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Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?

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    $\begingroup$ What is "the" nonsingular projective model? If you think of the canonical model, it lies in $\Bbb{P}^{g-1}$, which makes it untractable when the genus $g$ is not very small. $\endgroup$ – abx Nov 21 '14 at 7:14
  • $\begingroup$ Not sure if this is what the OP has in mind, but one could interpret this as asking for two affine charts and gluing data, which seems much more tractable than computing equations for the canonical model. $\endgroup$ – Daniel Litt Nov 21 '14 at 17:27
  • $\begingroup$ @DanielLitt, yes affine charts and gluing data would suffice. Is there a way to get that? $\endgroup$ – 352506 Nov 21 '14 at 18:26
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For a curve, desingularization means normalization.

If you can reduce yourself to the affine case, you can use the command normalization in Singular.

Another opition is the command integralClosure in Macaulay2.

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Magma has CanonicalMap and CanonicalImage, which can be used to compute the canonical model (which is smooth if the curve is not hyperelliptic). This should work for reasonable genus and size of coefficients. You can use IsHyperelliptic and IsGeometricallyHyperelliptic to get a hyperelliptic model (smooth in a weighted projective plane) when the curve is hyperelliptic.

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