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Given a divisor $D$ on an algebraic curve $X$, there is a corresponding map $\phi_D$ from $X$ to the projective space (of dimension $\dim L(D)-1$). In particular, we know that if $D$ is a very ample divisor, then this is a holomorphic embedding into projective space. When $D$ is not very ample, how does one read off information about the map?

Here is a specific problem from Miranda's textbook to provide context:

Let $X$ be an algebraic curve of genus $2$. Let $K$ be a canonical divisor on $X$, and let $\phi_{2K}$ be the associated map to projective space. Show that the image of $\phi_{2K}$ is a smooth projective plane conic, and that the map has degree $2$.

Here are my thoughts so far. We know that $\deg(2K) = 4g-4 = 4$. By Riemann-Roch, we also have $\deg L(2K) = \deg(2K) + 1 - g = 3$. This would suggest that the image is in $\mathbb(P)^2$, and by the degree it could be a double covering of a conic, an embedding as a degree $4$ curve, or perhaps a quadruple covering of a line?

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If you have a question, then please phrase it as a question, not as an assignment or an order to the group; if it is homework, consider using the [homework] tag. Whether it is or it is not homework, the polite thing to do is to post your query as a query, specifying why you are interested in the answer, and what your thoughts so far on the problem are. – Arturo Magidin Jan 12 '11 at 19:35
Sorry, I'm still used to problem forums where one just posts math problem statements. I will edit the question accordingly. Please don't view it as an "order" to anybody! – Tony Jan 13 '11 at 23:12

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