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?.

Shepherd-Barron and Taylor. $\endgroup$ – abx Feb 14 '18 at 5:31