The Hilbert-Chow algebra of a proper surface and Grojnowski calculus.

Abstract

We exhibit that the direct sum of the Chow groups of the Hilbert schemes of points on a proper smooth surface has a commutative ring structure, an action from the multiplicative semigroup of positive integers, and a self-action by differentials; and how they are compatible with each other. Moreover, for every curve in this surface, there are canonical homomorphisms from the ring of symmetric functions compatible with actions from the multiplicative semigroup of positive integers; and, for every point on this curve or rational family of curves, we describe canonical derivatives of this homomorphism. This structure allows us to have a unifying view of the Nakajima operators, and, in the projective plane case, the Mallavibarrena-Sols basis.