I'm reading Rick Miranda's Algebraic Curves and Riemann Surfaces and got some problems. Let $\mathbb{P}^n$ be the complex projective space of dimension $n$.
Let $X$ be a compact Riemann surface, $f_0, f_1, \dots, f_n$ meromorphic functions on $X$ that are not identically zero. Then these functions yields a holomorphic map $$ \phi: X\to\mathbb{P}^n, x\mapsto [f_0(x) : f_1(x) : \dots : f_n(x)]. $$ Of course $\phi(X)\subset\mathbb{P}^n$ is a Riemann surface. In particular, if $D$ is a positive divisor on $X$ and $$ L(D) = \{f\in \mathcal{M}(X): \mathrm{div}(f) + D\ge 0\},$$ where $\mathrm{div}(f)$ is the principal divisor obtained from $f$, then a basis $\{f_1, \cdots, f_n\}$ of $L(D)$ yields a holomoprhic map $X\to \mathbb{P}^n$; denote this map by $\phi_D$. I know that a compact Riemann surface is an algebraic curve, so my question is:
Is there a general way to find some homogeneous polynomials $F_1, \cdots, F_m$ s.t. $\phi(X)$ is the vanishing set of these polynomials?
For example, the problem VII.1:C of Miranda's book asks to prove that, for a compact Riemann surface $X$ of genus $2$, the map $\phi_{2K}$ obtained from the divisor $2K$ is a degree $2$ map, where $K$ is a canonical divisor on $X$, and $\phi_{2K}(X)$ is a conic (i.e., defined by a degree $2$ polynomial).
I can prove the statement, by using the fact that a compact Riemann surface of genus $2$ is hyperelliptic and write down the equation $y^2 = h(x)$ that defines $X$ explicitly. However, I want to prove this without writing down the equation. Here is what I have tried.
By Riemann-Roch, $\dim L(K) = 2$ and $\dim L(2K) = 3$. So pick a non-constant function $f\in L(K)$, then $1, f, f^2\in L(2K)$ and they are linearly independent, hence form a basis of $L(2K)$ and gives the map $$\phi_{2K}: X\to \mathbb{P}^2, \phi_{2K}(p) = [1: f(p): f(p)^2]. $$ It seems that $\phi_{2K}(X) = Y: y^2 = xz$, but I'm not sure, because $p$, as a point in $X$, is not "free", so I don't know if $\phi_{2K}$ maps $X$ surjectively onto $Y$. Also, I don't know how to see the degree of $\phi_{2K}$ in this case.
Any hint or answer will be appreciated!