Let $X$ be a projective $k$-scheme of pure dimension $n$ where $k$ is a field. Let $L$ be an ample line bundle on $X$. For all $x \in X$ there is some open neighborhood $U \subseteq X$ and an isomorphism $w: L_{\mid U} \cong \mathcal{O}_U$.
Show that there is some $n \geq 1$ and a global section $s \in L^{\otimes n}(X)$ such that $$ X_s = \{ x \in X \mid w_x(s_x) \in \mathcal{O}_{X,x}^\times \} $$ is everywhere dense, i.e. $s$ does not vanish identically on any of the irreducible components of $X$.
EDIT:
I want to emphasize that I am interested in an answer that will also work over a finite field $k$.
Note that some power of $L$ is very ample, i.e. you find a closed immersion $i: X \to \mathbb P^n_k$ with $i^*\mathcal O(1) = L^n$.
Now for each irreducible component $V_i$, pick a point $p_i \in V_i$. Certainly there is a global section of $\mathcal O(1)$ which does not vanish at all $i(p_i)$ (A hyperplane can always avoid finitely many points). Its pull-back to $X$ is a section of $L^n$ which does not vanish at all $p_i$.