Invertible top form on the Hilbert scheme of a plane in positive characteristic

TL;DR

This paper proves the existence of an invertible top differential form on the Hilbert scheme of a plane in positive characteristic, leading to new integrability results and a novel function related to the discriminant.

Contribution

It establishes the existence of an invertible top form in positive characteristic and introduces a new locally summable function related to the discriminant, extending characteristic zero results.

Findings

Existence of an invertible top differential form on the Hilbert scheme in positive characteristic

Definition of a new function on monic polynomials that is locally summable

Application to positive characteristic analog of Harish-Chandra's local integrability theorem

Abstract

We prove that the Hilbert scheme of the plane in positive characteristic admits an invertible top differential form. This implies certain integrability properties of the symmetric powers of the plane. This allows to define a function on the collection of monic polynomials over a local field which can be thought of as a variant of the inverse square root of the discriminant. In characteristic 0 it essentially coincides with this inverse square root, however in general it is quite different, and unlike this inverse square root, it is locally summable. In a sequel work [AGKS] we use this local summability in order to prove the positive characteristic analog of Harish-Chandra's local integrability theorem of characters of representations under certain conditions. The main results of this paper are known in characteristic zero. In fact a stronger result is known: there is a symplectic form on the Hilbert scheme of a plane.