# Topological properties of differentials with prescribed zeroes on an algebraic curve

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.

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.
• Is it also true in this generality that the dimension of the locus is $n-1+d$? – Fabian Ruoff Mar 2 at 15:03
• The locus has dimension at most $d$, thus that formula is certainly wrong. In any case, a simple formula does not exist in general: for instance, if you take the trivial partition $n=n$, the locus can be either empty or of dimension $0$ depending on $L$. Moreover, the locus can be not irreducible, and the various components can have different dimension. And even if you fix $L=\omega$, the dimension of the locus probably depends not only on the partition of $2g-2$ but on $C$, too. The dimension should be upper semicontinuous wrt $C$, though, you might try to compute it for a generic curve. – Giulio Bresciani Mar 2 at 15:27
• I should add, the dimension depends not only on $n$, $d$, $L$ but on the specific partition, too. All sorts of things can happen, but studying them should not be too hard. An useful point of view is the following. A partition gives a closed embedding $C^d\to C^n$, and there is a natural morphism $C^n\to Pic^n(C)$ which is very well understood. The line bundle $L$ gives you a point in $Pic^n(C)$, and the locus is the intersection in $C^n$ between the fiber of $L$ and $C^d$. – Giulio Bresciani Mar 2 at 15:40
• For every $i=1,\dots,d$ you have an integer $m_i$ given by the partition. Let $\Delta_i\subset C^d\times C$ be the pullback of the diagonal wrt to the projection $(a_i,b):C^d\times C\to C\times C$, then $M=b^*L\otimes O(\sum_i -m_i\Delta_i)$. $S$ is the support of a coherent sheaf, thus it is closed in $C^d$ and hence proper, the map $S\to\mathbb{P}(H^0(L))$ has automatically closed image. The image of $S$ contains the loci associated to the sub-partitions, if you remove those ones (which are closed for the same reason) you get a locally closed subset. – Giulio Bresciani Mar 3 at 14:30