We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections respectively. Let $f: X = \mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-2C_{\infty})) \to \mathbb{F}_1$. Notice that general element $C \in |2C_{\infty}|$ is a quadric and $\mathrm{dim}$ $\mathrm{H}^0(\mathbb{F}_1,\mathcal{O}(2C_{\infty}) ) = 6$. Let $\xi$ and $D$ be the zero section and the infinity section respectively on $X$. $f^*C_0 \cong \mathbb{P}^1 \times \mathbb{P}^1$ is the pull-back of divisor $C_0$ on $X$.
For $D^{\prime} \in |D|$ another infinity section of $X$, we have that $D^{\prime} \cap D = C$ the quadric curve on $\mathbb{F}_1$ since the normal bundle $\mathcal{N}_{D/X} = \mathcal{O}(2C_{\infty})$ (cf here for the formular of normal bundle of infinity section $D$ on a split rank 2 projective bundle). If $f^*C_0$ intersects $D$ properly, then $f^*C_0 \cap D = E$ is the zero section on $D$ ($\cong \mathbb{F}_1$) with self-intersection $-1$. On $f^*C_0$, self-intersection of $E$ is $0$ as it is also a zero section on $\mathbb{P}^1 \times \mathbb{P}^1$. Here I claim that $f^*C_0$ could not intersect all General elements of $|D|$ properly. Indeed, $\mathrm{dim}$ $\mathrm{H}^0(X,\mathcal{O}_X(D) ) = 7$ since the sequence:
$0 \to H^0(X,O_X) \to H^0(X,O_X(D)) \to H^0(D,O_X(D)|_D) \to 0$
is exact. If $f^*C_0$ intersects properly to all General Element of $|D|$, then the restriction map:
$H^0(X,O_X(D)) \to H^0(f^*C_0,f^*C_0 \cap D )$
provides that there exists general elements $D_1, D_2 \in |D|$ such that $D_1 \cap D_2 \supset f^*C_0 \cap D_1 = f^*C_0 \cap D_2$. This happens because $\mathrm{dim}$ $\mathrm{H}^0(f^*C_0,f^*C_0 \cap D ) = 2$. This contradicts to the fact that in general the scheme-theoretic intersection $D_1 \cap D_2 $ should be a quadric curve or an effective divisor that is equivalent to quadric curve. Thus there exists several infinity sections that do not intersect $f^*C_0$ properly.
In this situation I would want to ask if my claim about the non-proper intersection between $f^*C_0$ and General elements of $|D|$ is true. And if it is true, has it appeared anywhere in the literature? I would really want to know if there is any reference discusses about this problem. Thank you very much in advance.
