# Vanishing locus of global section of invertible sheaf has codimension one

Let $X$ be a projective scheme over some field $k$. Let $\mathcal{L}$ be an ample invertible sheaf on $X$ and let $s \in \mathcal{L}(X)$ denote a global section of $\mathcal{L}$. Let $X_s = \{P \in X \mid s_P\mathcal{O}_{X,P} \cong \mathcal{L}_P\}$ be open. I am looking for a reference that $\operatorname{codim}(X\setminus X_s, X) \leq 1$ holds.

• $X_s$ could be empty. – Mohan Sep 5 '18 at 18:24
• @Mohan Do you know a reference for the edited question/statement? – windsheaf Sep 6 '18 at 6:27
• the problem at hand had nothing to do with the scheme being projective or not .. – skeptic Sep 6 '18 at 7:15
• @skeptic Okay. However, do you know a reference? – windsheaf Sep 6 '18 at 7:17
• have a look at krulls principal ideal theorem – skeptic Sep 6 '18 at 7:24