Let $C$ be a quartic plane curve. Suppose that for a given coordinate system 
$C=(F\_4(x\_0,x\_1,x\_2)=0)$ where there the polynomial $F\_4$ factorizes as  

$$
F\_4(x\_0,x\_1,x\_2) = F\_3(x\_0,x\_1,x\_2)F\_1(x\_0,x\_1,x\_2)-F\_2(x\_0,x\_1,x\_2)^2
$$
If the coordinate system is fixed. How many factorization are there? 


More generally, Let $Hilb\_d \cong \mathbb{P}^{Nd}$ be the Hilbert scheme of curves of degree $d$ in $\mathbb{P}^2$. We can define the map
$$
Hilb\_3 \times Hilb\_1 \times Hilb\_2 \to Hilb\_4
$$
as 
$$
\{ 
F\_3(x\_0,x\_1,x\_2), F\_2(x\_0,x\_1,x\_2), F\_1(x\_0,x\_1,x\_2)
\}
\to 
F\_3F\_1-F\_2^2
$$
What can I say about the fibers of the map? 


I noticed that if $F\_1(x\_0,x\_1,x\_2)$ is a bitangent of the curve $C$, then I can start playing around... However, this seems to be an elementary question, and maybe I am missing something obvious.



factorizing a quartic plane curve as $f_3f_1-f_2^2$
Yes, if the quartic $C : F\_4=0$ is smooth then the line $l : f\_1=0$
must be one of the $28$ bitangents (because the restriction $F\_4|\_l$
is the square of $f\_2|\_l$), and then $f\_2$ restricted to $l$
is one of the square roots of $F\_4|\_l$, and each of the lifts of $f\_2|\_l$
lets you solve for $f\_3$.


Note that writing $F\_4$ as $f\_1 f\_3 - f\_2^2$ is not a factorization,
and a parameter count would already indicate that there should be
infinitely many such expressions: there are $3+6+10 = 19$ undetermined
coefficients in $f\_1,f\_2,f\_3$, and only $15$ coefficients to match in $F\_4$.
But there's also a $4$-dimensional group acting: multiply $f\_1,f\_3$ by
$\lambda,\lambda^{-1}$, or change $f\_2,f\_3$ to $f\_2 + g f\_1$ and
$f\_3 + 2 g f\_2 + g^2 f\_1$ for any linear form $g$.
So we expect finitely many solutions up to these transformations,
and indeed we get the $28$ pairs described in the first paragraph.


(Likewise for $F\_4 = b^2-ac$ with $a,b,c$ each of degree $2$
we have $2^6 - 1 = 63$ three-dimensional families, indexed by the nontrivial $2$-torsion
divisor classes $D$ in the Jacobian of $C$; the function $a/b=b/c$ on $F\_4$
is the quotient of two sections of $D+K$ where $K$ is the canonical divisor.)


