This is related to Hartshorne's book Algebraic Geometry, Lemma II.7.8, page 158.
Let $X$ be a nonsingular projective variety over an algebraically closed field $k$ and let $\mathcal{L}$ be an invertible sheaf on $X$. For any $s \in \Gamma(X, \mathcal{L})$, we have $\mathrm{Supp}(s)_{0} = (X_{s})^{c}$, where $\mathrm{Supp}(s)_{0}$ means the union of the prime divisors of the divisor of zeros $(s)_{0}$ of $s$ and $X_{s} = \{ P \in X \mid s_{p} \not\in \mathfrak{m}_{p}\mathcal{L}_{p} \}$.
Thanks.