I have two questions about the proof of Clifford's theorem in Algebraic Curves and Riemann Surfaces by Rick Miranda.
(a) In the beginning of the proof, a special divisor is defined as a divisor $D$ such that $D \geq 0$ and $L(K - D) \neq 0$. The author then writes
Note that $D$ is a special divisor if and only if both $\dim L(D) \geq 1$ and $\dim L(K - D) \geq 1$.
I don't see why the conditions $\dim L(D) \geq 1$ and $\dim L(K - D) \geq 1$ imply that $D \geq 0$.
(b) A little later in the proof, we have a divisor $D$ and a linear system $| M |$ with no base points. The author notes that we can find a divisor $E \in | M |$ such that $D$ and $E$ have disjoint support. If the support of $D$ consists of one point, this is of course trivially true, but I don't see why it holds if the support contains more points. Could anyone explain why this is so?
I've solved both questions. For those who are interested:
(a) If $\dim L(D) \geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D \geq 0$.
(b) Let $\{p_1, \ldots, p_n\}$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i \in L(M)$ such that $\text{ord}_{p_i}(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_i\neq 0$. Then $h = \sum_i \frac{1}{c_i} g_i$ has the property that $\text{ord}_{p_i}(h) = \text{ord}_{p_i}(g_i)$ for all $i$, so $\text{div}(h) + M$ has none of the $p_i$ in its support.