Timeline for Vector Bundles on the Moduli Stack of Elliptic Curves
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2010 at 17:34 | comment | added | Angelo | The fact that vector bundles on $\mathcal M_{1,1}$ split as sums of line bundles in characteristic larger than 3 can also be seen by the classical description of $\mathcal M_{1,1}$ as an open substack of the weighted projective stack $\mathbb P(4,6)$, coming from the Weierstrass form. Any locally free sheaf on $\mathcal M_{1,1}$ extends to a reflexive sheaf on $\mathbb P(4,6)$, which is locally free, because $\mathbb P(4,6)$ is regular of dimension 1. It is not hard to prove that any locally free sheaf on a weighted projective stack $\mathbb P(m,n)$ splits as a direct sum of line bundles. | |
| Apr 9, 2010 at 16:48 | comment | added | Tyler Lawson | @Cardano: Sure. | |
| Apr 9, 2010 at 16:32 | comment | added | BCnrd | @Charles: Good point, I should have noticed the link with the non-liftability to char. 0 for the Hasse invariant when $p = 3$ (and its square when $p=2$). Thanks. | |
| Apr 9, 2010 at 16:10 | comment | added | Lennart Meier | @Tyler Lawson: you use the covering $\mathcal{M}(4) \to \mathcal{M}$ and some flatness criterion. I can send you a pdf file with some details if you like. | |
| Apr 9, 2010 at 15:33 | comment | added | Charles Rezk | You can also deduce that $Ext^1(O, \omega^2)$ is non-trivial from the fact that there are non-trivial "mod p" modular forms of weight 2 when p=2 or 3; these are computed (for instance) in Deligne's note on "formulas, after Tate" in Antwerp 4. | |
| Apr 9, 2010 at 15:07 | comment | added | Tyler Lawson | @Brian: The computation of the Ext-groups of tensor powers of $\omega$ on this moduli stack is written up in the Bauer paper that was linked to under part (1). | |
| Apr 9, 2010 at 14:49 | comment | added | BCnrd | Since the coarse space is affine and the automorphism groups of the geometric points have order a power of 2 times a power of 3, and the Ext-group (or $\omega^{-1}$ by $\omega$, to be precise) is identified with degree-1 coherent cohomology (of $\omega^2$), its nontriviality is also not visible over $\mathbf{Z}[1/6]$. So can you also briefly indicate how you know it is nontrivial? | |
| Apr 9, 2010 at 12:55 | comment | added | Tyler Lawson | How did you prove that every vector bundle is an extension? | |
| Apr 9, 2010 at 10:06 | history | asked | Lennart Meier | CC BY-SA 2.5 |