Let $D\_1$, ... , $D\_n$ be a finite set of divisor classes on a nonsingular projective irreducible algebraic curve. We say that $D\_1\geq D\_n$ if the line bundle defined by $D\_1-D\_n$ has a section. This obviously satisfies the axioms of a partial order.


Suppose $\{x\_1,....,x\_n\}$ is a finite partially ordered set. Does there exist a (projective, nonsingular) algebraic curve of sufficiently high genus, and a set of divisors on it, that are isomorphic as a partially ordered set to $\{x\_1,...,x\_n\}$?



Can every finite poset be realized as divisors of an algebraic curve?
Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible sums with coefficients $0,1$ of these $n$ points. This poset is the same as the poset of subsets of an $n$-element set.


Since every finite poset seems to be a subposet of the poset of subsets of a finite set [EDIT: this is true-see the comments below], just embed your poset in a "power set poset" and remove the unwanted divisors, to deduce that what you want is true.


