Quadratic Euler Characteristic of Symmetric Powers of Curves
Lukas F. Bröring Anna M. Viergever

Low-mass dark sector searches with deuteron photodisintegration
    Marc Vanderhaeghen
    April 29, 2024
===============================================================




  We compute the quadratic Euler characteristic of the symmetric powers of a
  smooth, projective curve over any field k that is not of characteristic
  two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an
  application, we show over a
  field of characteristic zero that the power structure on the Grothendieck-Witt ring
  introduced by Pajwani-Pál computes the compactly supported 𝔸^1-Euler
  characteristic of symmetric powers for all curves.


tocsectionIntroduction



§ INTRODUCTION

One can assign a quadratic Euler characteristic to a smooth
projective scheme over a field k that is not of characteristic 2 using motivic
homotopy theory as introduced by Morel-Voevodsky. Quadratic Euler
characteristics were first introduced by Hoyois <cit.> and are
elements of the Grothendieck-Witt ring (k) of quadratic forms over
k. These quadratic forms carry a lot of information: If
k⊂ℝ, Levine <cit.> proved that the signature of the
quadratic Euler characteristic χ(X/k) of a smooth projective scheme X
over k is equal to the topological Euler characteristic of X(ℝ)
and the rank of χ(X/k) is the topological Euler characteristic of
X(ℂ). In practice however, χ(X/k) is generally hard to
compute. Quadratic Euler characteristics are often used in the program of
refined enumerative geometry, which aims to obtain “quadratically enriched”
versions of results in classical enumerative geometry.


Our main theorem is the following.

	Let C be a smooth, projective curve of genus g over a field k
        that is not of characteristic 2. Let n∈ℤ_≥ 1.
        Then if n = 2m is even, we have
	
    χ(^nC/k)
                 = ∑_i=0^mgi⟨ -1⟩^i +
                1/2(2g-2n -
                ∑_i=0^mgi)H∈(k)
       = 1/2(2g-2n + (-1)^mg-1m)·⟨ 1⟩
         + 1/2(2g-2n-(-1)^mg-1m)·⟨ -1⟩

	and if n is odd, we have
	
    χ(^nC/k) = -1/22g-2n H∈(k),

	where ⟨ a⟩∈(k) for a ∈ k^× denotes the quadratic
        form x ↦ ax^2 and H = ⟨ 1⟩ + ⟨ -1⟩∈(k) is the hyperbolic
        form.



  For a smooth, projective curve C, the schemes ^nC are again smooth and
  projective. Thus, the quadratic Euler characteristic of ^nC is
  well defined.



  MacDonald <cit.> computed the topological Euler
  characteristic of ^nC for a smooth, projective complex curve C to
  be
  
    χ^top(^nC) = (-1)^n2g-2n.

  This implies
  
    rank(χ(^nC/k)) = (-1)^n2g-2n

  for k a field of characteristic zero. Indeed, if C is a
  smooth, projective curve over a
  field k of characteristic zero, there exists a smooth, projective curve C_0 over a
  subfield k_0 ⊂ k with the following two properties: First, k_0 is
  finitely generated over  and, second, the base change (C_0)_k to k
  is isomorphic to C. Since the rank of the quadratic Euler characteristic
  is invariant under base-change and k_0 admits an embedding into , we
  get after choosing an embedding k_0 →
  
    rank(χ(^nC/k))    = rank(χ(^nC_0/k_0))
       =
        rank(χ(^n(C_0)_/))
       = (-1)^n2g-2n

  as desired.




  The formula in Theorem <ref> evaluates to zero for a smooth,
  projective curve C of genus g whenever n > 2g-2. If C has a rational
  point, this can also be seen with the following brief argument: Consider the map
  ^nC→Pic^n(C) sending a point on ^nC with
  associated divisor D to 𝒪_C(D). By for example the proof of
  Mustata <cit.>, this map realizes ^nC as
  a projective space bundle over Pic^n(C) and we can find an
  isomorphism Pic^n(C)≅Pic^0(C). Using the Motivic
  Gauss-Bonnet Theorem, the quadratic Euler characteristic of any
  Abelian variety is zero because its tangent bundle is trivial. Combining
  this with <cit.> yields the desired vanishing.



As an application, we extend a result on compatibility of power structures of
Pajwani and Pál <cit.>:
Over a field k of characteristic
zero, Arcila-Maya, Bethea, Opie, Wickelgren and Zakharevich
<cit.> extended the quadratic Euler characteristic to a motivic measure

    χ_c: K_0(Var_k)→(k)

called the compactly supported 𝔸^1-Euler characteristic. On K_0(Var_k), Gusein-Zade,
Luengo and Melle-Hernández <cit.> showed that
symmetric powers of schemes define a power structure. In <cit.>,
Pajwani and Pál constructed a power
structure a_∗ on (k) with the following property: If there exists a power
structure on (k) such that χ_c respects the power structures, then the power structure on (k) has to agree with
a_∗. It is still an open question,  whether there indeed is a power
structure on (k) such that χ_c respects the power structures. In
Theorem <ref>, we prove that if (k) is given the power structure a_*, then χ_c respects the power
structures for all curves over a field of characteristic zero.


At the moment, there is no general formula for the quadratic Euler
characteristic of a quotient scheme. Theorem <ref> is an example
result for this
and might be helpful for understanding the general case. Further work
on quadratic Euler characteristics of quotient schemes will appear in the
upcoming PhD thesis of the first named author.


The proof of Theorem <ref> uses an explicit computation in
conjunction with the Motivic
Gauss-Bonnet Theorem proven by Levine and Raksit <cit.>, which computes
the quadratic Euler characteristic of a smooth projective
scheme as a composition of a cup product and trace on Hodge cohomology.


In related work,
the Motivic Gauss-Bonnet Theorem was also used by Levine, Pepin Lehalleur, and Srinivas <cit.>
to compute the quadratic Euler characteristic of a smooth hypersurface in
ℙ^n and in the second named author's computation of the quadratic Euler
characteristic of a smooth same-degree
complete intersection in ℙ^n, see <cit.>.

Adjacent to our application, Pajwani and Pál <cit.> showed that their power structure is
respected for zero dimensional schemes. Work of Pajwani, Rohrbach, and the
second named author
<cit.> extended this to the class of “étale linear
varieties”, which is the subring of K_0(Var_k) generated by the class
of [𝔸^1] and the classes of the form [Spec(L)] where L is a finite étale algebra over
k. Since curves are almost always outside
of the class of “étale linear varieties” by <cit.>, Theorem <ref> yields a proper extension of the class of
varieties for which the power structure is respected.

We would like to highlight that Theorem <ref> has been proven independently by Simon Pepin Lehalleur and Lenny Taelman, as part of work in progress on the higher dimensional case.



 §.§ Acknowledgments

We would like to extend our sincere thanks to Marc Levine for suggesting this project to us, for helpful
discussions, and for useful comments on an earlier draft of this paper. We are very grateful to Lenny Taelman and Simon Pepin
Lehalleur for their helpful comments on an earlier draft of this paper. We further thank Jesse Pajwani for pointing us to the paper
<cit.> and for helpful comments on an earlier draft of this
paper. We also thank Herman Rohrbach for pointing us to the paper
<cit.> and for helpful discussions. In addition, we
would like to acknowledge Raymond Cheng and Hind Souly for helpful discussions.

The first named author was supported by the RTG 2553. The second named author was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 948066 (ERC-StG RationAlgic).



 §.§ Notation and Conventions

Throughout, we let k be a field which is not of characteristic 2. By a
curve, we mean a one-dimensional, geometrically connected, separated, finite-type scheme over k. For a
smooth projective scheme X over k, we write Ω_X for the sheaf of
Kähler differentials on X. For i∈ℤ_≥ 0, we write
Ω_X^i = Λ^iΩ_X.

For p,q, p', q' ∈_≥ 0, we denote the cup product of a ∈
H^q(X,Ω^p_X)  and b ∈ H^q'(X,Ω^p'_X) by ab := a· b :=
a ∪ b ∈ H^q+q'(X,Ω^p+p'_X).
This cup-product is graded commutative if we
consider a ∈ H^q(X,Ω^p_X) to be of degree q-p. That is,  we have for
a ∈ H^q(X,Ω^p_X) and b ∈ H^q'(X,Ω^p'_X) that ab =
(-1)^(q-p)(q'-p') ba.



§ QUADRATIC EULER CHARACTERISTIC

The quadratic Euler characteristic, which was first studied by Hoyois
<cit.>, is a refinement of the
topological Euler characteristic. It is constructed by applying the categorical Euler characteristic of Dold-Puppe <cit.> to the stable motivic
homotopy category (k) constructed by Morel-Voevodsky <cit.>. See
also Hoyois <cit.> for an introduction to (k). See <cit.> for a detailed exposition of the quadratic
Euler characteristic and its basic
properties.

One can assign a quadratic Euler characteristic χ(X/k) to any smooth
scheme X over
k. This is an element of the Grothendieck-Witt ring of quadratic forms over k,
which we introduce below.


  The Grothendieck-Witt ring (k) over k is
  defined as the group completion of the monoid of isometry classes of
  non-degenerate quadratic forms over k with respect to taking orthogonal
  direct sums.

  The group (k) inherits a ring structure from the tensor
  product of quadratic forms.
  


  Over a field of characteristic not 2, a non-degenerate quadratic form is
  the same as a non-degenerate symmetric bilinear form, and elementary linear
  algebra computations yield the following presentation. The
  quadratic forms ⟨ a ⟩:x ↦ ax^2 for x ∈ k^×
  generate (k) and they are subject to the following relations for
  a, b∈ k^×:
  
  
  * ⟨ a⟩·⟨ b⟩ = ⟨ a b ⟩,
  
  * ⟨ a⟩ + ⟨ b⟩ = ⟨ a+b⟩ + ⟨
    ab(a+b)⟩, whenever a+ b ∈ k^×,
  
  * ⟨ ab^2⟩ = ⟨ a⟩, and
  
  * ⟨ a⟩ + ⟨ -a⟩ = ⟨ 1⟩ + ⟨
    -1⟩ =: H
  
  The quadratic form H = ⟨ 1⟩ + ⟨ -1⟩ is called the
  hyperbolic form.

  Witt <cit.> first described this presentation. In the form
  presented here, it is
  <cit.>. There, the statement is deduced from the
  statement for Witt rings, proven by Milnor and Husemoller
  <cit.>.



  In <cit.>, Levine shows that for k ⊂ and X
  a smooth, projective scheme over k, the rank of χ(X/k) agrees with
  the topological Euler characteristic of X(ℂ) and the signature
  agrees with the topological Euler characteristic of X(). By a theorem of Saito <cit.>, the discriminant of χ(X/k) can also be interpreted in terms of the determinant of ℓ-adic cohomology for any ℓ which is coprime to the characteristic of k. See  <cit.> and the surrounding text, for a more detailed discussion. 


The Motivic Gauss-Bonnet Theorem of Levine-Raksit <cit.> provides a
way to compute the quadratic Euler characteristic using Hodge
cohomology. A more general version has been proved by
Déglise-Jin-Khan <cit.>. We do not state the theorem in full
generality here, but rather a simplified version of one of its corollaries,
which is enough to prove Theorem <ref>.


  Let X be a smooth, projective scheme over k.
  
  
  * If X is odd, then χ(X/k) = m· H for some m ∈.
  
  * If X = 2n is even, then χ (X/k) = Q + m· H for some m ∈
    Z where Q is the quadratic form given by the composition
    
    H^n(X,Ω^n_X)× H^n(X,Ω^n_X) 
          H^2n(X,Ω_X^2n)  k.

    Here, ∪ denotes the cup product and  the trace map from Serre
    duality.
  


Arcila-Maya, Bethea, Opie, Wickelgren, and Zakharevich <cit.> constructed a motivic
measure χ_c K_0(_k)→(k), called the compactly
supported ^1-Euler characteristic, which extends the quadratic Euler
characteristic of smooth, projective schemes
to varieties over k when k has characteristic 0. Levine, Pepin Lehalleur and
Srinivas <cit.> give a definition of χ_c in positive characteristic.



§ PROOF OF THEOREM <REF>

Let C be a smooth, projective curve of genus g over k and let
n∈ℤ_≥ 1. In order to prove Theorem <ref>, we first reduce to the case that k has
characteristic zero.


  Assume Theorem <ref> holds for all fields of characteristic
  zero. Then Theorem <ref> holds for all fields of characteristic
  not two.

  
    Assume that k has positive characteristic. Using <cit.>, we can assume without loss of generality that k is a
    perfect field. By for example
    <cit.>, we can find a complete
    discrete valuation ring A with residue field k and field of fractions
    K of characteristic zero. Now <cit.> proves that
    we can lift C to a smooth, projective curve C̃ over A, that is
    we get the following diagram of fibre squares:
    
    C[d, "p_k"] [r, "i"][dr, phantom, "⌜", very near start]    C̃[d, "p_A"]    C̃_K[l,
            "j"][d, "p_K"][dl, phantom, "⌝", very near start]
     k [r, "i"]     A     K[l, "j"].

    We now use the six functor formalism on (-) to prove the
    lemma, see for example <cit.>. For a scheme S, denote the unit object in (S) by 1_S.
    As
    ^n C̃ is smooth and projective,     (p_A)_!(1_^nC̃) =
    (p_A)_∗(1_^nC̃) is strongly dualisable by <cit.>. Thus, the compactly supported 𝔸^1-Euler
    characteristic of ^nC̃ is defined as the categorical Euler characteristic 
    
    χ^cat((p_A)_!(1_^nC̃)) ∈_(A)(1_A).


    Consider the following commutative diagram
    
    (^n C)[d, "(p_k)_!"]    (^nC̃)[d, "(p_A)_!"][l,
            "i^∗"][r, "j^∗"]    (^nC̃_K)[d, "(p_K)_!"]
    (k)    (A)[l, "i^∗"][r, "j^∗"]    (K).

    Since i^∗ and j^∗ are symmetric monoidal, we get using
    <cit.>
    
    χ^cat(^nC)    = χ^cat((p_k)_!1_^nC)
       =
          χ^cat((p_k)_!i^∗1_^nC̃) 
       =
          χ^cat(i^∗(p_A)_!1_^nC̃) 
       =
          i^∗(χ^cat((p_A)_!1_^nC̃)) 
       = i^∗(χ^cat(^nC̃))
          ∈_(k)(1_k)

    and similarly χ^cat(^nC̃_K) = j^∗(χ^cat(^nC̃))∈_(K)(1_K).

    Hornbostel <cit.> constructed a
    ℙ^1-ring spectrum KQ_S ∈(S) for S a regular
    scheme with 2 ∈𝒪_S(S)^× representing Hermitian K-theory.
    Using the unit map u 1_-→KQ_- of Hermitian K-theory, we can
    now map each of the above categorical Euler characteristics to the (0,0)-th stable homotopy group of
    KQ_-, which is the Grothendieck-Witt ring by for example <cit.>. That is, we
    get uχ^cat(^nC) = i^∗ (uχ^cat(^nC̃))∈KQ^0,0_k =
    (k) and uχ^cat(^nC̃_K) = j^∗ (uχ^cat(^nC̃))∈ KQ^0,0_K =
    (K), where we use Hoyois, Jelisiejew, Nardin, and Yakerson's result
    <cit.> that Hermitian K-theory is stable under base
    change. The map u is an
    isomorphism in (k), (A) and (K) by an argument of
    Bachmann-Hoyois <cit.>. These isomorphisms
    are compatible with Morel's identification of the endomorphisms of
    the unit in (-) with the Grothendieck-Witt ring <cit.>.

    We are now in the following situation:
    
    [column sep=small]
            (k) [d, phantom, "[origin=c]90∈"]    [l, "i^∗"'] 
            (A) [r, "j^∗"][d, phantom, "[origin=c]90∈"]    (K)[d, phantom, "[origin=c]90∈"]
    χ(^nC/k)    [l, mapsto] uχ^cat(^nC̃) [r, mapsto]    χ(^nC̃_K/K).

    By assumption, we know that χ(^nC̃_K/K) is given by the
    formula in Theorem <ref>. Since the map (A) →(K)
    is injective by for example <cit.>, this formula also computes uχ^cat(^nC̃), which in
    turn computes χ(^nC/k), proving the lemma.
  


 
  Since ^nC has odd dimension n,
  its quadratic Euler characteristic is a multiple of hyperbolic
  forms by the Motivic Gauss-Bonnet Theorem, and thus it is completely
  determined by its rank. Thus, Remark <ref> implies
  
    χ(^nC/k) = -1/22g-2n· H

  as desired.


From now on, assume that n=2m is even and that k has characteristic
zero. The only reason for the assumption that k is of characteristic zero
is to ensure that n! is invertible in k. We will compute χ(^nC/k) using the Motivic Gauss-Bonnet Theorem. MacDonald <cit.> has computed the topological Euler characteristic of the symmetric product of a curve over the complex numbers using an explicit basis computation. We follow the same strategy. 

 
   Let β^∨∈ H^0(C,𝒪_C) be the unit element with respect
   to the cup product and let β∈ H^1(C,Ω_C^1) be the dual element
   with respect to Serre duality. In particular, the trace map sends β
   to 1. Let α_1,…, α_g be a basis for H^0(C,Ω_C) and
   let
   α_1^∨, …, α_g^∨ be the dual basis for
   H^1(C,𝒪_C) with respect to Serre duality.

  Let π_j C^n → C be the projection on the j-th
  component. We write α_i^(j) := π_j^∗α_i and
  similarly α_i^∨, (j) := π_j^∗α_i^∨,
  β^(j) = π_j^∗β, and β^∨, (j) = π_j^∗β^∨.



  
  We have the following multiplication rules for the above generators. Let
  i,j∈{1,…, g} be such that i≠ j. Then
  
    β·β = 0       β·β^∨ = β      β^∨·β^∨ = β^∨
    β·α_i = 0       β·α_i^∨ = 0       α_i^∨·α_i = β
    α_i·α_j= 0       α_i^∨·α_j =0       α_i^∨·α_i^∨
                                                                    = 0
    α_i·α_i = 0       β^∨·α_i = α_i       β^∨·α^∨_i
                                                                                  = α^∨_i

  The symmetric group S_n acts on
  C^n by σβ^(j) = β^(σ(j)) and
  σα_i^(j) = α_i^(σ(j))  and similarly on the duals for σ∈ S_n, and
  this extends multiplicatively. 



We have that ^nC = C^n/S_n. Since ^nC is smooth, we have for
p,q∈_≥ 0 that
H^q(^nC, Ω^p_^nC) ≅
H^q(C^n,Ω^p_C^n)^S_n where the isomorphism is induced by the
quotient map C^n →^nC. In the following, we identify the two
k-vector spaces.


	We have H^n(C^n, Ω_C^n^n) ≅ k.
	A basis is given by β^(1)⋯β^(n). Therefore,
        H^n(^nC, Ω_^nC^n) ≅ k has basis
        
    ∑_σ∈ S_nβ^(σ(1))⋯β^(σ(n)) = n!β^(1)⋯β^(n).




  
  Let 0 ≤ν≤ m and let I = {i_1 < … < i_ν},J = {j_1< …<
  j_ν}⊂{1, …, g} be subsets. Define
  
    a_IJ := α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)·β^(2ν+1)⋯β^(m+ν)·β^∨, (m+ν+1)⋯β^∨, (n)

  in H^m(C^n, Ω_C^n^m)
  and
  
    α_IJ := ∑_σ∈ S_nσ a_IJ.

  Then, α_IJ is an elements of H^m(^nC,
  Ω_^nC^m). Furthermore, H^m(^nC, Ω_^nC^m) has basis given by
  α_IJ , where I,J runs over all same order subsets I,J ⊂{1, …, g} of order at most
  m.


 
  Let π_i: C^n→ C be the projection on the i'th coordinate. We 
  have
  
    Ω_C^n =
        π_1^*Ω_C⊕⋯⊕π_n^*Ω_C.

  This implies
  
    Ω_C^n^m    = Λ^mΩ_C^n
       = ⊕_i_1 + ⋯ + i_n = mΛ^i_1π_1^*Ω_C ⊗⋯⊗Λ^i_nπ_n^*Ω_C

  Using the Künneth formula, we find
  
    H^m(C^n,Ω_C^n^m)    = ⊕_i_1 + ⋯ + i_n = mH^m(C^n,Λ^i_1π_1^*Ω_C ⊗⋯⊗Λ^i_nπ_n^*Ω_C)
       = ⊕_i_1 + ⋯ + i_n = m⊕_j_1 + ⋯ + j_n = mH^j_1(C,Λ^i_1π_1^*Ω_C) ⊗⋯⊗ H^j_n(C, Λ^i_nπ_n^*Ω_C)
 
  We note that H^j(C,Λ^iΩ_C) has basis 
  
    β^∨    if  j = i = 0,
    α_1,…, α_g     if  j = 0, i = 1,
    α_1^∨,…, α_g^∨    if  j = 1, i = 0,
    β    if  j = i = 1.

  For I and J same-length tuples
  (i_1, …, i_ν), (j_1, … j_ν) with 0≤ i_1 ≤…≤ i_ν≤ g
  and 0 ≤ j_1 ≤…≤ j_ν≤ g and ν≤ m define
  
    a_IJ := α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)·β^(2ν+1)⋯β^(m+ν)·β^∨, (m+ν+1)⋯β^∨, (n).

  (The notation a_IJ for this is
  justified as it is a natural extension of the above definition.)
  Then, we get the basis (σ a_IJ)_I,J,σ for
  H^m(C^n,Ω_C^n^m) where I and J run over same-length tuples of the above form
  and σ runs over the elements of S_n not fixing a_IJ.

  Now, let I =(i_1, …, i_ν) and J = (j_1, …, j_ν) be tuples as
  above with ν≥ 2. Assume that i_μ = i_μ+1 for some
  μ. Then we note that σ a_IJ = -a_IJ for σ the
  transposition swapping μ and μ + 1.

  Thus, if v ∈ H^m(^nC, Ω^m_^nC) = H^m(C^n,
  Ω^m_C^n)^S_n, we have for such I,J that the a_IJ-coefficient of v
  vanishes. Analogously, we get the same vanishing if j_μ = j_μ+1.
  
  In order to construct a basis of H^m(^nC, Ω^m_^nC), we can
  therefore restrict to the a_IJ where i_1 < … < i_ν and j_1 <
  … < j_ν. Since, S_n acts freely on these elements, we take sums under all σ∈ S_n, to construct a
  basis of the S_n fixed-points. This is the desired basis.
 


  
  Let σ∈ S_n and I,J, I', J' ⊂{1, …, g}, with ν = |I| =
  |J| ≤ m and |I'| = |J'| ≤ m.
  In the notation of Lemma <ref>, we have
  
    a_IJ·σ a_I'J' = (-1)^νβ^(1)⋯β^(n)

  whenever I = J', J = I' and σ satisfies
  
  
  * σ(i) = ν+i and σ(ν+i) = i for 1 ≤ i ≤ν and
  
  * σ({2ν+1, …, m+ ν}) = {m+ ν+ 1, …, n}.
  
  Otherwise
  a_IJ·σ a_I'J' = 0.

  
    The last assertion follows from the multiplication table in Remark
    <ref>.

    Assume that I = J', J = I' and that σ satisfies the condition of
    the lemma. Then, we have
    
    σ a_JI = α_j_1^(ν + 1)⋯α_j_ν^(2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)·β^(σ(2ν+1))⋯β^(σ(m+ν))·β^∨, (σ(m+ν+1))⋯β^∨, (σ(n)).

    Since the degrees of β^(i) and β^∨,(i) are even, we can
    rearrange these without introducing a sign. Thus, the above equation
    can be rewritten as
    
    σ a_JI = α_j_1^(ν + 1)⋯α_j_ν^(2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)·β^∨, (2ν+1)⋯β^∨, (m+ν)·β^(m+ν+1)⋯β^(n).

    Define B' := β^∨, (2ν+1)⋯β^∨, (m+ν)·β^(m+ν+1)⋯β^(n) and,
    similarly, define B := β^(2ν+1)⋯β^(m+ν)·β^∨, (m+ν+1)⋯β^∨, (n).
    Then, since B and B' have even degree, we have that
    
    a_IJ·σ a_JI   = α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)· B
            ·α_j_1^(ν + 1)⋯α_j_ν^(2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)· B'
       = α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)·α_j_1^(ν + 1)⋯α_j_ν^(2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)· B· B'

    All α^(i)'s and α^∨, (i)'s have odd degree. Thus, if we rearrange these elements by a permutation
    σ, we have to multiply with (-1)^(σ). Therefore, if we
    rearrange
    
    α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)·α_j_1^(ν + 1)⋯α_j_ν^(2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)·
          B· B'

    as
    
    α_i_1^(1)⋯α_i_ν^(ν)·α_j_1^∨, (ν+1)⋯α_j_ν^∨, (2ν)·α_i_1^∨, (1)⋯α_i_ν^∨, (ν)·α_j_1^(ν + 1)⋯α_j_ν^(2ν)·
          B· B',

    we are swapping ν times, so this introduces the sign (-1)^ν. If we
    next rearrange this into 
    
    α_i_1^(1)α^∨,(1)_i_1⋯α_i_ν^(ν)α_i_ν^∨, (ν)·α_j_1^∨, (ν+1)α^(ν+1)_j_1⋯α_j_ν^∨, (2ν)α_j_ν^∨, (2ν)·
          B· B',

    we are moving α_i_1^∨, (1) by 2ν-1 elements to the
    left, then we are moving α_i_2^∨, (2) by 2ν-2 elements to the
    left, α_i_3^∨, (3) by 2ν-3 elements, and so on. Therefore,
    the resulting permutation has sign ∑_i=1^2ν 2ν -i =
    ∑_i=0^2ν-1 i = 2ν(2ν-1)/2 = ν(2ν -1). Thus, we
    have
    
    a_IJ·σ a_JI   = =1(-1)^ν(-1)^ν(2ν-1)·α_i_1^(1)α^∨,(1)_i_1⋯α_i_ν^(ν)α_i_ν^∨, (ν)·α_j_1^∨, (ν+1)α^(ν+1)_j_1⋯α_j_ν^∨, (2ν)α_j_ν^∨, (2ν)·
            B· B'
       = =β^(1)α_i_1^(1)α^∨,(1)_i_1⋯=β^(ν)α_i_ν^(ν)α_i_ν^∨, (ν)·=-β^(ν+1)α_j_1^∨, (ν+1)α^(ν+1)_j_1⋯=-β^(2ν)α_j_ν^∨, (2ν)α_j_ν^∨, (2ν)·
            B· B'
       = (-1)^νβ^(1)⋯β^(2ν)=
            β^(2ν+1)⋯β^(n)B· B'
       = (-1)^νβ^(1)⋯β^(n).

  



  
  Let f X → Y be a finite morphism of degree d between smooth,
  projective, n-dimensional schemes X and Y
  over k. Then, we have the commutative diagram
  
    H^n(X, Ω^n_X) [r, "_X/k"]     k.
    
          H^n(Y, Ω^n_X) [ur, "d·_Y/k"'][u,
          "f^∗"]


  
    This argument was explained to the first named author by Marc Levine. Let
    f_*: H^n(X,Ω^n_X)→ H^n(Y,Ω^n_Y) be the pushforward as
    constructed by Srinivas <cit.>. Then by <cit.>, the diagram
    
    H^n(X, Ω^n_X) [r, "_X/k"][d, "f_∗"]     k
    
          H^n(Y, Ω^n_Y) [ur, "_Y/k"']

  commutes and we have f_∗ f^∗ = d·id. This yields the desired result. 
  



  We will apply the Motivic Gauss-Bonnet Theorem and thus we need to compute the form 
  
    [row sep= small]
        H^m(^nC, Ω_^nC^m)× H^m(^nC,
        Ω_^nC^m) [d, "∪"]
    
        H^n(^nC,
        Ω_^nC^n) [d, ""]
    
        k.

  By Lemma <ref>, the vector space H^m(^nC, Ω_^nC^m) has basis α_IJ for
  I,J running over the same size subsets I,J
  ⊂{1, …, n} with |I| = |J| ≤ m. By Lemma
  <ref>, we know that α_IJα_I'J' = 0
  whenever I  J' or J  I'. Thus, it remains to compute
  the cup product α_IJα_JI. For this, note that
  
    α_IJα_JI = ∑_σ,τ∈ S_n (σ a_IJ)·
        (τ a_JI)= ∑_σ, τ∈ S_nσ (a_IJ·
        (σ^-1τ)a_JI) = ∑_σ, τ∈ S_nσ
        (a_IJ·τ a_JI)

  where in the last step, we replace σ^-1τ by τ. Since a_IJ·τ a_JI is a multiple of β^(1)⋯β^(n) and S_n acts trivial on this element, we get by Lemma <ref>
  
    α_IJα_JI   = ∑_σ, τ∈ S_nσ
        (a_IJ·τ a_JI)
       = n!·∑_τ∈ S_n
        a_IJ·τ a_JI
       = n!·∑_τ∈ S_n'
        (-1)^|I|β^(1)⋯β^(n)
       = (-1)^|I|· n!·
        |S_n'|·β^(1)⋯β^(n)

  where S_n' ⊂ S_n is the subset of all permutations satisfying the
  non-vanishing condition of Lemma <ref> for I and
  J. Thus, in order to compute the product, we have to determine the cardinality
  of S_n'. Note that
  
    S_n' = {τσ|σ∈ S_m-|I|×
        S_m-|I|},

  where we embed S_m-|I|×
  S_m-|I in S_n by letting it permute the last 2(m-|I|) elements of
  {1, …, n} and
  where τ∈ S_n is the permutation defined as follows:  We set τ(i) = ν+i and τ(ν+i) = i for 1 ≤ i ≤ |I|,
  and τ(2ν+j) = m+ ν+ j and τ(m+ν+j) = 2ν + j for 1 ≤
  j ≤ m-|I|.
  Thus, |S_n'| = |S_m-|I|× S_m-|I|| =
  ((m-|I|)!)^2, and we have
  
    α_IJα_JI = (-1)^|I|· n!·
        ((m-|I|)!)^2·β^(1)⋯β^(n).

  Since Lemma
  <ref> yields (β^(1)⋯β^(n)) = (n!)^-1, we obtain
  
    (α_IJα_I'J') =
        
          (-1)^|I|· ( (m-|I|)!)^2    if I=J'  and  J =
                                                 I'
    
          0    otherwise.

  Thus, the matrix representing the bilinear form with respect to the basis
  (α_IJ) takes the form of a diagonal block matrix with blocks
  A_{I,J} for I,J ⊂{1, …, g} same-size subsets of order at
  most m. Each block A_{I,J} takes the form
  
    A_{I,J} =
        [ (-1)^|I|· ((m-|I|)!)^2 ]

  if I = J and
  
    A_{I,J} =
        [                       0 (-1)^|I|· ( (m-|I|)!)^2; (-1)^|I|· ( (m-|I|)!)^2                       0 ]

  if I≠ J. The matrix A_{I,I} represents the form ⟨ (-1)^|I|·
  ( (m-|I|)!)^2⟩ = ⟨ (-1)^|I|⟩∈(k), and the
  matrix A_{I,J} with I  J represents the hyperbolic form in
  (k). Thus, we have that the quadratic form q in (<ref>) is given by
  
    q=∑_I ⊂{1, …, g}⟨ (-1)^|I|⟩ + l· H

  for some l ∈ where the sum only runs over subsets of order at most
  m. Since {1, …, g} contains g i subsets of order i, we
  get
  
    q = ∑_i=0^m gi ⟨ (-1)^i⟩ + lH.

  Combining this computation with Remark <ref> yields the first
  formula of the theorem. For the second formula, we plug in H = ⟨ 1⟩ + ⟨ -1⟩ and then note that
  
    ∑_i=0^mgi(-1)^i = (-1)^mg-1m.





§ COMPATIBILITY WITH THE POWER STRUCTURE

Recall that a power structure on a commutative ring R is a map
(1+tR[[t]])× R→ 1+tR[[t]] satisfying certain axioms, see for example
<cit.> for a precise definition. Under some
finiteness assumptions, it suffices to define (1-t)^-r = ∑_i=0^∞
a_n(r)t^n for r ∈ R, see <cit.>. That is, a power structure can be given by functions a_n: R→ R for n∈ℤ_≥ 0 such that:

	
  * a_0 = 1,
	
  * a_1 is the identity,
	
  * a_n(0) = 0 for all n≥ 1,
	
  * a_n(1) = 1 for all n≥ 1, and
	
  * a_n(r+s) = ∑_i=0^na_i(r)a_n-i(s) for all r,s∈ R.
 
A ring homomorphism ϕ: R→ R' between rings that both carry a power structure is said to respect the power structures if ϕ(a_n^R(r)) = a_n^R'(ϕ(r)) for all r∈ R and n∈ℤ_≥ 0. 

For the rest of this section, we assume that k is of characteristic zero.

There is a power structure on K_0(_k) given by a_n([X]) = [^nX]. This was proven by Gusein-Zade, Luengo and Melle-Hernández  <cit.>. 

Pajwani and Pál <cit.> constructed a power structure on
(k) given by

    a_n(⟨ a ⟩) = ⟨ a^n⟩ + n(n-1)/2(⟨ a
      ⟩ + ⟨ 2 ⟩ - ⟨ 1 ⟩ - ⟨ 2a ⟩).

 
  This power structure agrees with McGarraghy's non-factorial power structure
  <cit.> on (k) on the subgroup generated by all ⟨ a
  ⟩ satisfying a_n(⟨ a ⟩) = ⟨ a^n⟩, see
  <cit.>. The quadratic
  forms ⟨ 1 ⟩ and ⟨ -1 ⟩ are in this subgroup. 
 
Pajwani and Pál prove that χ_c: K_0(_k)→(k) respects the power structures for
zero dimensional schemes, i.e. χ_c(^nX/k) = a_n(χ_c(X/k)) for X a
zero dimensional scheme. This was extended to a larger class of schemes by
Pajwani, Rohrbach, and the second named author in <cit.>, who also showed
compatibility for curves of genus 0 and 1. We now show this result for
all curves, i.e. we check that all curves are symmetrizable in the sense of
<cit.>. 

For C a smooth, projective curve of genus g, we have χ(C/k) =
(1-g)H. We therefore need to evaluate a_n on negative multiples of
H. Note that McGarraghy <cit.> gave a
complete description of what happens for positive multiples.


	Let l∈ℤ_≥ 1. Then a_n(-lH) = (-1)^n
        ∑_i=0^nliln-i⟨ -1
        ⟩^n-i.
 


	One can prove by induction on l that 
	
    a_n(-l⟨ 1 ⟩ ) = (-1)^nln⟨ 1 ⟩
 
	and
	
    a_n(-l⟨ -1 ⟩ ) = (-1)^nln⟨ -1 ⟩^n.
 
	By combining (<ref>) and (<ref>), we obtain 
	
    a_n(-lH)    = ∑_i=0^na_i(-l⟨ 1 ⟩)a_n-i(-l⟨ -1 ⟩) 
       = (-1)^n∑_i=0^nliln-i⟨ -1⟩^n-i

	as desired. 



	We note that for l,n∈ℤ_≥ 1, the rank of a_n(-lH) equals 
	
    (-1)^n∑_i=0^nliln-i = (-1)^n2ln

	by the Chu-Vandermonde identity. Plugging in l = g-1, we find that
        the rank is precisely (-1)^n2g-2n. This matches with Theorem <ref>. 


There is another way to compute a_n(-lH), suggested by Marc Levine. 


	Let l∈ℤ_≥ 1. Then a_n(-lH) is the coefficient of t^n in the polynomial (⟨ 1 ⟩ -Ht+⟨ -1 ⟩ t^2)^l.
 


	For ⟨ a ⟩∈(k), let a_t(⟨ a ⟩) :=
        (1-t)^-⟨ a ⟩ = ∑_i=0^∞ a_i(⟨ a
        ⟩)t^i. Then a_t(⟨ a ⟩ + ⟨ b ⟩) =
        a_t(⟨ a⟩)a_t(⟨ b⟩). Now, we have a_t(-⟨ 1 ⟩) = 1-⟨ 1 ⟩ t
	and also 
	a_t(-⟨ -1 ⟩) = 1-⟨ -1 ⟩ t.
	This yields
	
    a_t(-lH)    = ((1-⟨ 1 ⟩ t)(1-⟨ -1⟩ t))^l 
       = (⟨ 1 ⟩ -Ht+⟨ -1 ⟩ t^2)^l
 
	as desired. 



	Let C be a smooth, projective curve of genus g and let
        n∈ℤ_≥ 0. Then 
    χ(^nC/k)=a_n(χ(C/k)).

 


	The two forms have the same rank by Remark <ref>. Given that they are both hyperbolic for n odd, this yields
        the result in that case. Thus, assume that n = 2m is even from now on.  
	 
	Note that both forms only consist of terms ⟨ 1 ⟩ and ⟨ -1 ⟩. Therefore, given that we know the ranks are the same, the two are the same if for both forms the difference between the number of ⟨ 1⟩'s and ⟨ -1⟩'s is the same. Note that if we work over the real numbers this difference is the signature of the quadratic form. By Theorem <ref>, this difference is 
	(-1)^mg-1m for χ(^2mC/k).

	On the other hand, denote this difference for a_n((1-g)H) by s. Lemma <ref> yields that a_n((1-g)H) is the coefficient of t^n in 
	
    (⟨ 1 ⟩ -Ht+⟨ -1 ⟩ t^2)^g-1   = ∑_i=0^g-1g-1i (⟨ 1 ⟩ + ⟨ -1 ⟩ t^2)^i (-Ht)^g-1-i.

	All summands with i≠ g-1 are a multiple of H and thus do not contribute to s. Thus, s is completely determined by the degree 2m-term of 
	
    (⟨ 1⟩ + ⟨ -1⟩ t^2)^g-1   = ∑_i=0^g-1g-1i⟨ -1 ⟩^it^2i,
 
	which is g-1m⟨ -1⟩^mt^2m. This shows s = (-1)^mg-1m, which proves the proposition. 


We can extend this to show that the power structure is respected for all curves. 


	Let C be a curve and let
        n∈ℤ_≥ 0. Then 
    χ_c(^nC/k) =
            a_n(χ_c(C/k)).




	Let C̃ be the normalization of C. Let S⊂ C be the
        locus of singular points on C and let S̃⊂C̃ be
        the points lying over S. Then, the normalization map π: C̃→ C is an isomorphism outside of the singular points of C and so 
	
    [C]  = [C̃] - [S̃] + [S]

	in K̃_̃0̃(Var_k). By Proposition <ref>, the power structure is respected on C̃. Also, as S and S̃ are zero dimensional, the power structure is respected on those by <cit.>. By <cit.>, the set of elements of K̃_̃0̃(Var_k) for which the power structure is respected forms a subgroup, implying that the power structure is respected for C.


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    Lukas F. Bröring 

    Universität Duisburg-Essen 

    Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany 

	E-Mail: mailto:lukas.broering@uni-due.de lukas.broering@uni-due.de


	Anna M. Viergever 

	Leibniz Universität Hannover

	Fakultät für Mathematik, Welfengarten 1, 30167 Hannover, Germany 

	E-Mail: mailto:viergever@math.uni-hannover.de viergever@math.uni-hannover.de 
 
 

	Keywords: motivic homotopy theory, refined enumerative geometry, other fields
 
	Mathematics Subject Classification: 14G27, 14N10, 14F42

