Parole chiave

• Hilbert scheme
• non-reduced component
• singular hypersurface

Abstract Hilbert schemes of points on a quasi-projective variety $X$ are classical objects in algebraic geometry. Roughly speaking, they parametrise ideals of a polynomial ring with complex coefficients having finite colength. Although Hilbert schemes always have a distinguished component called the smoothable component, their geometry is quite pathological and many of the open problems around them concern their irreducibility and their schematic structure. In a recent work with Lella, we generalise some of our previous results and provide a systematic way to build elementary and generically non-reduced components of the nested Hilbert scheme of points. Moreover, we study reducibility for those schemes. As a by-product of the theory we develop, we also get new information about the geometry of the classical Hilbert schemes of points on singular hypersurfaces of $\mathbb A^3$. In my talk I will present these results and I will give an idea of the main techniques we have used.