Timeline for Algebraic fiber bundles for various topologies

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Sep 24, 2013 at 18:53	comment	added	Laurent Moret-Bailly		No, since there are conic bundles without generic sections: take $Y=\mathbb{G}_m\times \mathbb{G}_m$ with coordinates $t$, $u$, and $X\subset Y\times \mathbb{P}^2_k$ given by $x^2+ty^2+uz^2=0$. It is an exercise to check that this equation has no nontrivial solutions with $x$, $y$, $z$ in $k(t,u)$.
Sep 23, 2013 at 20:46	comment	added	Julian Rosen		@LaurentMoret-Bailly Good point. Might it be true that etale $\mathbb{P}^1$ bundles are Zariski locally trivial when $S=\mathrm{Spec}(\overline{k})$?
Sep 23, 2013 at 20:29	comment	added	Laurent Moret-Bailly		If $Y$ is the spectrum of a field $k$ and $X$ is a $k$-conic without rational points, then $X$ is an étale $\mathbb{P}^1$-bundle which is not Zariski-locally trivial.
Sep 23, 2013 at 18:47	comment	added	Jason Starr		In characteristic $p$, the Frobenius morphism $F: \mathbb{P}^1\to \mathbb{P}^1$, defined by $[X,Y]\mapsto [X^p,Y^p]$, is fppf locally trivial with fiber $\text{Spec}k[\epsilon]/\langle \epsilon^p \rangle$, yet it is neither Zariski locally trivial nor 'etale locally trivial.
Sep 23, 2013 at 18:37	history	edited	Julian Rosen	CC BY-SA 3.0	added 4 characters in body
Sep 23, 2013 at 18:32	history	asked	Julian Rosen	CC BY-SA 3.0