Timeline for Rank 3 Lagrangian vector bundles on an elliptic curve
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Apr 7, 2020 at 11:02 | history | bounty ended | CommunityBot | ||
| S Apr 7, 2020 at 11:02 | history | notice removed | CommunityBot | ||
| Mar 31, 2020 at 21:30 | answer | added | Libli | timeline score: 1 | |
| S Mar 30, 2020 at 9:07 | history | bounty started | Jef | ||
| S Mar 30, 2020 at 9:07 | history | notice added | Jef | Draw attention | |
| Mar 27, 2020 at 11:32 | comment | added | Enrico | By Mukai-style I mean the vector bundle method (see for example section 3 of library.msri.org/books/Book28/files/mukai.pdf or section 5 of "Algebraic Geometry V"). Finding a rank 3 bundle with 6 sections on $E$ gives you a morphism to $Gr(3,6)$. Then your original $E$ can be recovered by taking zero locus of an appropriate number of sections of $\bigwedge^i R^{\vee}$. Note that $LG(3,6)$ is itself the zero locus of $\bigwedge^2 R^{\vee}$ on $Gr(3,6)$. | |
| Mar 27, 2020 at 11:24 | comment | added | Enrico | The Lagrangian Grassmannian $LG(k, 2k)$ is a special case of the symplectic Grassmannian $SG(k,n)$ (basically I typed the comment in a hurry using the notation I am most used to, sorry). | |
| Mar 27, 2020 at 9:29 | comment | added | Jef | Can you explain why the problem you pose is related to the one given here, and what $SG(3,6)$ stands for? I'm not really familiar to Mukai-style constructions. | |
| Mar 26, 2020 at 13:46 | comment | added | Enrico | Reversing the problem (since you are using a Mukai--style construction), you may ask for the existence of an homogeneous bundle $F$ on $SG(3,6)$ of rank 5 and $c_1(F)=4$. The only non--degenerate example I can come up with is $F= \bigwedge^2 R^{\vee} \oplus \mathcal{O}(1) \oplus \mathcal{O}(1)$, where $R$ is the rank 3-tautological. This corresponds to an elliptic curve as double (multi)-linear section of $(\mathbb{P}^1)^3$. Otherwise you can use $R^{\vee}$ (or $Q$) as a bundle, but then the embedding is degenerate in $SG(3,6)$, and you should embed $E$ directly in a 3-dimensional quadric. | |
| Mar 26, 2020 at 12:47 | history | asked | Jef | CC BY-SA 4.0 |