I am reading Hartshorne algebraic geometry.
Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed points of $A^2_k$ are in one-to-one correspondence with ordered pairs of elements of $k$. Futhermore, the set of all closed points of $A^2_k$, with the induced topology, is homeomorphic to the vareity called $A^2$ in Chapter I. In addition to the closed points, there is a generic point $\xi$, corresponding to the zero ideal of $k[x,y]$, whose closure is the whole space. Also, for each irreducible polynomial $f(x,y)$, there is a point $\eta$ whose closure consists of $\eta$ together with all closed points $(a,b)$ for which $f(a,b)=0$. We say that $\eta$ is a generic point of the curve $f(x,y)=0$.
I cannot understand the sentence for $\eta$. Can $\eta$ be expressed explicitly?
I think $\eta$ work like $\xi$ for the set of all closed points of $A^2_k$.
I should be greatly indebted if you help me.