As part of a project in the history of mathematics, I am trying to read a book by Jung from the 1920s, Algebraische Flächen. Nevertheless, I am struggling to understand his set up already in the introduction, where (in as far as I can tell) he is trying to define an algebraic surface in three-dimensional projective space in terms of its parametrization, but the precise mechanics of his set up I fail to understand.
He writes:
Das folgende Buch handelt von den algebraischen Flächen im engeren Sinne, also nicht von der Theorie der algebraischen Funktionen zweier unabhängigen Veränderlichen. Von dieser Theorie wird im wesentlichen nur der Satz benutzt, dass man die Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen in der Umgebung jeder Stelle als Quotienten zweier gewöhnlichen Potenzreihen zweier Hilfsgrößen $u,v$ darstellen kann. Wählt man irgend vier Funktionen des Körpers aus und setzt sie proportional zu den vier homogenen Koordinaten $x_{\alpha}$ eines Punktes, so stellt die Gesamtheit dieser Punkte eine algebraische Fläche $F$ dar. Die Koordinaten kann man dann für die Umgebung jeder Stelle $S$, da es auf einen gemeinsamen Faktor nicht ankommt, als gewöhnliche Potenzreihen zweier Veränderlichen $u,v$ darstellen. Man hat so eine ein-eindeutige Abbildung eines Teils der Fläche $F$ auf die Umgebung des Nullpunktes der $uv$-Ebene.
Which I would translate as:
The following book concerns itself with algebraic surfaces in a strict sense, that is to say, not with the theory of algebraic functions in two independent variables. From the latter theory we will essentially only use one theorem, that the functions of an algebraic field in two independent variables may, in the neighbourhood of each place, be expressed as the quotient of two ordinary power series in two auxiliary variables $u,v$. If one picks any four functions of the field and make them be proportional to the four homogeneneous coordinates $x_{\alpha}$ of a point, then the totality of these points form an algebraic surface $F$. Since a common factor is unimportant, the coordinates in the neighbourhood of each place $S$ can be represented as ordinary power series in two variables $u,v$. One thus has a one-to-one mapping from a part of the surface $F$ onto the neighbourhood of the origin in the $uv$-place.
It is the statement in bold that confuses me. Is he simply saying that, given $f_0 , f_1, f_2, f_3 \in K(u,v)$, $K(u,v)$ being some field of algebraic functions in two variables, one can define a surface $F$ as $$ F := \{ [f_0 (u,v) ; f_1 (u,v) ; f_2 (u,v) ; f_3 (u,v)] \in \mathbb{P}^3 | u,v \in K \} , $$ is he perhaps saying that you take a point $[x_0 ; x_1 ; x_2 ; x_3]$, set up a set of equations $f_i (u,v) = x_i$, $i=0,...,3$ and then solve for $u,v$? Alternatively, as has been suggested to me, is he suggesting that one in fact takes a point $[x_0 ; x_1 ; x_2 ; x_3]$, pick $A_i$ such that $A_i f_i (0,0) = x_i$, and then to find out the rest of the points simply vary the values of $u,v$?
What is going on here?
As always, look forward to what you might come up with!
EDIT: I think I'm going to go with the first interpretation after looking at the comments to this point. It is the only one that makes sense, although it is to be mentioned that after double-checking now with two German speaker (one of whom is a professor in algebraic geometry), they agreed that it was poorly and confusingly formulated by Jung. A more loose translation then, that would better encapsulate that Jung is trying to say, would be "If one picks any four functions of the field, then the totality of points whose four homogeneous coordinates $x_{\alpha}$ are proportional to the values of said functions form an algebraic surface $F$".