Suppose that you have a smooth projective variety $X$ and an invertible sheaf $L$ which is very ample. Let $F: X \rightarrow \mathbb{P}^{n}$ be the associated embedding. Suppose that $F(X)$ contains a plane of dimension $dim(X)-1$. Let $a \in Pic(X)$ be the class of this plane. Are there any obvious restrictions on $a^{\dim(X)} \in \mathbb{Z}$?
Background: I was reading the following text https://www.math.ens.fr/~debarre/ExposePoitiers2013.pdf and the following claim is made (page 3): Suppose $X$ is a smooth anti-canonically embedded Fano $3$-fold such that $Pic(X) = \mathbb{Z}[H]$ and $H^{3} \geq 4$. Then for any $a \in Pic(X)$ we have that $H^{3}$ divides $a^{3}$, hence the image of $X$ in the anti-canonical embedding cannot contain a plane. My main motivation was to understand this, although I asked a simpler question as to not clutter the discussion.
In the case of a Fano 3-fold you must have $a = kH$ for some $k \ge 1$. If $a$ is the class of a plane then $a\cdot H^2 = 1$, but on the other hand this is equal to $kH^3 \ge 4$.