Brill-Noether theory of stable curves

For a Riemann surface,
the classical Brill-Noether theory
concerns the subvarieties of the Picard scheme parametrizing
line bundles with unexpected global sections.
We shall present some work in progress extending Brill-Noether theory
to stable curves equipped with their compactified Picard scheme.
As an application, classical results about smooth curves admit a new,
simple, proof.