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Transforming genus zero curve to conic

asked 2023-10-05 18:46:12 +0200

azerbajdzan gravatar image

updated 2023-10-09 18:00:01 +0200

The following code outputs rational parametrization of genus 0 curve of degree 5:


x, y = QQ['x,y'].gens()
C=Curve(2*x^5 + x^2*y - 4*x^3*y + 2*x*y^2 + 2*x*y^3 + 1*y^5)
print(C.is_smooth())
print(C.genus())
C.rational_parameterization()

False
0
Scheme morphism:
  From: Affine Space of dimension 1 over Rational Field
  To:   Affine Plane Curve over Rational Field defined by 2*x^5 + y^5 - 4*x^3*y + 2*x*y^3 + x^2*y + 2*x*y^2
  Defn: Defined on coordinates by sending (t) to
        ((-2*t^2 - t)/(4*t^5 + 1), (4*t^4 + 2*t^3)/(4*t^5 + 1))

Is there a function that instead of parametrization outputs transformation x -> f1(u, v), y -> f2(u, v) so that the curve is transformed into a conic section curve (any curve of degree 2)?

The curve would look like this with given coefficients a1..a6.

a1*u^2 + a2*v^2 + a3*u*v + a4*u + a5*v + a6

UPDATE:

Here is an example of a curve of degree 3 and genus 0 that I was able to transform to conic - specifically parabola (with forward and backward transformations):

$$y^2=x^3-x^2,\left(x\to \frac{v}{u^2},y\to \frac{v}{u^3}\right)\\ v=u^2+1,\left(u\to \frac{x}{y},v\to \frac{x^3}{y^2}\right)$$


UPDATE2:

$$2 x^5-4 x^3 y+x^2 y+2 x y^3+2 x y^2+y^5=0,\left(x\to -\frac{8 u v^3-u+4 v^3-2 v}{16 v^5-1},y\to \frac{2 v \left(8 u v^3-u+4 v^3-2 v\right)}{16 v^5-1}\right)\\v-u^2=0,\left(u\to \frac{-2 x^5+2 x^3 y-x y^3-y^5}{2 x \left(x^3+y^3\right)},v\to -\frac{y}{2 x}\right)$$

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How the requested transformation is related to the rational parametrization?

Max Alekseyev gravatar imageMax Alekseyev ( 2023-10-06 00:22:59 +0200 )edit

If there does not exist rational parametrization of a curve then it is impossible to transform it birationally to conic curve. So this is the relation. I also think that from the rational parametrization it is possible to compute the transformation to conic, but I have not figured it out yet.

azerbajdzan gravatar imageazerbajdzan ( 2023-10-06 11:29:43 +0200 )edit

It does not look like just a technical task. You'd better ask a question about a suitable algorithm at https://math.stackexchange.com or alike.

Max Alekseyev gravatar imageMax Alekseyev ( 2023-10-06 13:58:08 +0200 )edit

It is likely that the function is already implemented in sage or that combination of two or three functions would produce the result, but it is hard to search the documentation with lots of functions. Wonder if there is some AI that would give reasonable answer to this question.

azerbajdzan gravatar imageazerbajdzan ( 2023-10-06 21:56:29 +0200 )edit

I see that you got a detailed answer at https://math.stackexchange.com/q/4784138 Now you can ask someone here to translate given Magma code to Sage.

Max Alekseyev gravatar imageMax Alekseyev ( 2023-10-11 02:10:43 +0200 )edit

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answered 2023-10-06 19:12:18 +0200

dan_fulea gravatar image

In case of an existing rational parametrization of the curve you need only one parameter, call it t - not two parameters, u, v as in the question. (A curve is something with dimension one.) So in our case:

R.<x,y> = PolynomialRing(QQ)
C = Curve(2*x^5 + x^2*y - 4*x^3*y + 2*x*y^2 + 2*x*y^3 + 1*y^5)    # please use spaces around the = sign
par = C.rational_parameterization()    # each time i type parametrization instead...

Now we can ask for par.domain() and par.codomain()

sage: par.domain()
Affine Space of dimension 1 over Rational Field
sage: par.codomain()
Affine Plane Curve over Rational Field defined by 2*x^5 + y^5 - 4*x^3*y + 2*x*y^3 + x^2*y + 2*x*y^2

There are many other things in there. But you want the parametrization, computed in a point $t$ (instead of a tuple / double (u, v)). We take this t as it comes and ask...

sage: par[0]
(-2*t^2 - t)/(4*t^5 + 1)
sage: par[1]
(4*t^4 + 2*t^3)/(4*t^5 + 1)
sage: list(par)
[(-2*t^2 - t)/(4*t^5 + 1), (4*t^4 + 2*t^3)/(4*t^5 + 1)]
sage: par(17)
(-85/811347, 49130/811347)

So the list conversion of the object par gives already the components. Note that t is not really defined, the print just involves it. So if we want to have the components in "our" (other) special ring with unknown / transcendental variable $T$...

sage: S.<T> = PolynomialRing(QQ)    # or use some t from somewhere else
sage: u, v = par
sage: u(T)
(-1/2*T^2 - 1/4*T)/(T^5 + 1/4)
sage: v(T)
(T^4 + 1/2*T^3)/(T^5 + 1/4)
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1

It does not seem to answer the question. Where is the conic in your answer?

Max Alekseyev gravatar imageMax Alekseyev ( 2023-10-06 19:25:46 +0200 )edit

@dan_fulea: Hello, I updated question with an example of degree 3 curve that was transformed to parabola. Can you do the same with my original equation of degree 5?

azerbajdzan gravatar imageazerbajdzan ( 2023-10-08 22:21:10 +0200 )edit

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Asked: 2023-10-05 18:46:12 +0200

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Last updated: Oct 09 '23