Given an isolated reduced complex plane curve singularity, we are naturally interested in studying its geometric and topological properties. A possible approach makes use of homological invariants of the algebraic link of the singularity, such as the triply-graded link homology introduced by Khovanov and Rozansky, and polynomial invariants built from such homology theories. However, the geometric understanding of link invariants is still very far, and punctual Hilbert schemes at the curve singularity come into play offering a fruitful geometric framework to give such link invariants an interpretation. In this direction, we can find specialized theorems for classical polynomial invariants by Campillo, Delgado and Gusein-Zade and by Maulik, as well as a conjecture proposed by Oblomkov, Rasmussen and Shende that suggests an interpretation of the Poincaré polynomial of the triply-graded link homology in terms of the geometry of weight polynomials of punctual Hilbert schemes. With this seminar, we present a first step towards the above conjecture in the case of so-called curvilinear Hilbert schemes, which seem to play an interesting role in the theory, and consequent extensions to the general setting involving the entire Hilbert scheme. Our work makes use of tools and techniques coming from the theory of p-adic integration.