Vector fields on bifurcation diagrams of quasi singularities

1.   Introduction

One of the core tools in singularity theory is to classify functions on a certain space equipped with a distinguished hyperspace in it. The infinitesimal level problems of this kind require finding diffeomorphisms of the ambient space such that this hypersurface is preserved. In order to construct these diffeomorphisms, it is necessary to provide a description of the generators of vector fields that are parallel to the hypersurface. Many authors have studied algorithm and algebraic aspects of such vector fields (see for example ^[1,2,3]) to classify singularities of maps (functions) between two manifolds that can be constructed from the differential geometry point of view (see e.g. ^[4,5]. Further motivations of the topics can be found in various relevant papers with differential geometry ^[6,7,8] and submanifolds theory ^[9,10,11]. The classification can help study manifolds via other functions such as the height function and distance squared function. In many cases, this hypersurface appears as a discriminant (or bifurcarion diagram) or caustics of versal deformation of classes with respect to a standard equivalence relation.

In a series of papers ^[12,13,14], a new non-standard equivalence relation, on a space $\mathbb R^n$ equipped with a variety $\Gamma$, are studied, and, consequently, simple classes were obtained. Classification of projections of Lagrangian manifolds endowed with a hypersurface $\Gamma$ is accomplished through the utilization of these classes. As a result of the classification, the bifurcation diagrams and caustics of versal unfolding of simple classes were described in ^[15], which were conduct in a different manner. In particular, let $G(z, u) = \widetilde G(z, u)+u_0$, with $z\in \mathbb R^n$ and $u = (u _0, u _1 \dots, u _ s)$ as parameters, be a versal unfolding of the simple $g(z) = G(z, 0)$ with respect to the quasi equivalence relation. Then, the respective bifurcation diagram in the space of parameters consists of two components $W_0$, which is the standard discriminant given by the equations $G = 0$ and $\frac{\partial G}{\partial z} = 0$ and $W_1$, which is contained in $W_0$ and it is determined by constraints that define $\Gamma$. The caustics is in the unfolding base $\widetilde u = (u _1, \dots, u _s)$ (which does not include $\lambda _0$), and consists of two parts $\Sigma _0$ which represents the singular set image of $W_0$ under the projection $\pi: u \to \widetilde u$ and $\Sigma _1 = \pi (W_1)$. The preceding construction yields that the bifurcation diagrams is a pair $W = (W_0, W_1)$, where $W_0$ is a hypersurface in $\mathbb R ^s_u$ and $W_1 \subset W_0$, while the caustics is the union $\Sigma ^* = \Sigma _0 \cup \Sigma _1$ with $dim(\Sigma _0) = dim (\Sigma _1)$.

In this work, in Section 2, we calculate the generators of the vector fields that are parallel to the quasi bifurcation diagrams and caustics, obtained in ^[15]. This implies that, for the bifurcation diagrams, we seek vector fields that preserve not only $W_0$ but also the points of $W_0$, and for the caustics, we seek vector fields that preserve both $\Sigma _0$ and $\Sigma _1$.

Singularity theory techniques and differential geometry tools can be employed to comprehend the geometry of an object by examining its interaction with planar objects, such as planes or lines. In order to determine the former, it is necessary to analyze the singularities of the height functions along particular directions, which define the object's contact with the plane orthogonal to that direction. Many authors have investigated the contact with planes of singular surfaces, including the cross-cap ^[16], the swallowtail ^[17], the cuspidal edge ^[18], and the folded cuspidal edges ^[19].

Thus, as an application, in Section 3, we consider a cuspidal edge $G$ equipped with a distinguished regular curve$B$ in it. The object appears as a bifurcation diagram of the quasi boundary class $B_3.$ We then apply the module of vector fields obtained in Section 2 to classify submersions on the pair $(G, B)$. Then, we use such classification to study the contact of a general cuspidal edge equipped with a regular curve in it by studying the singularities of height function on $(G, B)$. There are two distinguished regular curves, $\Sigma _G$ (the singular set) and $B$. Finally, we examine the duality of the two curves by describing the versal deformation of the generic submersions that are obtained.

2.   Preliminaries

Let $\mathbb{K}$ denote the real number $\mathbb{R}$ or the complex numbers $\mathbb{C}$ with local coordinates $z$. The set of all smooth function germs from $(\mathbb{K}^n, 0)$ to $\mathbb{K}$ is denoted by $\mathcal{E}_n$ (or $\mathcal{E}_z$), and the maximal ideal in this set is denoted by $\mathcal{M}_n$. Let $\theta _n$ represent the module over $\mathcal{E}_n$ consisting of all vector fields formed on $(\mathbb K^n, 0)$. Let $\mathbb{K}[z]$ be the polynomial ring or formal power series over $\mathbb{K}$.

Let $V \subset (\mathbb{K}^n, 0)$ be an analytic variety. The ideal of germs that vanish on $V$ is denoted by $I(V)$.

Definition 1. If $\xi (I(V)) \subseteq I(V),$ then a vector field $\xi \in \theta _n$ is considered to be tangent to $V$ or to preserve $V$. The module of vector fields of this nature is represented by $Der(-logV)$.

It is important to note that if $\xi \in Der(-logV)$, then it can be integrated to generate a diffeomorphism $\varphi :(\mathbb{K}^n, 0)\to (\mathbb{K}^n, 0)$ that preserves $V$, i.e., $\varphi (V) \subseteq V$.

Also note that Definition (1) implies that if $V = (h_1, h_2, \dots, h_s)$, where $h_i \in \mathcal{M}_n$, then

Definition 2. Let $\zeta$ be a vector field on $(\mathbb{K}^p)$ and $f:(\mathbb{K}^n, 0) \to (\mathbb{K}^p)$ be a smooth map germ. Then, $\zeta$ is said to be liftable over $f$ if there exist a vector field $\eta$ on $(\mathbb{K}^n)$ such that $df\circ \eta = \zeta \circ f$. In this case, $\eta$ is said to be lowerable.

The concepts of liftable and tangent vector fields on the discriminant are identical for stable map germs when $\mathbb{K} = \mathbb{C}$ (see ^[20]). In fact, Arnold, in ^[2], showed that there are liftable vector fields that are not tangent vector fields when $\mathbb{K} = \mathbb{R}$.

Let $V_1$ be an $\mathbb R-$analytic variety in $(\mathbb R ^r, 0)$ in local coordinates $(u_0, \dots, u_{r-1})$. Assume that $V_0 \subseteq V_1$ of codimension 1. Denote by $\widetilde V$ the pair consisting of $V_1$ and a distinguished sub-variety $V_0$ in it and set $\widetilde V = (V_1, V_0)$. Thus, we may assume that the pair represents a variety equipped with a boundary.

Definition 3. A diffeomorphism $\phi : (\mathbb{R}^r, 0) \to (\mathbb{R}^r, 0)$ will be said to preserve $\widetilde{V}$ if and only if $\phi (V_1) \subseteq V_1$ and $\phi (V_0) \subseteq V_0$.

Definition 4. A vector field $\xi \in \theta _r$ is considered tangent to $\widetilde V$ if and only if the following conditions are fulfilled.

(1) $\xi \left(I(V_1)\right)\subseteq I(V_1)$,

(2) $\xi \left(I(V_0)\right)\subseteq I(V_0)$.

The module of all vector fields satisfying the given conditions will be represented as $Der(-log \widetilde V)$ over the $\mathcal{E}_r$-module, that is

and it is commonly referred to as the stationary algebra of $\widetilde V$.

Remark 1. If $\xi$ belongs to $Der(-log\widetilde V)$, then $\zeta$ conserves $\widetilde V$ and, as a result, is tangent to it. Furthermore, $Der(-log \widetilde V)$ is the Lie algebra associated with the group of diffeomorphisms that preserve $(\widetilde V, 0)$ in the space $\left(\mathbb{R} ^n, 0\right)$.

3.   Vector fields on quasi bifurcation diagrams

Consider the coordinate space $\mathbb R ^n$ in local coordinates $z = (x, y_1, \dots, y_{n-1})$ equipped with a smooth hypersurface $\Gamma = \big\{ x = 0\big\}$, which is referred to as a boundary.

Recall from ^[12] that on the $\Gamma$, every simple function germ $g$ can be stably transformed via the quasi equivalence relation into one of the subsequent germs:

or

The tangent space to the quasi boundary equivalence singularity of $g$ at $g$ is

Let $G(z, u)$ be a deformation of $g: (\mathbb{R}^n, 0) \to (\mathbb{R}, 0)$, where $u = (u_0, u_1, \dots, u_{r-1}) \in \mathbb{R}^r$ are parameters. Set $G_u (z) = G(z, u)$; so that, $G_0 = g.$

The initial speeds of $G$ are defined by

The subsequent result is an adaptation of Theorem 3 from ^[16].

Proposition 1. A deformation $G$ of a function $g$ is considered versal in regard to the quasi equivalence if and only if

Assume that the elements $\omega _0, \dots, \omega _{r-1} \in {\mathcal E}_{z}$ form a basis of the quotient space ${\mathcal E}_{z}/TQ\Gamma _{g}$. Then, Proposition 1 implies that a miniversal deformation of a function germ $g$ may take the form:

Therefore, the formulas of quasi boundary versal deformations of $g_1\in B_k$ and $g_2 \in F_{p, k}$ are

and

respectively.

Remark 2. The versal deformation of the class $F_{2, 3}$ can be written equivalently as

Definition 5. ^[15] The quasi bifurcation diagram of a germ $g$ with $G(z, u)$ being its quasi versal deformation, is the pair $\mathcal{W}(g) = (\mathcal{W}_1, \mathcal{W}_0)$, where

and

Note that $\mathcal W_0$ is contained in $\mathcal W_1$ and it satisfies the constraint $x = 0$. Thus, in particular, the bifurcation diagrams of the classes $B_k$ is $\mathcal{W}(B_k) = (\mathcal{W}_1, \mathcal{W}_0)$, where

and

On the other hand, the bifurcation diagram of the class $F_{2, 3}$ is $\mathcal{W}(F_{2, 3}) = (\mathcal{W}_1, \mathcal{W}_0)$, where

and

Theorem 3.1. The stationary algebra of $\mathcal W (B_k)$, for $k = 2, 3, 4$, and $\mathcal W (F_{2, 3})$ is described as follows.

(1) $Der(-log\mathcal W (B_2))$ is generated by

(2) $Der(-log\mathcal W (B_3))$ is generated by

(3) $Der(-log\mathcal W (B_4))$ is generated by

(4) $Der(-log\mathcal W (F_{2, 3}))$ is generated by

Proof. Let $p_1, p_2 \in \mathcal E_r$. Assume that $p_1$ is the defining equation of $\mathcal W _1$ and $p_1, p_2$ are the defining equations of $\mathcal W _0$. Let $I(\mathcal W _1)$ be the ideal generated by $p_1$ and $I(\mathcal W _0)$ is the ideal generated by $p_1$ and $p_2$.

Let $\Theta (\mathcal W _1)$ be the module of all vector fields $\xi = \sum\limits _{i = 1}^r\xi_i\frac{\partial}{\partial u_i}$ on $\mathbb{R}^{r}$ such that $\xi \left(I(\mathcal W_1)\right)\subseteq I(\mathcal W_1)$. To find $\Theta (\mathcal W _1)$, we have to solve the equation

for $\xi _i$ and $q\in \mathcal E_r$. Now consider the map $\phi : \mathcal E_r ^{r+1} \to \mathbb R$, given by

where $\xi = (\xi _1, \dots, \xi _r) \in \mathcal E_r^r$ and $q \in \mathcal E_r^1$. Let $K = \ker\Phi$ and $\pi:\mathcal E_r^{r+1}\rightarrow \mathcal E_r^r$ be defined by $\pi(\xi, q) = \xi$. Then $\Theta(\mathcal W_1) = K$. Using the syzygies that are supplied in the Singular software package, we are able to obtain the $K$.

Next, we are looking for the module $\Theta (\mathcal W _1)$ of all vector fields such that $\xi \left(I(\mathcal W_0)\right)\subseteq I(\mathcal W_0)$. This implies that, for each $j = 1, 2$, we have to solve

for $\xi = \sum\limits _{i = 1}^r\xi_i\frac{\partial}{\partial u_i}$ and $q_i$.

For $j = 1, 2$, let $\Phi_j:\mathcal E_r^{r+2}\rightarrow \mathbb R$ be the map that is defined as

where $\xi = (\xi_1, \dots, \xi_r)\in\mathcal E_r^r$ and $\tilde q = (q_1, q_2)\in\mathcal E_r^2$. Let $K_j = \ker\Phi_j$. Set $\pi:\mathcal E_r^{r+2}\rightarrow \mathcal E_r^r$ be defined by $\pi(\xi, \tilde q) = \xi$. Let $S_j = \pi(K_j)$. Therefore,

Again we use the syzygies to obtain the $K _i$. Finally, we have

All of the vector fields $\xi$ that are created by this approach can be verified to be liftable. This means that there is a vector field $\eta$ on $\mathbb{R}^{r-1}$ that is such that $dp_i(\eta) = \xi\circ p_i$, and as a result, they are tangent to $\mathcal W$. □

In (3.1), if we set $\omega _0 = 1$ and $\omega _i\in {\mathcal M}_{z}$, then the space $\mathbb{R}^{r-1}$ in the local coordinates $u _1, \dots, u _{r - 1}$ is known as the principle of a shortened quasi-boundary versal deformation of $g$.

Consider the projection map $\pi : \mathbb R^r \to \mathbb R^{r-1}, (u_0, u_1, \dots u_{r-1}) \mapsto (u_1, \dots u_{r-1}).$

Definition 6. ^[15] The caustic of a a versal deformation of a function $g$ with respect to the quasi equivalence relation is a hypersurface in $\mathbb R^{r-1}$ which is a union

which will be denoted by $\Sigma ^*$, where $\Sigma _10 = \mbox{the $\pi$ -image of the singular points of $\mathcal{W}_1$ }$ and $\Sigma _0 = \pi (\mathcal{W}_0)$.

Recall from ^[15] (applying Definition [6]) that the caustic of $B_k$ is a union of two hypersurfaces which are tangent to each other, the first one is the set

that is a cylindrical generalized swallow tail over a line. The second one of them is the submanifold of the greatest dimension that crosses the edge of $\Sigma _0$., in particular:

Furthermore, the caustic of the class $F_{2, 3}$ is a union of the smooth surface

and Whitney Umbrella

Theorem 3.2. The module $Der(-log (\Sigma ^*)$ is generated by the vector fields

(1) For the caustic of $B_3$

(2) For the caustic of $B_4$

(3) For the caustic of $F_{2, 3}$

Proof. Assume that $\Theta (\Sigma _1)$ and $\Theta (\Sigma _0)$ are the modules of vector fields that are that are tangential to $\Sigma _1$ and $\Sigma _0$, respectively.

To determine $\Theta (\Sigma _i), i = 0, 1$, we employ similar processes as those used in the proof of Theorem 3.1 to determine $\Theta (\mathcal{W}_1)$.

Hence, we have,

All such vector fields $\xi \in \Theta (\Sigma ^*)$ are liftable and as a result, they are tangent to $\Sigma ^*$. □

Next, we investigate whether or not $\mathcal W$ and $\Sigma ^*$ are considered to be free divisors. Recall that a hypersurface $V = \{h = 0\} \subset (\mathbb{K}^n, 0)$ with a reduced defining ideal $I(V)$, is called free divisor, in the sense of Saito, if $Der(-logV)$ is a free $\mathcal{E}_n$-module, necessarily, so its rank is equal to $n$. The following criterion was established also by K. Saito and is now commonly referred to after him.

Proposition 2. ^[21][Saito's Criterion] Let $h \in \mathbb{K}[z]$ be reduced. Then, $h$ defines a free divisor if and only if there exists $n\times n$ matrix $H$ with entries in $\mathbb{K}[z]$ such that

Here, $\nabla h = \left (\frac{\partial h}{\partial z_1}, \dots, \frac{\partial h}{\partial z_n}\right)$ and the last condition just expresses that each entry of the (row) vector $(\nabla h)H$ is divisible by $h$ in $\mathbb{K}[z]$. The columns of $H$ can then be viewed as the coefficients of a basis, with respect to the partial derivatives $\frac{\partial }{\partial z_i}$, of the logarithmic vector fields along the divisor $h = 0$. The matrix $H$ is called a discriminant or Saito matrix of $h$.

The above discussion and Theorem 3.1 implies that the bifurcation diagrams of the classes $B_k, k = 2, 3, 4$ and class $F_{2, 3}$ are not free divisors due to the cardinality of a basis of $Der(-log \mathcal{W}(B_k))$, and $Der(-log \mathcal{W}(F_{2, 3}))$, respectively. Moreover,

Proposition 3. The caustics of $B_i, i = 3, 4$ are free divisors.

Proof. The caustic of the $B_3$ class is a union of the parabola $3u_1-u_2^2 = 0$ and the line $u_1 = 0$. Hence, the defining equation of $\Sigma ^*\subset \mathbb R^2$ for $B_3$ is $h_1 = u_1(3u_1-u_2^2).$ On the other hand, by similar consideration we find that the defining equation of the caustic $\Sigma ^*\subset \mathbb R^3$ for the class $B_4$ is $h_2 = u_1(108u_1^2+32u_2^3-108u_1u_2u_3-9u_2^2u_3^2+27u_1u_3^3).$ Applying Saito's Criterion and Theorem 3.2, one can easily show that the caustics of $B_i, i = 3, 4$ are free divisors. □

4.   Geometric cuspidal edge with smooth boundary

Let $G$ be a general cuspidal edge in $\mathbb{R}^3$ with local coordinates $z = (u, v, w)$, having the following parametrization at the origin as in [18]:

where $U$ is an open set in $\mathbb{R}^2$ and

and $a_{ij}, b_{ij} \in \mathbb{R}.$ Let $B \subseteq G$ be a distinguished smooth curve which is parametrized by:

i.e., the image of the line $s = t$. The pair $\widetilde{G} = (G, B)$ will be called a geometric cuspidal edge with a smooth boundary. Note that $\widetilde{G}$ is diffeomorphic to $\mathcal{W}(B_3)$.

The set of critical points of $G$ is $\{s = 0\}$, and hence the singular set of $G$ is the curve:

which will be denoted by $\Sigma$.

Denote by $\big (\kappa _{\Sigma }(0), \kappa _B(0)\big)$ and $\big (\tau _{\Sigma }(0), \tau _B(0)\big)$ the pairs of the curvatures and torsions of the pair of space curves $(\Sigma, B)$, respectively.

Proposition 4. (1) $\big (\kappa _{\Sigma }(0), \kappa _B(0)\big) = \big (\sqrt{a_{20}^2+b_{20}^2}, \sqrt{(a_{20}+1)^2+b_{20}^2} \big).$

(2) $\big (\tau _{\Sigma }(0), \tau _B(0)\big) = \big (\frac{a_{20}b_{30}-b_{20}a_{30}}{\kappa _\Sigma ^2(0) }, \frac{(a_{20}+1)(b_{30}+b_{03}+3b_{12})-b_{20}a_{30}}{\kappa ^2_B(0)}\big).$

(3) The osculating planes of $\Sigma$ and $B$ at the origin are orthogonal to the vectors $(0, -b_{20}, a_{20})$ and $(0, -b_{20}, a_{20}+1)$, respectively.

Proof. The results are obtained via the standard rules of calculating curvature and torsion on space curves. □

Note that the two curves $\Sigma$ and $B$ have a common tangent line, and hence the tangential direction, which will be denoted by $T_d$ at 0, is parallel to the vector $(1, 0, 0)$. Note that $w = 0$ is the plane that represents the tangent cone $L_d$ to $\widetilde{G}$.

4.1.   Functions on a cuspidal edge equipped with a smooth curve

Consider the pair $(\widetilde f, \widetilde \gamma _1)$ which consists of the $\mathcal A$-normal form of a cuspidal edge:

and the smooth curve $\widetilde \gamma _1 (t) = \widetilde f (t, t) = (t, 0, 0)$. Let $\mathcal{V}_1 = \widetilde f (U\subseteq \mathbb{R}^2)$ and $\mathcal{V}_0 = \widetilde \gamma _1(\mathbb{R})$. Then, the pair $\mathcal{V} = (\mathcal{V}_1, \mathcal{V}_0)$ will be considered as a model of a cuspidal edge with a smooth curve, which serves as a boundary. (See Figure 1).

Recall that the bifurcation diagrams of the $B_3$ class is the pair $\mathcal{W} = (\mathcal{W}_1, \mathcal{W}_0)$, where $\mathcal{W}_1$ is the cuspidal edge, which is parameterized by

and hence it is $\mathcal{A}$-equivalent to $\widetilde f$, via the right change of coordinates $t\mapsto t, s\mapsto s-\frac{1}{3}t$, followed by normalising the coefficients. Hence, the pair $\mathcal{V}$ is diffeomorphic to $\mathcal{W}$.

The defining equation of $\mathcal{V}_1$ is

while the defining equations of $\mathcal{V}_0$ are $K(z) = 0, v = 0$ and $w = 0$.

Definition 7. We say that $g_1, g_2 \in \mathcal E_z$ are $\mathcal{R}(\mathcal{V})$-equivalent whenever $g_2 = g_1\circ \Phi$ where $\Phi :(\mathbb{R}^3, 0) \to (\mathbb{R}^3, 0)$ is diffeomorphism-germ and $\Phi$ preserves $\mathcal{V}$, i.e., $\Phi (\mathcal{V}_1)\subseteq \mathcal{V}_1$ and $\Phi (\mathcal{V}_0)\subseteq \mathcal{V}_0$.

Define $\Theta (\mathcal{V})$ as the module over $\mathcal{E}_z$ of vector fields in $\mathbb{R}^3$ that is tangential to $\mathcal{V}$ and define

Then, the tangent space and the expanded tangent space to the orbit of $g$ at $g$ are

and

respectively.

The $\mathcal{R}_e^+$-codimension of $g$ is defined as $d(g, \mathcal{R}^+_e(V)) = dim_{\mathbb{R}}\left (\mathcal{M}_z/L_e\mathcal{R} (\mathcal V).g \right)$.

Theorem 4.1. The $\mathcal{E}_z$-module $\Theta (\mathcal{V})$ is generated by

Proof. By following the same procedures as that to prove Theorem 3.1. □

Corollary 1. For $g\in \mathcal{E}_z$, the tangent space to the orbit of $g$ at$g$ with respect to the $\mathcal{R}(\mathcal V)$-equivalence relation is defined as

We proceed to classify submersion-germs $g:(\mathbb{R}^3, 0) \to (\mathbb{R}, 0).$ $g$ from $(\mathbb{R}^3, 0)$ to $(\mathbb{R}, 0)$ in accordance with the $\mathcal{R}(\mathcal V)$-equivalence relation, where $d(g, \mathcal{R}^+_e(V)) \leq 2$. The classification method and prenormal forms are described in the following lemma and relies on Arnold's spectral sequence. At first, we go over several concepts from ^[22].

Let us assume that we have a certain appropriate Newton diagram $\Gamma$ that is a subset of the non-negative integers $\mathbb{Z}_{\geq 0}^n$. Each face $\Gamma _i$ of $\Gamma$ corresponds to a specific quasihomogeneity type $\boldsymbol{\alpha}_i = (\alpha _{i1}, \alpha _{i2}, \dots, \alpha _{in})$. In this type, the monomials $\boldsymbol{x^k} = x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$ with exponents lying on $\Gamma _i$ have a degree of one, meaning that $\left \langle \boldsymbol{k}, \boldsymbol{\alpha}_i\right \rangle = \alpha _{i1}k_1+\dots +\alpha _{in}k_n = 1.$

A monomial, denoted as $\boldsymbol{x^k}$, is considered to have a Newton degree of $d$ if $d$ is the minimum value obtained from the inner product of $\boldsymbol{k}$ and $\boldsymbol{\alpha}_j$. The monomials of Newton degree $d$ are precisely those whose exponents are contained in the diagram $d\Gamma$, which is created by scaling $\Gamma$ by a factor of $d$.

The smallest of the Newton degrees of the monomials that appear in a power series is known as the Newton order $d$. An ideal $\mathcal{S}_j$ in the ring $\mathcal{E}_x$ is formed by the series of order at least $d$. The Newton filtration in $\mathcal{E}_x$ is generated by the ideals $\mathcal{S}_j$. More precisely, $\mathcal{S}_0 = \mathcal{E}_x$, and $\mathcal{S}_k \subseteq \mathcal{S}_l$ whenever $k > l$.

The principal part of a power series $g$ of order $d$ is the sum of the terms of Newton degree $d$.

Let $g \in \mathcal{E}_z$. We may decompose $g$ into its principal portion $g_0$ of Newton degree being $N$ and greater order elements $\widetilde{g}$ as $g = g_0+\widetilde{g}$. It is assumed that the $\mathcal{R}_e^+$-codimension of $g_0$ is finite, meaning that $d(g_0, \mathcal{R}^+_e(\mathcal{V})) < \infty$. The subsequent result is a rendition of Lemma 8.1 in ^[12] and Lemma 2.10 in ^[13].

Lemma 1. Consider a monomial basis of the linear space $\mathcal{E} _z/L\mathcal{R}(\mathcal{V}).g_0$ and let $\rho _1(z), \rho _2(z),$ $\dots, \rho _s(z)$ be the subset of generators that have Newton degrees greater than $N$.

Assume that for every $\omega \in \mathcal{S}_{\beta }\setminus \mathcal{S}_{ > \beta},$ $\beta > N$:

(1) There is a vector field $\xi = \dot u\frac{\partial}{\partial u}+\dot v\frac{\partial}{\partial v}+\dot w\frac{\partial}{\partial w} \in \Theta (\mathcal{V})$, such that

where $\widehat\omega \in \mathcal{S} _{ > \beta}$ and $c_i \in \mathbb{R}.$

(2) Furthermore, given any $\delta,$ if $N < \delta < \beta,$ and for every $\psi \in \mathcal{S} _{\delta}$, the following statement

belongs to $\mathcal{S} _{\beta }$.

Then any germ $g = g_0+\widetilde g$ is $\mathcal{R}(\mathcal{V})$-equivalent to a germ $g_0+\sum\limits ^s_{i = 1} d_i \rho _i,$ where $d_i \in \mathbb R$.

Remark 3. By eliminating the prerequisite that $g_0$ has a finite codimension, the proof of Lemma 1 demonstrates that any function $g = g_0+\widetilde g$ is $\mathcal{R}(\mathcal{V})$-equivalent to a comparable form:

where $\Lambda$ is a member of a relatively large power of the maximal ideal. $

Definition 8. The families of germs of functions $H_1, H_2: \big (\mathbb{R}^3\times \mathbb{R}^l, (0, 0)\big) \to (\mathbb{R}, 0)$ are referred to as $P$-$\mathcal{R}^{+}(\mathcal{V}).$-equivalent if

where $\Phi :\big (\mathbb{R}^3\times \mathbb{R}^l, (0, 0)\big)\to \big (\mathbb{R}^3\times \mathbb{R}^l, (0, 0)\big)$ is a germ of diffeomorphism with $\Phi (z, u) = \left (\varphi (z, u), \chi (u) \right)$, and $C: (\mathbb{R}^l, 0) \to \mathbb{R}$ is a function germ.

Consider $G(z, u)$ as a deformation of $g\in \mathcal{E}_z$.Then, $G$ is referred to as versal with regard to the $\mathcal{R}^{+}(\mathcal{V})$-equivalence relation if, for whatever other form of deformation $H$ of $g$, there exists a map germ $\Phi (z, u)$ (which may not necessarily be a diffeomorphism) and $C$ as described above, satisfying

Proposition 5. A deformation $G$ of $g$ on $\mathcal{V}$ is $\mathcal{R}^{+}(\mathcal{V})$-versal provided

where $\dot G_i$ are the initial speeds of $G$.

The result that follows is a description of the classification of germs of submersions in accordance with the $\mathcal{R}(\mathcal{V})$-equivalence relation.

Theorem 4.2. Let $\mathcal{V}$ be a pair consisting of the cuspidal edge $\mathcal{V}_1$, which is represented by the parametrization $\widetilde f (t, s) = \big(t, \frac{1}{2}(s^2-t^2), s^3+t^3-2ts^2 \big)$, and a distinguished smooth curve $\mathcal{V}_0$ within it, represented by the parametrization $\widetilde \gamma(t)_1 = \widetilde f(t, t) = (t, 0, 0)$. Then, any function germ $g$ at 0 that has a $\mathcal{R}^{+}(\mathcal{V})$-codimension of not more than 2 (with moduli) is equivalent to a function germ mentioned in the Table 1.

Proof. The linear transformations of coordinates derived via the integration of the 1-jets of the vector fields in $\Theta (\mathcal{V})$ are:

where $c _i \in \mathbb{R}$.

Let $g$ be decomposed into its 1-jet $g_0 = au+bv+cw$, where $a, b, c \in \mathbb{R}$ and $\widetilde g \in \mathcal{M}_z^2$. Using $\varphi _i$, one can show that the orbits of the space of 1-jets are $\pm u$, $\pm v$, and $\pm w$.

The subsequent conclusions may be established by applying Lemma 1 and Remark 3.

Let $g_0 = \pm u$. Then, the tangent space to the orbit of $g_0$ at $g_0$ is

Clearly, we have $mod\; L\mathcal{R}(\mathcal{V}).g_0$: $u\equiv 0, v \equiv 0$ and $w \equiv 0$. Hence, Lemma 3 implies that $g$ is $\mathcal{R}(\mathcal{V})$-equivalent to its principal part $g_0 = \pm u$.

Next, consider the principal part $g_0 = \pm v$. Then,

Therefore, we have $mod\; L\mathcal{R}(\mathcal{V}).g_0$: $v\equiv 0$ and $w \equiv 0$. It follows that

Using Remark 3 and taking into account the constraints on $\widetilde g$, the germ $g$ is reduced to the form $h = \pm v+\widetilde h (u),$ where $\widetilde h\in \mathcal{M}^2_u$. Let $\widetilde h = d_2u^2+d_3u^3+\dots$, $d_i \in \mathbb{R}$. If $d_2 \ne 0$, then $h$ is $\mathcal{R}(\mathcal{V})$-equivalent to the germ $\pm v+\epsilon u^2$, where $0, \pm 1 \ne \epsilon \in \mathbb{R}$ (modulus) and its mini-versal deformation may be taken as $\pm v\pm u^2+\lambda _1u$. Next, if $d_2 = 0$ but $d_3 \ne 0$, then $h$ is $\mathcal{R}(\mathcal{V})$-equivalent to the germ $\pm v+\epsilon u^3$, where $0 \ne \epsilon \in \mathbb{R}$ (modulus) and its mini-versal deformation may be taken as $\pm v+\epsilon u^3+\lambda _1u+\lambda _2u^2$.

Finally, consider the 1-jet $g_0 = \pm w$. Then,

Therefore, we have $mod\; L\mathcal{R}(\mathcal{V}).g_0$:

and

Clearly, relations (4.2) and (4.3) are linearly independent. Hence, $v^2 \equiv 0$ and $vu^2\equiv 0$. Consequently

Using Remark 3 and taking into account the constraints on $\widetilde g$, the germ $g$ is reduced to the form $h = \pm w+a_2uv+\widetilde h (u),$ where $\widetilde h\in \mathcal{M}^2_u$. If $\widetilde h$ contains $\tilde d_3u^2$, where $0\ne \tilde d_3 \in \mathbb{R}$, then $h$ is equivalent to $\pm w+\epsilon u^2$, $0\ne \epsilon \in \mathbb{R}$ (modulus) and its mini-versal deformation may be taken as $\pm w+\epsilon u^2+\lambda _1u+\lambda _2v$. If $\tilde d_3 = 0$, then in the most degenerate case $h$ has codimension greater than 2. The proof of the theorem is now complete. □

4.2.   The discriminants of the deformations

Let $F:(\mathbb{R}^3\times \mathbb{R}^2, 0) \to \mathbb{R}; (z, \lambda)\mapsto F(z, \lambda)$ be a deformation of a germ $h(z)$ on $\mathcal{V}$ and consider the family $P(s, t, \lambda) = F(\widetilde f(t, s), \lambda)$, where $\widetilde f(s, t) = \big(t, \frac{1}{2}(s^2-t^2), s^3+t^3-2ts^2 \big)$. Then, we define the following types of discriminants:

(1) The discriminant of the family $P$, everywhere:

(2) The discriminant of $P$, restricted to $\Sigma$:

and

(3) The discriminant of $P$, restricted to the boundary $\mathcal{V}_0$:

We shall calculate $\mathbb{D}_i, i = 1, 2, 3$ for the mini-versal deformations $F(z, \lambda)$ of the submersions $g(z) = F(z, 0)$ in Table 1.

(1) $g(z) = u$. We have $F(z, \lambda) = u$, and hence $P = t$. Note that the fiber $g = 0$ is transverse for both $T_d$ and $L_d$. Clearly, $\mathbb{D}_i, i = 1, 2,$ are all empty sets.

(2) $g(z) = \pm v+\epsilon u^k, k = 2, 3$. Note that the tangent plane to the fiber $g = 0$ contains $T_d$ but is transverse $L_d$.

● For $k = 2$, we have $F(z, \lambda) = \pm v+\epsilon u^2+\lambda _1u,$ and hence $P = \pm (s^2-t^2)+\epsilon t^2+\lambda _1t.$ The $\mathbb{D}_1$ set is a smooth surface which is parametrized by

The $\mathbb{D}_2$ set coincides with $\mathbb{D}_1$. On the other hand, on the boundary we have $P = F(\widetilde f(t, t), \lambda) = \epsilon t^2+\lambda _1t$. Therefore, the $\mathbb{D}_3$ set is also a smooth surface which is parametrized by:

Note that $\mathbb{D}_1 = \mathbb{D}_2$ and $\mathbb{D}_3$ are tangent along the $\lambda _2$-axis.

● For $k = 3$, we have $F(z, \lambda) = \pm v+\epsilon u^3+\lambda _1u+\lambda _2u^2$. The $\mathbb{D}_1$ and $\mathbb{D}_2$ sets are coinciding cuspidal edge, which are parameterized by:

The $\mathbb{D}_3$ set is also cuspidal edge, that is parametrized by:

Note that $\mathbb{D}_1 = \mathbb{D}_2$ intersects $\mathbb{D}_3$ along a curve.

(3) $g = \pm w+\epsilon u^2$. We have $F = \pm w+\epsilon u^2+\lambda _2u+\lambda _2 v.$ We may consider the versal deformation:

The tangent plane to $g = 0$ includes both $T_d$ and $L_d$ in this scenario. Now, we have

Note that $\frac{\partial P}{\partial s} = 0$ if and only if $s = 0$ or $\lambda _2 = \frac{3}{2}s-2t$. Moreover, $\frac{\partial P}{\partial t} = 0$ if and only if $\lambda _1 = 2s^2\mp 3t^2-2(\epsilon +\lambda _2)t$. Hence, the $\mathbb{D}_1$ set consists of two components, the first one is a cuspidal edge which is parametrized by

and the second one is a smooth surface which is parametrized by

where $\lambda _1 = \pm 2s^2+(\mp 3-4)t^2-(2\epsilon +3s)t$ and $\lambda _2 = \frac{3}{2}s-2t$. The $\mathbb{D}_2$ set coincides with the first part of $\mathbb{D}_1$. The $\mathbb{D}_3$ is a regular surface that may be described by a parametrization

We condense the above calculation in the following.

Proposition 6. (1) The discriminants $\mathbb{D}_1$, $\mathbb{D}_2$ and $\mathbb{D}_3$ of the singularity $g = u$ are empty sets.

(2) The discriminants $\mathbb{D}_1$ and $\mathbb{D}_2$ of the singularity $g = \pm v +\epsilon u^2$ are coincident smooth surface, and the $\mathbb{D}_3$ is also a smooth surface that is tangent to $\mathbb{D}_1$ along a curve (Figures 2 and 3).

(3) The discriminants $\mathbb{D}_1$ and $\mathbb{D}_2$ of the singularity $g = \pm v +\epsilon u^3$ are coincident cuspidal edges, and the $\mathbb{D}_3$ is a different cuspidal edge that is tangent to $\mathbb{D}_1$ along a curve (Figure 4).

(4) The discriminants $\mathbb{D}_1$ of the singularity $g = \pm w +\epsilon u^2$ is a combination of two components: a cuspidal edge and a regular surface (Figure 5). The $\mathbb{D}_2$ is a cuspidal edge and coincides with one of the components of the $\mathbb{D}_1$. The $\mathbb{D}_3$ is a smooth surface (Figure 6).

4.3.   The height functions on a geometric cuspidal edge with a smooth curve

Let $H:\widetilde G \times S^2 \to \mathbb R, H((t, s), {\boldsymbol\eta}) = H_{{\boldsymbol\eta}}(t, s) = f (t, s)\cdot {{\boldsymbol\eta}}$, be a family of height functions on $\widetilde G$, where $S^2$ is the 2-sphere and ${\boldsymbol\eta} = (\eta _1, \eta _2, \eta _3) \in \mathbb R^3$. Then, $H_{{\boldsymbol\eta}}$ measures the contact of $\widetilde G$ with the plane $\pi _p$ which is orthogonal to the vector ${\boldsymbol\eta}$ at the point $p \in \widetilde G$. Generically, the submersions $g$, which are obtained in Theorem 4.2, describe explicitly such contact. The contact between the fiber $g = 0$ and the model $\mathcal{V}$ is equivalent to the contact between $\widetilde G$ and $\pi _ p$.

We discuss the contact of $\pi _p$ with $\widetilde G$ along $\Sigma$ and $B$ at the origin.

Note that the restriction of $H_{{\boldsymbol\eta}}$ along $\Sigma$ and $B$ are

and

respectively.

Clearly, $H_{\bf {\boldsymbol\eta}}$ is singular when $\eta _1 = 0$. Further, the contact of $\widetilde G$ with $\pi _p$ is measured by the zero of $g = u$ with $\mathcal{V}$ at the origin when $\pi _p$ is transverse to $T_d$. The following is a description of the remaining cases in which $\pi _p$ is a part of the pencil family of planes that is not transverse to $T_d$:

Theorem 4.3. If $\pi _p$ is not the tangential cone to $\widetilde G$, then the contact of $\pi _p$ with $\widetilde G$ is equivalent to that of the zero of $g = \pm v+\epsilon u^k$ (where $k = 2, 3$ and $\epsilon \ne 0, \pm 1$) with the representation $\mathcal{V}$. Moreover,

(1) the plane $\pi _p$ has an $A_1$-contact with $\Sigma$ and $B$ if and only if $\pi _p$ is not the osculating of neither $\Sigma$ nor $B$.

(2) $\pi _p$ has an $A_1$-contact with $\Sigma$ and an $A_2$-contact with $B$ if and only if and only if $\pi _p$ is not the osculating of $\Sigma$ but $\pi _p$ coincides with the osculating plane of $B$ and $\tau _B(0) \ne 0$.

Proof. Among the submersions listed in Theorem 4.2, the tangent plane to the zero fiber of $g = \pm v+\epsilon u^k,$ (where $k = 2, 3$), contains $T_d$ and is transverse to the tangent cone of $\widetilde G$. As a result, the contact of $\pi _p$ with $\widetilde G$ is equivalent to that of $g = 0$ with the representation $\mathcal{V}$.

(1) Let $k = 2$. Then, the contact of the tangential line along $\Sigma$ of and $B$ is measured by the type of

and

respectively. So, it is of type $A_1$ along the two curves. Now, consider the restrictions $H_{{\boldsymbol\eta}}(t, 0)$ and $H_{{\boldsymbol\eta}}(t, t)$ on $\Sigma$ and $B$, respectively. Then, the plane $\pi _p$ has an $A_1$-contact with $\Sigma _0$ if and only if $\eta _2a_{20}+\eta _3b_{20}\ne 0$, which implies that $(\eta _2, \eta _3) \ne (-b_{20}, a_{20})$. On the other hand, $\pi _p$ has an $A_1$-contact with $B$ if and only if $\eta _2(a_{20}+1)+\eta _3b_{20}\ne 0$, which implies that $(\eta _2, \eta _3) \ne (-b_{20}, a_{20}+1)$. Geometrically, this means that $\pi _p$ is not the osculating plane of neither $\Sigma$ nor $B$ at $p$.

(2) Let $k = 3$. Then, the contact of the tangential line along $\Sigma$ and $B$ is measured by the type of

and

respectively. So, it is of type $A_1$ along $\Sigma$ and of type $A_2$ along $B$. Now, consider the restriction $H_{{\boldsymbol\eta}}(t, t)$ along $B$. Then, the plane $\pi _p$ has an $A_2$-contact with $B$ if and only if

and

The constraint (4.4) implies that $(\eta _2, \eta _3) = (-b_{20}, a_{20}+1)$, which means that $\pi _p$ is the osculating plane of $B$. On the other hand, the constraint (4.5) becomes

which implies that $\tau _B(0)\ne 0$.

□

Theorem 4.4. If $\pi _p$ is the tangent cone to $\widetilde G$, then the contact of $\pi _p$ at $\widetilde G$ is equivalent to that of the zero fiber of $g = \pm w +\epsilon u^2$ ($\epsilon \ne 0, \pm 1$) with the representation $\mathcal{V}$. Furthermore, the plane $\pi _p$ has an $A_1$-contact with both $\Sigma$ and $B$, and it is not the osculating of neither $\Sigma$ nor $B$.

Proof. The tangent to the zero of $g = \pm w+\epsilon u^2,$ ($\epsilon \ne 0, \pm 1$), is the same as the tangent cone of $\widetilde G$ for the submersions shown in Theorem 1. Hence, the contact of $\pi _p$ with $\widetilde G$ is the same as that of $g = 0$ with the model $\mathcal{V}$. Note here that ${\boldsymbol\eta} = (0, 0, 1)$. On the other hand, the contact of the tangential line along $\Sigma$ and $B$ is measured by the singularity of

and

respectively, where $\epsilon \ne 0. \pm 1$. So, it is of type $A_1$. The corresponding height function restricted to $\Sigma$ and $B$ has an $A_1$ singularity if and only if $\eta _3b_{20}\ne 0$, which means that $b_{20}\ne 0$, and hence does not coincide with the osculating plane of both $\Sigma$ and $B$.□

4.4.   The dual of a geometric cuspidal edge with a smooth curve

The discriminants may be used for the examination of the dual of the cuspidal edge equipped with a smooth curve as explained below.

As pointed out in ^[23], an oriented plane in $\mathbb{R}^3$ in local coordinates $z = (u, v, w)$ is characterized by a unit vector ${\boldsymbol\eta}$ and a real number $c$. The equation of the plane can be expressed as $z\cdot {\boldsymbol\eta} = c$, where $\cdot$ represents the scalar product. It is important to observe that the pairs $({\boldsymbol\eta}, c)$ and $(-{\boldsymbol\eta}, -c)$ represent the same plane, but with opposing orientations. A unit space curve $\gamma (t)$ can be associated with an oriented tangent plane at $t_0 \in I \subset \mathbb R$ by a unit vector ${\boldsymbol\eta}$ that is perpendicular to the tangent vector $T(t)$ of $\gamma (t)$ at $t_0$. The equation of the tangent plane is given by $z\cdot {\boldsymbol\eta} = \gamma (t_0)\cdot {\boldsymbol\eta}$. The collection of all oriented tangent planes to the curve $\gamma (t)$ is referred to as "the dual" of $\gamma (t)$. Consequently, it is associated with the set defined as follows:

Define the following families:

and

Then, in accordance with ^[23], if the contact of $\widetilde G$ with $\pi _{{\boldsymbol\eta}}$ is characterized by that of the fiber $g = 0$ with $\mathcal{V}$, where $g$ is defined in Theorem (Classification of germs of submersions), then $\mathbb{D}_i(H)$ is diffeomorphic to $\mathbb{D}_i(F)$, where $F$ is a $\mathcal{R}^{+}(\mathcal{V}$)-versal unfolding of g with 2-parameters. Therefore, we have the following:

Proposition 7. Let $\widetilde G$ be a pair of a geometric cuspidal edge $G$ in $\mathbb R ^3$ equipped with a smooth $B$ in it. Then, the calculations and figures in Section 4.2 give the models, up to diffeomorphisms, of $\mathbb{D}_i(H)$, $i = 1, 2, 3$.

The above result implies that if $\pi _p$ is tangent to $T_d$ but it is transverse $L_d$, then $\mathbb{D}_1(H) = \mathbb{D}_2(H)$ and $\mathbb{D}_3(H)$ describe locally the dual of the curve $\Sigma$ and $B$, respectively. On the other hand, if $\pi_p$ coincides with $L_d$ and it is tangent to $T_d$, then $\mathbb{D}_1(H)$ composed of two parts: one is $\mathbb{D}_2(H)$ which is the dual of $\Sigma$) and the second is the proper dual of $G$ away from points of $\Sigma$, whereas the set $\mathbb{D}_3(H)$ describes locally the dual of the curve $B$.

5.   Conclusions

In this paper, we calculated the generators of the vector fields that are tangent to the bifurcation diagrams and caustics of the classes $B_k,$ $k = 2, 3, 4$ and $F_{2, 3}$ with respect to the quasi equivalence which is a non-standard equivalence relation. Consequently, we considered for application the generators of the $B_3$-class in which case the bifurcation diagram consists of two components: a cuspidal edge in $\mathbb{R}^3$ and a smooth curve in it, which serves as a boundary and denoted it by $V = (V_1, V_0)$. Then, we classified the submersion on $V$ with codimension less or equal 2. This model and classifications were used to study the geometry of the pair $\widetilde G = (G, B)$ of the geometric cuspidal edge $G$ equipped with a distinguished curve $B$ in it. Apart from the standard structure, $\widetilde G$ contains two curves: the singular pints (the ridge) $\Sigma$ and the smooth curve $B$. Thus, we discussed and described the contact of $\widetilde G$ with the plane $\pi _p$ at $p\in \widetilde G$ along the curves $\Sigma$ and $B$ via the height function on $\widetilde G$, using the zero fibers of the submesrion obtained on $V$. In particular, we distinguished two cases. First, if $\pi _p$ is the tangent cone to $\widetilde G$, then the contact is of type $A_1$ along both $\Sigma$ and $B$ if and only if $\pi _p$ is not the osculating plane of neither $\Sigma$ nor $B$, and of type $A_1$ along $\Sigma$ and $A_2$ along $B$ if and only if $\pi _p$ is not the osculating plane of $\Sigma$ but it coincides with the osculating plane of $B$ and $\tau _B(0) \ne 0$ (the torsion of $B$ at 0). Second, if $\pi _p$ is not the tangent cone to $\widetilde G$, then the contact is of type $A_2$ along both $\Sigma$ and $B$ if and only if $\pi _p$ is not the osculating plane of neither $\Sigma$ nor $B$.

Subsequent study extending beyond this work may involve examining the height function on other singular hypersurfaces in $\mathbb{R}^3$ characterized by a smooth or singular boundary. When the hypersurface is equipped with a distinguished singular curve, it is more intriguing as it may involve two transversal tangential directions, such as the situation of the cuspidal edge with a singular curve (cusp) in it.

Author contributions

Yanlin Li: Conceptualization, investigation, methodology, writing-review and editing; Fawaz Alharbi: Conceptualization, investigation, methodology, writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We wish to thank the anonymous reviewers for their insightful suggestions and careful reading of the manuscript.

The first author would like to express his gratitude to Raul Oset Sinha from the Universitat de València for the valuable discussion.

Conflict of interest

The authors declare no conflicts of interest.