We fix a binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree $d = 2g+2$, and consider the hyperelliptic curve

$$C_F: \displaystyle z^2 = F(x,y).$$

We say that a point $(x,y,z)$ is a *quadratic point* if there exists a quadratic extension $K/\mathbb{Q}$ over which $x,y,z$ are defined.

Then $C_F$ is easily seen to have infinitely many quadratic points: indeed, for any *rational integers* $x,y$ we obtain the quadratic point $(x,y, \sqrt{F(x,y)})$. These are the so-called obvious quadratic points. If we take the height of the point $(x,y,z) \in C_F$ to simply be the Weil height of the point $[x:y]$ on $\mathbb{P}^1$, then counting the obvious quadratic points of $C_F$ having bounded height $X$ is tantamount to counting the number of co-prime integers $x,y$ such $\max\{|x|, |y|\} \leq X$.

Do we expect this count to be dominant? That is, do the *non-obvious* quadratic points on $C_F$, i.e., those where $x,y,z$ are all in some quadratic extension and at least two of them not in $\mathbb{Q}$, more/less/equally dense than the obvious points?