How to give equations for $\phi_K(C)$, the canonical curve?

Question

So say we have a curve $C$ of genus $g\geq 4$ that is not hyperelliptic. Then we have the canonical embedding $\phi_K: C\longrightarrow \mathbb{P}^{g-1}$ and thus $\phi_K(C)\simeq C$. In Geometry of Algebraic curves, it is often asked for one to find the "equations of the canonical curve $\phi_K(C)$". Given that there is an isomorphism with $C$, what equations are being asked for? And how does one find them?

Answer 1

Question: "Given that there is an isomorphism with C, what equations are being asked for? And how does one find them?"

Answer: The canonical bundle $\omega_C:=\Omega^1_{C/k}$ is an invertible sheaf on $C$ (when $C$ is regular). In Proposition II.7.4 in Hartshorne you will find a relation between morphisms $\phi: C \rightarrow \mathbb{P}^n_k$ and global sections $s_0,..,s_n\in H^0(C, \omega_C)$ of $\omega_C$. There is a 1-1 correspondence between equivalence classes of surjections

$$\phi^*: \mathcal{O}_C^{n+1} \rightarrow \omega_C \rightarrow 0$$

and maps $\phi: C \rightarrow \mathbb{P}^n_k$ over $k$. With some extra conditions the map $\phi$ will be an embedding called the "canonical embedding" (HH, Chapter IV.5).

Note: The map $\phi$ is not canonical - given two sets of sections $S:s_0,..,s_n$ and $T:t_0,..,t_n$, the two corresponding maps $\phi_S, \phi_T$ will differ by a $k$-automorphism of projective space.

In Proposition HH.II.7.2.2 you construct (locally) the ideal $I(C) \subseteq k[x_0,..,x_n]$ of the embedded curve $\phi(C)$. It is locally the kernel $I(C)_i$ of the map

$$\rho_i: k[y_i] \rightarrow \Gamma(C_i, \mathcal{O}_{C_i})$$

defined by $\rho_i(y_j):=s_j/s_i$ and $I(C)_i:=ker(\rho_i)$. This gives (locally) equations defining the curve $\phi(C) \cap D(x_i) \subseteq \mathbb{P}^n_k$. Hence you must study the relation between invertible sheaves and maps to projective space in HH.CH.II.7. You calculate a set of global sections $s_i \in \Gamma(C, \omega_C)$ and construct the "quotients"

$$s_j/s_i \in \Gamma(C_i, \mathcal{O}_{C_i}).$$

This process involves choosing a local basis for the invertible sheaf $\omega_C$. Hence the sections $s_i$ are global sections of $\omega_C$, but you may construct in a well defined way sections of $\mathcal{O}_C$ over $C_i$ and this defines the map $\rho_i$.

Note: Your curve $C$ is either constructed via glueing or as an embedded curve in some projective space $C \subseteq^i \mathbb{P}^n_k$ but the canonical bundle $\omega_C$ is intrinsic and does not depend on the embedding $i$, hence the canonical embedding $\phi_K(C)$ is intrinsic to the curve - for this reason it is called the "canonical curve" of $C$.

Example: For the projective line $\mathbb{P}^1$, the invertible sheaf $\mathcal{O}(d)$ ($d\geq 2$) gives rise to the $d$-uple embedding

$$v_d: \mathbb{P}^1 \rightarrow \mathbb{P}^d.$$

Pointwise this is the map

$$v_d(a_0:a_1):=(a_0^d:a_0^{d-1}a_1:\cdots :a_1^d),$$

and you must calculate the corresponding ideal $I(\mathbb{P}^1) \subseteq k[x_0,..,x_d]$ using Prop.II.7.2. The canonical bundle of the projective line is $\mathcal{O}(-2)$ and hence there is no canonical embedding in this situation.

Here you find an elementary and "scheme theoretic" construction of the Veronese embedding using this principle:

An explicit construction of the "Veronese embedding".