Timeline for Is $\mathbb{C}[x^{\pm 1},y]/\langle y^3-(x^2+ax+b) \rangle$ a $n$-point ring?
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| Nov 6, 2018 at 20:47 | comment | added | Felipe | Hello! I'm still thinking about this question. Q1- When I go back to my first question, in the case $y^3=(x^2+ax+b)$, without using the notion of genus, how could I see that it isn't $n$-point? It looks like that the Mason-Stothers doesn't help us here. Q2- Using the notion of genus, the fact that $Frac(R)$ and $\mathbb{C}(t)$ are not isomorphic ensure us that if I'm in the case $y^m=p(t)$ (a superelliptic curve with $m\geq 3$) it has genus $\neq 0$? Q3- Do you have recommendations about exactly this topic that could help a beginner like me? | |
| Mar 9, 2018 at 12:19 | comment | added | David E Speyer | A pole of a rational function is an input which makes the rational function infinite. Factor $q(t) = \prod_{\alpha} (t-\alpha)^{m_{\alpha}}$ and $s(t) = \prod_{\alpha} (t-\alpha)^{n_{\alpha}}$. Then $y^2 - x^3=1$ forces $2m_{\alpha} = 3n_{\alpha}$. So $m_{\alpha}= 2 k_{\alpha}$ and $n_{\alpha}= 3 k_{\alpha}$ for some $k_{\alpha}$. Then $q(t) = \prod_{\alpha} (t-\alpha)^{3k_{\alpha}}$, $q(t) = \prod_{\alpha} (t-\alpha)^{3 k_{\alpha}}$ and we have $q=d^3$, $s=d^2$ with $d(t) = \prod_{\alpha} (t-\alpha)^{k_{\alpha}}$. | |
| Mar 9, 2018 at 10:26 | comment | added | Felipe | Thank you for answer. It is helping me a lot have this discussion here since I'm not an expert in algebraic geometry. I'm trying to understand better that to work on representations of Lie algebras. You claimed that there is some polynomial $d$ such that $q(t)=d(t)^3$ and $s(t)=d(t)^2$. Is this a Theorem or you choose $q$ and $s$ in order to satisfy this condition? What do you main about “Compairing poles”? | |
| Mar 8, 2018 at 18:32 | comment | added | David E Speyer | The field of rational functions on $\mathbb{CP}^1$ is $\mathbb{C}(t)$. You defined $n$-point rings as subrings of this. | |
| Mar 8, 2018 at 17:13 | comment | added | Felipe | Thank you for your so detailed answer. Could you just clarify why any $n$-point ring needs to be a sub ring of $\mathbb{C}(t)$? | |
| Mar 8, 2018 at 10:20 | vote | accept | Felipe | ||
| Mar 5, 2018 at 22:33 | history | answered | David E Speyer | CC BY-SA 3.0 |