Hilbert scheme of smooth projective curves with some easy examples

I will start with a brief introduction to the Hilbert scheme of smooth projective curves which is a classical object in curve theory.

I will continue to talk about two assertions of Severi regarding the irreducibility of the Hilbert scheme of smooth projective curves, which have been made more than a century ago. I will give easy examples violating the Severi's assertions and talk about the so-called modified assertion of Severi with some suggestions on how to handle the problem.

We will also discuss related problems concerning the existence of a component of the expected dimension, which is rather new.