Timeline for (crystalline cohomology version's) Tate's conjecture for K3 surfaces
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2023 at 12:32 | vote | accept | Yuan Yang | ||
| Jun 23, 2023 at 3:30 | answer | added | Keerthi Madapusi | timeline score: 3 | |
| Jun 10, 2023 at 0:52 | answer | added | Yuan Yang | timeline score: 1 | |
| Jun 9, 2023 at 22:40 | comment | added | Yuan Yang | If $\rho=22-2h$ is always true, then this formula should be written in the text book!! (Not just an inequality!!) | |
| Jun 9, 2023 at 22:28 | history | edited | Yuan Yang | CC BY-SA 4.0 |
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| Jun 9, 2023 at 22:26 | comment | added | Yuan Yang | Here is something else from Ogus' crystalline Toreli theorem for supersingular K3 paper, he said, over $W(\overline{\mathbb{F}_p})$, "the isomorphism class of a K3 crystal with $h\neq \infty$ is uniquely determined by $h$". So this means the space $\Phi=p$ is uniquely determined by $h$ and thus must have rank $22-2h$. Together with Tate's conjecture over finite fields, does this mean $\rho=22-2h$ (where $\rho$ is of course the geometric picard number) always hold? Or the F-crystal over $W(\mathbb{F}_q)$ behaves differently with F-crystals over $W(\overline{\mathbb{F}_p})$? | |
| Jun 9, 2023 at 22:17 | comment | added | Yuan Yang | @JasonStarr But this is very confusing: over $\overline{\mathbb{F}_p}$, a countable union of closed sets can cover the whole moduli space, which means it is possible that the locus $\rho=2$ in $M_{2d}^{1}$ is empty. So whether there are such K3 such that $\rho=2$ but $h=1$ is still not clear from his book. | |
| Jun 9, 2023 at 22:13 | comment | added | Yuan Yang | @JasonStarr I see what you mean. I should use the word 'general' rather than 'generic'. In Huybrechts' book he said "the condition $\rho(X)\geq\rho$ is not (a closed condition), it rather defines a countable union of closed sets" and "However, many more $(X, H)$ not contained in $M_{2d}^{10}$ satisfy $\rho(X) \leq 2$, in fact the general one should have this property". So apparently in his mind the equation $\rho=22-2h$ is false in general." | |
| Jun 9, 2023 at 19:40 | comment | added | Jason Starr | Perhaps what is going on is confusion about the word "generic." The point that I am making is that no K3 surface over a finite field is generic in the sense that its (geometric) Picard group is isomorphic to the geometric Picard group of the "generic" K3 surface in the sense of the fiber over the geometric generic point for a moduli space of (polarized) K3 surfaces. | |
| Jun 9, 2023 at 19:39 | comment | added | Yuan Yang | Let us continue this discussion in chat. | |
| Jun 9, 2023 at 19:11 | comment | added | Jason Starr | Elliptic curves over a finite field $\mathbb{F}_q$ always have an extra endomorphism: the Frobenius endomorphism relative to $\mathbb{F}_q$. So they do not satisfy the hypothesis in Shioda's article. Shioda also considers fields of positive characteristic that are not finite fields. Over some of these fields he produces examples of Picard rank $19$. | |
| Jun 9, 2023 at 19:04 | comment | added | Yuan Yang | @JasonStarr But for non-supersingular K3 the equation $\rho=22-2h$ should be false in general. But I couldn't see why this doesn't contradict Tate's conjecture for the structure of its F-crystal | |
| Jun 9, 2023 at 19:00 | comment | added | Yuan Yang | @JasonStarr Is that so? Even in characteristic 2? Check Shioda's paper 'Kummer surfaces in char 2' where the author constructed a K3 with $\rho=19$ | |
| Jun 9, 2023 at 19:00 | comment | added | Jason Starr | Also I just heard from an expert (who I will not name unless given permission) who tells me that also supersingular K3 surfaces have Picard rank 22. | |
| Jun 9, 2023 at 18:52 | comment | added | Jason Starr | What I know about this comes from the article "Constructing rational curves on K3 surfaces" by Bogomolov, Hassett, and Tschinkel: arxiv.org/abs/0907.3527 In that article they use Theorem 15 (due to many authors): every non-supersingular surface over $\overline{\mathbb{F}}_p$ has even Picard rank (so at least $2$). | |
| Jun 9, 2023 at 18:44 | comment | added | Yuan Yang | @JasonStarr I think my understanding for F-crystal must be wrong somewhere, I mean maybe isogenous F-crystals do not necessarily have the same rank for the $\Phi=p$ part. I don't know?? $H^2(X/W)$ should be isomorphic to a sub $W$-module of $H_{\lambda}\oplus H_1^{22-2h}\oplus H_{2-\lambda}$, so their $\Phi=p$ part shouldn't be different? I don't know any other examples of F-crystals | |
| Jun 9, 2023 at 18:39 | comment | added | Yuan Yang | @JasonStarr A generic K3 surface should have $h=1$? I mean the locus $M_{2d}^1$ where $h=1$ is an open subset of $M_{2d}$ right? According to Huybrechts' notes | |
| Jun 9, 2023 at 10:28 | comment | added | Jason Starr | Why do you write that $2$ is strictly less than $22-2h$? | |
| Jun 9, 2023 at 7:50 | comment | added | Yuan Yang | I must be apparently wrong somewhere, can someone tell me where am I wrong? | |
| Jun 9, 2023 at 7:41 | history | edited | Yuan Yang | CC BY-SA 4.0 |
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| Jun 9, 2023 at 7:34 | history | undeleted | Yuan Yang | ||
| Jun 9, 2023 at 7:34 | history | deleted | Yuan Yang | via Vote | |
| Jun 9, 2023 at 7:34 | history | asked | Yuan Yang | CC BY-SA 4.0 |