Let $C \subset \mathbb{P}^n$ be an one dimensional scheme, a curve, supporting a finite number of embedded points. Are those points necessarily located at singular points of $C$ ? I read a similar statement somewhere, but I cannot recall it or find the reference...

Thanks

Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded points) and a finite number of embedded points $l_p$ such that $C= \tilde{C} \cup l_p$. Are the points $l_P$ necessarily located at singular points of $\tilde{C}$ ? I read a similar statement somewhere, but I cannot recall it or find the reference...

Thanks