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Francesco Polizzi
Francesco Polizzi
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Euqations Equations for abelian coverings of $\mathbb{P^{1}}$

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Jack
Jack
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Cyclic coverings of $\mathbb{P^{1}}$ have thea 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?

Cyclic coverings of $\mathbb{P^{1}}$ have the 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?

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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Jack
Jack
  • 637
  • 3
  • 10

Euqations for abelian coverings of $\mathbb{P^{1}}$

Cyclic coverings of $\mathbb{P^{1}}$ have the 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?