Foliations by Curves with Curves as Singularities

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.

19 pages

References in corpus (1)

• On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations

Cited by in corpus (3)

• Analytic varieties invariant by foliations and Pfaff systems
• On the Milnor number of non-isolated singularities of holomorphic foliations and its topological invariance
• Foliations with isolated singularities on Hirzebruch surfaces