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.
Throughout the paper we fix a field F of arbitrary characteristic. We will also fix a natural number n.
1.1. The Hilbert Scheme. In order to formulate our results let us first recall the definition of the Hilbert scheme. where pr S is the projection.
Theorem 1.1.2 ( [Gro62], see also [BK05,Theorem 7.2.3]). If Z is a quasiprojective variety then the Hilbert functor Hilb n (Z) is representable by a scheme which we denote by Z [n] .
Theorem 1.1.3 (See e.g. [BK05, Theorem 7.4.1]). If Z is a smooth quasiprojective irreducible algebraic surface then Z [n] is a smooth irreducible variety of dimension 2n.
Theorem A. There exists an invertible top differential form on (A 2 ) [n] .
1.3. Relation to the singularities of the symmetric power of the plane. Theorem A is related to the singularities of the symmetric power of the plane. In order to formulate this relation we introduce some notations:
Definition 1.3.1. For a quasi projective algebraic variety Z define its symmetric power by: Z (n) := Z n //S n .
Here // denotes the categorical quotient. By Corollary 3.1.8 below this quotient exists.
Notation 1.3.2. Let Z be a quasi-projective variety. Let x ∈ Z [n] ( F ). It corresponds to a sheaf of ideals I x ⊂ O Z F . For any z ∈ Z( F ) denote
This gives a multiset in Z( F ) of size n. By Lemma 3.0.2 below, we can interpret this multiset as a point in Z (n) ( F ). Denote this point by H Z,n (x).
Theorem 1.3.3 ([Ive70, II.2,II.3], [BK05, Theorem 7.3.1]1 ). Let Z be a quasi-projective variety. There exists (and unique) a projective morphism H Z,n : Z [n] → Z (n) that gives on the level of F points the map H Z,n defined above. This morphism is called the Hilbert-Chow morphism.
Theorems 1.1.3 and 1.3.3 imply:
Corollary 1.3.4. Let Z be a (quasi-projective) smooth surface. Then the Hilbert-Chow map H Z,n : Z [n] → Z (n) is a resolution of singularities.
Theorem A is related to the properties of this resolution. In order to formulate these relations we make: Definition 1.3.5.
(i) We recall that a modification γ : Ṽ → V of algebraic varieties is a birational proper morphism. (ii) We call a modification γ : Ṽ → V of algebraic varieties integrable if for any open U ⊂ V and any top-form ω on the smooth locus of U the rational form γ * (ω) on γ -1 (U) is regular on the smooth locus of γ -1 (U). (iii) We call such modification sharply integrable if γ * (ω) vanishes only on γ -1 ( D) where D is the zero locus of ω. (iv) We call a variety (sharply) integrable if it admits a (sharply) integrable resolution of singularities.
Remark 1.3.6. In characteristic zero, one can show that TFAE:2 (a) the singularities of Z are rational, (b) Z is integrable and Cohen-Macaulay.
In positive characteristic there is no single accepted definition of rational singularities, and one can take condition (b) as a definition.
We will see that Theorem A follows from:
Theorem B. The Hilbert-Chow map
is a sharply integrable modification.
This implies:
Remark 1.3.7. In fact, it is easy to see that Theorem B and Theorem A are equivalent. So one could instead prove directly Theorem A and deduce Theorem B.
1.4. Background and motivation.
1.4.1. The characteristic zero case. The characteristic zero counterpart of the main results of this paper is well known. In fact, stronger results are known. Namely, the Hilbert-Chow map for smooth surfaces in characteristic 0 is a symplectic resolution (see e.g. [Nak99, Theorem 1.17]). This implies the characteristic zero counterpart of Theorem B. This also implies that in characteristic zero the Hilbert scheme of the plane is a symplectic variety. This in turn implies the characteristic 0 counterparts of Theorem A and Corollary C. In addition, (A 2 ) (n) is a quotient of algebraic variety by a finite group, therefore, by [Bou87,Corollaire], in characteristic zero its singularities are rational. As mentioned in Remark 1.3.6 this is equivalent to the fact that it is integrable and Cohen-Macaulay.
1.4.2. Relation to local finiteness of measures. If the field F is local, integrability of an algebraic variety Z implies that given a top form ω on its smooth locus, the corresponding measure |ω| on Z(F ) is locally finite. Therefore, given a (locally) dominant map ϕ : Z → Y to a smooth variety and a function f ∈ C ∞ c (Z(F )), the measure ϕ * (f |ω|) is also locally finite. Since it is also absolutely continuous (w.r.t. a smooth invertible measure on Y(F )), this measure has a locally summable density function.
Applying this consideration to the map (A 2 ) (n) → (A 1 ) (n) (induced by the projection A 2 → A 1 ) we get a locally summable density function η (defined up to multiplication by a smooth compactly supported function) on (A 1 ) (n) (F ). Note that (A 1 ) (n) is naturally identified with the space of monic polynomials of degree n.
Over C this function is the absolute value of the inverse square root of the discriminant -|∆| -1 2 . Over a general local field of characteristic zero, this function is bounded
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