Can someone point me to literature about the curve defined by $F(x,y,z):=(x+y+z)^3-27xyz$? I'm sure this curve must be well-studied, due to the remarkable property that $$ F(x^3,y^3,z^3) = \prod_{\alpha^3=\beta^3=1} (x+\alpha y+\beta z). $$ But I'm not sure how to search for information about this. Google tells me that this factorization is often assigned for homework in algebra classes, but I'm interested in the geometry of this curve.
$\begingroup$
$\endgroup$
7
-
1$\begingroup$ What would you like to know about the geometry of this curve? How did you become interested in that particular curve? $\endgroup$– André HenriquesMay 4, 2014 at 4:23
-
1$\begingroup$ This curve is of genus 0, isomorphic to $3X^2+Y^2-19X^2$? $\endgroup$– ConderMay 4, 2014 at 4:30
-
1$\begingroup$ A random sighting: Theorem 5b in "On the Foci of Circumparabolas" Forum Geometricorum Volume 11 (2011) 57–63. No special name was given, just saying "the cubic C" given by... $\endgroup$– ConderMay 4, 2014 at 4:39
-
2$\begingroup$ I guess it parametrised by $(t^3:(1-t)^3:-1)$, and the equivalent form is thus $X^{1/3}+Y^{1/3}+Z^{1/3}=0$. See for instance books.google.com/books?id=LG0NHn7bPz0C&pg=PA209 for some discussion of the $(X+Y+Z)^3=27\rho XYZ$ family. $\endgroup$– ConderMay 4, 2014 at 4:45
-
1$\begingroup$ Thanks for the references, especially to du Val's book. It looks like the pencil of cubic curves $(x+y+z)^3=mxyz$ is closely related to the Hesse pencil $x^3+y^3+z^3=cxyz$. This definitely gets me started towards tracking down what's known, thanks! $\endgroup$– Michael ZieveMay 4, 2014 at 5:56
|
Show 2 more comments