Let $ (X,F)$ a one-dimensional folication over a smooth variety $ X$ over $ \mathbb{Z}$ . Let $ (X_p,F_p)$ the modulus $ p$ reduction of $ (X,F)$ . We assume that $ (X_p,F_p)$ is a foliation in positive characteristic, for almost every prime $ p$ . The Ekedahl- Barron $ F$ -conjecture says that with it hypothesis, the leaves of $ (X,F)$ are algebraic curves.

Let $ L$ be a very ample line bundle on $ X$ . Let $ P$ a non-singular point of $ (X,F)$ . Let $ C$ a leave of $ (X,F)$ that contain $ P$ .

In https://drive.google.com/open?id=1_SNpE8FxC8BmO0n6sin8Ali8bKhtrL73 I have shown that if there existe a colection of $ F_p$ -invariant curves $ C_p$ with genus $ g_p$ for every prime $ p$ then:

$ $ \chi(C,L)(n)\leq \limsup_{p \text{ prime}} (g_p+h^0(X,L))n$ $

Were $ \chi(C,L)(n)$ is the Hilbert- Samuel polynomial. It means that if $ g_p$ is bounded for every $ p$ then the leaves of $ C$ are algebraic curves. It can prove it conjecture. My question is:

_ Is this inequality well known?.

_ There exist a way to bound the $ g_p$ ‘s?.

_ There exist more references about the Ekedahl-Barron $ F$ -Conjecture?.