Let's consider algebraic curves over fixed algebraicaly closed field $K$.

I's well known, that every smooth elliptic curve (genus $g = 1$) can be embedded in quadric surface in $\mathbb{P}^3$. This fact simply follows from Riemann–Roch theorem.

In more general, for smooth hyperelliptic curves of higher genus ($g \ge 2$) it's known that such curves can be embedded in quadric in weighted projective space $\mathbb{P}(1,1,g)$, see, for example, work of D. Eisenbud. (So, in the case $g=2$ we have embedding in $\mathbb{P}^4$).

But, does any smooth hyperelliptic curve $H$ can be embedded in quadric surface in $\mathbb{P}^3$?

It's natural question, because, by the definition we have mophfism $\phi:H \to \mathbb{P}^1$ of degree 2.

I think, this problem is connected with topics "Families of hyperelliptic curves and double covers of quadric surface" and Quotient Surface of A Hyperelliptic Involution, but I don't get it. (I'm interesting in the case of any algebraicaly closed field and in the case of finite field).