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Number of singularities of a foliation on ${\mathbb P}^n$

Author: Fernando Sancho de Salas
Journal: Proc. Amer. Math. Soc. 130 (2002), 69-72
MSC (2000): Primary 32S65, 14M12
Published electronically: June 6, 2001
MathSciNet review: 1855621
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Abstract: Let $\mathcal{D}$ be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of $\mathcal{D}$ are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let $m$ be the dimension of the singular locus of $\mathcal{D}$. We give an upper bound of the number of singularities of dimension $m$, counted with multiplicity and degree, that $\mathcal{D}$ may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.

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  • 1. P. Baum, R. Bott, On the zeroes of meromorphic vector fields, Essays on Topology and Related Topics, Mémoires dédiés à Georges de Rham, Springer, Berlin, 1970, pp. 29-47. MR 41:6248
  • 2. P. Baum, R. Bott, Singularities of holomorphic foliations, J. Differential Geometry 7 (1972), pp. 279-342. MR 51:14092
  • 3. W. Fulton, Intersection theory, Berlin, Heidelberg, New York, Springer, 1984. MR 85k:14004
  • 4. G. Kempf, D. Laksov, The determinantal formula of Schubert Calculus, Acta Math. 132 (1974), pp. 153-162. MR 49:2773
  • 5. F. Sancho, Milnor number of a vector field along a subscheme. Applications in desingularization, Advances in Mathematics 153 (2000), 299-324. CMP 2000:16

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Additional Information

Fernando Sancho de Salas
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain

Keywords: Singularities, foliations
Received by editor(s): May 13, 2000
Published electronically: June 6, 2001
Additional Notes: The author was supported in part by the Spanish DGES through the research project PB96-1305 and by the ‘Junta de Castilla y León’ through the research project SA27/98.
Communicated by: Michael Handel
Article copyright: © Copyright 2001 American Mathematical Society