Timeline for Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 12, 2020 at 9:01 | vote | accept | DDT | ||
| May 9, 2020 at 19:26 | comment | added | Abdelmalek Abdesselam | @WillSawin: Sorry I misread and missed the part about composition. | |
| May 9, 2020 at 19:22 | comment | added | Will Sawin | @AbdelmalekAbdesselam Sorry, I should have been more clear. There are actually two maps from biforms of bidegree $(2,2)$ to binary quartics which preserve the $j$ invariant. These are given by taking the discriminant in one of the two variables. They were mentioned as the "natural map" by Noam. To see what your map does to the $j$ invariant, we can compose it with either of them. Doing this gives the Hessian. Thus your map preserves $j$ if and only if the Hessian does. (But it could still help somehow even if it does not preserve $j$.) | |
| May 9, 2020 at 19:12 | comment | added | Abdelmalek Abdesselam | @WillSawin: my map is linear it sends the binary quartic $f(x_1,x_2)$ to $D^2 f$ where $D$ is the differential operator $y_1\frac{\partial}{\partial x_1}+y_2\frac{\partial}{\partial x_2}$. The result is a biform in $x$ and $y$ of bidegree $(2,2)$. What you wrote sounds more like the Hessian of $f$ which is another binary quartic. Sorry if my notations were confusing. | |
| May 9, 2020 at 19:07 | comment | added | Will Sawin | @AbdelmalekAbdesselam The composition of your map with Noam's is quadratic - it sends a binary quartic $f$ to $ (d^2f /dx_1^1) (d^2 f/dx_2^2) - (d^2 f/dx_1 dx_2)^2$, I think - so it shouldn't preserve the $j$ invariant. | |
| May 9, 2020 at 18:42 | history | edited | Noam D. Elkies | CC BY-SA 4.0 |
exhibit $j$ as a rational function of $J_2,J_3,J_4$; various local copy-edits
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| May 8, 2020 at 15:29 | comment | added | Noam D. Elkies | I'm not sure. (In any case the natural map goes the other way, from the moduli of $(2,2)$ curves to moduli of elliptic curves.) I did use the invariant theory of binary quartics to write the elliptic curve in terms of $J_2,J_3,J_4$. The $j$-invariant can be computed from my answer -- e.g. ellinit([0,-2*J2,0,J2^2-4*J4,J3^2]).j in gp -- though inevitably it's not very pretty. | |
| May 8, 2020 at 13:48 | comment | added | Abdelmalek Abdesselam | Nice answer. Does my polarization argument in my comments above help at all with the $j$ invariant question? It seems to provide an embedding of the moduli space of elliptic curves inside the OP's moduli space, but I don't know if the two ways computing $j$ coincide on this locus. | |
| May 8, 2020 at 1:22 | history | answered | Noam D. Elkies | CC BY-SA 4.0 |