On Embeddings Of Curves

[Tsakanikas Nickos]

In the following $X$ is a non-singular, projective curve over an algebraically closed field $k$.

1. Suppose that $X$ is an elliptic curve ($g=1$). Show that $X$ can be embedded in $ \mathbb{P}^{2}_{k} $ as a cubic curve.
2. Suppose that $X$ is of genus $g=2$. Show that $X$ can be embedded in $ \mathbb{P}^{3}_{k} $ as a curve of degree $5$.