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!

Edit: Apparently this is false, so the question is, what's 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?