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

Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Eitan Sayag

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.

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

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.
Paper Structure (17 sections, 33 theorems, 28 equations)

This paper contains 17 sections, 33 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The Hilbert Scheme
  3. Main results
  4. Relation to the singularities of the symmetric power of the plane
  5. Background and motivation
  6. The characteristic zero case
  7. Relation to local finiteness of measures
  8. Idea of the proof
  9. Structure of the paper
  10. Acknowledgments
  11. Conventions
  12. Factorizable actions
  13. Factorizabilty of quasi-projective varieties and compatibility with open embeddings
  14. Quotients by free actions
  15. Proof of \ref{['thm:int']}
  16. ...and 2 more sections

Key Result

Theorem 1.1.2

If $\mathbf{Z}$ is a quasi-projective variety then the Hilbert functor $Hilb_n(\mathbf{Z})$ is representable by a scheme which we denote by $\textcolor{DefColor}{\mathbf{Z}^{[n]}} \IfNoValueTF{-NoValue-}{}{}$.

Theorems & Definitions (54)

  • Definition 1.1.1
  • Theorem 1.1.2: Gro62, see also BK05
  • Theorem 1.1.3: See e.g. BK05
  • Theorem 1
  • Definition 1.3.1
  • Theorem 1.3.3: Ive, BK05
  • Corollary 1.3.4
  • Definition 1.3.5
  • Remark 1.3.6
  • Theorem 2
  • ...and 44 more