In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation on a compact manifold is not necessarily transverse to some compact Riemann surface. Indeed, the existence of such a compact transverse section often implies several restrictions on the foliation..."


My question is simply "What restrictions?" This is not really discussed in their paper nor the references, as they concentrate on real transversals. Also, most results I have seen are local, I'm interested in the global statement as phrased by Scardua. 


I'd be happy to even concentrate on the case of smooth projective surfaces and smooth transverse algebraic curves. The only general result of which I am aware is the result by Bogomolov/McQuillan in "Rational Curves on Foliated Varieties" and the follow-up article by Kebekus/Conde/Toma "Rationally connected foliations after Bogomolov and McQuillan", that if the transverse curve has positive self-intersection (i.e. is an ample divisor), then the foliation is a fibration with rational leaves. (I'm phrasing it in the case of dimension 2, similar results also hold in higher dimension, giving rational connectedness of the leaves, and with weaker hypotheses as to the nature of the curve and foliation singularities, amount of tangency, etc).


But how about everywhere transverse curves with non-positive self-intersection? What restrictions does this give on the foliation?



Holomorphic Foliations having transverse sections
If a smooth curve $C$ has zero self-intersection and is everywhere transverse to a foliation on a surface then there are also strong restrictions. 


If $C$ is rational then the foliation is a Riccati foliation (i.e. is birationally equivalent to the projectivization of a meromorphic flat connection on rank two bundle over a curve). 


For an arbitrary smooth curve $C$ everywhere transverse to a foliation $\mathcal F$ on a projective surface we can argue as follows. If $T^\* \mathcal F$ is not pseudo-effective then by Miyaoka's Theorem (Bogomolov-McQuillan is a generalization of Miyaoka's Theorem) then $\mathcal F$ is a foliation by rational curves. 


If instead $T^\*\mathcal F$ is pseudo-effective then the Zariski decomposition $P+N$ ($P$ nef $\mathbb Q$-divisor and $N$ effective contractible $\mathbb Q$-divisor) of $T^\* \mathcal F$ satisfies $P \cdot C = 0$. Hodge index theorem implies that $P$ is numerically equivalent to a rational multiple of $C$. This already imposes strong restriction on $\mathcal F$. It must be a foliation of special type (Kodaira dimension is not maximal). 




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**Synthesis.** Assume there exists a smooth curve C of zero self-intersection which is everywhere transverse to a foliation $\mathcal F$. I believe that going through the classification of foliations of special type we can proof that at least one of the following assertions holds true:


* The foliation $\mathcal F$ is a foliation by rational curves.
* The curve $C$ is a fiber of rational fibration and the foliation is transverse to the general fiber of it (Riccati foliation).
* The curve $C$ is a fiber of an isotrivial elliptic fibration and the foliation is transverse to the general fiber of it (turbulent foliation).
* The curve $C$ is a fiber of an isotrivial fibration with fibers of genus $\ge 2$ and the leaves of the foliation are all algebraic.


The main references here are McQuillan's paper Canonical models of foliations; and Brunella's book Birational geometry of foliations.




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**Examples.**
In contrast, it is rather easy to produce examples of foliations tranverse to curves of negative self-intersection. Start with a reduced foliation of general type on any surface and take a sufficiently general smooth curve. The tangencies between the foliation and the curve will be simple tangencies. Blowing-up each of the tangencies points twice will give rise to a curve everywhere transverse to the resulting foliation. 


