Let $X$ be a projective scheme over $S=spec A$, where A is a complete integrally closed noetherian local ring. $Y \subset X$ is a relative effective Catier divisor.
Then there exists a global section of $X$ over $S$ such that $Y$=image of this section. Is this true? If not then it is true in which case? Thank for your help.
I am not sure if this question is appropriate for MO, but here is an answer.
The relative dimension, over $S$, of the image of a section equals $0$. So, if $Y$ is the image of a section, then $X\to S$ is smooth of relative dimension $1$ at every point of $Y$. Moreover, $Y$ then has relative degree $1$ over $S$. Conversely, if $X\to S$ is smooth of relative dimension $1$ at every point of $Y$, and if $Y$ has relative degree $1$ over $S$, then $Y$ is the image of a section.
