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Jul 6, 2016 at 21:10 comment added Bernie @Jason: Thanks! This seems good. Since the birational map between the generic curves is induced by $\phi$, this birational map between the generic curves gives rise to the desired birational map between $C$ and $C'$.
Jul 6, 2016 at 21:09 comment added Bernie @Francesco: Thanks. I shortened the title a bit, because I did not want it to be too long. Maybe it got shortened too much. I hope now it is not misleading.
Jul 6, 2016 at 21:00 history edited Bernie CC BY-SA 3.0
Changed the misleading title
Jul 6, 2016 at 17:24 comment added Jason Starr If the birational transformation $X\dashrightarrow X'$ pulls back $a'$ to $a$, then the fibers of $C$, resp. $C'$, over the generic points of $X$, resp. $X'$ , are birational.
Jul 6, 2016 at 13:47 comment added Francesco Polizzi The title of the question is misleading (as it is written, the answer is surely no).
Jul 6, 2016 at 13:29 history asked Bernie CC BY-SA 3.0