Let $X$ be a scheme, $\mathcal{L}$ an invertible sheaf and $t \in \Gamma(X,\mathcal{L})$ a global section.
Consider the subset $X_t = \{x \in X \vert (t)_x \neq 0 \}$, or equivalently $(t)_x \not \in m_x\mathcal{L}_{X,x}$.
How to prove that if $\mathcal{L}$ ample (therefore the $X_s, s \in \Gamma(X, \mathcal{L}^{\otimes n}), n \ge 1$ form a base of topology of $X$),
then $X_t$ affine.
Ideas: I have to show that $\Gamma(X_t,\mathcal{O}_X)$ respects the localisations at prime ideals $\mathbb{p} \subset \Gamma(X_t,\mathcal{O}_X)$.
Therefore I have to show that $\mathcal{O}_{X, p} = \Gamma(X_t,\mathcal{O}_X)_p$ holds. No idea how to make next steps.