derivations {P_{i}, ·}, {Q_{i}, ·}, (1 ≤ i ≤ k) generate the tangent spaces to the leaves T^{∗}R^{k} × {q̄}. Since we have, for each j with 1 ≤ j ≤ n − k, {P_{i}, Q_{k+j}} = {Q_{i}, Q_{k+j}} = 0, (1 ≤ i ≤ k), we see each Q_{k+j} is independent of (q_{1}, p_{1}, …, q_{k}, p_{k}).

(3) ⇒ (4): Let Φ : (M, 0) → (M, 0)

Φ(q_{1}, p_{1}, …, q_{k}, p_{k}, q̄) = (Q_{1}, P_{1}, …, Q_{k}, P_{k}, \overline{Q}, \overline{P}),

\overline{P} = (P_{k+1}, …, P_{n}) be a diffeomorphism-germ covering ϕ with respect to π. The condition that Φ is a symplectomorphism is the existence of a smooth function E : (M, 0) → R, called a generating function of Φ, satisfying

Φ^{∗}θ − θ = dE,

where θ = ∑_{i=1}^{n} p_{i}dq_{i}. Then the condition is equivalent to that

\begin{matrix}∑_{i=1}^{k} P_{i} \frac{∂Q_{i}}{∂q_{j}} − p_{j} = \frac{∂E}{∂q_{j}} & (1 ≤ j ≤ k) \\ ∑_{i=1}^{k} P_{i} \frac{∂Q_{i}}{∂p_{j}} = \frac{∂E}{∂p_{j}} & (1 ≤ j ≤ k) \\ ∑_{i=1}^{k} P_{i} \frac{∂Q_{i}}{∂q_{j}} + ∑_{i=k+1}^{n} P_{i} \frac{∂Q_{i}}{∂q_{j}} − p_{j} = \frac{∂E}{∂q_{j}} & (k + 1 ≤ j ≤ n) \end{matrix}

for some function E : (N, 0) → R. The first and second equalities mean that Eq̄ is a generating function of the symplectomorphism ϕq̄ for each q̄ ∈ (R^{n−k}, 0). Therefore the function E : (N, 0) → R is uniquely determined by these two equalities up to the addition of functions on q̄. Then, by the last equality, the H-level preserving symplectic liftings Φ of ϕ is given by setting

\overline{P} = (\frac{∂E}{∂q̄} + p̄ − ∑_{i=1}^{k} P_{i} \frac{∂Q_{i}}{∂q̄} ) (\frac{∂Q_{ℓ}}{∂q_{j}} )_{k+1≤ℓ≤n,k+1≤j≤n}^{−1} ,

where \frac{∂E}{∂q̄} = (\frac{∂E}{∂q_{k+1}} , …, \frac{∂E}{∂q_{n}} ), and \frac{∂Q_{i}}{∂q̄} = (\frac{∂Q_{i}}{∂q_{k+1}} , …, \frac{∂Q_{i}}{∂q_{n}} ).

The implication (4) ⇒ (1) is trivial. □

Remark 3.3 In particular, in the case k = 1, the diffeomorphisms of types

ϕ(q_{1}, p_{1}, q̄) = (δ^{−1}q_{1} + β(q̄), δp_{1} + α(q), γ(q̄)),

and

ϕ(q_{1}, p_{1}, q̄) = (−p_{1}, q_{1}, q̄),

where α, β, γ are smooth functions, δ ∈ R − {0}, and q̄ = (q_{2}, …, q_{n}), and their any compositions are all symplectically liftable.

Remark 3.4 We have also the description of the infinitesimal deformations corresponding to liftable and lifted equivalences. Any vector field X over N generating a liftable equivalence is given by

X(q_{1}, p_{1}, …, q_{k}, p_{k}, q̄) = X_{hq̄}(q_{1}, p_{1}, …, q_{k}, p_{k}) + ∑_{i=1}^{n−k} a_{i}(q̄)\frac{∂}{∂q_{k+i}} ,

for some functions h(q_{1}, p_{1}, …, q_{k}, p_{k}, q̄) and a_{i}(q̄), 1 ≤ i ≤ n − k, where X_{hq̄} is the Hamiltonian vector field over T^{∗}R^{k} with the Hamiltonian hq̄ for each fixed q̄ ∈ (R^{n−k}, 0). The lifted Hamiltonian vector field X̃ over M = T^{∗}R^{n} has the Hamiltonian h̃ = h + ∑_{i=1}^{n−k} p_{k+i}a_{i}(q̄); X̃ = Xh̃.

Definition 3.5 Two isotropic map-germs L, L′ : (R^{n−k+1}, 0) → (M, 0) are called H-symplectically equivalent if there exist a symplectomorphism Φ :

(M, 0) → (M, 0) and diffeomorphisms ψ : (R^{n−k+1}, 0) → (R^{n−k+1}, 0), σ : (R^{n−k}, 0) → (R^{n−k}, 0) such that the following diagram commutes:

(R^{n−k+1}, 0) ⟶^{L} (M, 0) ⟶^{H} (R^{n−k}, 0)
ψ↓  ^{Φ}↓  ↓^{σ}
(R^{n−k+1}, 0) ⟶^{L′} (M, 0) ⟶^{H} (R^{n−k}, 0).

Then the following is clear:

Corollary 3.6 Let F, F′ : (R^{n−k+1}, 0) → (T^{∗}R^{k} × R^{n−k}, 0) be transverse map-germs. If F and F′ are liftably equivalent, then their isotropic liftings F̃ and F̃′ are H-symplectically equivalent.

4 Classification under the liftable equivalence.

In what follows, we concentrate on the symplectic bifurcation problem of curves on the symplectic plane (k = 1). Thus we consider map-germs F : (R^{n}, 0) → (R^{2} × R^{n−1}, 0).

Proposition 4.1 For a transverse map-germ F : (R^{n}, 0) → (N, 0) = (R^{2} × R^{n−1}, 0), we have the followings:

(a1) If F is an immersion at 0, then F is liftably equivalent to

(0, x_{1}, x_{2}, …, x_{n}).

(a2) Suppose F is not an immersion at 0. Then F is liftably equivalent to the germ ϕ ∘ F ∘ ψ such that the 2-jet j^{2}(ϕ ∘ F ∘ ψ) is equal to (x_{1}^{2}, x_{1}x_{2}, x′), (x_{1}^{2}, 0, x′), (x_{1}x_{2}, 0, x′) or (0, 0, x′), where x′ = (x_{2}, …, x_{n}).

Proof : (a1) follows from the Proposition 4.2 below.

(a2) By (a) of Propositon 2.5 and by right equivalence ψ we can assume

j^{2}F (0) = (c_{0}x_{1}^{2} + c_{1}(x′)x_{1} + c_{2}(x′), b_{0}x_{1}^{2} + b_{1}(x′)x_{1} + b_{2}(x′), x′),

where b_{1}(0) = 0, c_{1}(0) = 0 and x′ = (x_{2}, …, x_{n}).

Let c_{0} ≠ 0. Then by using right equivalence, we can assume c_{0} = 1, c_{1}(x′) = 0. Then, by the symplectically liftable diffeomorphism

(q_{1}, p_{1}, q′) ↦ (q_{1} − c_{2}(q′), p_{1} − b_{2}(q′), q′),

the 2-jet is transformed to (x_{1}^{2}, b_{0}x_{1}^{2} + b_{1}(x′)x_{1}, x′).

Moreover using the symplectically liftable diffeomorphism (q_{1}, p_{1}, q′) ↦ (q_{1}, p_{1} −b_{0}q_{1}, q′), we get (x_{1}^{2}, b_{1}(x′)x_{1}, x′) as the 2-jet. If the linear form b_{1}(x′) is not identically zero, by a coordinate change of x′, we obtain (x_{1}^{2}, x_{1}x_{2}, x′). If b_{1}(x′) = 0, then we have (x_{1}^{2}, 0, x′).

If c_{0} = 0 and b_{0} ≠ 0, we can perform similarly as above using the symplectically liftable diffeomorphism (q_{1}, p_{1}, q′) ↦ (−p_{1}, q_{1}, q′). If b_{0} = c_{0} = 0, then we get (x_{1}x_{2}, 0, x′) or (0, 0, x′) as the 2-jet within the liftable equivalence classes. □

Now we have the following prenormal form for F.

Proposition 4.2 Let F : (R^{n}, 0) → (N, 0) = (R^{2}×R^{n−1}, 0), F (x_{1}, …, x_{n}) = (q_{1}(x), p_{1}(x), q′(x)), be a smooth map-germ. Assume that F is transverse to R^{2}×{0} and that F is finite, namely, the ideal generated by components of F is of finite codimension. Then F is liftably equivalent to one of the following forms, for some m ≥ 2,

F_{m}(x) = (x_{1}^{m} + ∑_{i=1}^{m−2} a_{i}(x′)x_{1}^{i}, x_{1}c(x), x′),

where x′ = (x_{2}, …, x_{n}), and a_{i}(x′), c(x) are smooth function-germs.

Proof : By the transversality assumption, using right equivalence, one can reduce F to the form

(x_{1}, x′) ↦ (a(x_{1}, x′), c_{1}(x_{1}, x′), x′).

Since F is finite, we can assume, \frac{∂^{m}a}{∂x_{1}^{m}}(0) ≠ 0, for some m, up to liftable

equivalence. Then by the classification of A_{m}-type singularities of functions, we see the liftably equivalent form

(x_{1}, x′) ↦ (x_{1}^{m} + ∑_{i=1}^{m−2} a_{i}(x′)x_{1}^{i} + a_{0}(x′), c_{2}(x_{1}, x′), x′).

We write c_{2}(x_{1}, x′) = x_{1}c(x_{1}, x′) + c_{3}(x′). Then the liftable diffeomorphism (q_{1}, p_{1}, q′) ↦ (q_{1} − a_{0}(q′), p_{1} − c_{3}(q′), q′) yields the required form. □

Remark 4.3 In the case m = 2, we get the form

F_{2}(x_{1}, x′) = (x_{1}^{2}, x_{1}c(x_{1}, x′), x′).

Then, by setting c(x_{1}, x′) = x_{1}ϕ(x_{1}^{2}, x′) + ψ(x_{1}^{2}, x′) and by taking liftable diffeomorphism (q_{1}, p_{1}, q′) ↦ (q_{1}, p_{1} −ψ(q_{1}, q′), q′), we see F_{2} is liftable equivalent to the form

F′_{2}(x_{1}, x′) = (x_{1}^{2}, x_{1}ϕ(x_{1}^{2}, x′), x′).

For the case m = 2, we have the normal forms, by using the versality theorem in the symplectic case [8][9].

Proposition 4.4 Let F : (R^{n}, 0) → (R^{2}×R^{n−1}, 0) be a finite and transverse map-germ. Assume the 2-jet of F is equal to (x_{1}^{2}, x_{1}x_{2}, 0) or (x_{1}^{2}, 0, 0). Then F is liftably equivalent to

(q_{1}, p_{1}, q′) = (x_{1}^{2}, x_{1}^{2ℓ+1} + λ_{1}(x′)x_{1}^{2ℓ−1} + λ_{2}(x′)x_{1}^{2ℓ−3} + ⋯ + λ_{ℓ}(x′)x_{1}, x′),

for some positive integer ℓ, and for some functions λ_{1}(x′), …, λ_{ℓ}(x′) of x′ = (x_{2}, …, x_{n}) with λ_{j}(0) = 0, 1 ≤ j ≤ ℓ.

Proof : We may assume F |R × {0} is of type A_{2ℓ} : f(t) = (q_{1}, p_{1}) = (t^{2}, t^{2ℓ+1}), by using a liftable diffeomorphism. Remark that the right-left equivalence class and the symplectic equivalence class coincide for a plane curve of type A_{2ℓ}. See §10. Then

G(t, λ_{1}, …, λ_{ℓ}) = (t^{2}, t^{2ℓ+1} + λ_{1}t^{2ℓ−1} + λ_{2}t^{2ℓ−3} + ⋯ + λ_{ℓ}t, λ_{1}, …, λ_{ℓ})

is a versal unfolding of F |R × {0}. Then G is a symplectically versal unfolding of F |R × {0} (See §7). Let f : (R, 0) → (R^{2}, 0) be a plane curve-germ. An unfolding F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0) of f is called symplectically versal if, any unfolding G : (R×R^{s}, 0) → (R^{2} ×R^{s}, 0), of f is symplectically equivalent to φ^{∗}F for some smooth map-germ φ : (R^{s}, 0) → (R^{ℓ}, 0).

Therefore, there exists a smooth mapping x′ ↦ λ(x′) such that the pullback λ^{∗}G is liftably equivalent to F. □

5 Non-transverse bifurcations of curves.

In this section we treat map-germs F : (R^{2}, 0) → (R^{2} × R, 0), F (x_{1}, x_{2}) = (q_{1}(x), p_{1}(x), q_{2}(x)) such that q_{2} ∘ F : (R^{2}, 0) → (R, 0) is a submersion or a Morse function at 0. We call such map-germs of Morse type. We consider the generic classification problem of liftable germs among the class of map-germs of Morse type. Then we have:

Proposition 5.1 Let F : (R^{2}, 0) → (R^{2} × R, 0) be a generic liftable mapgerm of Morse type, namely, a generic one-parameter bifurcation problem of plane curves of Morse type. Then F is liftably equivalent to one of following types(Fig. 3):

(1) (x_{1}, x_{2}) ↦ (x_{1}, 0, x_{2}): The transverse immersion.

(2) (x_{1}, x_{2}) ↦ (x_{1}^{2}, x_{1}^{3} + x_{1}x_{2}, x_{2}): The transverse Whitney umbrella, or, the cusp bifurcation.

(3) (x_{1}, x_{2}) ↦ (x_{1}, x_{1}x_{2} + O(3), \frac{1}{2}(x_{1}^{2} − x_{2}^{2})): The hyperbolic Whitney umbrella, or, the X-pinch bifurcation.

(4) (x_{1}, x_{2}) ↦ (x_{1}, x_{1}x_{2} + O(3), \frac{1}{2}(x_{1}^{2} + x_{2}^{2})): The elliptic Whitney umbrella, or, the figure eight bifurcation.

Here O(3) means the terms in x_{1}, x_{2} at least of third order.

Moreover the germ F (x_{1}, x_{2}) = (x_{1}, x_{1}x_{2}+φ(x_{1}, x_{2}), \frac{1}{2}(x_{1}^{2}±x_{2}^{2})), ordφ ≥ 3

is liftable if and only if φ is of the form

φ(x_{1}, x_{2}) = ∫_{0}^{x_{2}} (∓x_{2} \frac{∂ψ}{∂x_{1}} + x_{1} \frac{∂ψ}{∂x_{2}} ) dx_{2} + κ(x_{1}),

for some smooth function ψ(x_{1}, x_{2}) of order ≥ 2 and κ(x_{1}) of order ≥ 3.

Proof : Let F be transverse. Then F is approximated by transverse immersions and transverse Whitney umbrella. A generic transversal Whitney umbrella is a versal one-parameter unfolding of plane curve of type A_{2}. Then F is liftably equivalent to the above normal form. If q_{2} ∘ F is a non-submersive

Morse function, then F is liftably equivalent to (q_{1}(x), p_{1}(x), \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})). Moreover by the genericity assumption we may assume q_{1}(x) = x_{1} using liftable equivalence. Then we see that F is liftably equivalent to (x_{1}, x_{2}) ↦ (x_{1}, ax_{1}x_{2} + O(3), \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})), for a ∈ R − \{0\}, which is liftably equivalent

to (x_{1}, x_{1}x_{2} + O(3), \frac{1}{2}(x_{1}^{2} ± x_{2}^{2})).

The last statement is clear: We see F is liftable if and only if there exists a function ϕ such that

x_{1} + \frac{∂φ}{∂x_{2}} ± x_{2} \frac{∂ϕ}{∂x_{1}} − x_{1} \frac{∂ϕ}{∂x_{2}} = 0.

If such ϕ exists, then ϕ must has the form x_{2} + ψ, ordψ ≥ 2. □

Figure 3: Liftable germs of Morse type

Remark that the lifting is an ordinary open Whitney umbrella in the case (2), while the lifting is an immersion in each case of (3) and (4). Also remark that the non-transverse immeresions are never liftable by Proposition 2.5. We do not have yet the exact normal forms for hyperbolic and elliptic Whitney umbrellas under the liftable equivalence.

6 Symplectic equivalence of plane curves.

A transversal map-germ F : (R^{n}, 0) → (R^{2} × R^{n−1}, 0) is liftably equivalent to an unfolding (t, λ) ↦ (f_{λ}(t), λ), where λ ∈ (R^{n−1}, 0) and f_{λ} is a family of parametrized curves in the symplectic plane R^{2}, t being the inner variable and λ = (λ_{1}, …, λ_{n−1}) the outer variable. Therefore we proceed to consider the classification problems of bifurcations (unfoldings) of curves on the symplectic plane.

Two families of plane curves f_{λ}, f′_{λ}, (λ ∈ (R^{ℓ}, 0)) are called symplectically equivalent if there exist a family of diffeomorphisms Σ = (σ_{λ}) : (R×R^{ℓ}, 0) → (R, 0), a family of symplectomorphisms T = (τ_{λ}) : (R^{2} × R^{ℓ}, 0) → (R^{2}, 0), and a diffeomorphism φ : (R^{ℓ}, 0) → (R^{ℓ}, 0) such that τ_{λ} ∘ f′_{λ} ∘ σ_{λ} = f_{φ(λ)}, for some representatives of germs. Then, setting F : (R×R^{ℓ}, 0) → (R^{2} ×R^{ℓ}, 0), F (t, λ) = (f_{λ}(t), λ) and F′ : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0), F (t, λ) = (f′_{λ}(t), λ), we see that if f_{λ} and f′_{λ} are symplectically equivalent then F and F′ are liftably equivalent.

In the ordinary singularity theory, the versal unfolding of a singularity dominates any other unfoldings. To seek the versal unfolding of curves on the symplectic plane for the symplectic equivalence, we must first study the symplectic classification problem of plane curves.

For example, consider the simple cusp (A_{2}) f = (t^{2}, t^{3}) : (R, 0) → (R^{2}, 0). Then the unfolding F : (R × R, 0) → (R^{2} × R, 0) defined by F (t, λ) = (t^{2}, t^{3} + λt, λ) is versal with respect to the right-left equivalence. Then we ask: Is it a symplectically versal unfolding?

Now first we consider the basic problem: Let C, C′ ⊂ (R^{2}, 0) be two curve-germs. Assume that there exist a diffeomorphism-germ σ : (R^{2}, 0) → (R^{2}, 0) with σ(C) = C′. Then does there exist a symplectic (area-preserving) diffeomorphism σ′ : (R^{2}, 0) → (R^{2}, 0) with σ′(C) = C′ ?

We call two map-germs f, f′ : (R, 0) → (R^{2}, 0) are isotopic (resp. equivalent) if there exist a smooth family τ_{t} : (R^{2}, 0) → (R^{2}, 0) of diffeomorphismgerms starting from the identity τ_{0} (resp. a diffeomorphism-germ τ : (R^{2}, 0) → (R^{2}, 0)) and a diffeomorphism-germ σ : (R, 0) → (R, 0) such that f′ ∘ σ = τ_{1} ∘ f (resp. f′ ∘ σ = τ ∘ f). Moreover f and f′ are called symplectically isotopic (resp. symplectically equivalent) if we can take, in the above definitions, τ_{t} (resp. τ) symplectic.

A map-germ f : (R, 0) → (R^{2}, 0), is called achiral (resp. chiral) if f and f̄ are isotopic (resp. non-isotopic). Here we denote by f̄ the map-germ (R, 0) → (R^{2}, 0) defined by f̄(t) = (f_{1}(t), −f_{2}(t)).

Here we give several illustrating examples for the notions introduced above.

Example 6.1 (About the definition of isotopy.) Consider curves f(t) = (t^{2}, t^{3}) and f′(t) = (t^{2}, −t^{3}) of type A_{2} (resp. f(t) = (t^{3}, t^{4}) and f′(t) = (t^{3}, −t^{4}) of type E_{6}) . Then we see f and f′ are symplectically isotopic, by just taking τ_{t} identity (resp. the rotation by tπ) and σ(t) = −t. However, there does not exist a smooth family of pairs of diffeomorphism-germs (σ_{t}, τ_{t}) starting from (id_{R}, id_{R^{2}}) with f′ ∘ σ_{1} = τ_{1} ∘ f. Therefore germs (t^{2}, t^{3}) and (t^{3}, t^{4}) are achiral.

Example 6.2 (The difference of equivalence and isotopy.) Consider curves f(t) = (t^{3}, t^{5}) and f′(t) = (t^{3}, −t^{5}) of type E_{8}. Then f and f′ are equivalent but not isotopic. Therefore the germ (t^{3}, t^{5}) is chiral.

Lemma 6.3 Let m, s be positive integers and s even. Then the two curvegerms f = (t^{m}, t^{m+s} + o(t^{m+s})) and f′ = (t^{m}, −t^{m+s} + o(t^{m+s})) are not isotopic.

Proof : Assume σ(t) = αt + ⋯, α ≠ 0, τ(p_{1}, q_{1}) = (ap_{1} + bq_{1} + ⋯, cp_{1} + dq_{1} + ⋯) and that τ ∘ f ∘ σ = f′. Then we see that, first, aα^{m} = 1, cα^{m} = 0. So c = 0, a = α^{−m}. Then we see α^{m+s}d = −1, so d = −α^{−m−s}. Therefore the linear term of τ must have the negative determinant −α^{−2m−s} < 0. Then it is impossible to connect τ and the identity by a smooth family of diffeomorphism-germs. □

Since any symplectomorphism-germ can be connected to the identity through symplectomorphism-germs, we see that f and f′ are symplectically isotopic if and only if they are symplectically equivalent. Therefore the following is clear.

Lemma 6.4 If f, f′ : (R, 0) → (R^{2}, 0) are symplectically equivalent, then they are isotopic.

Now naturally we are led to the following question: Are f, f′ : (R, 0) → (R^{2}, 0) symplectically equivalent if they are isotopic?

We answer it in detail in the following sections.

7 Symplectic versality and stability.

Let f : (R, 0) → (R^{2}, 0) be a map-germ. Recall the codimension of f is defined by

codim(f) := dim_{R} (V_{f}/tf(V_{1}) + wf(V_{2})) ,

where V_{f} := {v : (R, 0) → T R^{2} | π ∘ v = f} is the space of vector field-germs along f, V_{1} (resp. V_{2}) is the space of vector field-germs over (R, 0) (resp. (R^{2}, 0)), and tf : V_{1} → V_{f} (resp. wf : V_{2} → V_{f}) is the homomorphism defined by tf(ξ) := f_{∗}(ξ) (resp. wf(η) := η ∘ f). A plane curve f is called A-finite if codim(f) < ∞. Then f has a A-versal unfolding with the parameter dimension codim(f). If f is analytic, the condition of A-finiteness is equivalent to, for instance, that the complexification of f has an injective representative.

Moreover, in general, we define

sp-codim(f) := dim_{R} (V_{f}/tf(V_{1}) + wf(V H_{2})) ,

where V H_{2} ⊆ V_{2} the space of Hamiltonian vector field-germs over the symplectic plane (R^{2}, 0). Then clearly

sp-codim(f) ≥ codim(f).

Let f : (R, 0) → (R^{2}, 0) be a plane curve-germ. An unfolding F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0) of f is called symplectically versal if, any unfolding G : (R × R^{s}, 0) → (R^{2} × R^{s}, 0), of f is symplectically equivalent to φ^{∗}F for some smooth map-germ ϕ : (R^{s}, 0) → (R^{ℓ}, 0). The following result is a special case of the versality theorem in [9]:

Proposition 7.1 An unfolding F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0) of f : (R, 0) → (R^{2}, 0) is symplectically versal if and only if F is infinitesimally symplectically versal, that is,

V_{f} = ⟨ \frac{∂F̄}{∂λ_{1}} |_{R×0} , …, \frac{∂F̄}{∂λ_{ℓ}} |_{R×0}⟩_{R} + tf(V_{1}) + wf(V H_{2}).

Moreover two versal unfoldings F and F′ of f with the same parameter dimension are liftably equivalent.

A map-germ f : (R, 0) → (R^{2}, 0) has a symplectically versal unfolding if and only if sp-codim(f) < ∞.

Remark 7.2 By the Damon’s theory [9], we have the characterization of “symplectic finite determinacy”. A map-germ f : (R, 0) → (R^{2}, 0) is called symplectically finitely determined if there exists a positive integer k such that any f′ : (R, 0) → (R^{2}, 0) with j^{k}f′(0) = j^{k}f(0) is symplectically equivalent to f. Then f is symplectically finitely determined if and only if sp-codim(f) < ∞.

We have a close relation of symplectic versality and symplectic stability ([15]) via the notion of Lagrangian liftings.

Theorem 7.3 (Symplectic versality and stability.) Let F : (R × R^{ℓ}, 0) → (R^{2} ×R^{ℓ}, 0) be a symplectically versal unfolding. Then the Lagrangian lifting F̃ : (R×R^{ℓ}, 0) → (R^{2} ×R^{2ℓ}, 0) is symplectically stable, that is, any isotropic deformation of F̃ is trivialized by symplectic equivalences. Therefore F̃ is symplectically equivalent to an open Whitney umbrella. In particular F̃ has an injective representative.

Proof : To see the symplectic stability of F̃, we apply Proposition 5.1 of [15]. Then it suffices to show that F̃ is right-left equivalent to an analytic mapgerm, that RF̃ = F̃^{∗}E_{2+2ℓ} and that codim(SingF̃_{C}) ≥ 2. Here RF̃ is the space of function-germs h ∈ E_{1+ℓ} such that the exterior differential dh of h is a linear combination by the exterior differentials of components in F̃ with coefficients from E_{1+ℓ}.

Since f = F |_{R×0} is symplectically finitely determined, f (resp. F) is symplectically equivalent to a polynomial map-germ. Moreover we have

SingF̃_{C} = SingF_{C} = \{\frac{q_{1} ∘ F}{∂t} = \frac{p_{1} ∘ F}{∂t} = 0\} is of codimension 2.

Since F is finite map-germ of corank one, by Corollary 2.4 of [15], we have RF̃ = R_{F} is a finite E_{2+ℓ}-module via F. Therefore it is a finite E_{2+2ℓ}-module via F̃. On the other hand, we see that p_{j+1} ∘ F̃ is a generating function of

\frac{∂F̄}{∂λ_{j}} |_{R×0}, setting F (t, λ) = (F̄(t, λ), λ). Thus, by the symplectic versality of

F, we see RF̃ ⊆ F̃^{∗}E_{2+2ℓ}+m_{ℓ}R_{F} ⊆ F̃^{∗}E_{2+2ℓ}+m_{2+2ℓ}R_{F}. Then, by Nakayama’s lemma, we see that RF̃ ⊆ F̃^{∗}E_{2+2ℓ}. Since the converse inclusion is clear, we have the equality: RF̃ = F̃^{∗}E_{2+2ℓ}.

Thus we see F̃ is symplectically stable. Then, by Proposition 5.1 of [15], we see F̃ is symplectically equivalent to an open Whitney umbrella. □

For a mapping φ : (R^{s}, 0) → (R^{ℓ}, 0), and an unfolding F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0), F (t, λ) = (f_{λ}(t), λ), we define the pull-back unfolding φ^{∗}F : (R × R^{s}, 0) → (R^{2} × R^{s}, 0) by (φ^{∗}F)(t, μ) = (f_{φ(μ)}(t), μ).

Proposition 7.4 Let F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0) be an unfolding of f = F |_{R×0} and F̃ a Lagrangian lifting of F. Let φ : (R^{s}, 0) → (R^{ℓ}, 0) be a map-germ. Then the lifting φ̃^{∗}F : (R × R^{s}, 0) → (R^{2} × R^{s}, 0) of φ^{∗}F defined by

p_{j} := ∑_{1≤k≤ℓ} \frac{∂φ_{k}}{∂μ_{j}} (p_{k+1} ∘ F̃),

(2 ≤ j ≤ s + 1), is a Lagrangian lifting of φ^{∗}F. In fact we have

φ̃^{∗}F^{∗}θ_{R^{2}×R^{2s}} = (id_{R} × φ)^{∗}F̃^{∗}θ_{R^{2}×R^{2ℓ}},

for the Liouville form θ_{R^{2}×R^{2s}} (resp. θ_{R^{2}×R^{2ℓ}}) on R^{2} × R^{2s} = T^{∗}(R × R^{s}) (resp. R^{2} × R^{2ℓ} = T^{∗}(R × R^{ℓ})).

The above Proposition 7.4 means the Lagrangian lifting of the pull-back unfolding can be obtained by reduction from the Lagrangian lifting of the original unfolding.

In particular we have:

Corollary 7.5 Let G : (R^{n}, 0) → (N, 0) = (R^{2} × R^{n−1}, 0) be a tansverse map-germ to R^{2}×{0}. Assume the restriction (G^{−1}(R^{2}×{0}), 0) → R^{2}×{0} is A-finite, then G is obtained from an open Whitney umbrella by a reduction process.

Proof : We may suppose G is an unfolding of an A-finite map-germ g : (R, 0) → (R^{2}, 0). Let F : (R × R^{k}, 0) → (R^{2} × R^{k}, 0) be a symplectically versal unfolding of g. Then there exists a map-germ φ : (R^{n−1}, 0) → (R^{k}, 0) such that G is symplectically equivalent to the pull-back unfolding φ^{∗}F. Then the Lagrangian lifting G̃ is H-symplectically equivalent to φ̃^{∗}F, that is a reduction of the open Whitney umbrella F̃. □

8 Symplectic defect.

Set, for a map-germ f = (f_{1}, f_{2}) : (R, 0) → (R^{2}, 0),

\begin{aligned}G_{f} & := & \{h ∈ E_{1} | dh ∈ ⟨df_{1}, df_{2}⟩_{f^{∗}E_{2}}\}, \\ & = & \{h ∈ E_{1} | dh ∈ f^{∗}(Ω_{2}^{1})\}, \end{aligned}

where E_{1} (resp. E_{2}) is the R-algebra of C^{∞} map-germs on (R, 0) (resp. (R^{2}, 0)), Ω_{2}^{1} is the space of differential 1-forms on (R^{2}, 0) and the homomorphism f^{∗} : E_{2} → E_{1} (resp. f^{∗} : Ω_{2}^{1} → Ω_{1}^{1}) is defined by the pull-back by f. Moreover we set

R_{f} := {h ∈ E_{1} | dh ∈ ⟨df_{1}, df_{2}⟩_{E_{1}}},

(cf. [15]). Thus we have intrinsically the sequence of vector spaces:

E_{1} ⊇ R_{f} ⊇ G_{f} ⊇ f^{∗}E_{2},

for the right-left equivalence class of f.

For each element h ∈ G_{f}, the exterior differential dh is written as (b ∘ f)df_{1} −(a∘f)df_{2} = f^{∗}(bdq_{1} −adp_{1}) for some functions a, b ∈ E_{2}. Through the symplectic structure dp_{1}∧dq_{1} on the (q_{1}, p_{1})-plane R^{2}, the 1-form bdq_{1}−adp_{1}

on (R^{2}, 0) corresponds to the vector field η = a\frac{∂}{∂q_{1}} + b\frac{∂}{∂p_{1}} over (R^{2}, 0). The

vector field wf(η) along f is regarded as an infinitesimal isotropic deformation of f. In this case we say that h is a generating function of wf(η).

In general, any function h(t) is called a generating function of a vector

field v = v_{1}(t) (\frac{∂}{∂q} ∘ f)+v_{2}(t) (\frac{∂}{∂p} ∘ f) : (R, 0) → T R^{2} along f : (R, 0) →

(R^{2}, 0), if dh = v_{2}df_{1} − v_{1}df_{2}(= v^{∗}θ̃), the pull-back by the isotropic map v : (R, 0) → T R^{2} ≅ T^{∗}R^{2} of the Louville 1-form on T R^{2}.

Thus, G_{f} is the space of generating functions of infinitesimal deformations of f induced from diffeomorphisms on the plane R^{2}. Then we see that G_{f} is an R-vector subspace of E_{1} and that G_{f} contains f^{∗}E_{2}. Similarly, f^{∗}E_{2} is regarded as the space of generating functions of infinitesimal deformations of f induced from symplectomorphisms on the plane R^{2}. Moreover, R_{f} is the space of generating functions of all infinitesimal deformations of f.

The follwings are clear:

Lemma 8.1 Let f : (R, 0) → (R^{2}, 0) ba a map-germ. For a diffeomorphism τ : (R^{2}, 0) → (R^{2}, 0), we have R_{τ∘f} = R_{f} and G_{τ∘f} = G_{f}. Morover, for a diffeomorphism σ : (R, 0) → (R, 0), σ^{∗} : E_{1} → E_{1} maps R_{f} to R_{f∘σ} and G_{f} to G_{f∘σ} respectively.

The following is the main lemma of this paper:

Lemma 8.2 There exists a vector space isomorphism

\frac{tf(V_{1}) + wf(V_{2})}{tf(V_{1}) + wf(V H_{2})} ≅ \frac{G_{f}}{f^{∗}E_{2}} .

Proof : Taking generating functions (mod. f^{∗}E_{2}) we define a linear map Φ : tf(V_{1})+wf(V_{2}) → E_{1}/f^{∗}E_{2}. Remark that, for each tf(ξ) = (a(t)f′_{1}, a(t)f′_{2}) ∈ tf(V_{1}), we have a(t)f′_{2}df_{1} − a(t)f′_{1}df_{2} = 0, and tf(V_{1}) maps to 0 mod. f^{∗}E_{2}. (See also [15]). The image of Φ coincides with G_{f}/f^{∗}E_{2}. We will show that the kernel of Φ is equal to tf(V_{1}) + wf(V H_{2}). Using a symplectic equivalence, we may assume ord(f_{1}) < ord(f_{2}) ∉ Z(ord(f_{1})). Now suppose,

for η = a(q, p)\frac{∂}{∂q} + b(q, p)\frac{∂}{∂p} ∈ V_{2}, a generating function of wf(η) belongs

to f^{∗}E_{2}. This means b(f_{1}, f_{2})df_{1} − a(f_{1}, f_{2})df_{2} = dH(f_{1}, f_{2}) for some H ∈ E_{2}. Then b(f_{1}, f_{2})f′_{1} − a(f_{1}, f_{2})f′_{2} = H_{q}(f_{1}, f_{2})f′_{1} + H_{p}(f_{1}, f_{2})f′_{2}. So

b(f_{1}, f_{2}) = (a(f_{1}, f_{2}) + H_{p}(f_{1}, f_{2})) \frac{f′_{2}}{f′_{1}} + H_{q}(f_{1}, f_{2}).

Since ord (\frac{f′_{2}}{f′_{1}} ) = ord(f_{2})−ord(f_{1}) ≠ ord(f_{1}), we see that a(0, 0)+H_{p}(0, 0) =

0. So the quotient c(t) := \frac{a(f_{1}, f_{2}) + H_{p}(f_{1}, f_{2})}{f′_{1}} belongs to E_{1}. Then

( \begin{matrix}a(f_{1}, f_{2}) \\ b(f_{1}, f_{2}) \end{matrix} ) = c(t) ( \begin{matrix}f′_{1} \\ f′_{2} \end{matrix} ) + ( \begin{matrix}−H_{p} \\ H_{q} \end{matrix} ) ∈ tf(V_{1}) + wf(V H_{2}).

□

Remark that the dimension of G_{f}/f^{∗}E_{2} depends only on the right-left equivalence class of f. Thus we have

Theorem 8.3 Let f : (R, 0) → (R^{2}, 0) be an A-finite map-germ. Then the symplectic defect

sd(f) := sp-codim(f) − codim(f)

is equal to dim(G_{f}/f^{∗}E_{2}), and it depends only on the right-left equivalence class of f, that is, symplectic defect is an A-invariant. Hence sp-codim(f) is an A-invariant.

Remark 8.4 If f is A-finite, then, by Mather-Gaffney’s theorem, we see f is L-finite ([21], p.494). Then we have that the vector space E_{1}/f^{∗}E_{2} is of finite dimension. So, if f is A-finite, namely, if codim(f) is finite, then sp-codim(f) is necessarily finite.

Remark 8.5 The symplectic codimension is not an A-invariant (a diffeomorphism invariant) for map-germs R → R^{4}. For example, consider mapgerms

A_{2,0} : (q_{1} = t^{2}, p_{1} = t^{3}, q_{2} = 0, p_{2} = 0),

and

A_{2,1} : (q_{1} = t^{2}, p_{1} = t^{5}, q_{2} = t^{3}, p_{2} = 0),

from Arnold’s classification [3]. Then A_{2,0} and A_{2,1} are clearly A-equivalent. Besides we have sp-codim(A_{2,0}) = 3, and sp-codim(A_{2,1}) = 4. In fact, when f = A_{2,0} : (R.0) → (R^{4}, 0), we can take ^{t}(0, t, 0, 0),^{t}(0, 0, t, 0),^{t}(0, 0, 0, t) as a basis of the vector space V_{f}/(tf(V_{1}) + wf(V H_{4})). For f = A_{2,1} we need t(0, t^{2}, 0, 0) in addition. □

From the definition of the symplectic defect, we have:

Proposition 8.6 Let F : (R × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0) be an A-versal unfolding of f : (R, 0) → (R^{2}, 0) with sd(f) = 0. Then F is a symplectically versal unfolding of f.

Lastly we show that the symplectic codimension of an A-finite map-germ is, actually, equal to the classical δ-invariant.

Let f : (R, 0) → (R^{2}, 0) be an A-finite map-germ. Then we set δ(f) := dim_{R} E_{1}/f^{∗}E_{2}. Then we have:

Theorem 8.7 For an A-finite map-germ f : (R, 0) → (R^{2}, 0),

sp-codim(f) = δ(f).

Proof : We have an exact sequence of vector spaces:

0 → \frac{V′_{f}}{tf(V_{1})} → \frac{V_{f}}{tf(V_{1}) + wf(V H_{2})} → \frac{R_{f}}{f^{∗}E_{2}} → 0,

where V′_{f} is the space of vector fields along f having zero generating functions. Remark that wf(V H_{2}) ∩ V′_{f} ⊆ tf(V_{1}). Now we have

dim_{R} \frac{V′_{f}}{tf(V_{1})} = dim_{R} \frac{E_{1}}{R_{f}} .

To see this, we may assume f_{1} = t^{k} and ord(f_{2}) > k, for some positive integer. Then R_{f} = R + m_{1}^{k}. So dim_{R}(E_{1}/R_{f}) = k − 1. On the other hand, if we set φ(t) = f′_{2}/f′_{1}, the vector space V′_{f}/tf(V_{1}) has the basis

( \begin{matrix}1 \\ φ \end{matrix} ) , ( \begin{matrix}t \\ tφ \end{matrix} ) , …, ( \begin{matrix}t^{k−2} \\ t^{k−2}φ \end{matrix} ) .

Therefore we see that also dim_{R} V′_{f}/tf(V_{1}) is equal to k − 1.

Thus we have

sp-codim(f) = dim_{R} \frac{V_{f}}{tf(V_{1}) + wf(V H_{2})}
= dim_{R} \frac{V′_{f}}{tf(V_{1})} + dim_{R} \frac{R_{f}}{f^{∗}E_{2}}
= dim_{R} \frac{E_{1}}{R_{f}} + dim_{R} \frac{R_{f}}{f^{∗}E_{2}}
= dim_{R} \frac{E_{1}}{f^{∗}E_{2}} = δ(f).
□

Remark 8.8 The vector space V′_{f}/tf(V_{1}) has the clear geometric meaning: The space V′_{f} consists of vector fields v ∈ V_{f} along f such that, for any regular point t ∈ R of f, v(t) ∈ f_{∗}(T_{t}R). Such a vector field may not come from a vector filed over R via f_{∗}. Then V′_{f}/tf(V_{1}) measures its gap. Also it has the clear algebraic meaning as the cohomology of a complex 0 → V_{1} → V′_{f} → 0 defined by the Jacobi matrix of f.

Remark 8.9 For an A-finite map-germ f : (R, 0) → (R^{2}, 0) the Milnor number μ is equal to 2δ (cf. [20]). So we have sp-codim(f) = \frac{μ}{2}.

9 Symplectic defects and symplectically versal unfoldings of simple plane curves.

In this section, we calculate the symplectic defects, defined in the previous section, in several cases.

First we give examples of plane-curves without symplectic defect:

Proposition 9.1 If a plane curve is right-left equivalent to f : (R, 0) → (R^{2}, 0), f(t) = (t^{m}, t^{m+s}) for some positive integers m, s, then its symplectic defect is equal to zero.

Proof : It is sufficient to see sd(f) = 0. Let h ∈ G_{f}. Then

dh = a(t^{m}, t^{m+s})d(t^{m}) + b(t^{m}, t^{m+s})d(t^{m+s})

for some a, b ∈ E_{2}. Remark that, there is a positive integer k, such that f^{∗}E_{2} contains any functions with order ≥ k. Now it is easy to see that h is a function of t^{m}, t^{m+s} up to functions with order ≥ k. Therefore we have h ∈ f^{∗}E_{2}. Thus G_{f}/f^{∗}E_{2} = 0 and sd(f) = 0. □

Bruce and Gaffney [5] classified simple plane curves: The A-equivalence class of simple (0-modal) plane curves are classified into the following list:

A_{2ℓ} : t ↦ (t^{2}, t^{2ℓ+1});

E_{6ℓ} : t ↦ (t^{3}, t^{3ℓ+1} ± t^{3(ℓ+p)+2}), 0 ≤ p ≤ ℓ − 2; t ↦ (t^{3}, t^{3ℓ+1});

E_{6ℓ+2} : t ↦ (t^{3}, t^{3ℓ+2} ± t^{3(ℓ+p)+4}), 0 ≤ p ≤ ℓ − 2; t ↦ (t^{3}, t^{3ℓ+2});

W_{12} : t ↦ (t^{4}, t^{5} ± t^{7}); t ↦ (t^{4}, t^{5});

W_{18} : t ↦ (t^{4}, t^{7} ± t^{9}); t ↦ (t^{4}, t^{7} ± t^{13}); t ↦ (t^{4}, t^{7});

W_{1,2q−1}^{#} : t ↦ (t^{4}, t^{6} + t^{2q+5}), q ≥ 1.

Remark that, in the above list, the germs (t^{3}, t^{4}±t^{5}) and (t^{3}, t^{4}) of the type E_{6} (resp. (t^{3}, t^{5} ± t^{7}) and (t^{3}, t^{5}) of the type E_{8}) are actually A-equivalent. See also [4] pp. 57–59.

Then we have

Theorem 9.2 (1) If f is equivalent to A_{2ℓ}, E_{6}, E_{8} or E_{6ℓ} : (t^{3}, t^{3ℓ+1}); E_{6ℓ+2} : (t^{3}, t^{3ℓ+2}); W_{12} : (t^{4}, t^{5}); W_{18} : (t^{4}, t^{7}) then sd(f) = 0.

(2) If f is equivalent to E_{6ℓ} : (t^{3}, t^{3ℓ+1}±t^{3(ℓ+p)+2}), 0 ≤ p ≤ ℓ−2, ℓ ≥ 2, then t^{3(ℓ+p+1)+2}, …, t^{6ℓ−1} form a basis of G_{f}/f^{∗}E_{2} and sd(f) = ℓ−p−1. The family (t^{3}, (±1)^{ℓ+1}t^{3ℓ+1}+∑_{j=1}^{ℓ−1} λ_{j}t^{3(ℓ+j)−1}) contains all symplectic classes of type E_{6ℓ}. If f is equivalent to E_{6ℓ+2} : (t^{3}, t^{3ℓ+2} ± t^{3(ℓ+p)+4}), then t^{3(ℓ+p+1)+4}, …, t^{6ℓ+1} form a basis of G_{f}/f^{∗}E_{2} and sd(f) = ℓ − p − 1. The family (t^{3}, (±1)^{ℓ}t^{3ℓ+2} + ∑_{j=1}^{ℓ−1} λ_{j}t^{3(ℓ+j)+1}) contains all symplectic classes of type E_{6ℓ+2}.

(3) If f is equivalent to W_{12} : (t^{4}, t^{5}±t^{7}), then t^{11} forms a basis of G_{f}/f^{∗}E_{2} and sd(f) = 1. The family (t^{4}, t^{5} +λt^{7}) contains all symplectic classes of type W_{12}.

(4) If f is equivalent to W_{18} : (t^{4}, t^{7} ± t^{9}), then t^{13}, t^{17} form a basis of G_{f}/f^{∗}E_{2} and sd(f) = 2. If f is equivalent to W_{18} : (t^{4}, t^{7} ± t^{13}), then t^{17} forms a basis of G_{f}/f^{∗}E_{2} and sd(f) = 1. The family (t^{4}, t^{7} + λt^{9} + μt^{13}) contains all symplectic classes of type W_{18}.

(5) If f is equivalent to W_{1,2q−1}^{#} : (t^{4}, t^{6} ± t^{2q+5}), then t^{2q+9}, t^{2q+13} form a basis of G_{f}/f^{∗}E_{2} and sd(f) = 2. The family (t^{4}, ±t^{6} +λt^{2q+5} +μt^{2q+9}), λ ≠ 0, contains all symplectic classes of type W_{1,2q−1}^{#}.

To examine the symplectic equivalence classes, we first observe the following results:

Lemma 9.3 Let f : (R, 0) → (R^{2}, 0) be A-finite. Then we have: If ord(f) = m, then f is symplectically equivalent to (t^{m}, t^{mℓ+j} +o(t^{mℓ+j})) for some ℓ ≥ 1 and j with 1 ≤ j ≤ m − 1. In particular, we have: If ord(f) = 2, then f is symplectically equivalent to (t^{2}, t^{2ℓ+1} + o(t^{2ℓ+1})) for some ℓ ≥ 1. If ord(f) = 3, then f is symplectically equivalent to (t^{3}, t^{3ℓ+1} + o(t^{3ℓ+1})) or (t^{3}, t^{3ℓ+2} + o(t^{3ℓ+2})) for some ℓ ≥ 1.

If ord(f) = 4 and f is A-simple, then f is symplectically equivalent to (t^{4}, t^{5} + o(t^{5})), (t^{4}, t^{6} + o(t^{6})) or (t^{4}, t^{7} + o(t^{7})).

Remark 9.4 If f is symplectically equivalent to (t^{4}, t^{ℓ} + o(t^{ℓ})), ℓ ≥ 9, then f is not A-simple. Moreover if ord(f) ≥ 5, then f is not A-simple. See [5].

Proof of Theorem 9.2: We give the calculation for W_{18} in detail. Other cases can be treated in the similar way, so we omit the detail for them.

Let f : (R, 0) → (R^{2}, 0) be the map-germ defined by f(t) = (q_{1}, p_{1}) = (t^{4}, t^{7}+t^{13}) (i.e. most degenerate case of type W_{18}). We examine the quotient space E_{1}/f^{∗}E_{2}. The implicit equation of the curve is

y^{4} = x^{7} + 4x^{5}y^{2} − 2x^{10} + x^{13}.

Then examining monomials x^{i}y^{j}, i ≥ 0, 0 ≤ j ≤ 3 pulled back by f, we see

f^{∗}E_{2} = ⟨1, t^{4}, t^{8}, t^{12}, t^{16}, t^{14}, t^{15}, t^{7} + t^{13}, t^{11} + t^{17}⟩_{R} + m_{1}^{18}.

So the projections of t, t^{2}, t^{3}, t^{5}, t^{6}, t^{9}, t^{10}, t^{13}, t^{17} form a basis of E_{1}/f^{∗}E_{2} over R. Besides, we see

G_{f} = ⟨t^{11}, t^{17}⟩_{R} + f^{∗}E_{2}.

In fact, remark that a function h ∈ E_{1} with dh = (t^{7} + t^{13})d(t^{4}) belongs to

G_{f}. Thus h = \frac{4}{11}t^{11} + \frac{4}{17}t^{17} ∈ G_{f}. Since t^{11} + t^{17} ∈ f^{∗}E_{2} ⊆ G_{f}, we see both

t^{11} and t^{17} belong to G_{f}. Morever we easily see that any polynomial with just monomials t, t^{2}, t^{3}, t^{5}, t^{6}, t^{9}, t^{10} belonging to G_{f} must be zero. Therefore we see t^{17} forms a basis of G_{f}/f^{∗}E_{2}. So dim_{R} G_{f}/f^{∗}E_{2} = 1, and sd(f) = 1.

Similarly to above, for f = (t^{4}, t^{7} + t^{9}), we have

f^{∗}E_{2} = ⟨1, t^{4}, t^{8}, t^{12}, t^{16}, t^{14}, t^{7} + t^{9}, t^{11} + t^{13}, t^{15} + t^{17}⟩_{R} + m_{1}^{18},

and

G_{f} = ⟨t^{11}, t^{13}, t^{15}, t^{17}⟩_{R} + f^{∗}E_{2}.

Here we remark, in this case, ∫ (t^{7} + t^{9})d(t^{4}) = \frac{4}{11}t^{11} + \frac{4}{13}t^{13} ∈ G_{f}, and also

∫ t^{4}(t^{7} + t^{9})d(t^{4}) = \frac{4}{15}t^{15} + \frac{4}{17}t^{17} ∈ G_{f}. Thus we see t^{13}, t^{17} form a basis of

G_{f}/f^{∗}E_{2}, and sd(f) = 2.

For the case f = (t^{4}, t^{7}), we see sd(f) = 0 by Proposition 9.1.

Now consider the two-parameter family f_{λ,μ}(t) = (t^{4}, t^{7} + λt^{9} + μt^{13}). We show that, if a germ f is of type W_{18}, namely, if f is A-equivalent

to (t^{4}, t^{7}), (t^{4}, t^{7} + t^{9}) or (t^{4}, t^{7} + t^{13}), then f is symplectically equivalent to f_{λ,μ} = (t^{4}, t^{7} + λt^{9} + μt^{13}) for some (λ, μ) ∈ R^{2}. First if f is of type W_{18}. Then f is symplectically equivalent to f′ = (t^{4}, t^{7} +o(t^{7})) by Lemma 9.3. We write

f′(t) = (t^{4}, t^{7} + λt^{9} + μt^{13} + Φ(t)),

where Φ ∈ ⟨t^{8}, t^{10}, t^{11}, t^{12}⟩_{R} + m_{1}^{14}. Set v = Φ(t) (\frac{∂}{∂p} ∘ f′) ∈ V_{f′}. Consider

the generating function e ∈ m_{1} of v so that de = Φ(t)d(t^{4}). Then e ∈ ⟨t^{12}, t^{14}, t^{15}, t^{16}⟩_{R} + m_{1}^{18}. We remark that ⟨t^{12}, t^{14}, t^{15}, t^{16}⟩_{R} + m_{1}^{18} ⊆ f′^{∗}m_{2}^{2}. Therefore, e = f′^{∗}H, for some H ∈ m_{2}^{2}. Consider the Hamiltonian vector filed X_{H} with Hamiltonian H. Remark that X_{H} ∈ V H_{2} ∩ m_{2}V_{2}. Then the generating function of the vector field v − X_{H} ∘ f′ along f′ is zero. So, v − X_{H} ∘ f′ ∈ V′_{f′}. Moreover ord(v − X_{H} ∘ f′) ≥ 4. By Lemma 9.5 below, we see that there exists ξ ∈ m_{1}V_{1} such that v − X_{H} ∘ f′ = tf′(ξ). Thus we have v ∈ tf(m_{1}V_{1}) + wf(V H_{2} ∩ m_{2}V_{2}). Then, using the homotopy method we see f is symplectically equivalent to f_{λ,μ}. □

Lemma 9.5 Let w ∈ V′_{f}. If ord(w) ≥ ord(f) − 1, then w ∈ tf(V_{1}). If ord(w) ≥ ord(f), then w ∈ tf(m_{1}V_{1}).

Proof : Let w = a\frac{∂}{∂q} + b\frac{∂}{∂p}. Since w ∈ V′_{f}, we see b\frac{df_{1}}{dt} − a\frac{df_{2}}{dt} = 0. Set

c(t) = a/ (\frac{df_{1}}{dt} ). Then we see ^{t}(a, b) = c(t) ^{t}(\frac{df_{1}}{dt} , \frac{df_{1}}{dt}). □

In Theorem 9.2, each symplectic normal form (family) may have finite redundancy. In fact, we know, by the infinitesimal mathod, that the intersection of the family and an orbit under the symplectic equivalence forms, in a jet space, a zero-dimensional algebraic set, so a finite set.

Actually, by the direct and formal calculations, we have the exact description of symplectic moduli spaces of A-simple plane curves. We realize the exact determination of symplectic normal form turns out to be a surprisingly simple task, after the infinitesimal consideration stated in Theorem 9.2.

Theorem 9.6 (1) Let ℓ ≥ 2. Then any plane curve germ of type E_{6ℓ} is symplectically equivalent to

f_{λ} = (t^{3}, (±1)^{ℓ+1}t^{3ℓ+1} + ∑_{j=1}^{ℓ−1} λ_{j}t^{3(ℓ+j)−1}),

for some λ = (λ_{1}, …, λ_{ℓ−1}) ∈ R^{ℓ−1}. Moreover f_{λ} and f′_{λ} are symplectically equivalent if and only if λ′ = (±1)^{ℓ−1}λ.

(2) Let ℓ ≥ 2. Then any plane curve germ of type E_{6ℓ+2} is symplectically equivalent to

f_{λ} = (t^{3}, (±1)^{ℓ}t^{3ℓ+2} + ∑_{j=1}^{ℓ−1} λ_{j}t^{3(ℓ+j)+1}),

for some λ = (λ_{1}, …, λ_{ℓ−1}) ∈ R^{ℓ−1}. Moreover f_{λ} and f′_{λ} are symplectically equivalent if and only if λ′ = (±1)^{ℓ}λ.

(3) Any plane curve germ of type W_{12} is symplectically equivalent to

f_{λ} = (t^{4}, t^{5} + λt^{7})

for some λ ∈ R. Moreover f_{λ} and f′_{λ} are symplectically equivalent if and only if λ′ = λ.

(4) Any plane curve germ of type W_{18} is symplectically equivalent to

f_{λ,μ} = (t^{4}, t^{7} + λt^{9} + μt^{13})

for some (λ, μ) ∈ R^{2}. Moreover f_{λ,μ} and f_{λ′,μ′} are symplectically equivalent if and only if (λ′, μ′) = (λ, μ).

(5) Let q ≥ 1. Then any plane curve germ of type W_{1,2q−1}^{#} is symplectically equivalent to

f_{λ,μ} = (t^{4}, ±t^{6} + λt^{2q+5} + μt^{2q+9}),

for some (λ, μ) ∈ (R − {0}) × R. Moreover f_{λ,μ} and f_{λ′,μ′} are symplectically equivalent if and only if (λ′, μ′) = ±(λ, μ).

For the proof of Therem 9.6, we need the following lemma:

Lemma 9.7 Let Q = q +bp+h_{1}q^{2} +h_{2}qp+h_{3}p^{2} +…, P = p+k_{1}q^{2} +k_{2}qp+ k_{3}p^{2}+…, indicate the 2-jet of a symplectomorphism-germ (R^{2}, 0) → (R^{2}, 0). Then we have 2h_{1} − 2bk_{1} + k_{2} = 0 and h_{2} − bk_{2} + 2k_{3} = 0.

Proof : Since we have dP ∧dQ = (1+2h_{1}q +h_{2}p−2bk_{1}q +k_{2}q −bk_{2}p+2k_{3}p+ …)dp ∧ dq, the result is straightforward. □

Proof of Therem 9.6: We give the proof just in the case W_{18}. The remaing case, more or less, can be treated similarly.

Suppose (t^{4}, t^{7} +λt^{9} +μt^{13}) and (t^{4}, t^{7} +λ′t^{9} +μ′t^{13}) are symplectic equivalent by a difeomorphism-germ σ : (R, 0) → (R, 0), σ(t) = a_{1}t + a_{2}t^{2} + a_{3}t^{3} + ⋯, and a symplectomorphism-germ τ : (R^{2}, 0) → (R^{2}, 0), τ(p, q) = (P, Q) with Q = aq+bp+h_{1}q^{2}+h_{2}qp+h_{3}p^{2}+…, P = cq+ep+k_{1}q^{2}+k_{2}qp+k_{3}p^{2}+. . .. Namely we suppose

at^{4}+b(t^{7}+λt^{9}+μt^{13})+h_{1}t^{8}+h_{2}t^{4}(t^{7}+λt^{9}+μt^{13})+⋯ = (a_{1}t+a_{2}t^{2}+a_{3}t^{3}+⋯)^{4},

and

ct^{4} + e(t^{7} + λt^{9} + μt^{13}) + k_{1}t^{8} + k_{2}t^{4}(t^{7} + λt^{9} + μt^{13}) + ⋯
= (a_{1}t + a_{2}t^{2} + a_{3}t^{3} + ⋯)^{7} + λ′(a_{1}t + a_{2}t^{2} + a_{3}t^{3} + ⋯)^{9}
+μ′(a_{1}t + a_{2}t^{2} + a_{3}t^{3} + ⋯)^{13}.

Then we have a = a_{1}^{4}, c = 0, e = a_{1}^{7}. Thus we have 1 = ae = a_{1}^{11}, so we have a_{1} = 1 and a = 1, c = 0, e = 1. In particular, the 2-jet of τ has the form as in Lemma 9.7. Then, from the first equation, we see a_{2} = 0, a_{3} = 0, b = 4a_{4}, h_{1} = 4a_{5}, bλ = 4a_{6}. Moreover, from the second equation, we have k_{1} = 0, λ = λ′, a_{4} = 0. Then we have k_{2} = 7a_{5} and μ + k_{2}λ = μ′ + 9a_{5}λ′, besides b = 0. By Lemma 9.7, we see 0 = 2h_{1} −2bk_{1} +k_{2} = 8a_{5} +7a_{5} = 15a_{5}, hence a_{5} = 0, as well as k_{1} = 0. Therefore we have λ = λ′ and μ = μ′. □

We summarize the result in the table.

DIFF. NORMAL FORM  DIFECT  SYM. NORMAL FORM
A_{2ℓ}  (t^{2}, t^{2ℓ+1})  0  (t^{2}, t^{2ℓ+1})
E_{6}  (t^{3}, t^{4})  0  (t^{3}, t^{4})
(t^{3}, t^{3ℓ+1} ± t^{3(ℓ+p)+2}), 0 ≤ p ≤ ℓ − 2 ℓ − p − 1
E_{6ℓ}(ℓ ≥ 2)  (t^{3}, (±1)^{ℓ+1}t^{3ℓ+1} + Σ_{j=1}^{ℓ−1}λ_{j}t^{3(ℓ+j)−1})
(t^{3}, t^{3ℓ+1})  0
E_{8}  (t^{3}, t^{5})  0  (t^{3}, t^{5})
(t^{3}, t^{3ℓ+2} ± t^{3(ℓ+p)+4}), 0 ≤ p ≤ ℓ − 2 ℓ − p − 1
E_{6ℓ+2}(ℓ ≥ 2)  (t^{3}, (±1)^{ℓ}t^{3ℓ+2} + Σ_{j=1}^{ℓ−1}λ_{j}t^{3(ℓ+j)+1})
(t^{3}, t^{3ℓ+2})  0
(t^{4}, t^{5} ± t^{7})  1
W_{12}  (t^{4}, t^{5} + λt^{7})
(t^{4}, t^{5})  0
(t^{4}, t^{7} ± t^{9})  2
W_{18}  (t^{4}, t^{7} ± t^{13})  1  (t^{4}, t^{7} + λt^{9} + μt^{13})
(t^{4}, t^{7})  0
W_{1,2q−1}^{#}  (t^{4}, t^{6} + t^{2q+5}), q ≥ 1  2  (t^{4}, ±t^{6} + λt^{2q+5} + μt^{2q+9})

Table 1: The symplectic classification of simple plane curves.

In particular we have the following:

Corollary 9.8 The symplectically simple (0-modal) plane curves are symplectically classified into A_{2ℓ}, ℓ = 1, 2, 3, …; E_{6} : t ↦ (t^{3}, t^{4}); E_{8}^{±} : t ↦ (t^{3}, ±t^{5}).

Proof : By Theorem 9.2, the germs of type E_{6ℓ} (ℓ ≥ 2), E_{6ℓ+2} (ℓ ≥ 2), W_{12}, W_{18}, or W_{1,2q−1}^{#} is never symplectically simple. Since A_{2ℓ} is adjacent to just A_{2m} with m ≤ ℓ, E_{6} is adjacent to just E_{6}, A_{4}, A_{2}, A_{0}, and E_{8}^{±} is adjacent to just E_{8}^{±}, E_{6}, A_{4}, A_{2}, A_{0}. So they have nearby just finitely many symplectic equivalence classes. □

For the symplectically versal unfoldings we have:

Proposition 9.9 The symplectically versal unfolding with the minimal number of parameters for each A-simple plane curve is given by:

A_{2ℓ} (sp-codim = ℓ) :

(t^{2}, t^{2ℓ+1} + ∑_{j=1}^{ℓ} λ_{j}t^{2ℓ−2j+1}),

(λ_{1}, …, λ_{ℓ}) ∈ (R^{ℓ}, 0).

E_{6ℓ} (sp-codim = 3ℓ) :

( \begin{matrix}t^{3} + λt, \\ (±1)^{ℓ+1}t^{3ℓ+1} + ∑_{j=1}^{ℓ} μ_{j}t^{3ℓ−3j+1} + ∑_{j=1}^{2ℓ−1} ν_{j}t^{6ℓ−3j−1} \end{matrix} ) ,

(ν_{1}, …, ν_{ℓ−1}) ∈ R^{ℓ−1}, (λ, μ_{1}, …, μ_{ℓ}, ν_{ℓ}, …, ν_{2ℓ−1}) ∈ (R^{2ℓ+1}, 0).

E_{6ℓ+2} (sp-codim = 3ℓ + 1) :

( \begin{matrix}t^{3} + λt, \\ (±1)^{ℓ}t^{3ℓ+2} + ∑_{j=1}^{ℓ} μ_{j}t^{3ℓ−3j+2} + ∑_{j=1}^{2ℓ} ν_{j}t^{6ℓ−3j+1} \end{matrix} ) ,

(ν_{1}, …, ν_{ℓ−1}) ∈ R^{ℓ−1}, (λ, μ_{1}, …, μ_{ℓ}, ν_{ℓ}, …, ν_{2ℓ}) ∈ (R^{2ℓ+2}, 0).

W_{12} (sp-codim = 6) :

( \begin{matrix}t^{4} + λ_{1}t^{2} + λ_{2}t, \\ t^{5} + μ_{1}t^{7} + μ_{2}t^{3} + μ_{3}t^{2} + μ_{4}t \end{matrix} ) ,

μ_{1} ∈ R, (λ_{1}, λ_{2}, μ_{2}, μ_{3}, μ_{4}) ∈ (R^{5}, 0).

W_{18} (sp-codim = 9) :

( \begin{matrix}t^{4} + λ_{1}t^{2} + λ_{2}t, \\ t^{7} + μ_{1}t^{13} + μ_{2}t^{9} + μ_{3}t^{6} + μ_{4}t^{5} + μ_{5}t^{3} + μ_{6}t^{2} + μ_{7}t \end{matrix} ) ,

(μ_{1}, μ_{2}) ∈ R^{2}, (λ_{1}, λ_{2}, μ_{3}, μ_{4}, μ_{5}, μ_{6}, μ_{7}) ∈ (R^{7}, 0).

W_{1,2q−1}^{#} (sp-codim = q + 7) :

( \begin{matrix}t^{4} + λt^{2} + ρt, \\ ±t^{6} + t^{2q+5} + μt^{2q+9} + ∑_{j=0}^{q+2} ν_{j}t^{2q+5−2j} + θt^{2} + ρt^{2q+2} \end{matrix} ) ,

(ν_{0}, μ) ∈ R^{2}, ν_{0} ≠ −1, (λ, ν_{1}, …, ν_{q+2}, θ, ρ) ∈ (R^{q+5}, 0).

Remarkable thing is that the symplectic versal unfolding can be taken uniformly for each class of simple plane curves; this is not the case for the A-versal unfolding. This is natural because the A-E-W -W^{#}-classification is based on the constancy of the Milnor number μ, and the μ-constant strata coincide with the sp-codim constant strata (cf. Theorem 8.7).

In particular we have:

Proposition 9.10 Let F : (R^{n}, 0) → (R^{2} × R^{n−1}) be a symplectically versal unfolding of A_{2ℓ} : (t^{2}, t^{2ℓ+1}), ℓ ≤ n − 1. Then F is liftable equivalent to

\begin{matrix}(x_{1}, x_{2}, …, x_{n}) = (t, λ_{1}, …, λ_{n−1}) ↦ \\ (q_{1}, p_{1}, q_{2}, …, q_{n}) = (t^{2}, t^{2ℓ+1} + λ_{1}t^{2ℓ−1} + λ_{2}t^{2ℓ−3} + ⋯ + λ_{ℓ}t, λ_{1}, …, λ_{n−1}). \end{matrix}

Example 9.11 (The opening of Whitney umbrella.) Any symplectically versal unfolding of A_{1} : (t^{2}, t^{3}) is liftable equivalent to

\begin{matrix}(x_{1}, x_{2}, …, x_{n}) = (t, λ_{1}, …, λ_{n−1}) ↦ \\ (q_{1}, p_{1}, q_{2}, …, q_{n}) = (t^{2}, t^{3} + λ_{1}t, λ_{1}, …, λ_{n−1}). \end{matrix}

The Lagrangian lifting is symplectically equivalent to

\begin{matrix}(x_{1}, x_{2}, …, x_{n}) = (t, λ_{1}, …, λ_{n−1}) ↦ \\ (q_{1}, p_{1}, q_{2}, …, q_{n}, p_{2}, …, p_{n}) = (t^{2}, t^{3} + λ_{1}t, λ_{1}, \frac{2}{3}t^{3}, 0, …, 0). \end{matrix}

10 Isotopy and symplectic classifications.

We show that if the symplectic defect vanishes, then the classifications by isotopy and by symplectomorphism coincide.

Lemma 10.1 There are isomorphisms of the vector spaces:

\frac{tf(m_{1}V_{1}) + wf(m_{2}V_{2})}{tf(m_{1}V_{1}) + wf(m_{2}V_{2} ∩ V H_{2})} ≅ \frac{G′_{f}}{f^{∗}m_{2}^{2}} ≅ \frac{G_{f}}{f^{∗}E_{2}} ,

where m_{1} (resp. m_{2}) is the maximal ideal of E_{1} (resp. E_{2}) consisting of functions H with H(0) = 0, and G′_{f} = {h ∈ m_{1} | dh ∈ ⟨df_{1}, df_{2}⟩_{f^{∗}m_{2}}}.

Proof : In the proof of Lemma 8.2, assume a(0, 0) = b(0, 0) = 0. Then, from that a(0, 0) = 0, we have that H_{p}(0, 0) = 0, and from b(0, 0) = 0, we have that H_{q}(0, 0) = 0. So we have c(0) = 0. Therefore we have the first isomorphism.

To get the second isomorphism, first we remark that if we have (A ∘ f)df_{1} + (B ∘ f)df_{2} = 0, then we have A(0, 0) = B(0, 0) = 0. (It is proved by comparing orders of terms easily).

Then we show G′_{f} ∩ f^{∗}E = f^{∗}m_{2}^{2}. In fact, the inclusion G′_{f} ∩ f^{∗}E ⊇ f^{∗}m_{2}^{2} is clear. Let h = a(f_{1}, f_{2}) ∈ G′_{f} ∩f^{∗}E. Then dh = (C ∘f)df_{1} +(D ∘f)df_{2} with

C(0, 0) = 0, D(0, 0) = 0. Besides we have dh = (\frac{∂a}{∂q} ∘ f) df_{1}+(\frac{∂a}{∂p} ∘ f) df_{2}.

So we have \{(\frac{∂a}{∂q} − C) ∘ f\} df_{1} + \{(\frac{∂a}{∂p} − D) ∘ f\} df_{2} = 0. Therefore we

have \frac{∂a}{∂q}(0, 0) = \frac{∂a}{∂p}(0, 0) = 0, namely a ∈ m_{2}^{2}. So h = a ∘ f ∈ f^{∗}m_{2}^{2}.

Lastly we show for any h ∈ G_{f}, there exist α, β, γ ∈ R with h − (α + βf_{1} + γf_{2}) ∈ G′_{f}. In fact, dh = (R ∘ f)df_{1} + (S ∘ f)df_{2}, for some R, S ∈ E_{2}. Then d(h − h(0) − R(0, 0)f_{1} − S(0, 0)f_{2}) ∈ ⟨df_{1}, df_{2}⟩_{f^{∗}m_{2}}. □

Corollary 10.2 The symplectic defect of a plane curve-germ measures the codimension of the symplectic equivalence orbit in the A-equivalence orbit of the germ (in the jet space of sufficiently high order).

Remark 10.3 Let f : (R, 0) → (R^{2}, 0) be a map-germ. Then, for diffeomorphisms τ : (R^{2}, 0) → (R^{2}, 0) and σ : (R, 0) → (R, 0), we have G′_{τ∘f} = G′_{f} and σ^{∗}(G′_{f}) = G′_{f∘σ}.

Theorem 10.4 Let f : (R, 0) → (R^{2}, 0) be an A-finite map-germ with sd(f) = 0. If a map-germ f′ : (R, 0) → (R^{2}, 0) is isotopic to f, then f′ is symplectically equivalent to f.

Proof : Using a right equivalence, we may assume f′ = τ_{1} ∘ f, for a family of diffeomorphisms τ_{s} : (R^{2}, 0) → (R^{2}, 0) starting from τ_{0} = id_{R}. Set f_{s} = τ_{s} ∘ f : (R, 0) → (R^{2}, 0). Then sd(f_{s}) = 0. Thus tf_{s}(m_{1}V_{1}) + wf_{s}(m_{2}V_{2}) ⊆ tf_{s}(m_{1}V_{1}) + wf_{s}(m_{2}V_{2} ∩ V H_{2}). By the homotopy method, we have the required result. □

Example 10.5 The plane curves of type E_{8} are classified up to isotopy into E_{8}^{+} : t ↦ (t^{3}, t^{5}) and E_{8}^{−} : t ↦ (t^{3}, −t^{5}), because they are chiral. Then, since the symplectic defect vanishes in this case, this gives also the symplectic classification.

11 Lagrangian liftings of the swallowtails.

Let M^{2k} be the space of polynomials of degree 2k + 1 of the form (cf. [2])

M^{2k} = \{\frac{x^{2k+1}}{(2k + 1)!} + q_{1} \frac{x^{2k−1}}{(2k − 1)!} + … + q_{k} \frac{x^{k}}{(k)!} − p_{k} \frac{x^{k−1}}{(k − 1)!} + … + (−1)^{k}p_{1}\}

endowed with the symplectic Darboux form ∑_{i=1}^{k} dp_{i} ∧ dq_{i} (reduction of the sl_{2}-invariant symplectic form on the space of binary forms of 2k + 3-degree).

The canonical projection into N is given by the derivative

D^{k−1} = \frac{d^{k−1}}{dx^{k−1}} ,

which projects M^{2k} into the space of polynomials

N = \{\frac{x^{k+2}}{(k + 2)!} + q_{1} \frac{x^{k}}{(k)!} + … + q_{k}x − p_{k}\}.

The standard (generalized) swallowtail in N is defined as the space Σ_{k} ⊂ N of polynomials having at least one root of multiplicity ≥ 2.

The derivative \frac{d}{dx} of the polynomial decreases the multiplicities of its roots, however the difference of the degree of polynomial and the multiplicity of the root, called co-multiplicity, is not affected by the derivative. So the polynomials of Σ_{k} have the roots of comultiplicity ≤ k.

The canonical Lagrangian variety, which is a Lagrangian lifting of Σ_{k} is defined after V.I. Arnold as the space Σ̃_{k} of polynomials in M^{2k} having root of multiplicity at least k + 1. This lifting is most regular (stabilisation in the sense of Arnold) because the multiplicity is at least k + 1 and degree of the polynomial is 2k + 1 and finally the polynomials of Σ̃_{k} have only unique root of this multiplicity. So the intersection points of Σ_{k} are avoided.

A parametrisation of Σ_{k} is given in the form

F : (R^{k}, 0) → (N, 0)

F (s) = (s_{1}, …, s_{k−1}, −\frac{s_{k}^{k+1}}{(k + 1)!} − ∑_{i=1}^{k−1} s_{i} \frac{s_{k}^{k−i}}{(k − i)!},

−\frac{s_{k}^{k+2}}{(k + 2)k!} − ∑_{i=1}^{k−1} s_{k−i} \frac{s_{k}^{k−i+1}}{(k − i + 1)(k − i − 1)!}).

Its Lagrangian lifting Σ̃_{k}, F̃ : (R^{k}, 0) → (M^{2k}, ω) is generated by the following generating family (cf. [16], p. 106),

P_{k}(q, λ) = \frac{1}{2} ∫_{0}^{λ}(\frac{k + 2}{(k + 1)!}x^{k+1} + ∑_{i=1}^{k} q_{i} \frac{x^{k−i}}{(k − i)!})^{2}dx.

Thus the associated symplectic bifurcating family of curves (swallowtail bifurcation family) in (R^{2}, dp_{k} ∧ dq_{k}) is defined by:

q_{k} = −\frac{k + 2}{(k + 1)!}x^{k+1} − ∑_{i=1}^{k−1} \frac{1}{(k − i)!}q_{i}x^{k−i},

p_{k} = −\frac{1}{k!}x^{k+2} − ∑_{i=1}^{k−1} \frac{1}{(k − i + 1)(k − i − 1)!}q_{i}x^{k−i}.

where x is the curve parameter and (q_{1}, …, q_{k−1}) are the bifurcation parameters of the family. We see that this is an unfolding of the curve

(q_{k}, p_{k}) = (−\frac{k + 2}{(k + 1)!}x^{k+1}, −\frac{1}{k!}x^{k+2}).

More general setting may be formulated in the following way.

Proposition 11.1 Let G : (R × N^{k+1}, 0) → R be a function family germ with A_{k+1}-type singularity. Let Σ_{k} be the discriminant set of G,

Σ_{k} = {u ∈ N^{k+1} | G(x, u) = 0, G′_{x}(x, u) = 0, for some x ∈ (R, 0)}.

Then there exist a symplectic space (R^{2k}, ω) and an isotropic fibration

π : ((R^{2k}, ω), 0) → (N^{k+1}, 0); (p, q) ↦ (q_{1}, … q_{k}, p_{k})

and the Lagrangian lifting Σ̃ of Σ. Moreover Σ̃ is uniquely defined by the conditions:

Σ̃ = \{ū ∈ R^{2k} | D^{−(k−l)}G(x, u) + ∑_{i=1}^{k−l}(−1)^{i−1}p_{k−i} \frac{x^{k−i−l}}{(k − i − l)!} = 0, 1 ≤ ℓ ≤ k\},

where D^{0}G(x, u) = G(x, u) and ū = (u, p_{1}, …, p_{k−1}).

12 Frontal-symplectic versality and open swallowtails.

In the case k = 2, we interpret the Givental’s construction from the versality viewpoint of “frontal-symplectic” category, based on the fact that the swallowtail surface provides the versal unfolding of plane curve of type E_{6} : t ↦ (t^{3}, t^{4}), among wave-front curves.

Here we give a direct method to construct a versal unfolding in the frontalsymplectic category.

Let f : (R, 0) → (R^{2}, 0) be a non-flat map-germ. After using a symplectomorphism of R^{2}, we assume ordf_{1} < ordf_{2}. Let (f, φ) : (R, 0) → (R^{3}, 0) be the Legendrian liftings of f for the contact form α = dy − pdx. In fact

φ = (\frac{df_{2}}{dt})/(\frac{df_{1}}{dt}), and then df_{2}−φdf_{1} = 0. Remark that ordφ = ordf_{2}−ordf_{1}. Let w = (f, φ; ξ, η, ψ) : (R, 0) → T R^{3} be an infinitesimal deformation of (f, φ) among Legendrian (integral) mappings. Then dη − ψdf_{1} − φdξ = 0, that is, d(η − φξ) = −ξdφ + ψdf_{1}. Set k = η − φξ. Then k − k(0) has order ≥ min{ordf_{1}, ordφ}. For the induced infinitesimal deformation v = (f; ξ, η) : (R, 0) → T R^{2} of f, take a function h with dh = ηdf_{1} − ξdf_{2}, a generating function of v. Then dh = ηdf_{1} − ξφdf_{1} = (η − ξφ)df_{1} = kdf_{1}. So h − h(0) is a sum of a monomial of order ordf_{1} and a function of order ≥ min{2ordf_{1}, ordf_{2}}.

Set m = ordf_{1}, k = min{2ordf_{1}, ordf_{2}} and set

S = R + Rt^{m} + m_{1}^{k}.

Then S is a vector subspace of E_{1} containing f^{∗}E_{2}.

Lemma 12.1 Let F : (R^{1} × R^{ℓ}, 0) → (R^{2} × R^{ℓ}, 0), F (t, λ) = (F̄(t, λ), λ) be a frontal unfolding of a non-flat map-germ f : (R, 0) → (R^{2}, 0). Assume

ord(f_{1}) < ord(f_{2}). If \frac{∂F̄}{∂λ_{1}} |_{R×0} , …, \frac{∂F̄}{∂λ_{ℓ}} |_{R×0} generate S/f^{∗}E_{2} via generat-

ing functions over R, and also generate vector fileds t^{i} \frac{∂}{∂q_{1}} ∘ f + φt^{i} \frac{∂}{∂p_{1}} ∘

f, (2ord(f_{1}) − ord(f_{2}) ≤ i ≤ ord(f_{1}) − 2) over R, then F is a frontalsymplectically versal unfolding of f. Frontal-symplectically versal unfoldings are unique up to liftable equivalence.

Example 12.2 (The open swallowtail.) Let f : (R, 0) → (R^{2}, 0), f(t) = (t^{3}, t^{4}) be a map-germ of type E_{6}. Then the one-parameter unfolding F :

(R^{2}, 0) → (R^{3}, 0), F (t, λ) = (q_{1}, p_{1}, q_{2}) = (t^{3} + 3λt, t^{4} + 2λt^{2}, λ) of f is a frontal-symplectic versal unfolding of f. The image of F is the swallowtail surface and has the double point locus. The Lagrangian lifting

F̃(t, λ) = (q_{1}, p_{1}, q_{2}, p_{2}) = (t^{3} + 3λt, t^{4} + 2λt^{2}, λ, \frac{6}{5}t^{5} + 2λt^{3}),

of F coincides with the open swallowtail surface, which has no self-intersections.

Example 12.3 (The open folded umbrella.) Let f : (R, 0) → (R^{2}, 0), f(t) = (t^{2}, t^{5}) be a map-germ of type A_{4}. Then the one-parameter unfolding F : (R^{2}, 0) → (R^{3}, 0), F (t, λ) = (q_{1}, p_{1}, q_{2}) = (t^{2}, t^{5} + λt^{3}, λ) of f is a frontalsymplectic versal unfolding of f. The image of F is the folded umbrella and has the double point locus. The Lagrangian lifting

F̃(t, λ) = (q_{1}, p_{1}, q_{2}, p_{2}) = (t^{2}, t^{5} + λt^{3}, λ, \frac{2}{5}t^{5}),

of F has no selfintersections and may be called “the open folded umbrella”.

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Goo ISHIKAWA, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan. e-mail: ishikawa@math.sci.hokudai.ac.jp

Stanislaw JANECZKO, Institute of Mathematics, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland. e-mail: Janeczko@ise.pw.edu.pl and Max-Planck-Institut fur Mathematik Vivatsgasse 7, 53111 Bonn, Germany