Let $ X/k$ be a smooth projective variety over an algebraically closed field $ k$ .

Q1: For any $ f \in k(X)$ , is there a prime divisor $ D_{\infty,f}$ and $ g \in k(X)$ such that $ fg$ has poles only at $ D_{\infty,f}$ , i.e. $ (fg)=\sum n_iD_i-mD_{\infty,f}$ for $ n_i,m>0$ ?

Q2: Is there a prime divisor $ D_\infty$ , such that for any $ f\in k(X)$ , there exists $ g \in k(X)$ such that $ fg$ has poles only at $ D_{\infty}$ , i.e. $ (fg)=\sum n_iD_i-mD_{\infty}$ for $ n_i,m>0$ ?

This can be easily done if $ X=\mathbb{P}^n$ , simply because there are prime divisors of degree $ =1$ , hence the statements are true immediately for $ X$ that has a prime divisor of degree $ =1$ (in the sense of ex 6.2, pg 146 of Hartshorne AG).

In the case where the statements are not true, what is the next best thing we can do? (eg: poles at at most 2 prime divisors?)