Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces

We study holomorphic foliations on projective spaces that leave smooth projective complete intersection surfaces $M$ invariant. We determine precisely for which degrees such foliations on $M$ exist. As a consequence, we obtain new bounds for the classical Poincar'e problem for smooth projective complete intersection surfaces and prove that previously known bounds for smooth hypersurfaces in ${\mathbb{P}}^3$ are optimal. Furthermore, for a foliation $[s]$ on $M$ with isolated singularities and for degrees beyond an explicit bound that we provide, we show that a section $s^{\mathrm{\prime }}$ has singular scheme containing that of $s$ if and only if $s^{\mathrm{\prime }}=\varphi (s)$ for some global endomorphism $\varphi$ of the tangent bundle of $M$.