Abstract

The interaction between curves, their Jacobians, and maps to projective space is one of the cornerstones of classical algebraic geometry. In its modern version, this interaction is expressed as a relationship between moduli spaces; in the case of curves and maps, it is the relationship between the moduli space of curves and Gromov-Witten theory, a particularly rich and well-studied field. On the other hand, the connection between curves and Jacobians is substantially less developed, largely due to the difficulties surrounding compactifying families of Jacobians over families of nodal curves.

In this talk, I will survey some recent results which, by addressing some of these difficulties, lead to a perspective that strengthens the above interaction and to the solutions of previously unresolved problems regarding stable maps. The talk is based partially on results with Holmes-Pandharipande-Pixton-Schmitt and partially with Bae-Pixton.