Timeline for Gonality of specific Riemann surfaces $y^k=\tfrac{z^k-1}{z^k+1}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Dec 6, 2023 at 14:06 | history | bounty ended | CommunityBot | ||
| S Dec 6, 2023 at 14:06 | history | notice removed | CommunityBot | ||
| Nov 30, 2023 at 8:16 | answer | added | Sebastian | timeline score: 4 | |
| Nov 28, 2023 at 12:56 | history | edited | Sebastian |
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| S Nov 28, 2023 at 12:56 | history | bounty started | Sebastian | ||
| S Nov 28, 2023 at 12:56 | history | notice added | Sebastian | Draw attention | |
| Nov 27, 2023 at 6:47 | comment | added | Sebastian | I want the Riemann surface to be compact (otherwise, the above definition of gonality does not make sense). I therefore add these points, in the analogous way you do add two points for hyperelliptic curve determined by the algebraic equation $y^2=(z-z_1)\dots (z-z_{2g+2})$. Note that these curves are (in general) not smooth in $\mathbb CP^2$, as their genus $g=(k-1)(l-1)$ is quite arbitrary. | |
| Nov 27, 2023 at 5:44 | comment | added | Daniel Asimov | Is that because you want your surfaces to be projective, i.e., complex submanifolds of ℂℙ^2. or something else? | |
| Nov 27, 2023 at 1:24 | comment | added | Sebastian | Of course, the equations define a complex 1-dimensional submanifold in $\mathbb C^2$, but then you add some points ($k$ many over $v=\infty$ and $l$ many over $u=\infty$) to obtain the compact Riemann surface $\Sigma_{k,l}$. | |
| Nov 26, 2023 at 18:43 | comment | added | Daniel Asimov | Your algebraic equation appears to define your surface 𝜮_𝒌,𝒍 as a submanifold of ℂ^2. Is that right? | |
| Nov 26, 2023 at 10:58 | history | asked | Sebastian | CC BY-SA 4.0 |