Notation: If $E$ is a compact divisor on a Riemann surface and $p\in X$, I'll use the notation $E^{(p)}$ to denote the coefficient of the point $p$ in $E$.
Let $D\geq 0$ be a divisor on a compact Riemann surface $X$. I'll define the complete linear series of $D$ as:
$$|D|:=\{D'\in \text{Div}(X):D'\sim D,\ D'\geq 0\}$$
where $\sim$ is linear equivalence. I also define the complex vector space:
$$\mathscr{L}(D):=\{f\in\mathscr{M}(X):(f)+D\geq 0\}$$
The notes I'm using define the base divisor of $D$ as
$$\mathbf{B}=\text{gcd}(|D|)$$
but they are not very detailed about what the $\text{gcd}$ of a set of divisors actually is!
I think it's reasonable to define it as follows (correct me if it's wrong):
$$\text{gcd}(|D|)^{(p)}:=\min\{E^{(p)}:E\in |D|\}.$$
The points that appear with a non-zero coefficient in $\mathbf{B}$ (the support of $\mathbf{B}$) are called base points of $D$.
Then my notes state the following equality that kinda baffles me: $$\mathbf{B}=\text{lcm}\{(f)_{\infty}:f\in \mathscr{L}(D)\}$$ where I guess that the definition of $\text{lcm}$ is: $$\text{lcm}\{(f)_{\infty}:f\in \mathscr{L}(D)\}^{(p)}:=\max\{(f)^{(p)}_{\infty}:f\in \mathscr{L}(D)\}.$$
This equality just seems blatantly false (if the two definitions I tried to guess are actually correct). If the equality was true then for any $p\in X$:
$$\min\{E^{(p)}:E\in |D|\}=\max\{(f)^{(p)}_{\infty}:f\in \mathscr{L}(D)\}$$
that is to say:
$$\min\{D^{(p)}+(f)^{(p)}:f\in \mathscr{L}(D)\}=\max\{(f)^{(p)}_{\infty}:f\in \mathscr{L}(D)\}$$
On both sides the optimum is reached when the order of the pole of $f$ in $p$ is maximum. Let's say that the maximum possible order of that pole is $M>0$. Then:
$$D^{(p)}-M=M$$ And there is simply no reason for this to be true! I think the correct equality should be:
$$D-\mathbf{B}=\text{lcm}\{(f)_{\infty}:f\in \mathscr{L}(D)\}$$
Is this correct?