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

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

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


>But, can any smooth hyperelliptic curve $H$ be embedded in a 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"][2] and [Quotient Surface of A Hyperelliptic Involution][3], but I don't get it. (I'm interested in the case of any algebraicaly closed field and in the case of finite field).


  [1]: http://www.msri.org/~de/papers/pdfs/1980-004.pdf
  [2]: https://mathoverflow.net/questions/61005/families-of-hyperelliptic-curves-and-double-covers-of-quadric-surface
  [3]: https://mathoverflow.net/questions/59843/quotient-surface-of-a-hyperelliptic-involution