Timeline for Bounds on Betti numbers of subvarieties?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2017 at 19:26 | comment | added | Mikhail Bondarko | If the codimension of $X$ in $\mathbb{CP}^n$ is "small" (less than $n/3$) then the lower cohomology of $X$ is isomorphic to that of $\mathbb{CP}^n$ (by Barth's theorem); hence there is a simlar statement for its higher cohomology (given by duality). However, I don't know whether a result that tells nothing about the "middle" cohomology is interesting to you. | |
| May 27, 2017 at 6:09 | vote | accept | dhy | ||
| May 26, 2017 at 16:06 | comment | added | Jason Starr | I believe their method gives a bound on the Betti numbers for any projective manifold $X^n\subset \mathbb{P}^N$ in terms of $N$, $n$, and the least degree $d$ such that for the ideal sheaf $\mathcal{I}$, $\mathcal{I}(d)$ is globally generated. The bound should be a polynomial in $d$ whose degree is a linear function of $N$ and $n$. | |
| May 26, 2017 at 15:25 | comment | added | dhy | @JasonStarr Does the Milnor-Thom approach provide a bound for any subvariety? I was under the impression that their theorem only covered the cases of hypersurfaces. | |
| May 26, 2017 at 13:32 | comment | added | Jason Starr | There is a bound on the sum of the Betti numbers due to Thom and Milnor. | |
| May 26, 2017 at 12:26 | comment | added | Libli | might be the case | |
| May 26, 2017 at 12:20 | comment | added | dhy | @Libli: The issue is that the Hilbert scheme may become highly disconnected when one removes points corresponding to singular varieties - I admittedly don't have an example of the top of my head, but I would be extremely surprised if one did not exist. | |
| May 26, 2017 at 11:39 | comment | added | Libli | By the way, a result of Hartschorne says that the Hilbert scheme of subschemes of projective space is connected (see : numdam.org/article/PMIHES_1966__29__5_0.pdf). So, if I am not mistaken, the total Betti number should not vary too much along the Hibert scheme... | |
| May 26, 2017 at 11:07 | answer | added | Libli | timeline score: 17 | |
| May 26, 2017 at 10:26 | history | asked | dhy | CC BY-SA 3.0 |