The other component arises if you "flip" the smallest rhombus, collapsing the blue point to one of the other joints.pic.twitter.com/Jaew3aF7Vo
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The other component arises if you "flip" the smallest rhombus, collapsing the blue point to one of the other joints.pic.twitter.com/Jaew3aF7Vo
And unless I made a typo, the curve seems to be genus 3, unfortunately: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens() qc = QuarticCurve(x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4) qc.genus() ## result: 3
But now I'm a bit confused (and out of my depth): qc.geometric_genus() ## result: 0
Yes, that's the sort of thing I feared: there's probably a lot of fine print in Sage's genus computing commands about what it computes exactly and how. (Me, I can't even ever remember which is which between geometric and arithmetic genus, so…)
I believe the geometric genus is the one that matters re: parametrization; for some reason the QuarticCurve.genus() method gives the arithmetic genus (contradicting the behavior of Curve.genus(), of course...). Anyways, it turns out Sage has Curve.rational_parametrization() ...
And the output is: Scheme morphism: From: Projective Space of dimension 1 over Rational Field To: Projective Plane Curve over Rational Field defined by x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4 ...
Defn: Defined on coordinates by sending (s : t) to (54*s^4 - 36*s^2*t^2 - 16*s*t^3 - 2*t^4 : -63*s^4 - 84*s^3*t - 26*s^2*t^2 - 20*s*t^3 + t^4 : 45*s^4 - 12*s^3*t + 22*s^2*t^2 + 4*s*t^3 + 5*t^4)
A final picture: parametric_plot( (2*(27 - 18*t^2 - 8*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4), -(63 + 84*t + 26*t^2 + 20*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4) ), (t,-1e5,1e5), plot_points=1e6)pic.twitter.com/2Wt2w12BBE
Just for fun, here's a "conformal neighborhood" of the curve. The small gap in the upper left is where |t|>100. I'm impressed that you guys could come up with both implicit and parametric formulae for the curve. I didn't know the latter was even possible.pic.twitter.com/r0pH4tRMSL
We got lucky, I think. The algebraic curve just happened to be genus 0 which means that it's secretly a (projective) line in disguise: https://en.wikipedia.org/wiki/Algebraic_curve#Rational_curves … I do wonder whether we could have figured out the genus from the linkage or some other way without relying on Sage.
Maybe there's an argument that uses the combinatorics of the linkage to get the degree of the polynomial in x&y; then we observe from geometry that one irreducible component is a circle (the "collapsed" picture above) and that what's left over must be a quartic plane curve...
Then we would "just" have to be able to say something about the singularities of that leftover quartic curve, which maybe one can do from looking at the linkage carefully?
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