Cell decompositions of Hilbert schemes

Abstract

Motivated by work of Negut, and others (see [GN15], [AGH+12]), Gorsky, Hawkes, Schilling, and Rainbolt [GHSR20] studied a fixed-point localization formula for generalized q, t Catalan polynomials which computes equivariant Euler characteristics. Negut, conjectured that, under certain conditions, the coefficients of this polynomial are nonnegative. This would provide evidence to support the vanishing of higher cohomology. More generally, it is an open problem to provide a purely combinatorial method which can recover the polynomial given in [GHSR20]. Using Bialynicki-Birula cell decomposition, one can decompose the Hilbert scheme of points in C^2 into affine cells, which can be studied combinatorially using Young diagrams. Given a monomial ideal I corresponding to a partition λ ⊢ n, we consider the subvariety of the Hilbert scheme of points in C^2 consisting of ideals that contain I. Intersecting this subvariety with the Bialynicki-Birula cells yields subdivision. We wish to know when these subdivisions form a cell decomposition (i.e. each piece is homeomorphic to affine complex space). We study this question for a series of concave partitions. This work then leads to a conjecture for a combinatorial statistic which can recover the polynomial: the dimension of those affine complex spaces. We then provide computational evidence which supports a specialized case of the conjecture given by Negut,.