제목: Axis-like curves, their Jacobians, and the Torelli map on moduli spaces.
초록: Given a smooth curve C of genus g, its Jacobian, a moduli space of degree zero line bundles on C, is a principally polarized abelian variety of dimension g. It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian, extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev; in other words, stable curves (which are nodal) admit compactified Jacobians as limits of Jacobians of smooth curves. By Alexeev and Brunyate, the Torelli map cannot be extended to alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with seminormal singularities (i.e. axis-like singularities)? As a joint work with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with axis-like singularities (with topological constraints), where the Torelli map extends; this implies the existence of compactified Jacobian on axis-like curves, as a limit of Jacobians of smooth curves.
