The Bourbaki number of a plane curve

The study of free plane algebraic curves has a long history in algebraic geometry and commutative algebra, going back to the work of Saito in the 1980s. After briefly revising this theory, I will introduce the Bourbaki number of a plane curve, and invariant that measures how far from being free a given curve is. I will also discuss three 0-dimensional schemes naturally associated with a curve and a syzygy for its jacobian, explaining their geometry and how they are related.