I'd like to know when a scheme is consisted of closed points and generic points.
For example, if $X$ is an integral separated scheme of finite type over an algebraically closed field $k$ with generic point $\eta$, then any point $p$ in $X-\eta$ is closed?
If $X$ is as above, then I think $p$ is closed iff $p$ is closed in some open affine neighborhood $\text{Spec} A$ of $p$, so it suffices to prove that any non-minimal prime ideal in $A$ is maximal. This is equivalent to $\dim A=1$. But I don't think $\dim A=1$ is true...
A scheme $X$ satisfying the above conditions is called a variety on page 105 of Hartshorne, but I always have the feeling that schemes are generalized of varieties by adding generic points, that is if $X$ is an integral separated scheme of finite type over an algebraically closed field $k$ scheme, then by deleting all its closed points we can get the unique generic point. But I can't see why such a scheme is consisted of closed points and generic points... Where I am wrong?
By the way, in Hartshorne's definition, separatedness means $X\rightarrow X\times_kX$ is closed embedding or $X\rightarrow X\times_{\mathbb{Z}}X$ is closed embedding? I am not sure.
Thanks in advance.