Timeline for rational curves over K3 surfaces over $\mathbb{Q}$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2022 at 8:18 | comment | added | Ariyan Javanpeykar | If $X$ is a K3 surface over a number field $K$ with $Aut(X)$ infinite (i.e., there is an automorphism $\sigma$ of infinite order), then there is a number field $L/K$ such that $X_L$ contains infinitely many pairwise distinct (split) rational curves. (That is, there are infinitely many pairwise distinct morphisms $\mathbb{P}^1_L\to X_L$.) This was proven by Bogomolov-Tschinkel. | |
| Mar 10, 2022 at 20:01 | comment | added | Jason Starr | Welcome new contributor. What makes you believe there is any rational curve defined over $\mathbb{Q}$? A quartic Fermat hypersurface in projective $3$-space has no $\mathbb{Q}$-points, thus no copy of $\mathbb{P}^1_{\mathbb{Q}}$. Moreover, for the "generic" quartic surface, every zero-cycle on every curve has degree divisible by $4$. Since the anticanonical divisor class has degree $2$, there is no genus $0$ curve. | |
| S Mar 10, 2022 at 17:50 | review | First questions | |||
| Mar 10, 2022 at 18:51 | |||||
| S Mar 10, 2022 at 17:50 | history | asked | did | CC BY-SA 4.0 |