Broadly I would like to know what is the connection between homogeneous binary quartic forms and elliptic curves.
I see that the invariants on the first space under the action of $SL(2,\mathbb{C})$ are precisely the functions $g_2$ and $g_3$. Also given such a quartic form on $\mathbb{C}^2$, if one thinks its 0-set as the intersection of two conics in $\mathbb{C}^3$ then the determinant of the complex symmetric matrix representing the most general conic passing through this intersection is precisely the "RHS" of the elliptic curve equation.
What does this determinant mean geometrically?
More specifically consider the $SL(2,\mathbb{C})$ action on $\mathbb{C}^2$ (spanned by $x$ and $y$) and define $U = x^2$, $V = 2xy$ and $W=y^2$.
If the action is by the matrix $ \left [ \begin{array}{c c } a & b \\ c & d \\ \end{array}\right ]$ ($ad-bc=1$) of $SL(2,\mathbb{C})$ then the following transformation is affected on $\mathbb{C}^2$,
$$ \left [ \begin{array}{c} x\\ y\\ \end{array} \right ] \rightarrow \left [ \begin{array}{c} ax + by\\ cx + dy\\ \end{array} \right ] $$
This in turn implies the following transformation on $\mathbb{C}^3$ (defining the matrices, $X$ and $A$),
$$X = \left [ \begin{array}{c} U\\ V\\ W \end{array} \right ] \rightarrow A.X = \left [ \begin{array}{c c c} a^2 & ab & b^2\\ 2ac & (ad+bc) & 2bd \\ c^2 & cd & d^2\\ \end{array} \right ] \left [ \begin{array}{c} U\\ V\\ W \end{array} \right ] $$
$4UW-V^2$ being identically equal to $0$ is obviously kept invariant by the above transformation.
The invariance of $4UW-V^2$ can be thought of as $SL(2,\mathbb{C})$ preserving an inner-product on $\mathbb{C}^3$ with signature, $\left [ \begin{array}{c c c} & & 2 \\ & -1 & \\ 2 & & \\ \end{array} \right ]$
- But $A$ does not seem to be an orthogonal matrix though it satisfies the property of $det(A)=1$. Then why is it being said in the books that the $SL(2,\mathbb{C})$ action on $\mathbb{C}^2$ (spanned by $x$ and $y$) induces an action of $SO(3)$ on $\mathbb{C}^3$ (spanned by $U$, $V$ and $W$) with respect to the inner product $ \left [ \begin{array}{c c c} & & 2 \\ & -1 & \\ 2 & & \\ \end{array}\right ]$ ?