Timeline for Nakano vanishing in positive characteristic
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2019 at 22:03 | vote | accept | pgraf | ||
| Mar 7, 2019 at 4:58 | answer | added | Jorge Vitório Pereira | timeline score: 4 | |
| Mar 6, 2019 at 18:30 | comment | added | Takumi Murayama | You may already be aware of this, but I believe that Shepherd-Barron in the proof of Theorem 7 in "Unstable vector bundles and linear systems on surfaces in characteristic $p$" shows that on surfaces, the vanishing $(\ast)$ holds for $p = 1$ (with the weaker assumption that the divisor class $L$ of $\mathscr{L}$ is in "positive cone" $C_{++}(X)$) as long as the surface is neither quasi-elliptic nor of general type. In the general type case, if $(\ast)$ does not hold, the proof seems to say that the surface is purely inseparably uniruled. | |
| Mar 6, 2019 at 13:00 | comment | added | Frank | Probably the best place to look for a counterexample is in the Lauritzen-Rao example (as appearing for example in Sandor's paper arxiv.org/abs/1703.02080) - it's a 6fold Fano in characteristic 2 whose construction is rather explicit, so it should be possible to check if the above vanishing holds. There's also the Totaro examples arxiv.org/abs/1710.04364 but checking here might involve more work | |
| Mar 6, 2019 at 11:40 | comment | added | Jason Starr | By Koll\'ar - Miyaoka - Mori, every Fano manifold is rationally chain connected. By the Bloch - Srinivas argument, there exists a "decomposition of the diagonal" after inverting an integer $N$ that Chatzistamatiou and Levine call the "torsion order". If $N$ is prime to $\text{char}(k)$, then regardless of liftings mod $p^2$, the cohomology groups above vanish. This is proved by Totaro in his article about stable irrationality of hypersurfaces. The torsion order divides all "enumerative 2-point, genus-0 Gromov-Witten invariants" and so can be bounded, cf. my work with Z. Tian and R. Zong. | |
| Mar 6, 2019 at 9:07 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
added 27 characters in body
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| Mar 6, 2019 at 9:05 | answer | added | Francesco Polizzi | timeline score: 2 | |
| Mar 6, 2019 at 1:38 | history | asked | pgraf | CC BY-SA 4.0 |