数理解析研究所講究録
第 1868 巻 2013 年 59-62

59Singularities of the discriminant of pairs of regularfoliations in$\mathbb{R}^{3}$

Luciana F. Martins and Ana Claudia Nabarro
Departamento de Matem\’atica, Instituto de Bioci\^encias, Letras

e Ci\^encias Exatas, Univ Estadual Paulista, UNESP, Campus de S\~ao Jos\’e do Rio Preto
lmartins@ibilce.unesp.br
Departamento de Matem\’atica, Instituto de Ci\^encias Matem\’aticas e de Computa\ca\~ao,
Univ de S\~ao Paulo, USP-C\^ampus de S\~ao Carlos
anaclana@icmc.usp.br1

IntroductionAn important feature of pairs of codimension one regular foliations in
is its discriminant
which is the locus of points where the foliations are tangent. Assuming that the foliations
are the leaves of germs of differential 1-forms and , then the discriminant
ofpairthe, that is, the locus where the 1-form is a multiple
is the zero locus ofof . This is generically a germ of a space curve. In local coordinates, the discriminant isgiven by the fibre of a map-germ (the discriminant map-germ). In[2] the authors show that the discriminant
determines the local topological typeof the pairand obtain a complete list of discrete topological models.
More precisely, assume that the discriminantis transverse away from to thepair of foliations. When the number of branches of the discriminant in each half-regiondelimited by the leaf of (or ) through is at most 2, then it is showed in [2] that thelocal topological type is determined by their configuration. The authors also remark that
if the number of the branches of the discriminant in a half-region delimited by the leaf of(or ) through is more than 2, then there is no discrete topological model,Since the discriminant plays a key role in the topological classification of the pair
, it is of interest to analyse its singularities. This is carried out in [2, 3]. We give inthis paper a r\’esume on the various classifications that are given in those articles.
$\mathbb{R}^{3}$$\omega$$(\omega, \eta)$$D(\omega, \eta)$$\eta$$\omega\wedge\eta$$\omega$$\eta$$F:(\mathbb{R}^{3},0)arrow(\mathbb{R}^{2},0)$$D(\omega, \eta)$$(\omega, \eta)$$D(\omega, \eta)$$\omega$$\omega$$\eta$

$0$

$0$

$0$

$\eta$$(\omega, \eta)$

2Classifications

In the coordinates system $(x, y, z)$ , we can set $\omega=df$ and $\eta=dz$ , whereis a germ of a smooth function. Thenis the zero flbre of the map-germ:
given by $F_{\omega,\eta}(x, y, z)=(f_{x}(x, y, z), f_{y}(x, y, z))$ .
Let
be the local ring of germs of functions
andits maximal ideal(which is the subset of germs that vanish at the origin). Denote bythe -tuples$f:\mathbb{R}^{3},0arrow \mathbb{R},$ $0$$D(\omega, \eta)$$F_{\omega,\eta}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2}$$\mathcal{E}_{n}$$\mathbb{R}^{n},$$0arrow \mathbb{R}$$m_{n}$$\mathcal{E}(n,p)$$p$

60of elements in$\mathcal{E}_{n}$. Let$\mathcal{A}=\mathcal{R}\cross \mathcal{L}=Diff(\mathbb{R}^{n}, 0)\cross Diff(\mathbb{R}^{p}, 0)$

denotes the group ofby $(h, k).G=k\circ G\circ h^{-1}$ . We
right-left equivalence which acts smoothly on
have another group of interest, namely the contact group . The group is the set ofgerms of diffeomorphisms $H=(h, H_{1})\in Diff(\mathbb{R}^{n+p}, 0)$ , with $h\in Diff(\mathbb{R}^{n}, 0)$ . Then $H$
as follows: $G=H.F$ if and only if $H(x, F(x))=(h(x), G(h(x)))$ .
acts on$rn_{n}.\mathcal{E}(n,p)$$\mathcal{K}$$\mathcal{K}$$m_{n}.\mathcal{E}(n,p)$-singularities of the discriminant
is a natural one to use when one seeks to understand theThe action of the group2.1$\mathcal{K}$$\mathcal{K}$.
singularities of the zero fibres of germs in
When at least one foliation is regular, the description of the singularities of the disreduces tovia an action of the contact group oncriminant of the pairan action on families of matrices. The corank 2 simple germs are given in Table 1 ([1]).
See [2], Sect. 5.1 for details.
$m_{n}\mathcal{E}(n, p)$$\mathcal{K}$$(\omega, \eta)$

Table 1: The$\mathcal{K}$-simples singularities of corank 2 germs$\prime rr\iota_{3}.\mathcal{E}(3,2)$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$([1]).

$\overline{\frac{NameNorma1form\mathcal{K}_{e}-codimension}{P_{2,k}(\prime xy,\prime x^{k}+y^{2}+z^{2})),k\geq 2k+1}}$$P_{3,3} (\prime xy, \prime x^{3}+y^{3}+z^{2}) 5$$P_{3,4} (xy, x^{4}+y^{3}+z^{2}) 6$$P_{3,5} (xy, x^{5}+y^{3}+z^{2}) 7$$J_{7} (y^{2}+xz, x^{3}+yz) 5$$J_{8} (y^{2}+xz, x^{2}y+yz) 6$$J_{9} (y^{2}+xz, x^{4}+yz) 7$$K_{8} (y^{2}+xz, x^{3}+z^{2}) 6$$K_{9} (y^{2}+xz, x^{2}y+z^{2}) 7$$G_{10} (y^{2}+x^{3}, x^{3}+z^{2}) 7$$G_{11} (y^{2}+x^{3}, x^{2}y+z^{2}) 8$

2.2$\mathcal{K}_{V}$-singularities of the discriminant, where the changes of coordinates inof on
The action of the subgroupthe source preserve the variety given by $z=0$ , preserves the smooth type of the zero
fibre ofas well as the leaf $z=0$ of $dz$ . Since the key ingredient in the
topological classification of the pairs $(df, dz)$ is the number of branches of the discriminant
in each semi-space delimited by the plane $z=0$ , it is natural to seek a classification ofunder such action of$\mathcal{K}$$\mathcal{K}_{V}$$\prime rrb_{3}.\mathcal{E}(3,2)$$V$$F:\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$$F$$\mathcal{K}_{V}.$

Theorem 2.1 (Theorem 5.1, [2]) The
Table 2.

$\mathcal{K}_{V}$-simple map-germ$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$

are given in-simple singularities of the discriminant $D(df, dz)$ of pairs $(df, dz)$
It follows that theof preserves the number of branches,
are given in Table 2. The action of the subgroupin each semi-space $z>0$ and $z<0$ , of the zero fibre of any map-germ$\mathcal{K}_{V}$$\mathcal{K}_{V}$$\mathcal{K}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0.$61Table 2: Normal forms of$\mathcal{K}_{V}$-simple map-germs

Normal form$F$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0.$$d_{e}(F, \mathcal{K}_{V})$$(x, y) 0$$(x, z+y^{k}), k\geq 2 k-2$$(x, y^{2}+z^{k}), k\geq 2 k-1$$(x, y^{2}-z^{2k}), k\geq 1 2k-1$$(x, yz+y^{k}), k\geq 3 k-1$$(\prime x, z^{2}+y^{3}) 3$

2.3$\mathcal{K}^{*}$-singularities of the discriminantIt is of interest to classify the singularities of the discriminant up to an equivalence that
preserves the leaves of the foliation .or . See [2], Sect. 5.3. Let
be the subgroup(and indeed ofof) where the changes of coordinates in the source preserve thefoliation defined by $dz$ , i.e., the horizontal planes. Recall that the discriminant is thezero fibre of map germ:, so we are seeking a classification of thesingularities of map-germs
under the action of the subgroupwhich is aDamon geometric subgroup. The classification of the -simple map-germs
coincide with the-simple singularities.
$\mathcal{K}^{*}$$\omega$$\mathcal{K}$$\eta$$\mathcal{K}_{V}$$F_{\omega,\eta}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$$\mathcal{K}_{)}^{*}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$$\mathcal{K}^{*}$$\mathcal{K}_{V}$Theorem 2.2 (Theorem 5.4, [2])map-germ
is-simple if and only if-simple. So theit-simple germs are also those given in Table 2.
$A$$ls\mathcal{K}_{V}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$$\mathcal{K}^{*}$$\mathcal{K}^{*}$The action
does not only preserves the number of branches in each semi-space$z>0$ and $z<0$ of the zero fibre of any map-germ, but also the contact of
the zero fibre with the horizontal planes.
$\mathcal{K}^{*}$$\mathbb{R}^{3},0arrow \mathbb{R}^{2},0$2.4-singularities of the discriminant$\mathcal{B}$We also can study the singularities of the discriminant up to an action that preserves
the leaf of $\eta=dz$ through the origin considering the subgroup of which consists ofpairs of diffeomorphisms $(h, k)$ , where preserves the plane $z=0$ in the source and isany diffeomorphism in the target. The group acts on
and is a Damon
geometric subgroup. Theorem 2.3 models codimension less than or equal to four simple
singularities of the discriminant map-germ up to$\mathcal{B}$$\mathcal{A}$

$h$

$k$

$\mathcal{B}$$\prime rr\iota(x, y, z).\mathcal{E}(3,2)$$subg\backslash roup\mathcal{B}.$

Theorem 2.3 (Theorem 1.1, [3]) The -simple map-germs
at most 1 andcodimension
are given in Table 3.
$\mathcal{B}$$\mathcal{B}_{e}-$$\leq 4$$(\mathbb{R}^{3},0)arrow(\mathbb{R}^{2},0)$of comnk

62
Table 3: Norm$al$forms ofNormal form$\mathcal{B}_{e}-$codimension$\mathcal{B}_{e}$$\leq 4(\epsilon, \epsilon_{1}=\pm 1)$.

-codimension$(x,y)0$$(x, z+xy+y^{3}) 0$$(x, z+\epsilon y^{2}) 0$$(x, z+y^{3}+\epsilon^{k-1}x^{k}y), k\geq 2 k-1$$(x, z+xy+\epsilon y^{4}) 1$$(x, z+xy+y^{5}+\epsilon y^{7}) 2$$(x, z+xy+y^{5}) 3$$(x, z+xy+\epsilon y^{6}+y^{9}) 4$$(x, z+xy^{2}+\epsilon y^{4}+y^{2k+1}), k\geq 2 k$$(x, z+xy^{2}+y^{5}+\epsilon y^{6}) 3$$(x, z+xy^{2}+y^{5}+\epsilon y^{9}) 4$$(x, z+x^{2}y+\epsilon y^{4}+\epsilon_{1}y^{5}) 3$$(x, z+x^{2}y+\epsilon y^{4}) 4$$(x, y^{2}+\epsilon z^{2}+\epsilon_{1^{k}}x^{k-1}z), k\geq 2 k-2$$(x, y^{2}+xz+\epsilon z^{3}) 1$$(x, y^{2}+xz+\epsilon z^{4}+\epsilon_{1}z^{b}) , 2$$(x, y^{2}+xz+\epsilon z^{4}) 3$$(x, yz+xy+\epsilon y^{3}) 1$$(x, yz+xy+y^{4}+\epsilon y^{6}) 2$$(x, yz+xy+y^{4}) 3$$(x, xy+z^{2}+y^{3}+\epsilon y^{k}z), k\geq 2 k$

We observe that the classification under the action of the group also models theto planes and those ofsingularities of projections of hypersurfaces with boundary in. See [3] for details.
invariant map-germs$\mathcal{B}$$\mathbb{R}^{4}$$\mathbb{R}^{3}arrow \mathbb{R}^{2}$

References[1] A. A. du Plessis and C. T. C. Wall, The geometry of topological stability. London
Mathematical Society Monographs. New Series, 9. Oxford Science Publications. Oxford University Press, 1995.
[2] L. F. Martins, R. D. S. Oliveira, and F. Tari, “On pairs of regular foliations inand singularities of map-germs”, Geometriae Dedicata, 135 (2008), 103-118.

[3] L. F. Martins and A. C. Nabarro, “Projections of hypersurfaces into planes”, to appear in Glasgow Mathematical Joumal, 2013.

$\mathbb{R}^{4}$$\mathbb{R}^{3}$

with boundary