Number of singularities of a foliation on

Author:
Fernando Sancho de Salas

Journal:
Proc. Amer. Math. Soc. **130** (2002), 69-72

MSC (2000):
Primary 32S65, 14M12

DOI:
https://doi.org/10.1090/S0002-9939-01-06149-4

Published electronically:
June 6, 2001

MathSciNet review:
1855621

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Abstract | References | Similar Articles | Additional Information

Abstract: Let 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 are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let be the dimension of the singular locus of . We give an upper bound of the number of singularities of dimension , counted with multiplicity and degree, that 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.

**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

Email:
fsancho@gugu.usal.es

DOI:
https://doi.org/10.1090/S0002-9939-01-06149-4

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