# Software for surface parameterization/rationality?

Is there software (preferably open source) besides Magma for:

(1) Deciding if a surface is rational

(2) Compute a parameterization of the surface

And

(3) Relatively easy to code algorithm for (1) or (2).

I found the main paper for (3) Josef Schicho. Rational Parameterization of Surfaces', but this algorithm doesn't appear easy to code for me.

One can use IsRational(Scheme)' in the Magma web calculator - it timeouts for complicated examples because of limitation of 60 seconds of execution.

The motivation is just doing few numerical experiments.

EDIT Per M.P.'s request here are the surfaces I am currently interested in (non trivial rational points or parameterization)

2*x^4*y^3*z^3 - x^4*y^2*z^4 + x^2*y^4*z^4 + 2*x^4*y^3*z + 2*x^4*y^2*z^2 - 2*x^2*y^4*z^2 - 2*x^4*y*z^3 + 4*x^2*y^3*z^3 - x^4*y^2 + x^2*y^4 - 2*x^4*y*z + 4*x^2*y^3*z - 4*x^2*y*z^3 + 2*y^3*z^3 + x^2*z^4 - y^2*z^4 - 4*x^2*y*z + 2*y^3*z - 2*x^2*z^2 + 2*y^2*z^2 - 2*y*z^3 + x^2 - y^2 - 2*y*z = 0


Almost certainly the second is not rational:

x^4*y^2*z^4 + x^2*y^4*z^4 - 2*x^4*y^2*z^2 - 2*x^2*y^4*z^2 - 4*x^2*y^2*z^4 + x^4*y^2 + x^2*y^4 - 8*x^2*y^2*z^2 + x^2*z^4 + y^2*z^4 - 4*x^2*y^2 - 2*x^2*z^2 - 2*y^2*z^2 + x^2 + y^2 = 0

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## 1 Answer

A possible algorithm to decide if a surface $S$ is rational is to use Castelnuovo's Rationality Criterion: check that the both the irregularity and the second plurigenus of $S$ vanish. In classical notation, this is equivalent to the vanishing of $q(S)$ and of $P_2(S)$. In less classical terms, a necessary and sufficient condition for a smooth projective surface to be rational is the vanishing of ${\rm H}^1(S , \mathcal{O}_S)$ and of ${\rm H}^0(S , \omega_S^{\otimes 2})$, where $\omega_S$ denotes the canonical bundle of $S$. Both these conditions are algorithmically testable in terms of an explicit presentation by equations in projective space of your surface. I suspect that the implemented algorithm on MAGMA checks exactly this.

On the other hand, if your equations are complicated, then proceeding in this way might be very tricky, and there are lots of ad hoc ways of proving that a surface is rational that tend to work better in general: can you give us more details on the surfaces you are interested in?

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@M P Thanks! Gave 2 examples per your request. I am mainly interested in nontrivial rational points. –  jorox Aug 21 '11 at 10:59
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