Number of singularities of a foliation on ℙⁿ

Sancho de Salas, Fernando

doi:10.1090/S0002-9939-01-06149-4

Number of singularities of a foliation on ${\mathbb P}^n$
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by Fernando Sancho de Salas PDF

Proc. Amer. Math. Soc. 130 (2002), 69-72 Request permission

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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