What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:11:56Z http://mathoverflow.net/feeds/question/63745 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63745/what-is-the-minimal-degree-of-a-smooth-projective-embedding-of-a-hyperelliptic-cu What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? Felipe Voloch 2011-05-02T21:53:15Z 2011-05-02T23:41:07Z <p>(partly motivated by this question, but different: <a href="http://mathoverflow.net/questions/63707/degree-of-a-variety" rel="nofollow">http://mathoverflow.net/questions/63707/degree-of-a-variety</a>)</p> <p>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 <b>smooth</b> curve of degree $d$? Does it depend only on $g$? Can anything be said for an arbitrary curve?</p> http://mathoverflow.net/questions/63745/what-is-the-minimal-degree-of-a-smooth-projective-embedding-of-a-hyperelliptic-cu/63751#63751 Answer by Yusuf Mustopa for What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? Yusuf Mustopa 2011-05-02T23:16:27Z 2011-05-02T23:16:27Z <p>According to Castelnuovo's bound (e.g. Theorem 6.4 in Chapter IV of Hartshorne), the degree of a smooth projective curve of genus $g$ in $\mathbb{P}^{3}$ (hyperelliptic or otherwise) is of degree at least $2\sqrt{g}+2$ (if the degree is even) or $\sqrt{4g+1}+2$ (if the degree is odd). </p> http://mathoverflow.net/questions/63745/what-is-the-minimal-degree-of-a-smooth-projective-embedding-of-a-hyperelliptic-cu/63752#63752 Answer by Tom Graber for What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? Tom Graber 2011-05-02T23:26:22Z 2011-05-02T23:26:22Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/63745/what-is-the-minimal-degree-of-a-smooth-projective-embedding-of-a-hyperelliptic-cu/63753#63753 Answer by David Speyer for What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? David Speyer 2011-05-02T23:41:07Z 2011-05-02T23:41:07Z <p>I agree with Tom G in the case of hyperelliptic curves. Interestingly, I think the bound for a general curve of degree $g$ should be $d = (3/4)g+3$. More specifically, the <a href="http://www.ams.org/mathscinet-getitem?mr=563378" rel="nofollow">Brill-Noether theorem</a> tells us that, for this $d$, there is a line bundle with $\dim H^0(X, L) =4$. I would guess (but don't know a reference) that, for generic $X$, this line bundle gives an embedding $X \to \mathbb{P}^3$.</p>