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Let $k$ be a field. Let's say that $k$ has property $(∗)$ iff for a scheme $X$ and morphisms $f_1,f_2:X\rightarrow \mathrm{Spec}\,k$, geometric irreducibility of $f_2$ is implied by geometric irreducibility of $f_1$. Does there exist a non-pseudo-algebraically closed field with property $(∗)$?

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    $\begingroup$ As written, your question is unclear; are you asking whether this is true for any $X, f_1, f_2$, or whether there are two families $f_1, f_2$ depending on $X$ so that this is true, or something else? $\endgroup$ – user44191 Mar 13 at 10:12
  • $\begingroup$ @user44191 I thought that this was perfectly clear. In case it was not, the question is for any $X$, any $f_1$, any $f_2$ $\endgroup$ – rori Mar 13 at 10:41
  • $\begingroup$ Welcome new contributor. Are you the same user who has posted other questions with the username "rori"? Do you have more than one MathOverflow account with the name "rori"? There is a way for the administrators to merge your accounts if you ask them. $\endgroup$ – Jason Starr Mar 13 at 14:03

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