In Hartshorne's proof of the Bertini Theorem (II, Thm. 8.18) he shows that the set $B_x$ is a linear system of dimension $n-r-1$ and uses this fact to conclude that the set $B$ is irreducible. However I don't see how this would work if $\dim X = n-1$ since then $B_x$ would have dimension $0$. I would be very happy for some hints as to why this is not a problem in this case.
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Why do you think this would be a problem? This question shows that a proper morphism to an irreducible variety with nonempty irreducible equidimensional fibers implies the source is irreducible. That's exactly this situation. – KReiser Mar 12 '24 at 22:54
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In some other proof, they handled the case $r = n-1$ separately and claimed that then $U = \check{\mathbb{P}^n} \setminus X_{\text{dual}}$, so I thought that was what you had to do. But I guess I just didn’t think about the fact that being $0$-dimensional is no problem for irreducibility – jb1403 Mar 13 '24 at 11:23