Timeline for Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2023 at 23:40 | answer | added | Will Sawin | timeline score: 9 | |
| May 29, 2023 at 15:22 | answer | added | R. van Dobben de Bruyn | timeline score: 12 | |
| May 29, 2023 at 12:53 | comment | added | a_g | Thank you for the example! Absolutely. I'm mostly interested in the $\mathbb{F}_q$-case, but I'd be excited to see what happens on $\overline{\mathbb{F}}_q$ too. | |
| May 29, 2023 at 12:43 | comment | added | R. van Dobben de Bruyn | First example: set $X = \operatorname{Spec} \mathbf Z[x]/(f)$ and $Y = \operatorname{Spec}\mathbf Z[x]/(g)$ for monic irreducible polynomials $f,g \in \mathbf Z[x]$ of the same degree. Then $X_{\mathbf C} \cong Y_{\mathbf C}$, and $X_{\mathbf Q}\cong Y_{\mathbf Q}$ if and only if $X_{\mathbf F_p} \cong Y_{\mathbf F_p}$ for a density $1$ set of primes. So besides 'spreading out' there is also a 'Galois twist' thing going on. The answer would be very different if you replace $\mathbf F_p$ by $\bar{\mathbf F}_p$. Are you interested in the arithmetic direction, or only the geometric version? | |
| May 29, 2023 at 12:05 | history | asked | a_g | CC BY-SA 4.0 |