Nested Hilbert schemes and the nested q,t-Catalan series

TL;DR

This paper investigates the geometry of nested Hilbert schemes, derives a formula for Euler characteristics of equivariant sheaves, and introduces a conjectural polynomial series related to the nested q,t-Catalan series.

Contribution

It provides a general formula for Euler characteristics on nested Hilbert schemes and introduces the nested q,t-Catalan series with conjectural positivity and properties.

Findings

Derived a formula for Euler characteristics of sheaves on nested Hilbert schemes

Conjectured the positivity and polynomial nature of the nested q,t-Catalan series

Identified properties analogous to the classical q,t-Catalan series

Abstract

In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\TT^2$-equivariant locally free sheaf on $\Hil{n,n-1}$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as \textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series.