incidence correspondence of bitangents In the accepted answer to this question, an example of an incidence correspondence was given as:
$$ I =\{(Q,L) : L \text{ is bitangent to } Q\} \subset \mathbb{P}^{16}.$$
However I did not understand why $I$ is a closed subset of $\mathbb{P}^{16}$, so why we can express the condition "$L$ is bitangent to $Q$" through a set of homogeneous polynomials in the coefficients of $Q$ and $L$.
This is what I tried so far:
Suppose that $L$ is given by the equation $ax + by + cz = 0$, then suppose $c = 1$ so $z = -ax - by$ on $L$.
Then if $Q$ is given by the quartic polynomial $f$, the restriction of $f$ to the line $L$ must have two double points (the bitangent points), so $f(x,y,-ax-by) = (\alpha x + \beta y)^2 (\gamma x + \delta y)^2$ for some $\alpha,\beta,\gamma,\delta$. Then if you expand out the right hand side, you get 5 equations in terms of the coefficients of $f$ and of $a,b,\alpha,\beta,\gamma,\delta$. Then if you eliminate $\alpha, \beta,\gamma,\delta$ you get an equation solely in terms of the coefficients of $f$ and $a,b$.
Then repeat this process again for $a = 1$ (so $x = -by-cz$) and $b = 1$ (so $y = -ax - cz$), and take the union of all equations that you get.
But I do not know if the equations that we get by eliminating $\alpha,\beta,\gamma,\delta$ are homogeneous.
Also I was wondering if $L$ being bitangent to $Q$ for this incidence correspondence means that $L$ and $Q$ must have two distinct points of contact $P_1$ and $P_2$, or if these points may coincide.
 A: Let's define incident variety 
$$ J =\{(Q,L,x,y):L \text{ is tangent to } Q \text{ at} \  x, y \} \subset \mathbb{P}^{14}\times (\mathbb P^{2})^*\times \mathbb P^2\times \mathbb P^2.$$
This is the common zero locus of the set of equations $x\in Q, x\in L, L\subset T_xQ$ together with the same set of equations on $y$, so it is closed. 
$J=J_1\cup J_2$ has two irreducible components. The general element of $J_1$ is a line tangent at two distinct points on $Q$, and $J_2$ is the set of the triple $(Q,L,x)$ such that $L$ tangent to $Q$ at $x$. By projection to the $Q$-factor, it is not hard to compute $\dim J_1=14$ and $\dim J_2=15$.
Now, the incidence variety 
$$ I =\{(Q,L) : L \text{ is bitangent to } Q\} \subset \mathbb{P}^{14}\times (\mathbb P^{2})^*$$
can be seen as projection of $J_1$ onto the first two coordinates.
Lastly, when the two bitangent points come together, the tangent line has intersection multiplicity 4 with the quartic curve. It is called a hyperflex (for example see this paper). These points are special bitangents and forms a codimension one locus in $J_1$.
