I have a simple and basic question concerning degree of projective curves and I'm referring to something I've read on Miranda's book, Algebraic curves and Riemann Surfaces, Chapter VII, 3. The Degree of Projective Curves.
Suppose that $X\subset \mathbb P^n$ is a smooth nondegenerate curve such that $\text{deg}(X)=d$. This means that given any hyperplane $H\subset \mathbb P^n$, the hyperplane divisor $D:=\text{div}(H)$ on $X$ has exactly degree $d$. Take the complete linear system $|D|$, we want to conclude that $\dim |D|\geq n$. So, let $Q$ be the set of hyperplane divisors on $X$: $$Q:=\{\text{div}(L)\in \text{Div}(X): L \text{ is an hyperplane}\}.$$
Now Miranda says that $Q$ is indeed a subsystem of the complete linear system $|D|$. It's clear that $Q\subseteq |D|$, but I don't understand why $Q$ is a subsystem. To see that it is a subsystem, I should show that there exists a linear subspace $V$ of $L(D)$ such that $Q=S(\mathbb P(V))$ where $$S\colon \mathbb P(L(D))\to |D|, [f]\mapsto \text{div}(f)+D$$ is the standard bijection between these two spaces. Who is $V$? Is there a more direct way to say that $Q$ is a subsystem? I'm sure this is really simple, but I'm quite new to Algebraic Geometry so if someone could help me I would appreciate a lot.
To avoid notational confusion: by $\mathbb PW^\vee$ I mean the set of one-dimensional linear subspaces of $W$, namely the quotient $(W\setminus\{0\})/k^\times$.
If I am not mistaken, in your notation $D=H\cap X$, a hyperplane section. Your $V$ is then the image of the restriction map $\rho:H^0(\mathbb P^n,\mathscr O_{\mathbb P^n}(1))\to H^0(X,\mathscr O_X(1))$. This map $\rho$ takes a degree one form $h$ on $\mathbb P^n$ (i.e. a linear homogeneous polynomial $h=\sum_{0\leq i\leq n} a_ix_i$) to its restriction to $X$.
The inclusion $V\subset H^0(X,\mathscr O_X(1))$ gives a surjection $H^0(X,\mathscr O_X(1))^\vee\to V^\vee$ and hence a closed immersion $$\mathbb PV^\vee\subset \mathbb PH^0(X,\mathscr O_X(1))^\vee,$$ which is your $Q\subset |D|$.
Added. Disregard all the above! Now, $|D|$ is the projective space whose points correspond to effective divisors on $X$ which are linearly equivalent to $D=H\cap X$. It is the projectivisation of (the dual of) $L(D)$. Now, in $L(D)$ you can consider the subspace $V\subset L(D)$ consisting of elements arising as restrictions of homogeneous degree one forms $$h=\sum_{0\leq i\leq n} a_ix_i.$$ Note that such forms (which live on $\mathbb P^n$) are exactly those whose zero loci are hyperplanes $L\subset\mathbb P^n$. But this says exactly that $Q$ is the projectivisation of (the dual) of $V$.