Canonical embedding

Question

I'm reading the second chapter of this paper and I need help to understand what exactly this canonical embedding is:

Remark: $C$ is a smooth non-hyperelliptic complete irreducible algebraic curve of genus $g\ge 4$.

Answer 1

The comment of Adeel refers to the correct sources, but this sums it up: If $C$ is a curve, the canonical sheaf is the sheaf $\Omega^1_{C/k}$. Global sections of this sheaf are everywhere regular differential forms $C$.

The number of sections of $\Omega^1_{C/k}$ is by definition the genus of the curve. The degree of $\Omega_{C/k}^1$ (as a divisor) is by Riemann-Roch $2g-2$.

Now, if $\omega := \Omega_{C/k}^1$ is very ample (which is the case if the genus is high enough and is the case here), then we get a map $C \to \mathbb P^{g-1}$ given by sending a point $x \in C$ to the point $(\omega_1(x):\omega_2(x):\ldots :\omega_g(x)) \in \mathbb P^{g-1}$ where $\omega_1,\cdots,\omega_g$ is a fixed basis of $\Gamma(C,\omega)$.

Answer 2

When I started reading stuffs for my master thesis I was also puzzled about this.

Hopefully the introduction chapter of my thesis will be insightful for you! There are some drawings and examples which could be useful, especially if you want to get some geometrical intuition about what's going on.

You can find it here. Enjoy!