The extra-nice dimensions

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.


Introduction
Around the middle of last century Whitney formulated the concept of stability of smooth maps and characterized stable singularities in dimensions (n, p) with p ≥ 2n, (n, 2n − 1), (2,2) showing that in these cases stable maps are dense in the space of C ∞ maps. He then conjectured that this holds in any pair (n, p). Thom showed that this is not the case (see [34]) by giving an example in (9,9) of a singularity which appears generically in a 1-parameter family of maps. This singularity has A e -codimension 1 and is not simple. A germ is simple if there are only a finite number of orbits nearby, therefore, in the pair of dimensions (9,9) not all maps can be approximated by stable maps and so the stable maps are not dense. He then conjectured that topologically stable maps are always dense and this was proved by Mather ( [20]).
In his well known series of papers about stability of C ∞ maps Mather showed that the set of stable maps f : N n → P p is dense in C ∞ pr (N, P ) (proper C ∞ maps) with the Whitney strong topology if and only if the pair of dimensions (n, p) is in the nice dimensions ( [18]), which he determined completely in [19]. Mather gave a stratification of the set J k (n, p) of k-jets of smooth maps by K -orbits and characterized stability in terms of transversality of the jet extension of the map to this stratification. More precisely, he defined the nice dimensions as the pairs (n, p) such that there exists a Zariski closed K -invariant set Π(n, p) in J k (n, p), for sufficiently high k, of codimension bigger than n such that its complement in J k (n, p) is the union of finitely many K -orbits.
When the pair (n, p) is in the nice dimensions and the source manifold N is compact, an important problem in the applications of singularity theory to differential topology is the characterization of the simplest existing paths between two stable maps. A 1-parameter family F : N × [0, 1] → P connecting two non-equivalent stable maps always intersects the set of non stable maps for a finite number of values of the parameter, the bifurcation points. The classification of singularities of bifurcation points in generic families of maps is a fundamental step in results on elimination of singularities, which is still an active field of research ( [15], [1]), and on various results about the topology of the space of smooth maps such as work by J. Cerf ([3]) or K. Igusa ([14]) or even Vassiliev's theory of topological invariants ( [35]).
We say that a family F : N ×[0, 1] → P is a locally stable family if F t : N → P is a stable map for all t ∈ [0, 1] except possibly for a finite number of values {t 1 , . . . , t k } and the non-stable singularities of F t i are a finite number of points x j ∈ N, and the map (F, t) : N × [0, 1] → P × [0, 1] is a locally A e -versal unfolding of F t i for all non-stable points x j . This definition implies that the non-stable singularities of F t i are A-finitely determined and their A e -codimension is equal to 1.
In this paper we obtain a parallel result to Mather's characterization of the nice dimensions. First we define the extra-nice dimensions as the pairs (n, p) where there exists a smallest Zariski closed A -invariant set Λ(n, p) in J k (n, p), for sufficiently high k, of codimension greater than n + 1 whose complement is a finite number of A -orbits. Then we prove that the subset of stable 1-parameter families in C ∞ (N ×[0, 1], P ) is dense if and only if the pair (n, p) is in the extra-nice dimensions.
In the nice dimensions all the A e -codimension 1 singularities are simple (see Proposition 3.4). Goryunov ([9]), Cooper, Mond and Wik Atique ( [4]) and Houston ( [12]) studied the classification of germs and multigerms of A e -codimension 1, corank 1. A recent work by Oset Sinha, Ruas and Wik Atique ( [26]) defined operations that allow the classification of A e -codimension 2 multigerms in the nice dimensions and a natural question arises: are all of these simple? In Section 3 we prove that all corank 1 A e -codimension 2 monogerms in (n, p) are simple when (n + 1, p + 1) is in the nice dimensions. We give a sufficient condition for any A e -codimension 2 germ to be simple. This condition is related to stable germs in one dimension higher having A e -codimension 1 hyperplane sections. We also give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple.
In Section 4 we define the extra-nice dimensions, we relate them to the simplicity of A e -codimension 2 germs and we characterize them by stable germs in one dimension higher having A e -codimension 1 hyperplane sections (the sufficient condition in Section 3). In Section 5 we determine the boundary of the extra-nice dimensions completely. Figure 1 shows this boundary and compares it to the boundary of the nice dimensions. In Section 6 we characterize the extra-nice dimensions by the density of locally stable 1-parameter families.
Section 2 establishes the necessary notation and basic results. Finally, in Section 7, we answer a question posed by Wall about the codimension of non-simple maps. We define further refinements of the nice dimensions and give an example in the equidimensional case of a stratification in terms of increasing codimension of the subset of non-simple maps.
Acknowledgements: The authors thank J. J. Nuño-Ballesteros for spotting errors in previous versions of the manuscript and T. Nishimura for helpful discussions, both helping to improve the presentation of the results.

Notation
We consider map-germs f : (K n , S) → (K p , 0), where K = R or K = C, and S ⊂ K n a finite subset. For simplicity, we will say that f is smooth if it is smooth (i.e. C ∞ ) when K = R or holomorphic when K = C. We denote by O n = O K n ,S and O p = O K p ,0 the rings of smooth function germs in the source and target respectively, by M n and M p the maximal ideals of O n and O p respectively and by θ n = θ K n ,S and θ p = θ K p ,0 the corresponding modules of vector field germs. The module of vector fields along f will be denoted by θ(f ). Associated with θ(f ) we have two morphisms tf : θ n → θ(f ), given by tf (χ) = df • χ, and wf : θ p → θ(f ), given by wf (η) = η • f . Let f * : O p → O n be the induced map of f given by composition with f on the right. Let G = A e , A , K e , K . The G -tangent space and the G -codimension of f are defined respectively as It follows from Mather's infinitesimal stability criterion [16] that a germ is stable if and only if its A e -codimension is 0. We refer to Wall's survey paper [36] and Nuño-Ballesteros and Mond book [23] for general background on the theory of singularities of mappings. There are some relations between the different codimensions. One between the A -codimension and the A e -codimension due to L. Wilson [38] (a proof can be found in [29]): if f has finite A e -codimension and is not stable, where r = |S| is the number of branches. And another one between the K -codimension and the K e -codimension, which can be found in [36]: We say that f : (K n , 0) → (K p , 0) has corank r if df (0) has rank min n, p − r.
We say that f has finite singularity type or it is K -finite if K e -cod(f ) < ∞. Another remarkable result of Mather is that f has finite singularity type if and only if it admits an s-parameter stable unfolding (see [36]). We recall that an s-parameter unfolding of f is another map-germ Along the paper, we use the notation of small letters x 1 , . . . , x n , λ 1 , . . . , λ r for the coordinates in K n ×K r and capital letters X 1 , . . . , X p , Λ 1 , . . . , Λ r for the coordinates in , λ) and f 0 = f there exists a sufficiently small neighbourhood U of S × {0} such that for every (y 1 , λ), . . . , (y r , λ) ∈ U where F (y 1 , λ) = . . . = F (y r , λ) the multigerm f λ : (K n , {y 1 , . . . , y r }) → (K p , f λ (y i )) lies in one of those finite classes.
Definition 2.1. Let f : (K n , S) → (K p , 0) be a smooth map-germ. A vector field germ η ∈ θ p is called liftable over f , if there exists ξ ∈ θ n such that df • ξ = η • f (i.e., tf (ξ) = wf (η)). The set of vector field germs liftable over f is denoted by Lift(f ) and is an O p -submodule of θ p .
When K = C and f has finite singularity type, we always have the inclusion Lift(f ) ⊆ Derlog(∆(f )), where ∆(f ) is the discriminant of f (i.e., the image of non submersive points of f ) and Derlog(∆(f )) is the submodule of θ p of vector fields which are tangent to ∆(f ). Moreover, we have the equality Lift(f ) = Derlog(∆(f )) in case f has finite A e -codimension (see [5,23]). Definition 2.2. Let h : (K n , S) → (K p , 0) be a map-germ with a 1-parameter stable unfolding H(x, λ) = (h λ (x), λ). Let g : (K q , 0) → (K, 0) be a functiongerm. Then, the augmentation of h by H and g is the map A H,g (h) given by (x, z) → (h g(z) (x), z). A map which is not an augmentation is called primitive.
In Theorem 4.4 in [11] it is proved that where τ is the Tjurina number of the function g. Equality is reached if g is quasihomogeneous or H is a substantial unfolding (see [11]). In this paper we will only use the particular case when g is a Morse function and so A e -cod(A H,g (h)) = A e -cod(h).
Following Mather in [17] if f : (K n , 0) → (K p , 0) has finite singularity type then there is a stable germ F : (K n+s , 0) → (K p+s , 0) and a germ of an immersion i : (K n , 0) → (K p+s , 0), i ⋔ F, such that f is the pull-back of F by i in the diagram: Any germ f is a pull-back of a stable s-parameter unfolding F by the natural inclusion i. Damon ([5], for K = C) and Houston ([11], for K = R) proved that .
When s = 1 we say that L defines the hyperplane section f of F . Besides, to obtain a hyperplane section of A e -codimension 1 it is sufficient to consider the 1-jet of L (see for example [22]). Also notice that if F is minimal, ∆(F ) K e -cod(L) = 1 if and only if M p+1 θ(L) ⊂ tL(Lift(F )) + L * M s θ(L).

On simplicity of codimension 2 germs in the nice dimensions
We need the following characterisation of the openness of an A -orbit in the K -orbit.
Theorem 3.1. ( [32], [33]) Let f : (K n , 0) → (K p , 0) be a K -finite germ and let {v 1 , . . . , v r } be a basis for By the formulas relating the G -codimension to the G e -codimension in the previous section, this is equivalent to A e -cod(f ) = K e -cod(f ) − p, so basically, a non stable germ has the A -orbit open in the K -orbit if and only if there is no stable germ in the K -orbit before the versal unfolding.
Example 3.2. The germ (x 5 + yx, y) has A e -codimension 3 but admits a 2parameter stable unfolding, so its A -orbit is not open in the K -orbit. However, (x 5 + yx + x 7 , y) has A e -codimension 2 and its A -orbit is open in the K -orbit (see [28]).
In Subsection 3.9 we will see an example of a K -orbit which does not admit an open A -orbit.
The germs of A e -codimension 1 and corank 1 in the nice dimensions are well known and are hyperplane linear sections of stable germs. We review them here for sake of completeness. The case of hyperplane sections of stable corank 2 germs in (n, n + 1) has been studied in [8]. Proof. Suppose first that F is minimal.
1) Case n ≥ p. Let g 0 : (K m , 0) → (K, 0) be a simple function singularity of type A k , D k , E 6 , E 7 or E 8 , and let ϕ 1 = 1, ϕ 2 , . . . , ϕ µ be a homogeneous basis for On Jg 0 where ϕ µ is the unique highest weight term. Then, by [17], the map germs are stable minimal corank 1 germs. Moreover, any stable minimal corank 1 germ is A -equivalent to one of such germs. The section u µ = 0 defines an A e -codimension 1 section, see [9].
If F is not minimal, then F is A -equivalent to Id K r ×F ′ where F ′ : (K n−r , 0) −→ (K p−r , 0) is minimal. So there exists an A e -codimension 1 hyperplane section f ′ : we obtain a germ f : (K n , 0) −→ (K p , 0) of A e -codimension 1. From the normal forms showed above all corank 1 A e -codimension 1 germs in the nice dimensions are simple. We give here a proof of this fact for any corank. Proof. Let f : (K n , 0) −→ (K p , 0) be an A e -codimension 1 germ.
Recall from [4] and [11] that f is either primitive or a quadratic augmentation, that is, an augmentation of a primitive A e -codimension 1 germ h : (K n−l , 0) −→ (K p−l , 0) by a Morse function g.
For germs of A e -codimension 1, the A -orbit is open in its K -orbit if and only if the germ is primitive (see [13] or [33]). In fact, if f is primitive then its miniversal unfolding F is minimal. We have This dimension cannot be 0 because F is minimal, therefore tf (θ n ) + wf (θ p ) + f * M p θ(f ) ⊂ tf (θ n ) + wf (θ p ) and so, by Theorem 3.1, the A -orbit of f is open in the K -orbit of f , i.e. A -cod(f ) = K -cod(f ). Suppose f is non simple, then there must be a modal stratum Y with cod J k (n,p) (Y ) ≤ A -cod(f ) − 1 = A e -cod(f ) + n − 1 = n. So we cannot find a subset of J k (n, p) with codimension greater than n whose complement is a finite union of K -orbits and this contradicts the fact that we are in the nice dimensions. When f is an augmentation then f is A -equivalent to A H,g(z) (h) where H(x, λ) = (h λ (x), λ) is the versal unfolding of the primitive germ h and g is a Morse function. Therefore (x, z, u) → (h g(z)+u (x), z, u) is a versal unfolding of f and f can be deformed only in a finite number of A -classes.
iv) There exists a linear map L : Proof. i) if and only if ii) can be found in [33] or [13]. Notice that since F is minimal, the K -codimension of F (and of f ) must be n + 1. If the A -orbit of f is open in its K -orbit, then f has A -codimension n + 1 and thus A e -codimension 1. ii) if and only if iii) is trivial by Damon's Theorem on transverse fibre squares and the fact that F is minimal. iii) if and only if iv) follows directly from Damon's result A e -cod(f ) = ∆(F ) K e -cod(L) and the fact that i(K p ) = L −1 (0).
Proof. Suppose we have a corank 1 germ f : Since f does not admit a 1-parameter stable unfolding this dimension cannot be 1, therefore it must be 2. This implies that tf (θ p ) + wf (θ n ) + f * (M p )θ(f ) ⊂ tf (θ p ) + wf (θ n ) and by Theorem 3.1 the A -orbit is open in the K -orbit, so the codimension of the K -orbit is equal to n + 2. Since the codimension of the modal stratum is n + 1 we have a 1-parameter family of K -orbits of codimension n + 2. So in (n + 1, p + 1) we have modality of K -orbits of codimension n + 2, which contradicts that (n + 1, p + 1) are nice dimensions. Now suppose f admits a 1-parameter stable unfolding F : (K n × K, 0) −→ (K p × K, 0). By Proposition 3.3, there exists f ′ in the same K -orbit such that A e -cod(f ′ ) = 1. So cod J k (n,p) (X ′ ) = n + 1, where X ′ denotes the A -orbit of f ′ . As the codimension of the A -orbit of f ′ is greater than or equal to the codimension of the modal stratum, then f ′ is not simple, which contradicts Proposition 3.4.
Notice that in the above proof the hypothesis of corank 1 is only used to ensure the existence of A e -codimension 1 hyperplane sections so the above result can be rephrased as We believe that the results in this section hold also for multigerms, but in this paper we are concerned only with monogerms.
3.1. An example of a corank 2 codimension 2 germ which is not simple. From Theorem 3.6 in order to find A e -codimension 2 non simple germs we have two possibilities, either it has corank greater than 1, or it is has corank 1 and is just below the boundary of the nice dimensions. In the latter case it must come from a section of an A e -codimension 1 non simple germ in the boundary of the nice dimensions and must have A -orbits open in the K -orbits.
From Proposition 3.5 it follows that a stable germ F : (K n+1 , S) −→ (K p+1 , 0) admits an A e -codimension 1 hyperplane section if and only if there exists a linear map L such that M p+1 ⊂ L(Lift(F )) + L .
Consider the stable germ F 3,3 : (K 6 , 0) → (K 6 , 0) given by Proof. F 3,3 is a free divisor so Lift(F 3,3 ) is generated by 6 vector fields. The Euler vector field is clearly liftable and the other 5 are all linearly independent and liftable by the following lowerable vector fields: ) + L and so F 3,3 does not admit a codimension 1 hyperplane section.
Proof. The idea is to use Damon's Theorem relating A e -codimension and V K ecodimension where V is the discriminant of F 3,3 . By integrating the linear parts of the vector fields in Lift(F 3,3 ) we obtain the linear parts of diffeomorphisms in V K , which are: and by using η 1 we get j 1 L = U 3 + λU 4 where λ is a modulus. A complete 2transversal is given by U 2 [2]). Rescaling we set L = U 3 + λU 4 + U 2 4 . If λ = 0, this germ has V K e -codimension 2 and is not simple, so the A e -codimension of f is 2 and it is not simple. By Proposition 3.5, this is an example of a K -orbit which does not admit an open A -orbit.

The extra-nice dimensions
Mather gave a stratification of the set J k (n, p) of k-jets of smooth mappings by K -orbits. This induces a partition of J k (N, P ) by K -orbit bundles. Mather characterized stability in terms of transversality of the k-jet extension j k f : N → J k (N, P ) to this stratification. He showed that there exists a smallest Zariski is the union of finitely many K k -orbits. The codimension of Π k (n, p) decreases as k increases. Moreover, there exists a big enough k for which the codimension of Π k (n, p) attains its minimum. For this k the codimension of the bad set Π(n, p) is denoted by σ(n, p.) When σ(n, p) > n, then the k-jet of a generic map does not meet the set Π(n, p) and therefore it is transversal to Mather's stratification in J k (N, P ) and hence it is stable. He defined the nice dimensions as the pairs (n, p) such that σ(n, p) > n. See [6] for the notion of semi-nice dimensions, where 2-modality of K -orbits appears.
As a consequence of Proposition 3.4, the fact that Mather's bad set has codimension greater than n means that one can detect lack of simplicity at the A ecodimension 1 level. Take also into account that K -orbits of codimension less than or equal to n have stable representatives of A -codimension less than or equal to n (i.e. there is an open A -orbit in the K -orbit). If we want to refine this definition to detect lack of simplicity in the A e -codimension 2 level we must consider bad sets of codimension greater than n + 1. Furthermore, since A -orbits of non stable germs may or may not be open in their K -orbit, we must consider stratification by A -orbits instead of K -orbits. This leads to the following Definition 4.1. The pair (n, p) is said to be in the extra-nice dimensions if, for large enough l, there is a Zariski closed A -invariant subset Λ of J l (n, p), of codimension greater than n + 1, whose complement is a finite union of A -orbits.
It follows from the definition that Proof. Suppose we have an A e -codimension 2 non simple germ. Then there is a 1-parameter family of A -orbits of codimension n + 2, so we cannot find and Ainvariant subset Λ of J l (n, p), of codimension greater than n+1, whose complement is a finite union of A -orbits.
The converse of Proposition 4.2 is not true as we shall see in a further example (Proposition 5.8).
The subset Λ in the definition can be constructed containing all non-simple Aorbits and all A -orbits of codimension greater than or equal to n + 2. In order to be in the extra-nice dimensions, the codimension of this set must be greater than or equal to n + 2 (see Section 6, Proposition 6.2). It is not contained and it does not contain Mather's bad set, because in a K -orbit of codimension less than or equal to n there can be an infinite number of simple A -orbits (for example, augmentations), and in order to have finite A -orbits in the complement of our bad set Λ we must include in Λ some of these A -orbits (i.e. we include the ones of codimension greater than or equal to n + 2). In Remark 5.3 we compare Mather's bad set and ours for some examples.
The previous definition is well defined because there is an estimate (depending on n and p) for the degree of determinacy of A e -codimension 1 germs. Namely, results by Mather and Gaffney which can be found in [36] and [23] state that if

in particular any
A e -codimension 1 (and therefore A -codimension n + 1) germ f : (R n , 0) → (R p , 0) is (2(p + n + 1) 2 − 1)-A -determined. Proof. By definition there exists an A -invariant subset Λ of J l (n, p), of codimension greater than n + 1, whose complement is a finite union of A -orbits. For each one of those A -orbits, consider the K -orbit which contains it. We therefore have a finite number of K -orbits of codimension less than or equal to n + 1 which may include some A -orbits which were originally in Λ. The complement of this finite number of K -orbits is included in Λ, and therefore the codimension of this complement is greater than n + 1. This complement contains all the K -orbits of codimension greater than or equal to n + 2 and is Zariski closed. In conclusion, there exists a K -invariant subset Λ ′ of codimension greater than n + 1 such that its complement is a finite number of K -orbits. The codimensions of these strata are the same in (n + 1, p + 1) and so (n + 1, p + 1) is in the nice dimensions.
This means that if (n + 1, p + 1) is not in the nice dimensions, then (n, p) is not in the extra-nice dimensions.
It is obvious from the definition of nice dimensions that if a pair (n + 1, p + 1) is in the nice dimensions, then (n, p) is in the nice dimensions, since the codimension of K -orbits is invariant under unfoldings. However, this is not so obvious for the extra-nice dimensions: Proof. Suppose that (n, p) is not in the extra-nice dimensions. Then, for any Zariski closed A -invariant subset Γ of J l (n, p) of codimension greater than n + 1, its complement is an infinite number of A -orbits.
Let Γ be the union of all the A -orbits of codimension greater than or equal to n + 2. If the codimension of Γ is greater than or equal to n + 2, since it is an A -invariant set, its complement is an infinite number of A -orbits. On the other hand, the complement of Γ is the union of all A -orbits of codimension less than or equal to n + 1. In the nice dimensions there are a finite number of stable and A ecodimension 1 orbits, therefore we have a contradiction. Hence, the codimension of Γ must be less than or equal to n + 1. This means that there is a Zarisky open set in Γ foliated by an infinite number of A -orbits of the same codimension (greater than n + 1, i.e. A e -codimension greater than 1).
If these infinite A -orbits are contained in a single K -orbit X then cod J k (n,p) X ≤ cod J k (n,p) Γ ≤ n + 1. Therefore the K -orbit has a stable germ F : (K n+1 , 0) → (K p+1 , 0) which is a 1-parameter stable unfolding of the germs of the A -orbits. By augmenting these germs by F and a Morse function we obtain a stratum of codimension less than or equal to n + 1 of germs of A e -codimension greater than 1 in (n + 1, p + 1), and so (n + 1, p + 1) is not extra-nice dimensions.
If no single K -orbit contains the infinite A -orbits, there is a stratum of codimension equal to the codimension of Γ with modality of K -orbits. This implies that (n+1, p+1) cannot be nice dimensions. Therefore, by Proposition 4 .3, (n+1, p+1) is not extra-nice dimensions.
Remark 4.5. The above proof implies that if a pair (n, p) is in the nice dimensions but not in the extra-nice dimensions, then there exists a K -orbit which contains a stratum of codimension less than or equal to n + 1 of non-simple germs. In fact, the codimension is equal to n + 1 (see Section 7). Proof. 1) Firstly we are going to prove that if (n, p) is in the extra-nice dimensions then every stable germ F : (K n+1 , 0) → (K p+1 , 0) of K -codimension n + 1 admits a hyperplane section of A e -codimension 1.
Suppose that there exists a stable germ F : (K n+1 , 0) → (K p+1 , 0) of Kcodimension n + 1 which does not admit an A e -codimension 1 hyperplane section. Then there exists a k > 1 and a section f ′ : (K n , 0) → (K p , 0) of A e -codimension k such that all sections of F have A e -codimension greater than or equal to k. Therefore the A -orbit of f ′ is not open in its K -orbit, so the union of a (k − 1)parameter family of A -orbits of codimension n + k is a Zariski open subset of the K -orbit. Then this K -orbit is a codimension n + 1 subset of J l (n, p) which is not a finite union of A -orbits of minimal codimension, so there does not exist a subset of codimension greater than n + 1 such that its complement is a finite number of A -orbits, contradicting the fact that (n, p) is in the extra-nice dimensions.
Lets consider now F : (K n+1 , 0) → (K p+1 , 0) a stable germ of K -codimension n−r, r ≥ 0. Then F is a trivial unfolding of a minimal stable germF : (K n−r , 0) → (K p−r , 0). As (n, p) is in the extra-nice dimensions hence, by Proposition 4.4, (n−r− 1, p − r − 1) also is and we can apply the above argument to find an A e -codimension 1 hyperplane sectionf ofF . Then the augmentation off is an A e -codimension 1 hyperplane section of F (see [4]).
2) Suppose that (n, p) is not in the extra-nice dimensions. Following the proof of Proposition 4.4 there exists Γ of codimension less than or equal to n + 1 with an open Zariski subset foliated by infinite A -orbits of codimension greater than n + 1.
If no single K -orbit contains the infinite A -orbits, there is a stratum of codimension equal to the codimension of Γ with modality of K -orbits, contradicting the fact that that (n+1, p+1) is nice dimensions. Therefore, there exists a K -orbit which contains the infinite A -orbits. This K -orbit has codimension less than or equal to n + 1, therefore there exists a stable germ F : (K n+1 , 0) → (K p+1 , 0) (not necessarily minimal) such that all non stable hyperplane sections f of F are not A -simple. Then by Proposition 3.4 A e -cod(f ) > 1.
In order to determine when a pair (n, p) is in the extra-nice dimensions, by Theorem 4.6, we need to know the stable germs (K n+1 , 0) → (K p+1 , 0), so we describe Mather's procedure to obtain the normal forms for all stable germs. Start with a germ f 0 : (K s , 0) → (K t , 0) of rank 0 and find a basis {φ 1 , . . . , φ d } of Mnθ(f 0 ) T Kef 0 . Then, F : (K s+d , 0) → (K t+d , 0) given by is stable. Furthermore, any stable germ can be obtained by this procedure. By Proposition 3.3, all stable corank 1 germs admit an A e -codimension 1 hyperplane section, so we need to study simple germs f 0 of corank at least 2. The rank 0 germs f 0 have been classified by several authors such as Mather, Arnol'd, Giusti, Wirthmüller, Damon, du Plessis, Wall and Gibson. A good account can be found in the book by du Plessis and Wall [7].
In the following discussion of simple algebras of corank greater than 1, there are two of corank 2 which play a special role: B p,q = (x p + y q , xy), B ′ p,q = (x p , y q , xy).
Notice that B ′ p,q is obtained from B p,q by adding its jacobian.

4.1.
Algebras. We describe the algebras by the number of variables and generators. We only list the least degenerate ones (which will be enough for our purpose as we will see in the next section). Case n ≥ p: (1) With 1 generator and any number of variables: A k , D k , E 6 , E 7 and E 8 , which are of corank 1. (2) With 2 generators and 2 variables: B p,q . The stable germs in these algebras are in (p + q, p + q). (3) With 2 generators and 3 variables: P p,q = B p,q + (z 2 , 0). The stable germs in these algebras are in (p + q + 2, p + q + 1). (4) With 2 generators and more than 3 variables there are no simple algebras.

The boundary of the extra-nice dimensions
Many of the calculations which did not make it to the final version of this article were either done or double-checked using an algorithm implemented in the computer package Singular developed by Hernandes, Miranda and Peñafort-Sanchis in [10]. The calculations which appear in this section have been done by hand unless otherwise stated. 5.1. The case n = p. In Theorem 3.9 we obtain a non-simple A e -codimension 2 germs in (5, 5) of type B 3,3 . By Proposition 3.7 this means that there exists a stable germ in (6, 6) which does not admit a hyperplane section of A e -codimension 1. Therefore, by Theorem 4.6, (5,5) is not in the extra-nice dimensions. Since (5,5) is in the nice dimensions we have the converse of Theorem 4.6, so to establish the boundary of the extra-nice dimensions we must verify if all stable germs in (5, 5) admit hyperplane sections of A e -codimension 1. Taking into account the adjacencies of discrete algebra types we only have to investigate the stable germ in B 3,2 : Lemma 5.1. Lift(F 3,2 ) is generated by is generated by 5 vector fields. The Euler vector field is clearly liftable and the other 4 are all linearly independent and liftable by the following lowerable vector fields: Proposition 5.2. When n = p, (5,5) is the boundary of the extra-nice dimensions.
Proof. It remains to prove the existence of a hyperplane section of A e -codimension 1 of F 3,2 . Let L : K 5 → K be given by L(X, Y, U 1 , U 2 , U 3 ) = U 3 . Then M 5 ⊂ L(Lift(F 3,2 )) and by Proposition 3.5 iv), F 3,2 admits a hyperplane section of A ecodimension 1, so (4,4) is in the extra-nice dimensions.
Remark 5.3. The pair (5,5) is in the nice dimensions but not in the extra-nice dimensions. In J l (5, 5) for a sufficiently high l Mather's bad set Π(5, 5) is a codimension 6 K -invariant set given by Π(5, 5) = A 6 ∪ B 3,3 . Its complement is composed by the K -orbits A i , i = 1, . . . , 5, B 2,2 and B 3,2 , all of which have a stable representative in (5,5). In the K -orbit of B 3,3 there is a 1-parameter family of A -orbits of A e -codimension 2 (i.e. A -codimension 7) which is dense in the K -orbit (Theorem 3.9). Therefore we cannot find an A -invariant set of codimension greater than or equal to 7 such that the complement is a finite number of A -orbits, and hence (5,5) is not in the extra-nice dimensions.
On the other hand, in (4, 4) the set Π(4, 4) is a codimension 5 K -invariant set in J l (4, 4) given by Π(4, 4) = A 5 ∪ B 3,2 . Its complement is composed by the K -orbits A i , i = 1, . . . , 4 and B 2,2 . Both K -orbits A 5 and B 3,2 have an open A -orbit. Let Λ be the union of the closure of the complement of these open orbits in A 5 and B 3,2 and of all the A -orbits of codimension greater than or equal to 6 in all the K -orbits. This is a codimension 6 A -invariant subset such that the complement of it is a finite number of A -orbits and hence (4,4) is in the extra-nice dimensions.

5.2.
How to go from (n, p) (n ≤ p) to (n, p + 1) by adding a 0 component.
Theorem 5.6. If F 0 admits a hyperplane A e -codimension 1 section then F also does. The converse is true if the image of the set of linear part of vector fields in Lift(F ) by π 2 is a subset of M Z,W θ(π 2 ), where π 2 : (K p+k+r+1 , 0) −→ (K r+1 , 0) is the projection π 2 (X, U, Z, W ) = (Z, W ).
Let h : (K p+k+r+1 , 0) → (K, 0) be given by h(X, U, Z, W ) = h 0 (X, U ) + W r . We are going to prove that h = 0 defines an A e -codimension 1 hyperplane section of the discriminant V ⊂ (K p+k+r+1 , 0) of F , that is, h has V K e -codimension 1.
As F 0 and F are both minimal stable unfoldings, all liftable vector fields in both cases vanish at zero.
Therefore, A e -cod(i * (F ′ 3,3 )) = 1 if and only if the rank of N F ′ 3,3 is 11. In fact, the same argument proves that A e -cod(i * (F ′ 3,3 )) = k if and only if the rank of N F ′ 3,3 is 12 − k for k ≥ 1.
are the coordinates of the target.
Proof. η 1 is the Euler vector field and lowerable vector fields for the remaining ones are , where X = x 3 +u 1 x+u 2 y+u 3 y 2 , Y = y 3 +v 1 x+v 2 y+v 3 x 2 and Z = xy+w 1 x+w 2 y.
The next result shows a counterexample of the converse of Proposition 4.2.
Proposition 5.8. Any hyperplane section of F ′ 3,3 has A e -codimension greater than 2.
Proof. Analyzing the linear parts of Lift(F ′ 3,3 ) we can see that the rank of N F ′ 3,3 is 8 so the best possible hyperplane section has A e -codimension 4.
This means that in (9,10) all A e -codimension 2 germs are simple, but (9,10) is not in the extra-nice dimensions.
Lemma 5.11. The generators of Lift(P 2,2 ) are Proof. The discriminant (image of critical points set) V of P 2,2 is a free divisor so Derlog(V ) is generated by 5 vector fields. In order to obtain them we calculate the order 5 minors of the differential and eliminate the variables x, y, z from the ideal generated by 2x 2 − 2y 2 + u 1 x − u 2 y, u 1 u 3 + 2U 3 x − 2yz, u 2 u 3 − 2u 3 y − 2xz, X − (x 2 + y 2 + z 2 + u 1 x + u 2 y), Y − (xy + u 3 z) (the first three are the minors). Using the computer package Singular we obtain the defining equation of V. Again using Singular we can compute with syzygies the generators of Derlog(V ). It can be seen that all of them are linearly independent and liftable.
Notice that there are only 4 vector fields with non zero linear parts.
Proof. Analyzing the vector fields in Lemma 5.11 and by Proposition 3.5 iv), P 2,2 does not admit a hyperplane section of A e -codimension 1, so (5,4) is not in the extra-nice dimensions. In fact, it does not admit an A e -codimension 2 section either. Since P 2,2 is the simplest corank 2 algebra in (n + 1, n) and (5, 4) is in the nice dimensions, by the converse of Theorem 4.6, (4, 3) is in the extra-nice dimensions and so (5,4) is the boundary of the extra-nice dimensions.
5.6. The case n > p + 1. From the discussion of the algebras, we know that there are no rank 0 simple algebras in this case. Here the boundary of the nice dimensions is given by the non simple A e -codimension 1 section of the stable unfolding of the simplest rank 0 germ for each case (n + k, n), k > 2.
On the other hand, for (n+k, n) with k > 2, the boundary of the nice dimensions is given by the non simple A e -codimension 1 section of the stable unfolding of x 3 + y 3 + z 3 + λxyz + Σ k i=1 w 2 i , whose stable unfolding is in (10 + k, 8), and so this boundary is (9 + k, 7). By Proposition 4.3 (8 + k, 6) is not in the extra-nice dimensions. Since there are no stable corank 2 germs in (8 + k, 6) (or below) the boundary of the extra-nice dimensions is (8+k, 6). We remark that in (8+k, 6) there may be a non simple A e -codimension 2 corank 1 germ obtained as a section (by a codimension 2 plane) of the stable unfolding of the germ x 3 +y 3 +z 3 +λxyz+Σ k i=1 w 2 i . Proof. If the codimension of Λ(n, p) is less than or equal to n + 1, then it is foliated by infinite A -orbits and you cannot find a subset of codimension greater than n + 1 such that its complement is a finite number of A -orbits. The "if" part is trivial by definition since in the nice dimensions there are a finite number of A -orbits of codimension less than or equal to n + 1.
Consider a 1-parameter family F : N × [0, 1] → P and consider the partial jet extension If F is a locally stable 1-parameter family, by definition (F, t) is an A e -versal unfolding of F t i at the points where it has non stable singularities (and these are of A e -codimension 1 only). By the above result this means that j k 1 F is transversal to all A -orbits of codimension less than or equal to n + 1. Since F does not have singularities of A e -codimension greater than 1 for any t, j k 1 F is also transversal to Λ(N, P ). Therefore, j k 1 F is transversal to the stratification S(N, P ). Conversely, if (n, p) is in the extra-nice dimensions, Λ(N, P ) has codimension greater than or equal to n + 2, so transversality to the stratification S(N, P ) means that j k 1 F is transversal to all A -orbits of codimension less than or equal to n + 1 and avoids Λ(N, P ). Therefore, (F, t) unfolds versally all the F t i which have non stable singularities of A e -codimension 1. Since N is compact the points x j where F t i is non stable are finite, so F is a locally stable 1-parameter family. Proof. The pair (n, p) is in the extra-nice dimensions if, by definition, there exists a bad set Λ(n, p) ⊂ J k (n, p) of codimension greater than n + 1 such that its complement is a finite number of A -orbits. This set induces a bad set Λ(N, P ) in J k (N, P ) of codimension greater than n+1. Thus, for a generic family F ∈ C ∞ (N ×[0, 1], P ), j k 1 F (N × [0, 1]) ∩ Λ(N, P ) = ∅. By Thom's transversality theorem, the set of families F such that j k 1 F is transversal to any A -orbit of codimension less than or equal to n + 1 is also a residual set. Since there are finite A -orbits of codimension less than or equal to n + 1, the set of families F such that their partial jet extension is transverse to S(N, P ) is a finite intersection of residual sets and so is residual. Equivalently, by Lemma 6.3, the set of locally stable 1-parameter families in C ∞ (N × [0, 1], P ) is dense. Now suppose (n, p) is not in the extra-nice dimensions. Then the codimension of Λ(N, P ) is less than or equal to n + 1. In Section 5, for any (n, p) in the boundary of the extra-nice dimensions such that (n + 1, p + 1) is nice dimensions, we construct a stable germ F in (n + 1, p + 1) such that any section of it f 0 is not simple and such that F = (F, t) is a 1-parameter stable unfolding of f 0 . By taking a trivial unfolding of these stable germs we obtain stable germs with this property in any pair of dimensions outside the extra-nice dimensions. Considering one of the sections of these stable germs in (n, p) and its deformation F , F is clearly not a locally stable 1-parameter family but j k 1 F (N × [0, 1]) ∩ Λ(N, P ) = ∅ and j k 1 F is transversal to S(N, P ). There is a sufficiently small neighbourhood of F in C ∞ (N × [0, 1], P ) such that any G in that neighbourhood satisfies that j k 1 G is transversal to S(N, P ) and j k 1 G(N × [0, 1]) ∩ Λ(N, P ) = ∅. Therefore, there is an open set of non locally stable 1-parameter families and so the set of locally stable 1-parameter families is not dense.

Codimension of non-simple germs
In this section we answer partially a question posed by Wall to the first author during his talk at the workshop on "Singularities in Generic Geometry and Applications IV" held in Edinburgh in 2013: what is the codimension of the non-simple germs?
Let N S denote the A -invariant subset of J l (n, p) composed of all non A -simple orbits. If (n, p) is in the nice dimensions, by Proposition 3.4, all A e -codimension 1 germs are simple, so if a germ is not simple its A e -codimension is at least 2 (A -codimension n + 2). Therefore cod J l (n,p) (N S) ≥ n + 1. Similarly, from the definition of the extra-nice dimensions, if (n, p) is in the extra nice dimensions, then cod J l (n,p) (N S) ≥ n + 2. In fact, if (n, p) is in the nice dimensions but not in the extra-nice dimensions, Remark 4.5 shows that cod J l (n,p) (N S) ≤ n + 1, so cod J l (n,p) (N S) = n + 1.
This naturally leads to the following definition Definition 7.1. The pair (n, p) is said to be in the ∆ m -nice dimensions if, for large enough k, there is an A -invariant subset Λ of J l (n, p), of codimension greater than n + m, whose complement is a finite union of A -orbits.
∆ 1 -nice dimensions are the extra-nice dimensions and ∆ 0 -nice dimensions are the nice dimensions. With this definition, if (n, p) is in the ∆ m -nice dimensions but not in the ∆ m+1 -nice dimensions, then cod J l (n,p) (N S) = n + m + 1.
Example 7.2. In the case n = p, the Thom-Levine example in the introduction is an example in the boundary of the ∆ 0 -nice dimensions of an A e -codimension 1 germ which is not simple. It has corank 3. Inside the ∆ 0 -nice dimensions all A e -codimension 1 germs are simple.
From Theorem 3.9 we have an example in the boundary of the ∆ 1 -nice dimensions of an A e -codimension 2 germ which is not simple. It has corank 2. Inside the ∆ 1 -nice dimensions all A e -codimension 2 germs are simple.
From [32] and [24] we know that if n ≥ 3, then a germ f : (K n , 0) → (K n , 0) with m 0 (f ) ≥ n + 3 is not simple. In fact, in (3, 3) we find the germ (x, y, z 6 + yz 2 + xz) which has A e -codimension 3 and is not simple with 1 modal parameter. It has corank 1. From [28] and [30] we know that all A e -codimension 3 germs in (2, 2) are simple. This means that (3,3) is the boundary of the ∆ 2 -nice dimensions. Inside the ∆ 2 -nice dimensions all A e -codimension 3 germs are simple.