Timeline for Deforming to decompose vector bundles
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2013 at 17:43 | comment | added | Jason Starr | Correction: Replace "semistable" in last sentence of previous comment by "isomorphic to locally free sheaf in the bounded family. For the corresponding set to be open, please assume the original bounded family is versal (which I am sure is a hypothesis anyway). | |
| May 29, 2013 at 22:08 | comment | added | Jason Starr |
Here's another argument. For any bounded family of vector bundles on $X$ of rank $r$ with fixed determinant $L\vee$, for every ample invertible sheaf $A^\vee$, for all integers $n$ sufficiently positive, every vector bundle $E$ in the family fits into a short exact sequence $$0\to L\otimes A^{\otimes n(r+1)} \to (A^{\otimes n})^{\oplus (r+1)} \to E \to 0.$$ Thus, there is a surjective morphism to the moduli space from an open subset of $\text{Hom}_{\mathcal{O}_X}(L\otimes A^{\otimes n(r+1)},(A^n)^{\oplus(r+1)})$ -- the open set parameterizing injective maps with semistable cokernel.
|
|
| May 29, 2013 at 16:24 | history | answered | David E Speyer | CC BY-SA 3.0 |