Timeline for Taking quotient of a variety by the additive group
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2021 at 20:40 | comment | added | Will Sawin | @MikhailBorovoi Probably, but I'm not sure where that exact statement appears. It follows quickly from stacks.math.columbia.edu/tag/039P in the stacks project: Take a section of $\mathbb A^n_R$ over $R$ containing $\pi(x)$, and then pull back along $\pi$. | |
| May 1, 2021 at 18:30 | comment | added | Mikhail Borovoi | Thank you! Is your argument for "In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere" written somewhere, so that I could refer? | |
| May 1, 2021 at 18:14 | comment | added | Will Sawin | @MikhailBorovoi Yes. Locally trivializable in the etale topology implies locally trivializable in the flat, complex analytic, and several other topologies. In fact, the comparison between etale and Zariski cohomology for a quasicoherent sheaf implies that it is locally trivializable in the Zariski topology as well. The only subtlety I can think of, which applies equally to all these topologies, is that you want the schematic fibers to be orbits, i.e. you want the fibers to be reduced. | |
| May 1, 2021 at 18:05 | comment | added | Mikhail Borovoi | Does it follow that a morphism $\varphi$ as in my question induces a locally trivial fibre bundle of complex analytic manifolds (with the usual topology)? | |
| Apr 29, 2021 at 19:15 | vote | accept | Mikhail Borovoi | ||
| Apr 29, 2021 at 17:28 | comment | added | Will Sawin | @MikhailBorovoi of $X$. For example, take $y\in Y$, $x \in X$ with $\varphi(x)=y$, and choose any function on (a neighborhood of $x$ in) $X$ vanishing at $x$ and whose first derivative is nonzero on the relative tangent space of $X$ over $Y$. The vanishing locus of this function is still smooth over $Y$ at the point $x$, thus smooth over $Y$ in a neighborhood of $x$. | |
| Apr 29, 2021 at 16:51 | comment | added | Mikhail Borovoi | Thank you, Will, for a prompt answer! Could you please add details to the proof for 1? A generic hypersurface section of what? | |
| Apr 29, 2021 at 16:19 | history | answered | Will Sawin | CC BY-SA 4.0 |