Consider two global sections of a same line bundle $\mathcal O(D)$ on a projective, smooth, geometrically connected curve $X$ over a field $k$. They induce two scheme morphisms $f,g:X\longrightarrow\mathbb A^1\longrightarrow \mathbb P^1_k$, sending $k[x]\longrightarrow \mathcal O(D)$, $x\mapsto f$. Now we can take linear combinations of the sections, e.g. $f+g$: this is again a global section and therefore induces a morphism in $\mathbb P^1$.
What is the degree of the induced map?
I would claim that, if $\rm deg\ f\neq \rm deg\ g$, the degree is the maximum, as happens for polynomials. The problem is that the degree of a map in $\mathbb P^1$ is not the one of the polynomial that induces it…
Remark: I call degree of a scheme morphism $f$ the number $[k(X):k(Y)]$ (seeing the second immersed the first by $f^*$) if $f$ is dominant, $0$ otherwise.
ADDED LATER: With this kind of result I actually hope to prove that, if the global sections of the line bundle contain maps of several degree, then the dimension of the vector space of global sections must be high… Is this true? Does it have any links with the presence of maps of several distinct degrees?
Thank you in advance.