Bulletin of the Korean Mathematical Society (대한수학회보)

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ON THE SINGULAR LOCUS OF FOLIATIONS OVER ℙ2

  • Shi Xu (School of Mathematical Sciences East China Normal University)
  • Received : 2023.10.17
  • Accepted : 2024.06.05
  • Published : 2024.07.31
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Abstract

For a foliation 𝓕 of degree r over ℙ2, we can regard it as a maximal invertible sheaf N∨𝓕 of Ωℙ2, which is represented by a section s ∈ H0(Ωℙ2 (r+2)). The singular locus Sing𝓕 of 𝓕 is the zero dimensional subscheme Z(s) of ℙ2 defined by s. Campillo and Olivares have given some characterizations of the singular locus by using some cohomology groups. In this paper, we will give some different characterizations. For example, the singular locus of a foliation over ℙ2 can be characterized as the residual subscheme of r collinear points in a complete intersection of two curves of degree r + 1.

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Acknowledgement

The author very grateful to professors Shengli Tan, Jun Lu and Xin Lu for many useful discussions and suggestions. He would like to thank the referees sincerely for pointing out mistakes and useful detailed suggestions, which helped to improve the paper.

References

  1. F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izv. 13 (1978), 499-555. https://doi.org/10.1070/IM1979v013n03ABEH002076
  2. R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203-248. https://doi.org/10.2307/1969996
  3. M. Brunella, Birational Geometry of Foliations, IMPA Monographs, 1, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-14310-1
  4. A. Campillo and J. Olivares, Polarity with respect to a foliation and Cayley-Bacharach theorems, J. Reine Angew. Math. 534 (2001), 95-118. https://doi.org/10.1515/crll.2001.036
  5. D. Eisenbud, M. Green, and J. Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295-324. https://doi.org/10.1090/S0273-0979-96-00666-0
  6. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer, New York, 1977.
  7. R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121-176. https://doi.org/10.1007/BF01467074
  8. C. Okonek, M. H. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, corrected reprint of the 1988 edition, Modern Birkhauser Classics, Birkhauser/Springer Basel AG, Basel, 2011.
  9. S. L. Tan and E. Viehweg, A note on the Cayley-Bacharach property for vector bundles, Complex analysis and algebraic geometry, 361-373, De Gruyter, Berlin, 2000.
  10. A. Tyurin, Cycles, curves and vector bundles on an algebraic surface, Duke Math. J. 54 (1987), no. 1, 1-26. https://doi.org/10.1215/S0012-7094-87-05402-0