On non-Kupka points of codimension one foliations on ℙ<sup>3</sup>

We study the singular set of a codimension one holomorphic foliation on ℙ ³ . We find a local normal form for these foliations near a codimension two component of the singular set that is not of Kupka type. We also determine the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on ℙ ³.

holomorphic foliations; Kupka sets; non-Kupka points

Authorship

OMEGAR CALVO-ANDRADE

Centro de Investigaciones en Matemáticas/CIMAT, A.C., Jalisco, s/n, Col. Valenciana CP 36023 Guanajuato, Gto, MéxicoCentro de Investigación en MatemáticasMexicoGuanajuato, Guanajuato, MexicoCentro de Investigaciones en Matemáticas/CIMAT, A.C., Jalisco, s/n, Col. Valenciana CP 36023 Guanajuato, Gto, México

MAURÍCIO CÔRREA ^**Correspondence to: Maurício Côrrea E-mail: mauriciomatufmg@gmail.com

Universidade Federal de Minas Gerais, Departamento de Matemática, Av. Antônio Carlos, 6627, 31270-901 Belo Horizonte, MG, BrazilUniversidade Federal de Minas GeraisBrazilBelo Horizonte, MG, BrazilUniversidade Federal de Minas Gerais, Departamento de Matemática, Av. Antônio Carlos, 6627, 31270-901 Belo Horizonte, MG, Brazil

ARTURO FERNÁNDEZ-PÉREZ

*Correspondence to: Maurício Côrrea E-mail: mauriciomatufmg@gmail.com

Dedicated to José Seade in his 60 birthday.

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (69)

(4.1)

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(A.1)

\psi_{{\alpha\beta}}=\psi_{{\beta}{\alpha}}^{-1},\quad f_{{\beta}}|_{U_{\alpha%}\cap U_{\beta}}=\psi_{{\beta}{\alpha}}\circ f_{{\alpha}}|_{U_{\alpha}\cap U_{%\beta}}\quad\mbox{and}\quad\psi_{{\alpha}{\gamma}}=\psi_{{\alpha\beta}}\circ%\psi_{{\beta}{\gamma}}.

0\rightarrow F\rightarrow TM\xrightarrow{\{df_{{\alpha}}\}}N\rightarrow 0,% \quad 0\rightarrow{\mathcal{F}}\rightarrow\Theta\xrightarrow{\{df_{{\alpha}}\}% }\mathcal{N}\rightarrow 0,\quad[{\mathcal{F}},{\mathcal{F}}]\subset{\mathcal{F}}

\wedge^{n}TM^{\ast}=det(F^{\ast})\otimes N^{\ast},\qquad{\Omega}^{n}_{M}:=K_{M% }=det({\mathcal{F}}^{\ast})\otimes\mathcal{N}^{\ast},\quad n=dim(M).

0\rightarrow{\mathcal{F}}\rightarrow\Theta\rightarrow\mathcal{N}\rightarrow 0,% \quad[{\mathcal{F}},{\mathcal{F}}]\subset{\mathcal{F}}

\omega_{{\alpha}}\in{\Omega}^{1}(U_{{\alpha}}),\quad\omega_{{\alpha}}={\lambda%}_{{\alpha\beta}}\omega_{{\beta}}\mbox{ in }U_{{\alpha\beta}}={U_{\alpha}\cap U%_{\beta}},\quad L=[{\lambda}_{{\alpha\beta}}]\in\check{H}^{1}(\mathfrak{U},{%\mathcal{O}}^{\ast}).

K_{M}={\Omega}^{n}_{M}=det({\mathcal{F}}^{\ast})\otimes\mathcal{N}^{\ast}=K_{{%\mathcal{F}}}\otimes L^{-1},\quad det(N)\simeq L

\begin{array}[]{rcl}\mathcal{F}(M,L)&=&\{[\omega]\in{\mathbb{P}}H^{0}(M,{% \Omega}^{1}(L))\,|\,\textrm{codim}(S_{\omega})\geq 2,\quad\omega\wedge d\omega% =0\,\}\\ \mathcal{F}(n,d)&=&\{[\omega]\in{\mathbb{P}}H^{0}({\mathbb{P}}^{n},{\Omega}^{1% }(d+2))\,|\,\textrm{codim}(S_{\omega})\geq 2,\quad\omega\wedge d\omega=0\}.% \end{array}

S=\bigcup_{k=2}^{n}S_{k}\quad\text{where}\quad\textrm{codim}(S_{k})=k.

K(\omega)=\{p\in M\,|\>\omega(p)=0,\,d\omega(p)\neq 0\}.

N\!K(\omega)=\{p\in M|\omega(p)=0,\,d\omega(p)=0\}\supset S_{3}\cup\cdots\cup S% _{n}.

j^{1}_{0}(\omega)=x_{1}dx_{2}+x_{2}dx_{1}

\omega=x_{1}dx_{1}+g_{1}(x_{2})(1+x_{1}g_{2}(x_{2}))dx_{2},

\omega=xdx+(1+xf)df

\omega=xdx+2yz(1+xy^{2}z)dy+y^{2}(1+xy^{2}z)dz.

\varphi(x,y,z)=(x,t).

\deg(D_{\omega})=\deg(K_{{\mathcal{F}}})-\deg(K_{Z}).

S=\bigcup_{j=2}^{n}S_{j}\qquad\mbox{ where }\qquad\textrm{codim}(S_{j})=j.

\omega=gdf\quad\text{on a neighborhood of}\,\,\,0\in{\mathbb{C}}^{n}.

0=\omega(\mathbf{X})=g\cdot\left(\sum_{i=1}^{n}(\partial f/\partial z_{i})dz_{%i}(\partial/\partial z_{n})\right)=g\cdot(\partial f/\partial z_{n}),\mbox{ %therefore }\,\partial f/\partial z_{n}\equiv 0,

(4.1)

\{d\omega_{{\alpha}}\}\in H^{0}(S,({\Omega}^{2}_{M}\otimes L)|_{S}).

K(\omega)=\{p\in M\,|\,\omega(p)=0,\quad d\omega(p)\neq 0\,\}.

\wedge^{2}N_{K(\omega)}=L|_{K(\omega)},\quad K_{K(\omega)}=(K_{M}\otimes L)|_{% K(\omega)}=K_{{\mathcal{F}}}|_{K(\omega)}.

0\longrightarrow{\mathcal{O}}\xrightarrow{\,\,{\sigma}\,\,}V\longrightarrow{%\mathcal{J}}_{K}(d+2)\rightarrow 0\quad\mbox{with}\quad\{{\sigma}=0\}=K

c(V)=1+(d+2)\cdot\mathbf{h}+\deg(K(\omega))\mathbf{h}^{2}\in H^{\ast}({\mathbb%{P}}^{n},{\mathbb{Z}})\simeq{\mathbb{Z}}[\mathbf{h}]/\mathbf{h}^{n+1}.

c(V)=1+(d+2)\cdot\mathbf{h}+\frac{(d+2)^{2}}{4}\cdot\mathbf{h}^{2}=\left(1+%\frac{(d+2)\cdot\mathbf{h}}{2}\right)^{2}\in H^{\ast}({\mathbb{P}}^{n},{%\mathbb{Z}})\simeq{\mathbb{Z}}[\mathbf{h}]/\mathbf{h}^{n+1}.

\xi^{\ast}E\simeq{\mathcal{O}}_{{\mathbb{P}}^{2}}\oplus{\mathcal{O}}_{{\mathbb%{P}}^{2}}

E\simeq{\mathcal{O}}_{{\mathbb{P}}^{n}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{n}}

N\!K(\omega)=S_{3}\cup\cdots\cup S_{n}.

\ell(S_{n})=\sum_{p\in S_{n}}\mu(\omega_{p},p),\quad\mu(\omega,p)=dim_{{%\mathbb{C}}}\frac{{\mathcal{O}}_{p}}{(\omega_{1},\dots,\omega_{n})},\quad%\omega_{p}=\sum_{i=1}^{n}\omega_{i}dz_{i}.

\omega=\omega_{1}+\cdots,\qquad d\omega=d\omega_{1}+\cdots=0,

\omega_{1}=x_{1}dx_{2}+x_{2}dx_{1}.

(5.1)

\omega=xdx+[g(f(\zeta))+xh(f(\zeta))]df(\zeta).

{S_{\omega}}_{,p}=\{x=\zeta_{1}=0\}=\{x=g(f(\zeta))=0\}\cup\left\{x=\frac{% \partial{f}}{{\partial{\zeta_{1}}}}=\cdots=\frac{\partial{f}}{{\partial{\zeta_% {n-1}}}}=0\right\}.

G(x,\zeta)=(x,\zeta_{1}\psi^{1/k}(\zeta),\zeta_{2},\ldots,\zeta_{n})=(x,y,% \zeta_{2},\ldots,\zeta_{n})

G_{*}(\omega)=xdx+(g(y^{k})+xh(y^{k}))ky^{k-1}dy=xdx+(g_{1}(y)+xh_{1}(y))dy,

(5.2)

\tilde{\omega}=xdx+(g_{1}(y)+xh_{1}(y))dy\quad\text{with}\quad S_{\tilde{% \omega}}=\{x=g_{1}(y)=0\}.

\tilde{\omega}=xdx+g_{1}(y)(1+x(g_{1}(y))^{m-1}H(y))dy=xdx+g_{1}(y)(1+xg_{2}(y%))dy,

(5.3)

\omega={\varphi}^{\ast}(xdx+g_{1}(y)(1+xg_{2}(y))dy).

(5.4)

\omega=xdx+(g(\zeta^{k}_{1}\psi)+xh(\zeta^{k}_{1}\psi))d(\zeta^{k}_{1}\psi),

(5.5)

\omega=xdx+(g(\zeta^{k}_{1}\psi)+xh(\zeta^{k}_{1}\psi))\zeta^{k-1}_{1}(k\psi d% \zeta_{1}+\zeta_{1}d\psi).

{\varphi}(x,\zeta)=(x,\zeta^{k}_{1}\psi(\zeta))=(x,t),

\omega={\varphi}^{\ast}(\eta),

0\to\mathcal{G}\to\Theta_{M}\xrightarrow{\omega}\mathcal{N}_{\mathcal{G}}\to 0%,\quad\mathcal{N}_{\mathcal{G}}\simeq\mathcal{J}_{S}\otimes L.

d\omega_{\alpha}=\theta_{\alpha}\wedge\omega_{\alpha}

\textrm{BB}(\mathcal{G},Z)=\frac{1}{(2\pi i)^{2}}\int_{L_{p}}\theta\wedge d\theta.

c_{1}(L)^{2}=c^{2}_{1}(\mathcal{N}_{\mathcal{G}})=\sum_{Z\subset S_{2}}\textrm% {BB}(\mathcal{G},Z)[Z],

\sum_{Z\subset S_{2}}\textrm{BB}(\mathcal{G},Z)\deg[Z]=(d+2)^{2}.

\omega|_{U}=P(x,y)dy-Q(x,y)dx

\theta=\frac{(\frac{\partial{P}}{\partial x}+\frac{\partial{Q}}{\partial y})}{% |P|^{2}+|Q|^{2}}(\bar{P}dx+\bar{Q}dy).

(5.6)

\textrm{BB}(\mathcal{G},Z)=\frac{1}{(2\pi i)^{2}}\int_{L_{p}}\theta\wedge d% \theta=\mbox{Res}_{0}\displaystyle\left\{\frac{{\textrm{Tr}}\big(D\mathbf{X}% \big)\,dx\wedge dy}{PQ}\right\},

\textrm{BB}(\mathcal{G},Z)=\frac{{\textrm{Tr}}(D\mathbf{X}(p))^{2}}{\det(D% \mathbf{X}(p))}.

\mathcal{G}(U)=\left\langle\mathbf{X}=P(x,y)\frac{\partial}{\partial x}+Q(x,y)% \frac{\partial}{\partial y},\frac{\partial}{\partial z_{3}},\dots,\frac{% \partial}{\partial z_{n}}\right\rangle

\omega=xdx+g_{1}(y)(1+xg_{2}(y))dy,

\textrm{BB}(\mathcal{F}_{\omega},Z)={\textrm{Res}}_{t=0}\left[\frac{(g_{1}(t)g% _{2}(t))^{2}dt}{g_{1}(t)}\right]={\textrm{Res}}_{t=0}\left[g_{1}(t)(g_{2}(t))^% {2}\right].

0<(d+2)^{2}=\sum_{Z\subset S_{2}}\textrm{BB}(\mathcal{F}_{\omega},Z)\deg[Z]=0

rot(\omega)=\mathbf{X}\Longleftrightarrow d\omega=\imath_{\mathbf{X}}dx\wedgedy%\wedge dz,

\mathbb{K}(\mathbf{X})_{0}:0\to{\Omega}^{3}_{{\mathbb{C}}^{3},0}\xrightarrow{% \imath_{\mathbf{X}}}\Omega^{2}_{{\mathbb{C}}^{3},0}\xrightarrow{\imath_{% \mathbf{X}}}\Omega^{1}_{{\mathbb{C}}^{3},0}\xrightarrow{\imath_{\mathbf{X}}}{% \mathcal{O}}_{{\mathbb{C}}^{3},0}\to 0

\omega=\imath_{\mathbf{X}}\theta,\quad\mbox{and}\quad\theta=\imath_{\mathbf{S}% }dx\wedge dy\wedge dz,\quad\mbox{implies}\quad\omega=\imath_{\mathbf{X}}\theta% =\imath_{\mathbf{X}}\imath_{\mathbf{S}}dx\wedge dy\wedge dz

0\rightarrow\mathcal{J}/\mathcal{J}^{2}\rightarrow{\Omega}^{1}\otimes{\mathcal% {O}}_{Z}\rightarrow{\Omega}^{1}_{Z}\rightarrow 0

0\rightarrow\wedge^{2}\mathcal{J}/\mathcal{J}^{2}\otimes L\rightarrow{\Omega}^% {2}_{{\mathbb{P}}^{3}}|_{Z}\otimes L\rightarrow\cdots

d\omega|_{Z}\in H^{0}(Z,\wedge^{2}(\mathcal{J}/\mathcal{J}^{2})\otimes L)

K_{Z}^{-1}\otimes K_{{\mathbb{P}}^{3}}\simeq\wedge^{2}(\mathcal{J}/\mathcal{J}% ^{2}),\quad\mbox{and}\quad L\simeq K_{{\mathbb{P}}^{3}}^{-1}\otimes K_{{% \mathcal{F}}}

H^{0}(Z,\wedge^{2}(\mathcal{J}/\mathcal{J}^{2})\otimes L)=H^{0}(Z,K^{-1}_{Z}% \otimes K_{{\mathcal{F}}}|_{Z}),

D_{\omega}=\sum_{p\in Z}ord_{p}(d\omega)

\deg(D_{\omega})=\deg(K_{{\mathcal{F}}})-\deg(K_{Z}),

|N\!K(\omega)\cap S_{2}|=deg(D_{\omega})=\deg(K_{{\mathcal{F}}})-\deg(K_{S_{2}%})=-\chi(S_{2})=2.

\chi(F)=h^{0}(F)-h^{1}(F)+h^{2}(F)=[ch(F)\cdot Td({\mathbb{P}}^{2})]_{2}=2,

h^{0}(F)+h^{2}(F)=[ch(F)\cdot Td({\mathbb{P}}^{2})]_{2}+h^{1}(F)\geq[ch(F)%\cdot Td({\mathbb{P}}^{2})]_{2}=2

(A.1)

0\longrightarrow{\mathcal{O}}\stackrel{\cdot\tau}{\longrightarrow}F%\longrightarrow\mathcal{Q}\longrightarrow 0\quad\mbox{with}\quad\mathcal{Q}=F/%\mathcal{O}.

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