Ramification of Linear Series      
Linear series of degeree $ d$ and dimension $ k+1$:

$ L\ \subset\ H^0({\mathbb{P}}^1,{\mathcal O}(d))$     with   dim$ L=k+1$.

At $ p\in{\mathbb{P}}^1$, L has ramification
$\displaystyle \lambda(p)\ \colon\ \lambda_0(p)\geq \lambda_1(p)\geq \cdots \geq \lambda_k(p)\ \geq\ 0 $
$ \to$ unique non-decreasing sequence such that $ L$ has section $ f_i$ with
   ord$\displaystyle _p(f_i)\ =\ i + \lambda_{k-i}(p)\,. $

For most $ p$, $ \lambda(p)=(0,\ldots,0)$.

$ L$ is ramified at $ p\in{\mathbb{P}}^1$ if $ \lambda(p)\neq (0,\ldots,0)$.