Sheaves on nodal singularities I

We study two families of moduli spaces of sheaves attached to the node, namely the punctual Hilbert schemes of the non-reduced plane curve $x^uy^v=0$, and the punctual Quot schemes of the module $k[[x]]^u\oplus k[[y]]^v$ over the reduced node $xy=0$. We compute their motivic generating functions for all $u,v$ and over every field. Both are specializations of a single rational function whose numerator counts matrices by rank, and they are related by the substitution $\mathbb{L}\mapsto Q\mathbb{L}$. This proves the Hilb--Quot correspondence for the node with independent ranks on the two branches. The main geometric ingredient is an elementary modification of an ideal along a branch of the curve, whose fibres are affine spaces of dimension equal to a contact length. We also construct stratifications of the Hilbert schemes into products of affine spaces and tori, and we formulate a branchwise Hilb--Quot--HOMFLY conjecture, which we test on the colored Hopf link.