Linear series on smooth curves, classically known as $g^r_d$’s, correspond to non-degenerate maps to projective space up to a change of coordinates. Special linear series appear in subtle configurations, related to the projective geometry of the curve.
I will recall the determinantal construction of the moduli spaces of $g^r_d$’s and their infinitesimal study. After summarising some classical results about them, I would like to introduce a degeneration technique that was used in the proof of the dimension theorem by Eisenbud and Harris.
Applications include the minimal degree of an embedding of a curve, existence of Weierstrass points, and the construction of meaningful cycles on the moduli space of stable curves (e.g. allowing the proof that $overlinemathcal M_g$ is not unirational for $ggeq 23$). Time permitting, I will mention the first one.