Let, $S$ and $X$ be two hypersurfaces in $\mathbb{P}^{3}$ and we denote Sing($S $) := singular points of $S$.
Then my question is the following : Is the inclusion, Sing$(S\cap X) \subset $ Sing $X$ always true?
My attempt:Let, $S =Z(f)$ and $X = Z(g)$, then we have $Z(f) \cap Z(g) = Z((f) \cup (g))$.Now if $p$ is a singular point for any element of the ideal $(f) \cup (g)$, then it's also a singular point for any point of the ideal $(g)$, and therefore $p \in $Sing$X$.
Is the following argument correct?Any help from anyone is welcome.