In this talk, we will discuss work in two distinct directions, in which the unifying theme is that both depend on ramification-related phenomena. The first part is joint with Rafael da Silva Barbosa (UFF).
We will first propose a candidate for the (3,\gamma)-hyperelliptic semigroup of maximal weight, when the genus g of the semigroup is large relative to \gamma.
We will then explain how to probe the (local) compactified Jacobian of a curve singularity C, which parametrizes isomorphism classes of torsion-free modules of rank one over C. Topologically, the compactified Jacobian is a linear section of a Grassmannian G, and from G it inherits a stratification into Schubert varieties indexed by semimodules over the value semigroup of C. We will explain how to compute the Schubert stratification combinatorially, and discuss some open problems.
