GAAG seminar - The geometry of the Hilbert scheme of points and its variants

Hilbert schemes of points on a quasi-projective variety X are classical objects in algebraic geometry. Roughly speaking, they parametrise zero-dimensional closed subschemes of X of fixed length. The geometry of these schemes is quite pathological. For instance they might be reducible as soon as dim X > 2 and non-reduced as soon as dim X > 3. The situation becomes even more complicated if one considers nested versions of Hilbert schemes of points or, even worse, nested Quot schemes. In my talk I will firstily give  a gentle introduction to these phenomena and explain some of my contributions to the subject. Then, I will explain the main techniques I've used to detect irreducible components of the Hilbert schemes of points.