Hilbert schemes, polygraphs and the Macdonald positivity conjecture (Q2723517)

Let \(H_n=\text{Hilb}^n(\mathbb{C}^2)\) be the Hilbert scheme which parametrizes the subschemes \(S\) of length \(n\) of \(\mathbb{C}^2\). To each such subscheme \(S\) corresponds a unordered \(n\)-tuple with possible repetitions \(\sigma(S)=[[P_1,...,P_n]]\) of points of \(\mathbb{C}^2\). There exists an algebraic variety \(X_n\) (called the isospectral Hilbert scheme) which is finite over \(H_n\) and which consists of all ordered \(n\)-tuples \((P_1,...,P_n)\in(\mathbb{C}^2)^n\) whose underlying unordered \(n\)-tuple is \(\sigma(S)\). The main aim of the paper under review is to study the geometry of \(X_n\), which is more complicated than the geometry of \(H_n\). For instance, a classical result of J. Fogarty asserts that \(H_n\) is irreducible and non-singular. The main result of the paper under review asserts that \(X_n\) is normal and Gorenstein (in particular, Cohen-Macaulay). Earlier work of the author indicated that there is a far-reaching correspondence between the geometry and sheaf cohomology of \(H_n\) and \(X_n\) on the one hand, and the theory of Macdonald polynomials on the other hand. The link between Macdonald polynomials and Hilbert schemes comes from some recent work [see \textit{A. M. Garsia} and \textit{M. Haiman}, Proc. Nat. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)].NEWLINENEWLINENEWLINEThe main result proved in this paper is expected to be an important step toward the proof of the so-called \(n!\)-conjecture and Macdonald positivity conjecture. The main theorem is based on a technical result (theorem 4.1) which asserts that the coordinate ring of a certain type of subspace arrangement is a free module over the polynomial ring generated by some of the coordinates.