On Saturday, 28 October 2023 at 15:26:35 UTC-7 Kwankyu wrote:

f1, f2, f3 are univariate polynomials (say in y) over rational function 
field (say in x). Then x and y are not always the variables X and Y of the 
coordinate ring of the affine plane. Things are more complicated if the 
curve is in space (of dim > 2).
 

Sure. But you did get x,y somehow from the model with which C was given. 
Most likely, the A2 on which x,y are coordinates is obtained via projection 
from whatever model C was given by? Even if it is not a linear map, there 
are still expressions through which you can pull back x,y to the original 
model. You're right that the representations may get rather bad, but I'm 
not sure there's really something you can do about that. 

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