# Foliations by curves with curves as singularities

• [1] ICEx - UFMG Departamento de Matemática Av. Antônio Carlos 6627 30123-970 Belo Horizonte MG (Brazil)
• Volume: 64, Issue: 4, page 1781-1805
• ISSN: 0373-0956

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## Abstract

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Let $ℱ$ be a holomorphic one-dimensional foliation on ${ℙ}^{n}$ such that the components of its singular locus $\Sigma$ are curves ${C}_{i}$ and points ${p}_{j}$. We determine the number of ${p}_{j}$, counted with multiplicities, in terms of invariants of $ℱ$ and ${C}_{i}$, assuming that $ℱ$ is special along the ${C}_{i}$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.

## How to cite

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Corrêa Jr, M., et al. "Foliations by curves with curves as singularities." Annales de l’institut Fourier 64.4 (2014): 1781-1805. <http://eudml.org/doc/275642>.

@article{CorrêaJr2014,
abstract = {Let $\mathcal\{F\}$ be a holomorphic one-dimensional foliation on $\{\mathbb\{P\}^n\}$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal\{F\}$ and $C_i$, assuming that $\mathcal\{F\}$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.},
affiliation = {ICEx - UFMG Departamento de Matemática Av. Antônio Carlos 6627 30123-970 Belo Horizonte MG (Brazil); ICEx - UFMG Departamento de Matemática Av. Antônio Carlos 6627 30123-970 Belo Horizonte MG (Brazil); ICEx - UFMG Departamento de Matemática Av. Antônio Carlos 6627 30123-970 Belo Horizonte MG (Brazil); ICEx - UFMG Departamento de Matemática Av. Antônio Carlos 6627 30123-970 Belo Horizonte MG (Brazil)},
author = {Corrêa Jr, M., Fernández-Pérez, A., Nonato Costa, G., Vidal Martins, R.},
journal = {Annales de l’institut Fourier},
keywords = {holomorphic foliations; non-isolated singularities},
language = {eng},
number = {4},
pages = {1781-1805},
publisher = {Association des Annales de l’institut Fourier},
title = {Foliations by curves with curves as singularities},
url = {http://eudml.org/doc/275642},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Corrêa Jr, M.
AU - Fernández-Pérez, A.
AU - Nonato Costa, G.
AU - Vidal Martins, R.
TI - Foliations by curves with curves as singularities
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1781
EP - 1805
AB - Let $\mathcal{F}$ be a holomorphic one-dimensional foliation on ${\mathbb{P}^n}$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal{F}$ and $C_i$, assuming that $\mathcal{F}$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.
LA - eng
KW - holomorphic foliations; non-isolated singularities
UR - http://eudml.org/doc/275642
ER -

## References

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1. Carolina Araujo, Maurício Corrêa Jr., On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations, Math. Z. 276 (2014), 505-515 Zbl1285.14016MR3150215
2. P. Baum, R. Bott, On the zeros of meromorphic vector-fields, (1970), Springer-Verlag, Berlin Zbl0193.52201MR261635
3. A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorem, a theorem of Severi, and the equations defining projectives varieties, J. Amer. Math. Soc. 4 (1991), 587-602 Zbl0762.14012MR1092845
4. A. Campillo, J. Olivares, On sections with isolated singularities of twisted bundles and applications to foliations by curves, Math. Res. Lett. 10 (2003), 651-658 Zbl1046.32008MR2024722
5. X. Gomez-Mont, G. Kempf, Stability of meromorphic vector fields in projective spaces, Comment. Math. Helv. 64 (1989), 462-473 Zbl0709.14008MR998859
6. R. Lazarsfeld, Positivity in algebraic geometry, I, II, Springer (2004) Zbl1093.14500MR2095471
7. G. Nonato Costa, Holomorphic foliations by curves on ${ℙ}^{3}$ with non-isolated singularities, Ann. Fac. Sci. Toulouse, Math. (6) 15 (2006), 297-321 Zbl1129.32018MR2244219
8. I. R. Porteous, Blowing up Chern class, Proc. Cambridge Phil. Soc. 56 (1960), 118-124 Zbl0166.16701MR121813

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