(partly motivated by this question, but different: [Degree of a Variety](https://mathoverflow.net/questions/63707/degree-of-a-variety))


For a hyperelliptic curve $C$ of genus $g$ (over an algebraically closed field of characteristic not two) what is the smallest $d$ for which $C$ can be embedded in some $\mathbb{P}^n$ (I guess $n=3$ wlog) as a **smooth** curve of degree $d$? Does it depend only on $g$? Can anything be said for an arbitrary curve?



What is the minimal degree of a smooth projective embedding of a hyperelliptic curve?
Felipe, I believe the answer here is d=g+3. To see that you can embed your curve in this degree is straightforward - just choose a generic line bundle of degree g+3 and it will work.


In the case of hyperelliptic curves, I don't think you can do better. The key point is that any special linear series on a hyperelliptic curve comes from taking a multiple of the pullback of O(1) on P^1 together with some base points. (You can find this fact in Arbarello-Cornalba-Griffiths-Harris.) Since these cannot give rise to embeddings (either have base points or the associated map factors through the hyperelliptic involution) we conclude the the embedding line bundle has no H^1. The Riemann-Roch gives g+3 as the lower bound for having 4 sections.


