Invariant hypersurfaces and nodal components for codimension one singular foliations

Abstract

It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after reduction of singularities of a germ of singular foliation in $({\mathbb {C}}^2,0)$. Here, we state and prove a generalization of this property to any ambient dimension.

Appendices

Appendix I: Simply connectedness

Let us give in this Appendix an outline of a proof of Proposition 8. We consider a combinatorial strata structure ${{\mathcal {H}}}\subset {{\mathcal {P}}}(I)$ and we have to prove

• a) The topological space $\varOmega _{{\mathcal {H}}}$ is connected if and only if ${{\mathcal {H}}}$ is 1-connected.

• b) The combinatorial strata structure ${\mathcal {H}}$ is simply connected if and only if $\varOmega _{{{\mathcal {H}}}_3}$ is a simply connected topological space.

Let us prove a). The simplicial complex $\varOmega _{{\mathcal {H}}}$ is connected if and only if any two given vertices $\xi _i$ and $\xi _j$ may be connected by a topological path; indeed any point is $\varOmega _{{\mathcal {H}}}$ is connected with a vertex by a topological path. This already shows that if ${\mathcal {H}}$ is 1-connected, then $\varOmega _{{\mathcal {H}}}$ is a connected topological space. Conversely, assume that ${\mathcal {H}}$ is not 1-connected, and consider the decomposition in connected components ${{\mathcal {H}}}=\cup _{\lambda \in \varLambda } {{\mathcal {H}}}_\lambda$ as in Remark 8. We have that

$$\begin{aligned} \varOmega _{{\mathcal {H}}}=\cup _{\lambda \in \varLambda }\varOmega _{{{\mathcal {H}}}_\lambda };\quad \varOmega _{{{\mathcal {H}}}_\lambda }\cap \varOmega _{{{\mathcal {H}}}_{\lambda '}}=\emptyset , \text { if }\lambda \ne \lambda '. \end{aligned}$$

Hence $\varOmega _{{\mathcal {H}}}$ is not connected, since it is a disjoint union of finitely many (at least two) simplicial complexes.

Let us prove b). In view of Remark , we assume that ${{\mathcal {H}}}={{\mathcal {H}}}_3$ and in view of part a) we also assume that ${\mathcal {H}}$ and hence $\varOmega _{{\mathcal {H}}}$ are connected. Let us consider the combinatorial fundamental group $\pi _1({{\mathcal {H}}},\{i_0\})$, whose elements are the homotopy classes of loops, that is constructed “mutatis mutandis” as the classical Poincaré group. Now it is enough to prove that

$$\begin{aligned} \pi _1({{\mathcal {H}}},\{i_0\})=\pi _1(\varOmega _{{\mathcal {H}}}, \xi _{i_0}). \end{aligned}$$

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We prove the equality in Eq. (19) by induction on the lexicographical counter $(\sharp I,\sharp {{\mathcal {H}}})$. The starting case (1, 1) corresponds to a single point and in this case both groups are trivial. In order to do the induction step, let us separate the cases ${{\mathcal {H}}}(3)=\emptyset$ and ${{\mathcal {H}}}(3)\ne \emptyset$.

Assume that ${{\mathcal {H}}}(3)=\emptyset$ and $\sharp I\ge 2$. Note that in this case $\varOmega _{{\mathcal {H}}}$ is a connected union of linear segments. Given $i\in I$, denote ${\text {Star}}({\mathcal {H}},i)$ the set

$$\begin{aligned} {\text {Star}}({\mathcal {H}},i)=\{j\in I{\setminus }\{i\}; \{i,j\}\in {\mathcal {H}}\}. \end{aligned}$$

In there is $i\in I$ such that $\sharp {\text {Star}}({\mathcal {H}},i)=1$, we can consider

$$\begin{aligned} {{\mathcal {H}}}'={{\mathcal {H}}}\cap {{\mathcal {P}}}(I{\setminus }\{i\}). \end{aligned}$$

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Taking $i_0\ne i$, we see directly that

$$\begin{aligned} \pi _1({{\mathcal {H}}}',\{i_0\})=\pi _1({{\mathcal {H}}},\{i_0\}),\quad \pi _1(\varOmega _{{{\mathcal {H}}}'},\xi _{i_0})=\pi _1(\varOmega _{{{\mathcal {H}}}},\xi _{i_0}); \end{aligned}$$

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we are done by induction. Assume that $\sharp {\text {Star}}({\mathcal {H}},i)\ge 2$ for any $i\in I$. Let us choose $i_1,i_0\in I$, with $i_1\ne i_0$. Consider ${{\mathcal {H}}}'$ defined as follows:

$$\begin{aligned} {{\mathcal {H}}}'=({{\mathcal {H}}}\cap {{\mathcal {P}}}(I{\setminus }\{i_1\}))\cup {{\mathcal {P}}}({\text {Star}}({\mathcal {H}},i_1))_2. \end{aligned}$$

That is, we eliminate $i_1$ and we add all the two by two connections between the elements of ${\text {Star}}({\mathcal {H}},i_1)$. We see in a direct way that $\pi _1({{\mathcal {H}}}',\{i_0\})=\pi _1({{\mathcal {H}}},\xi _{i_0})$ and, by means of a deformation retract, that $\pi _1(\varOmega _{{{\mathcal {H}}}'},\{i_0\})=\pi _1(\varOmega _{{\mathcal {H}}},\xi _{i_0})$; as before, we are done by induction.

Assume that ${{\mathcal {H}}}(3)\ne \emptyset$. There is a stratum $J=\{i_1,i_2,i_3\}\in {{\mathcal {H}}}(3)$. In this case, we have that ${{\mathcal {H}}}= {{\mathcal {H}}}'\cup {{\mathcal {P}}}(J)$, where ${{\mathcal {H}}}'={{\mathcal {H}}}{\setminus }\{J\}$. Hence we have that

$$\begin{aligned} \varOmega _{{\mathcal {H}}}=\varOmega _{{{\mathcal {H}}}'}\cup \varDelta _J, \end{aligned}$$

where $\varOmega _{{{\mathcal {H}}}'}=\varDelta _{J}$ is connected and, more precisely, it is the frontier $\partial \varDelta _J$ of $\varDelta _J$. By Classical Seifert–Van Kampen theorem, we know that $\pi _1(\varOmega _{{\mathcal {H}}},\xi _{\{i_1\}})$ is isomorphic to $\pi _1(\varOmega _{{\mathcal {H}}}',\xi _{\{i_1\}})$ quotient by the normal subgroup generated by a single loop $\sigma$ supported by $\partial \varDelta _J$. We use Seifert–Van Kampen theorem [22], applied to the decomposition ${{\mathcal {H}}}= {{\mathcal {H}}}'\cup {{\mathcal {P}}}(J)$, where ${{\mathcal {H}}}'={{\mathcal {H}}}{\setminus }\{J\}$, to show that $\pi _1({{\mathcal {H}}},\{i_1\})$ is the quotient of $\pi _1({{\mathcal {H}}}',\{i_1\})$ by the normal subgroup generated by the single loop $\sigma =(\{i_1\},\{i_2\},\{i_3\},\{i_1\})$, In this way, we end by induction.

Appendix II: Strong desingularization in dimension three

Let us give here an outline for a proof of the statement in Remark 2 in the case of a three-dimensional ambient space. This approach is based on the classical methods by Hironaka, Abhyankar and others for reduction of singularities in small dimensions, presented in [13] (See also [1, 17]).

We start with a pair $({{\mathcal {M}}},{{\mathcal {L}}})$, where ${{\mathcal {M}}}=(M,E;K)$ is a three-dimensional ambient space and ${{\mathcal {L}}}$ is a finite list of irreducible hypersurfaces not contained in E. Denote by H the union of the hypersurfaces in ${{\mathcal {L}}}$ and take a point p in K. We consider the following local invariants:

• 1. The multiplicity $\nu _p(H)$ of H at the point p.

• 2. The dimension $d_p(H)$ of Hironaka’s strict tangent space $T_pH$.

• 3. The number $e_p(E)$ of irreducible components of E through p.

• 4. The “encombrement” $t_p(H, E)$ of $T_pH$ with respect to E.

The reader is supposed to be familiar with the multiplicity $\nu _p(H)$. Let us note that $\nu _p(H)>0$ if and only if $p\in H$. The strict tangent space $T_pH$ of H in a point $p\in H$ is the ${\mathbb {C}}$-vector subspace of $T_pM$ whose elements are the vectors leaving the tangent cone $C_pH$ invariant by translation. In other words, if $\xi _1,\xi _2,\ldots ,\xi _s$ is a basis for the orthogonal $T_pH^\vee \subset T^*_pM$, then $C_pH$ has an equation of the form

$$\begin{aligned} \phi (\xi _1,\xi _2,\ldots ,\xi _s)=0, \end{aligned}$$

where $\phi$ is an homogeneous polynomial of degree $\nu _p(H)$.

Let us give the definition of $t_p(T_pH, E)$. Choose local coordinates

$$\begin{aligned} x=(x_1,x_2,\ldots ,x_n) \end{aligned}$$

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such that $E=(x_1x_2\cdots x_e=0)$, locally at p. Recall that $n=3$ in our case, although several concepts are valid for any dimension n. Note that each $x_i$ gives a cotangent vector ${\bar{x}}_i\in T^*_pM$. Given a subset $J\subset \{1,2,\ldots ,e\}$, we define $t_J$ to be the dimension of the ${\mathbb {C}}$-vector subspace $T_J(H,E;p)$ of $T^*_pM$ given by

$$\begin{aligned} T_J(H,E;p)=T^\vee _pH\cap <{\bar{x}}_i;i\in J> . \end{aligned}$$

For each maximal sequence $\sigma : \{1,2,\ldots ,e\}=J_1\supsetneq J_2\supsetneq \cdots \supsetneq J_e\supsetneq \emptyset ,$ we put $t_\sigma =(t_{J_1}, t_{J_2},\ldots ,t_{J_e})\in {{\mathbb {Z}}}_{\ge 0}^e$. Put $\theta _p(H,E)=t_{J_1}$. Note that

$$\begin{aligned} n-d_p(H)\ge {\theta _p(H,E)}\ge t_{J_2}\ge \cdots \ge t_{J_e}\ge 0. \end{aligned}$$

In particular ${\theta _p(H,E)}=0$ if and only if $t_p(\sigma )=(0,0,\ldots ,0)$ for any $\sigma$ (in an equivalent way: for one $\sigma$). Finally, we define $t_p(H,E)$ to be the maximum for the lexicographical ordering of the sequences $t_\sigma$, when $\sigma$ varies.

Let us introduce the number $\zeta _p(H,E)$, obtained from $d_p(H)$, $t_p(H,E)$ and $e_p(E)$. If If $d_p(H)\in \{0,1\}$, we put $\zeta _p(H,E)=0$. If $d_p(H)=2$, then $\zeta _p(H,E)$ takes one of the four possible values 0, 1, 2, 3 defined as follows:

• $\zeta _p(H,E)=0$ if and only if $\theta _p(H,E)=0$.

• $\zeta _p(H,E)=1$ if and only if $t_p(H,E)=(1,0,0)$.

• $\zeta _p(H,E)=2$ if and only if $t_p(H,E)=(1,0)$ or $t_p(H,E)=(1,1,0)$.

• $\zeta _p(H,E)=3$ if and only if $t_p(H,E)=(1,1)$ or $t_p(H,E)=(1,1,1)$.

The main invariant of control $I_p(H,E)$ is the lexicographical invariant

$$\begin{aligned} I_p(H,E)=(\nu _p(H),d_p(H), \zeta _p(H,E)). \end{aligned}$$

By convention, we put $I_p(H,E)=(0,0,0)$ when $\nu _p(H)=0$. The following results of stability are implicitly contained in the above cited works (the reader can prove them by taking a Weierstrass–Tchirnhausen preparation of a local equation of H):

Lemma 15

(Horizontal stability) The invariant $I_p(H,E)$ is analytically upper semicontinuous.

Lemma 16

(Vertical stability) Let $\pi : ({{\mathcal {M}}}',{{\mathcal {L}}}')\rightarrow ({{\mathcal {M}}},{{\mathcal {L}}})$ be an admissible blowing-up with an equimultiple center Y and consider a point $p\in Y$. Let us recall that $\pi ^{-1}(p)={\mathbb {P}}\mathrm{roj }(T_pM/T_pY)$. Then $T_pY\subset T_pH$. Moreover, for any $p'\in \pi ^{-1}(p)$, we have that $I_{p'}(H',E')\le I_p(H,E)$ for the lexicographical ordering and if $\nu _{p'}(H')=\nu _p(H)$, then $p'\in {\mathbb {P}}\mathrm{roj }(T_pH/T_pY)$.

When $({{\mathcal {M}}},{{\mathcal {L}}})$ is locally simple, we can make it simple just by blowing-up the points where the strata are not connected. Thus our objective is to get a locally simple pair after a suitable admissible transformation.

Let us denote by ${\text {Imax}}(H,E)$ the maximum of the invariants $I_p(H,E)$ for $p\in K$. We have that $({{\mathcal {M}}},{{\mathcal {L}}})$ is locally simple if and only if

$$\begin{aligned} {\text {Imax}}(H,E)\le (1,2,0). \end{aligned}$$

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By an elementary induction, our objective is reached if we show how to get a new $({{\mathcal {M}}}',{{\mathcal {L}}}')$ such that ${\text {Imax}}(H',E')< {\text {Imax}}(H,E)$, when we start with ${\text {Imax}}(H,E)> (1,2,0)$. Assume thus that ${\text {Imax}}(H,E)=(r, d, \zeta )>(1,2,0)$ and consider the analytic subset

$$\begin{aligned} {\text {Sam}}_{r,d,\zeta }(H,E)=\{p\in M;\; I_p(H,E)=(r,d,\zeta )\}. \end{aligned}$$

We have to obtain that ${\text {Sam}}_{r,d,\zeta }(H,E)=\emptyset$ by means of an admissible transformation with equimultiple centers.

$\bullet$ Let us describe first how to proceed in the cases with $d\le 1$. Note that in this cases we have $r\ge 2$ and thus the set $Eq_r(H)$ of r-multiplicity defined by

$$\begin{aligned} Eq_r(H)=\{p\in M; \nu _p(H)\ge r\}= \{p\in M; \nu _p(H)=r\} \end{aligned}$$

is a finite union of points and curves, that contains ${\text {Sam}}_{r,d,\zeta }(H,E)$.

Assume we are in the case $d=0$. In this case $(r,d,\zeta )=(r,0,0)$. The sets and $Eq_r(H,E)$ and ${\text {Sam}}_{r,0,0}(H,E)$ coincide and they consist in a finite union of points. We blow-up all these points (at the same time or one after the other) and we are done by Lemma 16.

Assume we are in the case $d=1$. In this case $(r,d,\zeta )=(r,1,0)$. Any curve $\varGamma$ in $Eq_r(H)$ is also contained in ${\text {Sam}}_{r,1,0}(H,E)$. In fact, a non isolated point in $Eq_r(H)$ cannot have 0-dimensional strict tangent space. We can proceed by induction on the number $\alpha$ of irreducible curves contained in ${\text {Sam}}_{r,1,0}(H,E)$ which is the same one as the number of r-equimultiple curves.

If $\alpha =0$, then ${\text {Sam}}_{r,1,0}(H,E)$ is a finite set of points. We blow-up one of such points. By Lemma 16, at most a new point of multiplicity r may appear, if it does not appear, the number of points in ${\text {Sam}}_{r,1,0}(H,E)$ decreases by a unit; if it appears in a persistent way, we detect an equimultiple curve,what is impossible.

Assume that $\alpha >0$. By blowing-up points, we obtain that ${\text {Sam}}_{r,1,0}(H,E)$ has strong normal crossings with E, since the curves in ${\text {Sam}}_{r,1,0}(H',E')$ are the strict transforms of the curves in ${\text {Sam}}_{r,1,0}(H,E)$. We can assume this property an choose an r-equimultiple curve $\varGamma$ as center. By Lemma 16, no r-multiple point appears over a point in $\varGamma$. Then $\alpha '=\alpha -1$ and we are done.

$\bullet$ Let us consider now the cases with $d=2$. There are four possible situations following the value of $\zeta \in \{0,1,2,3\}$.

Case $\zeta =0$: Note that in this case we have $r\ge 2$. We start by getting strong normal crossings between ${\text {Eq}}_r(H)$ and E by means of a finite number of blow-ups centered at points in ${\text {Eq}}_r(H)$. This property is stable under new punctual blowing-ups.

In this situation, if we blow-up a curve $\varGamma \subset {\text {Eq}}_r(H)$, we do not destroy the property that ${\text {Eq}}_r(H)$ and E have strong normal crossings. To see this, we can use the existence of local maximal contact provided by a Weierstrass–Tchirnhausen presentation of an equation of H. More precisely, given a point $p\in {\text {Eq}}_r(H)$, there are local coordinates $(x_1,x_2,z)$ such that $E\subset (x_1x_2=0)$, an equation of H has the form

$$\begin{aligned} z^r+f_2(x_1,x_2)z^{r-2}+f_3(x_1,x_2)z^{r-3}+\cdots +f_r(x_1,x_2)=0 \end{aligned}$$

and ${\text {Eq}}_r(H)\subset (z=x_1=0)\cup (z=x_2=0)$. If we blow-up $z=x_1=0$ and it is contained in ${\text {Eq}}_r(H)\subset (z=x_1=0)\cup (z=x_2=0)$, then the new ${\text {Eq}}_r(H')$ is contained in the intersection with the exceptional divisor of the strict transform of the maximal contact surface $z=0$.

The global strategy is as follows: if there is a curve $\varGamma$ contained in ${\text {Eq}}_r(H)$ (intersecting ${\text {Sam}}_{r,2,0}(H,E)$, but this is not essential); then blow-up one of such $\varGamma$. Otherwise, blow-up a point in ${\text {Sam}}_{r,2,0}(H,E)$. To show that this procedure ends in a finite number of steps, the reader may follow the ideas in Hironaka’s Bowdoin College Seminar [13]. Roughly speaking, we reduce the global control to a local one along “bamboes” and after this, we use the properties of the evolution of the characteristic polygon to show the finiteness.

Case $\zeta =1$: There are only finitely many points p in ${\text {Sam}}_{r,2,1}(H,E)$, since $e_p(E)=3$ for each of such points p. Consider local coordinates $(x_1,x_2,x_3)$ at p such that $E=(x_1x_2x_3=0)$. The initial part of a local equation of H has the form

$$\begin{aligned} ( x_1+\alpha x_2+\beta x_3)^r;\quad \alpha \beta \ne 0. \end{aligned}$$

Then, after the blowing-up of p, each point $p'$ in the exceptional divisor with $r=\nu _{p'}(H')$ and $2=d_{p'}(H)$ has $\zeta _{p'}(H',E')=0$. Thus, we end by blowing-up one by one the points in ${\text {Sam}}_{r,2,1}(H,E)$.

Case $\zeta =2$: Recall that any point $p\in {\text {Sam}}_{r,2,2}(H,E)$ has $e_p(E)\ge 2$. Thus ${\text {Sam}}_{r,2,2}(H,E)$ is contained in the union of the curves $E_{ij}=E_i\cap E_j$, with $i\ne j$. We follow the following strategy: if there is an $E_{ij}\subset {\text {Eq}}_r(H)$ with $E_{ij}\cap {\text {Sam}}_{r,2,2}(H,E)\ne \emptyset$, then blow-up one of such $E_{ij}$. Otherwise, blow-up a point $p\in {\text {Sam}}_{r,2,2}(H,E)$. Note that the centers have strong normal crossings with E, since they are points or curves of the type $E_{ij}$.

Let us see what happens when we blow-up a curve $E_{ij}\subset {\text {Eq}}_r(H)$ containing a point $p\in {\text {Sam}}_{r,2,2}(H,E)$. Take local coordinates $(x_1,x_2,y)$ such that

$$\begin{aligned} (x_1x_2=0)\subset E\subset (x_1x_2y=0), \quad E_{ij}=(x_1=x_2=0). \end{aligned}$$

A local equation h of H has the form $h=(\alpha x_1+\beta x_2)^r+{\tilde{h}}$, $\alpha \beta \ne 0$, where ${\tilde{h}}$ has generic order $\ge r$ along $x_1=x_2$. After the blow-up of $E_{ij}$, there are no points $p'$ over p with $I_{p'}(H',E')=(r,2,2)$. Then all that curves disappear after finitely many steps.

We are then in a situation without r-equimultiple curves $E_{ij}$ that intersect ${\text {Sam}}_{r,2,2}(H,E)$. We blow then a point $p\in {\text {Sam}}_{r,2,2}(H,E)$. In local coordinates, the initial part of an equation of H has the form $(\alpha x_1+\beta x_2)^r$, with $\alpha \beta \ne 0$ and $(x_1x_2=0)\subset E$. The only possible point $p'\in {\text {Sam}}_{r,2,2}(H',E')$ over p corresponds to the intersection with the exceptional divisor of the strict transform of $x_1=x_2=0$ and moreover, no new r-equimultiple curves of the type $E'_{i'j'}$ will appear. We found that this points disappear after finitely many steps, since otherwise we find that $x_1=x_2=0$ should be r-equimultiple.

Case $\zeta =3$: Each point $p\in {\text {Sam}}_{r,2,3}(H,E)$ selects an irreducible component E(p) of the divisor E, given by the following property: there are local coordinates (x, y, z) such that the initial part of an equation of H has the form $x^r$ and $E(p)=(x=0)$. These E(p) act as maximal contact surfaces. More precisely, consider an admissible blow-up $\pi :({{\mathcal {M}}}',{{\mathcal {L}}}')\rightarrow ({{\mathcal {M}}},{{\mathcal {L}}})$ centered in Y, with $p\in Y$. Then $Y\subset E(p)$ and any

$$\begin{aligned} p'\in \pi ^{-1}(p)\cap {\text {Sam}}_{r,2,3}(H',E') \end{aligned}$$

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satisfies that $E'(p')$ is the strict transform of E(p). In particular, the number of possible E(p) is not increased. Thus, by finite induction it is enough to eliminate one of them. We select an irreducible component D of E and we consider the set

$$\begin{aligned} {\text {Sam}}^D_{r,2,3}(H,E)=\{p\in {\text {Sam}}_{r,2,3}(H,E); E(p)=D \}. \end{aligned}$$

We want to make disappear this set after finitely many blow-ups. The first step is to obtain that E and $D\cap {\text {Eq}}_r(H)$ do have strong normal crossings by blowing-up points. This property comes by classical two dimensional arguments, taking D as a new ambient space. The property is stable under blow-up centered in points or in r-equimultiple curves contained in D. We take now the strategy of blowing-up first the curves $\varGamma \subset D\cap {\text {Eq}}_r(H)$ that intersect ${\text {Sam}}^D_{r,2,3}(H,E)$ and when there is no one, we chose as center a point in ${\text {Sam}}^D_{r,2,3}(H,E)$. The classical control by the characteristic polygon assures that a procedure following this strategy stops in finitely many steps.