Given a field of characteristic not 2 or 3 containing a primitive third root of unity, is it true that every nonsingular cubic curve, i.e. a curve defined by one homogeneous form of degree 3 in 3 variables, equivalent (up to a linear change of variables) to something of the form $X^3+b Y^3+c Z^3=d X Y Z$? If not then what is the necessary and sufficient condition for the cubic curve to be equivalent to something of this form? What is the canonical form for cubic curves?
1 Answer
Over the complex numbers, there are several normal forms. A ternary cubic can be written as $$ \lambda XYZ-(X+Y+Z)^3 $$ with a parameter $\lambda$. This normal form is invariant under the symmetric group $S_3$ of permutations of variables. This leads to the so-called Hesse normal form, given by $$ X^3+Y^3+Z^3-3\mu XYZ. $$ It has the property that the Hessian of any of its members is of the same form.
Hesse showed that the inflection points of a ternary cubic over the rationals are defined over a solvable extension. As a result, any ternary cubic can be brought to the Hesse normal form over a solvable extension of the base field. More precise details can be found in the reference: Ternary Cubic Forms and Etale Algebras, by Raczek and Tignol.