Let $C$ be an algebraic curve (one dimensional projective regular connected scheme of finite type) of genus $g$ over an algebraically closed field $k$ with structure morphism $\pi$. By Riemann-Roch, the global sections of the sheaf of differentials $\Omega$ are a $g$-dimensional vectorspace over $k$ and every element $\omega \in \Omega(C)$ has exactly $2g-2$ zeroes (counted with multiplicity).
Therefore the vector bundle $V = \mathcal{Spec} (\pi\_\* \Omega)$ over $\operatorname{Spec}(k)$ can be naturally decomposed by the order of the zeroes of the differentials, i.e. for every partition of $2g-2$ into nonnegative integers one can form the subset of differentials such that the orders of the zeroes give the same partition.


My guess is, that this is not just a decomposition set theoretically, but actually a decomposition in locally closed subsets and therefore a stratification (using the induced reduced structure). Naively for every subset one wants that some zeros agree (a closed condition) and some do not agree (an open condition) but I seem to be unable to make this rigorous.



Topological properties of differentials with prescribed zeroes on an algebraic curve
This probably belongs to MSE. In any case, it works for any line bundle $L$, there is nothing special about $\omega$ for this problem. Let $n$ be the degree of $L$ and choose a partition of $n$ with $d$ parts, a point $p$ of $C^d$ gives a set $Z\_p\subset C$ of zeroes with multiplicities. You can construct a line bundle $M$ on $C^d\times C$ such that the restriction of $M$ to $p\times C$ is $L(-Z\_p)$.


Now let $a:C^d\times C\to C^d$, $b:C^d\times C\to C$ be the projections and consider $a\_\*M$, it is a coherent sheaf whose support $S\subset C^d$ is precisely the closed subset of points $p\in C^d$ such that $L$ has a global section whose zeroes are $Z\_p$. There is a natural map $M\to b^\*L$, its pushforward to $C^d$ is a map $a\_\*M\to a\_\*b^\*L=H^0(L)$ where the latter is a constant vector bundle. Since the restriction of $\pi\_\*M$ to $S$ is a line bundle, you get a natural map $S\to\mathbb{P}(H^0(L))$ whose image is the desired locus.


