Is it possible to check two curves on birational equivalence by some computer algebra system? - MathOverflow

Is it possible to check two curves on birational equivalence by some computer algebra system?

2011-11-08T22:59:42Z

I have two curves, for example hyperelliptic:

\begin{align} &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\ &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1 \end{align}

Is it possible to check them on birational equivalence (is able one curve be birationally transformed to another?) via some computer algebra system (like GAP, Sage, Magma, Maple, Maxima or something)?

It would be great if such system be free, but it is almost OK if It isn't.

Answer by Jack Huizenga for Is it possible to check two curves on birational equivalence by some computer algebra system?

2011-11-09T00:33:03Z

Your example is a bit of a red herring, as this is relatively easy for hyperelliptic curves. A hyperelliptic curve can be reconstructed uniquely from the data of the branch divisor of the degree $2$ map to $\mathbb{P}^1$. Furthermore, isomorphisms of hyperelliptic curves commute with the degree $2$ map to $\mathbb{P}^1$. Thus for two hyperelliptic curves, the only issue is whether or not the branch divisors are projectively equivalent, and this is quite straightforward to check.

Answer by joro for Is it possible to check two curves on birational equivalence by some computer algebra system?

2011-11-09T06:46:41Z

I suppose Magma's IsIsomorphic will do the job.

From the documentation

IsIsomorphic(C, D) : Crv, Crv -> BoolElt,MapSch Given irreducible curves C and D this function returns true is C and D are isomorphic over their common base field. If so, it also returns a scheme map giving an isomorphism between them. The curves C and D must be reduced. Currently the function requires that the curves are not both genus 0 nor both genus 1 unless the base field is finite.

Example code in the web calculator

K<x,y>:=AffineSpace(Rationals(),2);
C1A:=Curve(K,x^10-1-y^2);
C2A:=Curve(K,x^10-2^10-y^2);
C1:=ProjectiveClosure(C1A);
C2:=ProjectiveClosure(C2A);
IsIsomorphic(C1,C2);

true Mapping from: CrvPln: C1 to CrvPln: C2
with equations :
-2*$.1
32*$.2
$.3

Wish this is implemented in sage.