The first paragraph of this question shows the construction of the first counter example to Hilbert's 14th Problem. There, we start from a prime field $P$ of arbitrary characteristic, i.e., $P=\Bbb Q$ or $P=\Bbb F_p$ for some prime $p$, and adjoin 48 algebraically independent elements $a_{ij}$, $i=1,2,3$, $j=1,...,16$, of some field $k$ containing $P$.

Now for the proof of the example to work, Nagata uses the following geometric lemma:

Let $p_1,...,p_{16}$ be

independent generic pointsof the projective plane $S$over the prime field$P$. For any curve $C$ of degree $d$, the sum of the multiplicities of $p_i$ on $C$ is less than $4d$.

At first I had some problems with the word "generic" here, but it seems to mean "general" as in "in general position", since this paper is from the fifties and hence some uncommon and no-longer-used terminology occurs. I can imagine what this means in this case, but I'm not really sure about the exact definition; 4 points of a projective plane are in general position if no 3 of them lie on a line, this is what I know. I also found a definition for 6 points, then additionally no 6 of them shall lie on a conic. How do these generalize to even more points?

Then, I find "independent" confusing there. Is it maybe that "independent generic" = "in general position", i.e., the two words only occur together?

I'd also have thought that we're talking about the projective plane $\Bbb P_P^2$ because of "over the prime field $P$"; then the points $p_i$ would be of the form $(a_0:a_1:a_2)$ with $a_j\in P$. But in the actual proof, we come across the $k$-algebra $k[y_1,y_2,y_3]$ (for the exact construction you could read the linked question, but I think it has no real impact on my actual problem here), and interpret it as a homogeneous coordinate ring of "the projective plane $S$", and consider the sixteen points $p_j=(a_{1j}:a_{2j}:a_{3j})$. I quote: *Then the* $p_i$ *are independent generic points of* $S$ *over the prime field* $P$.

What does this mean? To me, $k[y_1,y_2,y_3]$ can be considered as homogeneous coordinate ring of the projective plane $\Bbb P_k^2$, and the $p_i$ lie in there. What does the "over the prime field $P$" part do there? The $p_i$ are no points of $\Bbb P_P^2$ at least.

Any help in the fight against my confusion is greatly appreciated!