Hilbert scheme of points on cyclic quotient singularities of type {\((p,1)\)}
Let \(X\) be a complex quasi-projective surface and let \(n\) be a positive integer. The Hilbert scheme \(\hbox{Hilb}^n(X)\) parametrizes the zero-dimensional subschemes of \(X\) of length \(n\). The author studies the topological Euler characteristics of these Hilbert schemes, which he collects in a generating series \[ Z_X(q) = \sum_{n \geq 0} q^n \chi(\hbox{Hilb}^n(X)). \] When \(X\) is a smooth surface, these have been carefully studied by Fogarty, and Göttsche described the generating series of the Poincaré polynomial of these Hilbert schemes in terms of the Betti numbers of \(X\). To extend this, the author considers a particular action of \(\mathbb Z_p\) (\(p\) a positive integer) on \(\mathbb C^2\) involving another integer \(q\) coprime to \(p\), and denotes by \(X(p,q)\) the quotient variety. The main result of this paper is a representation of \(Z_{X(p,1)}(q)\) as a coefficient of a two-variable generating function, obtained by studying a torus action on \(X(p,1)\). The author links the combinatorics of this problem to \(p\)-fountains, a generalization of the notion of a fountain of coins.
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