This talk aims to provide a simple introduction on how to probe the

explicit geometry of certain moduli schemes arising in enumerative

geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of

curves used in Donaldson-Thomas theory. In particular, they exclude

curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced

structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct

ribbons (and higher non-reduced structures) from the data of line

bundles on a reduced scheme. With this approach, we can consider stable

pairs whose underlying curve is a ribbon: the remaining data is

determined by allowing devenerations of the line bundle defining the

double structure.