Do all complex curves of genus two admit a map to the projective line with the following properties? Degree is 16; there are 5 points with ramification index 3, and 6 points with ramification index 5. If not all of them admit such a map, is there an explicit description of those that do?
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$\begingroup$ Why do you want to have such a strange map? $\endgroup$– SashaCommented Feb 25, 2018 at 14:09
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$\begingroup$ 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 $28$. $\endgroup$– Jason StarrCommented Feb 25, 2018 at 15:18
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$\begingroup$ 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. $\endgroup$– Jason StarrCommented Feb 25, 2018 at 15:35
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$\begingroup$ 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. $\endgroup$– user105346Commented Feb 26, 2018 at 13:09
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