Cyclic coverings of $\mathbb{P^{1}}$ have a simple (affine) equation, namely the formula, $y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{n})^{t_{n}}$. Is there such a nice equation for abelian non-cyclic coverings of $\mathbb{P^{1}}$? At least in the case where the covering group $G= (\mathbb{Z}/m_{1}\mathbb{Z})\times .... \times (\mathbb{Z}/m_{r}\mathbb{Z})$ is there such a trim general equation for the abelian non-cyclic covers?
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3 Answers
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Can't you just write :
$y_1^{m_1}=(x-a_{11})^{t_{11}}\dots (x-a_{1n})^{t_{1n}}$
$\dots$
$y_r^{m_r}=(x-a_{r1})^{t_{r1}}\dots (x-a_{rn})^{t_{rn}}$
since every abelian cover is a fiber product of cyclic covers?
answered Jul 27, 2012 at 23:04
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$\begingroup$ I don't know, my own guess was that the formula should look like this. But is it a well known theorem that every abelian covers are fiber product of the cyclic ones? $\endgroup$– JackJul 28, 2012 at 8:14
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1$\begingroup$ Yes, since intermediate covers correspond to subgroups of the Galois group. Take the kernel onto the projection of each cyclic factor, and then take the fiber product. $\endgroup$ Jul 28, 2012 at 15:20
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There is a beautiful theorem of Rita Pardini regarding abelian covers. She gives a set of data that describes the cover. If I am not mistaken the relevant paper is this: R. Pardini Abelian covers of algebraic varieties J. Reine Angew. Math. 417 (1991), 191–213. (In principle this is available online, but the journal website timed out when I tried. I may include a link later, but you should be able to find it easily). I imagine that if the target is $\mathbb P^1$ then the computation is somewhat simplified.
There are many more nice results related to this, both regarding covers and applications. Just find this paper on MathSciNet and click the reference link.
answered Jul 27, 2012 at 20:04
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$\begingroup$ Dear Sandor-- Thank you very much for your hints, but as you predicted I could not find a useful link to this article (there were some, but I could not open them or dwonload) $\endgroup$– JackJul 28, 2012 at 10:05
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1$\begingroup$ Here is a link that schould work : digizeitschriften.de/en/dms/toc/?PPN=PPN243919689_0417 $\endgroup$– BS.Jul 28, 2012 at 14:19
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Dear Jack,
In general it is not possible to write down nice equations for abelian coverings of the Riemann sphere, but this is not the end of the world: as it is explained in details in Section 2 of this paper of Alex Wright (http://arxiv.org/abs/1203.2683), one can still reasonably understand such abelian coverings by putting our hands on them by analyzing all intermediate cyclic coverings.
Best,
Matheus
