I refer to this answer for definitions.
Let $X$ be a smooth projective uniruled variety covered by lines, $Pic X=\mathbb Z$, and let $L=\mathcal O(1)$ (if it's simpler, $X$ is just an isotropic Grassmannian). Consider the incidence diagram $$ X \longleftarrow I=\{(x,l) \in X \times G(\mathbb P^1,X): x \in l\} \longrightarrow G(\mathbb P^1,X) $$ where $G(\mathbb P^1,X)$ is the variety parametrizing the lines contained in $X$. We denote the projections by $p:I \to X$ and $q: I \to G(\mathbb P^1,X)$. Now, suppose that $s \in \Gamma(X,L)$ is a regular section, that is such that $$ codim (Z(s) \subset X)=1 $$ where $Z(s)=\{x \in X: s(x)=0\}$. I know that $q_* p^* L=Q$ where $Q$ is the dual of the tautological bundle in the Grothendieck notation, I want to study the codimension of $$ Z(q_* p^* s) \subset G(\mathbb P^1, X). $$ Any idea or hint on how to prove or disprove that such a codimension is 2?