General ($ {\mathbb{C}}$omplex) Ramification      
Multiplicity of ramification $ \vert\alpha\vert:=\sum_i \alpha_i$
Theorem. (Eisenbud-Harris)     $ {\displaystyle \sum_{p\in{\mathbb{P}}^1} \vert\alpha(p)\vert = (k+1)(d-k) . }$
Enumeration: Given $ p_1,\ldots,p_s\in{\mathbb{P}}^1$ and prescribed ramification $ \alpha(p_1),\ldots,\alpha(p_s)$
where $ \sum_i\alpha(p_i)=(k+1)(d-k)$, how many linear series have this ramification ?
Example $ d=4$ with 6 flexes: (using MAPLE )       [Number = 5]
$ \bullet$   (Huber, S.-, Sturmfels, Verschelde)   Numerical methods to compute these linear series:
      $ \{$linear series with ramification $ \alpha$ at $ p$   $ \leadsto$   The Schubert variety in Grass $ (k,{\mathbb{P}}^d)$
        defined by $ \alpha$ and flag osculating rational normal curve at $ p$.