Is it possible to check two curves on birational equivalence by some computer algebra system? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:52:20Z http://mathoverflow.net/feeds/question/80433 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80433/is-it-possible-to-check-two-curves-on-birational-equivalence-by-some-computer-alg Is it possible to check two curves on birational equivalence by some computer algebra system? Maxim 2011-11-08T22:59:42Z 2011-11-09T12:22:48Z <p>I have two curves, for example hyperelliptic:</p> <p>\begin{align} &amp;y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\ &amp;y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1 \end{align}</p> <p>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)?</p> <p>It would be great if such system be free, but it is almost OK if It isn't.</p> http://mathoverflow.net/questions/80433/is-it-possible-to-check-two-curves-on-birational-equivalence-by-some-computer-alg/80440#80440 Answer by Jack Huizenga for Is it possible to check two curves on birational equivalence by some computer algebra system? Jack Huizenga 2011-11-09T00:33:03Z 2011-11-09T12:22:48Z <p>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.</p> http://mathoverflow.net/questions/80433/is-it-possible-to-check-two-curves-on-birational-equivalence-by-some-computer-alg/80456#80456 Answer by joro for Is it possible to check two curves on birational equivalence by some computer algebra system? joro 2011-11-09T06:46:41Z 2011-11-09T06:46:41Z <p>I suppose Magma's <code>IsIsomorphic</code> will do the job.</p> <p>From the <a href="http://magma.maths.usyd.edu.au/magma/handbook/text/1238#13406" rel="nofollow">documentation</a></p> <blockquote> <p>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. </p> </blockquote> <p>Example code in the <a href="http://magma.maths.usyd.edu.au/calc/" rel="nofollow">web calculator</a> </p> <pre><code>K&lt;x,y&gt;:=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 </code></pre> <p>Wish this is implemented in sage.</p>