**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!



Is very ampleness of a divisor on a curve determined entirely by degree and genus?
Let $C\_1$ be a hyperelliptic curve of genus $g \geq 3$ (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 $C\_1 \rightarrow P^2$ associated to $K\_1$ is a double cover of a rational curve (a conic, in fact) in $P^2$, rather than an embedding.


