# 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.

## 1 Answer

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
• First of all sorry for my late reply and thank you for your long answer. When I posted the formula above I was thinking about a different setting (families of curves), you are completely right in that it makes no sense here. I was confident that I would either understand the details you left out myself or find a reference using the more general point of view you have given me but neither seems to be true. Do you know of any reference where this stratification is discussed in detail, maybe even in a more general setting? – Fabian Ruoff Mar 3 at 8:55
• 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