Timeline for Connectedness of degeneracy loci
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 6 at 11:23 | answer | added | Jason Starr | timeline score: 0 | |
| Dec 5 at 21:06 | answer | added | Cob | timeline score: 3 | |
| Dec 5 at 10:37 | comment | added | Enrico | In general, you should be aware of relaxing these hypothesis too much. If you take $X=Gr(n,2n)$, $E= \mathcal{O}_G, F= Sym^2(U^*)$ (with $U$ the tautological bundle), you get a zero locus (in this case the only degeneracy loci) which has two connected components. In this case $ E^* \otimes F= Sym^2(U^*)$ is definitely globally generated, but I am not sure about its bigness. | |
| Dec 4 at 23:45 | comment | added | Jason Starr | I suspect this can be proved using Minoccheri’s connectedness theorem. I will try to write it up as an answer soon. | |
| Dec 4 at 17:14 | comment | added | Cob | @JasonStarr I am assuming that $E^* \otimes F$ is globally generated, hence yes, it is a quotient of $r$ copies of the structure sheaf ${\mathcal O}_X$. I am also assuming that $E^* \otimes F$ is big, in the sense that its tautological line bundle ${\mathcal O}(1)$ is big on ${\mathbb P}(E^* \otimes F)$. This implies that the determinant of $E^* \otimes F$ is big, hence the image of the associated morphism has dimension $n$, hence also the image of the morphism into the Grassmannian has dimension $n$. But, in general, it might not be birational. | |
| Dec 4 at 17:03 | comment | added | Cob | @J. W. Tanner Thanks! | |
| S Dec 4 at 16:57 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
I think I corrected spelling in title
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| Dec 4 at 16:38 | comment | added | Jason Starr | Do you assume that $E^*\otimes F$ is a quotient of a direct sum of a finite number (say $r$) of copies of the structure sheaf on a smooth, projective variety $X$? Do you assume that the induced morphism from $X$ to the Grassmannian $\text{Grass}(ef,r)$ is birational to its image? | |
| Dec 4 at 16:17 | review | Suggested edits | |||
| S Dec 4 at 16:57 | |||||
| Dec 4 at 14:39 | history | asked | Cob | CC BY-SA 4.0 |