Timeline for Are unirational K3 surfaces defined over finite fields?
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| Nov 1, 2017 at 9:51 | comment | added | Ariyan Javanpeykar | @ViniciusM. If you want a more explicit example, find a supersingular abelian surface $A$ over $k(t)$ (with $k$ some finite field) which can't be defined over $\overline{\mathbb{F}_p}$, and then let $X$ be the associated Kummer surface. Probably, $X$ is a (supersingular) K3 surface which can't be defined over a finite field; see arxiv.org/pdf/1412.7107.pdf for examples of such abelian surfaces | |
| Nov 1, 2017 at 9:46 | comment | added | Ariyan Javanpeykar | @ViniciusM. Let $K$ be the function field of a one-dimensional subspace of the moduli $M_{ss}$ of supersingular (polarized) K3 surfaces. There is a finite field extension $K'/K$ and a "natural" morphism Spec $K'\to M_{ss}$. The universal family over $M_{ss}$ induces a polarized supersingular K3 surface $X$ over $K'$. Note that $K'$ is the function field of a curve over $\mathbb{F}_q$, and that $X$ can't be defined over a finite field. Finally, by construction, $X$ is supersingular, hence $\rho = 22$ and $X$ admits an elliptic fibration. Is this good enough? | |
| Oct 31, 2017 at 21:18 | comment | added | Vinicius M. | @AriyanJavanpeykar and WillSawin, Thanks for the comment. Ok, I see, but what I'd like to know is if there exists a known example of a elliptic surface over a function field that is K3 and supersingular (as a surface, $\rho = 22$). All papers I found treated special cases of supersingular k3 elliptic surfaces, but all are defined over a finite field.. | |
| Oct 31, 2017 at 19:21 | comment | added | Ariyan Javanpeykar | You are probably thinking of the analogy with elliptic curves. A supersingular elliptic curve is defined over a finite field "because" the moduli space of supersingular elliptic curves is zero-dimensional. However, the moduli space of supersingular princ. pol'd abelian surfaces is positive dimensional. As in Will's comment, a generic point of (a positive-dimensional component of) this moduli space will not be defined over a finite field (although all supersingular abelian surfaces over an algebraically closed field $k$ are $k$-isogenous to a product of supersingular elliptic curves). | |
| Oct 31, 2017 at 12:35 | comment | added | Will Sawin | Doesn't the proof of unirationality involve studying the moduli space of supersingular K3 surfaces, which is ten-dimensional or something? A generic point of the moduli space will not be defined over any finite field. | |
| Oct 31, 2017 at 11:52 | history | asked | Vinicius M. | CC BY-SA 3.0 |