Scheme has no Embedded Points

Question

Let $X$ be scheme and I want to know what are the weakest conditions for $X$ such that $X$ has no embedded points.

The embedded points are defined by:

$$\operatorname{Emb}(\mathcal{O}_C):= \{x \in X \ | \ m_x \in \operatorname{Ass}_{\mathcal{O}_{X,x}}(\mathcal{O}_{X,x})~\text{and}~x~\text{is not generic}\} $$

I think that it's clear that integral schemes don't have embedded points (because localisations of integral domains are integral domains), right?

Do reduced schemes have embedded points? Because the problem is local, equivalent question: Why a reduced ring has no embedded points?

Answer 1

It's been a while since this question has been asked, nevertheless, let me try to give an answer:

Indeed, a reduced scheme has no embedded points. This has already been answered e.g. here. In particular, an integral scheme does not have embedded points.

You asked for some sort of "weakest condition" for having no embedded points. If $X$ is a locally Noetherian scheme of dimension $\leq 1$ there is a nice answer:

Theorem. The following are equivalent:

1. $X$ has no embedded points
2. $X$ is Cohen-Macaulay

See this Stacks Project entry for a reference.