Question

Edit: Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not?

Question as originally stated:

Suppose X is a complete nonsingular curve (smooth proper integral scheme of dimension 1 over C) and D ∈ DivX.

I have heard that very ampleness of D is determined entirely by its degree and the genus of X. How can this be done explicitly?

Thanks!

Answer 1

Let C_1 be a hyperelliptic curve of genus g >= 3 (example: y^7 = x^2 + 1 for g = 3), and C_2 be a non-hyperelliptic curve of the same genus g (for example, the Klein quartic with g= 3 again: I'll use it in the form y^7 = x^2(x-1)).

Then let K_1, K_2 be the canonical divisors of C_1, C_2 respectively (In the example above: K_1 = 4[infty_1] and K_2 = 4[infty_2], where infty_1, infty_2 are the "points at infinity" on the curves above: note that the projective curves corresponding to the equations above both have singularities at infinity, but in both cases these singularities can be resolved to give a single point at infinity on each curve.) Then K_1 is not very ample but K_2 is: this is a standard application of Riemann-Roch (see also Hartshorne, Chapter IV, Proposition 5.2.) The reason that K_1 is not very ample is that the map C_1 -> P^2 associated to K_1 is a double cover of a rational curve (a conic, in fact) in P^2, rather than an embedding.

Answer 2

Not entirely by its degree; see Hartshorne Chapter 4, section 3 (Prop. 3.1 and corollary 3.2) and section 5.

Answer 3

I'm late to the game, but I would like to point out that the answer is systematically no. One class of examples. Suppose g>2 for simplicity. In that case any general line bundle of degree 2g is very ample and special ones are not. This can be seen by using the criteria that a line bundle is very ample iff for any effective divisor $D$ of degree 2, $h^0(L(-D) = h^0(L)-2$. One checks using R-R that this holds iff $L$ is not of the form $L= K_C(D)$ where $D$ is an effective divisor of degree 2. Line bundles of the form $L= K_C(D)$ are a 2 dimensional subset of the g dimensional (Picard) variety. This can be expanded upon.

Answer 4

The other way to construct examples is just by considering curves in a given projective space. For example, a plane curve $C$ of degree $d$ has genus $g = \frac{(d-1)(d-2)}2$. One can, of course, rephrase this as saying that $\mathcal{O}_C(1)$ is a very ample divisor of degree $d$ on $C$. On the other hand, as $d < g$, a general divisor of degree $d$ on $C$ does not even have a single global section.