There is a result which states that:
A line bundle $L$ on a scheme $S$ is ample if and only if there exists an $n \in \mathbb{N}$ and global sections $\sigma_1, \dots, \sigma_n \in \Gamma(S, L^{\otimes n})$ such that the schemes $S_{\sigma_i}$ are affine and the schemes $S_{\sigma_i}$ cover $S$.
Here, $S_{\sigma_i}$ is defined as $$ S_{\sigma_i} := \{ x \in S \mid \sigma_{i, x} \notin \text{ maximal ideal of } (L^{\otimes n})_x \}. $$
I'm trying to think of an example of a non-ample line bundle. I suspect that $\mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}$ might work, and I am trying to show this rigorously using the above result (though I'd also be interested in hearing other ways to show/see that this isn't ample). My main problem is calculating $$ (\mathbb{P}^1 \sqcup \, \mathbb{P}^1)_\sigma = \{ x \in \mathbb{P}^1 \sqcup \, \mathbb{P}^1 \mid \sigma_x \notin \text{ maximal ideal of } \mathcal{O}^{\otimes n}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1,x} \cong \mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1,x} \} $$ where $ \sigma \in \Gamma(\mathbb{P}^1 \sqcup \, \mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1})$. I'm not really sure where to start with this, since I'm not even sure what the maximal ideals of the stalks of this structure sheaf are. The stalks are local rings, so there will be a unique maximal ideal, and I suppose this maximal ideal corresponds to a point in $\mathbb{P}^1 \sqcup \, \mathbb{P}^1$, so a point in precisely one of the copies. So it looks to me like $(\mathbb{P}^1 \sqcup \, \mathbb{P}^1)_\sigma$ will turn out to be one of the copies of $\mathbb{P}^1$, and this is not affine, so $\mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}$ is not ample? Though I'm not sure about this.
My intuition tells me that it's not ample, because one of the characterisations of ampleness (the definition in my case) states that $L$ on $X$ is ample if for all coherent sheaves $F$, $F \otimes L^{\otimes n}$ is generated by global sections for $n$ large enough, i.e. there exists an $r$ such that we have a surjection $$ \bigoplus_r \mathcal{O}_X \longrightarrow F \otimes L^{\otimes n} $$ for $n$ large enough. If we write $X = \mathbb{P}^1 \sqcup \, \mathbb{P}^1$ and $L = \mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}$, then it seems like $$ \bigoplus_r \mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1} \longrightarrow F \cong F \otimes \mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}^{\otimes n} $$ for $F$ coherent on $\mathbb{P}^1 \sqcup \, \mathbb{P}^1$ cannot be a surjection, since it might fail to be a surjection on the level of stalks (though this is purely speculation).
Thank you for any answers.
Edit: Since my notion of $\mathbb{P}^1 \sqcup \, \mathbb{P}^1$ may be rather ambiguous, perhaps I should say that I'm gluing the two copies of $\mathbb{P}^1$ along the empty set/scheme.