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This has been answered at MO.

I am interested what is the type of the surfaces over the rationals $$ x^5 - y^5 + z^2 + x=0$$

and

$$ x^5 - y^5 + z^2 + x+1=0$$

Magma's KodairaEnriquesType(S : CheckADE:=true); fails to compute it.

According to Magma they are not rational.

Partial answers (e.g. it is not $X$) or approaches how to compute with a CAS are welcome.

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  • $\begingroup$ have you tried Wolfram Alpha? try taking $z=+\sqrt{y^5-x^5-x}$ to start with $\endgroup$ Commented Aug 18, 2015 at 9:43
  • $\begingroup$ @DavidQuinn No. What is the function for surface classification? $\endgroup$
    – joro
    Commented Aug 18, 2015 at 9:54
  • $\begingroup$ A small comment: you can easily see by hand that this thing does not have ADE singularities. Indeed, its projectivisation $x^5-y^5+z^2t^3+xt^4$ has one singular point, at $x=y=t=0$, and this is a triple point, whereas ADE singularities are double points. $\endgroup$
    – Schemer
    Commented Aug 18, 2015 at 15:09
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    $\begingroup$ Anyway, these questions seem surprisingly difficult and frustrating. I guess there is no slick way to do this for arbitrary singular surfaces --- if there was, it would probably already be implemented in computer algebra! $\endgroup$
    – Schemer
    Commented Aug 18, 2015 at 15:11
  • $\begingroup$ @Relapsarian Thanks. This well might be a Magma bug, do you need exact code for Magma Online? This might be not be implemented in Magma yet and the documentation to be not entirely correct. $\endgroup$
    – joro
    Commented Aug 18, 2015 at 15:15

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