I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing sequences $a_i^{l_1}(p)= a_i^{l_2}(p)$ for all $i\in\{0,\dots,r\}$and for all $p\in C$, i.e all their ramification points as well as their ramification types at those points are the same, is it true that $l_1=l_2$?
If one specifies all ramification points of a linear series then the adjusted Brill-Noether number is very negative, so one would not expect to find any linear series of that prescribed ramification at all. So, if we have two linear series of that ramification type, then the chances are "not low" that they are equal. Or maybe the statement just holds for a general curve, I don't know.
Any ideas would be appreciated.
