Exercise 111.46.7. Choose your favorite algebraically closed field $k$. As best as you can determine all possible $\mathfrak g^ r_ d$ that can exist on some curve of genus $7$. While doing this also try to

determine in which cases the $\mathfrak g^ r_ d$ is base point free, and

determine in which cases the $\mathfrak g^ r_ d$ gives a closed embedding in $\mathbf{P}^ r$.

Do the same thing if you assume your curve is “general” (make up your own notion of general – this may be easier than the question above). Do the same thing if you assume your curve is hyperelliptic. Do the same thing if you assume your curve is trigonal (and not hyperelliptic). Etc.

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