Timeline for Some rational functions on genus two curves
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2018 at 13:09 | comment | added | user105346 | Thank you. Yes, this is indeed my convention. I realized that I actually need an extra condition: all 5 points with ramification index 3 being mapped to the same point in P^1. My motivation: a possible counterexample to the Jacobian Conjecture. | |
| Feb 25, 2018 at 15:35 | comment | added | Jason Starr | I just realized that your convention for ramification index is that "simple ramification" is "ramification index $2$", rather than "ramification index $1$". With that convention, the codimension of the ramification conditions equals $5\times 1 + 6\times 3 = 23$, not $34$. Thus, you do expect a $\mathfrak{g}^1_{16}$ with ramification indices at least as large as you specify. Probably you can prove this by computing the degree of the locus in $\mathcal{G}^r_d(C)$ via Thom-Porteous, as in the thesis of Rebecca Lehman. | |
| Feb 25, 2018 at 15:18 | comment | added | Jason Starr | A parameter count suggests the answer is "no". For every smooth, projective curve $C$ of genus $g$, for every $d>2g-2,$ the parameter space $\mathcal{G}^r_{d}(C)$ of $\mathfrak{g}^r_{d}$ linear series on $C$ has dimension equal to the expected dimension, $\rho(g,r,d)=g+(r+1)(d-g-r)$, (the space is empty if $g+r>d$). For $g=2$, $r=1$, and $d=16$, this is $\rho(2,1,16)=28$. On the other hand, for every $d\gg 0$, the ramification conditions you impose on a $\mathfrak{g}^1_d$ are independent of total codimension $5\times 2 + 6\times 4 = 34$. Finally, $34$ is greater than $28$. | |
| Feb 25, 2018 at 14:09 | comment | added | Sasha | Why do you want to have such a strange map? | |
| Feb 25, 2018 at 2:14 | review | First posts | |||
| Feb 25, 2018 at 8:09 | |||||
| Feb 25, 2018 at 2:12 | history | asked | user105346 | CC BY-SA 3.0 |