2.1 Cycle modules

Let F be a field, and let $\textrm{T}^{\bullet}(F)=\oplus_{i} (F^*)^{\otimes i}$ be the tensor algebra over F. We will write $\lbrace a_1, \ldots, a_n \rbrace$ for the element $a_1 \otimes \cdots \otimes a_n \in \textrm{T}^{\bullet}(F)$ , so that the multiplication reads $\lbrace a_1, \ldots, a_r \rbrace \cdot \lbrace b_1, \ldots, b_d \rbrace = \lbrace a_1, \ldots, a_r, b_1, \ldots, b_d \rbrace$ .

Milnor’s K-theory ring of F is the graded ring $\textrm{K}^{\bullet}_\textrm{Mil}(F)=\textrm{T}^{\bullet}(F)/I$ , where I is the bilateral ideal generated by elements $\lbrace a_1, \ldots, a_r \rbrace$ such that $a_1 + \cdots + a_r = 1$ . Some properties of $\textrm{K}_{\textrm{Mil}}^{\bullet}(F)$ we will need are as follows:

• – $\textrm{K}_{\textrm{Mil}}^{\bullet}(F)$ is graded-commutative, i.e.

  \[\lbrace a_1, \ldots , a_r, b_1, \ldots, b_d \rbrace = (-1)^{rd} \lbrace b_1, \ldots, b_d, a_1, \ldots , a_r \rbrace;\]

• – $\lbrace a_1, \ldots, a_n \rbrace$ is zero if $a_1 + \cdots + a_n = 0$ ;

• – $\lbrace a, a \rbrace = \lbrace -1, a \rbrace.$

Crucially, Milnor’s K-theory induces a functor from the category of finitely generated field extensions of $\mathbf{k}$ , and it has a series of important functorial properties.

1. (i) For any finitely generated extension $f:F \to F'$ there exists a map

   \[f^*:\textrm{K}^{\bullet}_\textrm{Mil}(F) \to \textrm{K}^{\bullet}_\textrm{Mil}(F'), \quad f^*\lbrace a \rbrace = \lbrace f(a) \rbrace.\]

2. (ii) For any finite extension $f:F \to F'$ there exists a map

   \[f_*:\textrm{K}^{\bullet}_\textrm{Mil}(F') \to \textrm{K}^{\bullet}_\textrm{Mil}(F), \quad f_*\lbrace a \rbrace = \lbrace \textrm{N}^{F'}_F(a) \rbrace.\]

3. (iii) For a DVR (R,v) with quotient field $\mathbf{k}(R)$ and residue field $\mathbf{k}_v$ there exists a ramification map

   \[\partial_v:\textrm{K}^{\bullet}_\textrm{Mil}(\mathbf{k}(R)) \to \textrm{K}^{\bullet}_\textrm{Mil}(\mathbf{k}_v), \quad \partial_v(\lbrace \pi, a_1, \ldots, a_r \rbrace)=\lbrace \overline{a_1}, \ldots, \overline{a_r} \rbrace, \]
   where $\pi$ is a uniformizer for R, which lowers the degree by one.

The maps above satisfy the expected functorial compatibilities and a projection formula $f_*(f^*(\lbrace a_1, \ldots, a_r\rbrace))=\textrm{deg}(F'/F)\lbrace a_1, \ldots, a_r\rbrace$ . Moreover, we have the following exact sequence for Milnor’s K-theory of a purely transcendental extension:

\[0 \to \textrm{K}_{\textrm{Mil}}^{\bullet}(F) \to \textrm{K}_{\textrm{Mil}}^{\bullet}(F(t)) \xrightarrow{\partial} \bigoplus_{x \in {\mathbb{A}}^1_F} \textrm{K}_{\textrm{Mil}}^{\bullet}(\mathbf{k}(x)) \to 0 \]

where the map $\partial$ is given by the sum of all ramification maps coming from points of codimension one in ${\mathbb{A}}^1_F$ .

A cycle module $\textrm{M}$ is a functor from the category $(\mathcal{F}/\mathbf{k})$ of finitely generated extensions of $\mathbf{k}$ to graded groups, such that $\textrm{M}^{\bullet}(F)$ is a graded $\textrm{K}^{\bullet}_\textrm{Mil}(F)$ -module, that is equipped with all of the maps and properties above and compatible with the respective maps and properties for Milnor’s K-theory. The crucial examples to keep in mind are:

• – $\textrm{K}_{\ell}^{\bullet}(F) = \textrm{K}_\textrm{Mil}^{\bullet}(F)/\ell$ , where $\ell$ is a positive integer coprime to $\textrm{char}(\mathbf{k})$ ; and

• – $\textrm{H}_D^{\bullet}(F) = \bigoplus_i \textrm{H}^i(F,D(i))$ , where D is a $\ell$ -torsion Galois module over $\mathbf{k}$ and $D(i)=D\otimes \mu_\ell^{\otimes i}$ .

The norm-residue isomorphism shows that $\textrm{K}_\ell = \textrm{H}_{\mathbb{Z}/\ell\mathbb{Z}}$ , which also explains the $\textrm{K}_\textrm{Mil}$ -module structure of $\textrm{H}_{D}$ as coming from cup product.

Remark 2.1. Note that if $x \in \textrm{M}^{\bullet}(F)$ is an $\ell$ -torsion element and $\alpha \in \textrm{K}^{\bullet}(F)$ , then the product $\alpha \cdot x$ only depends on the class of $\alpha$ in $\textrm{K}^{\bullet}_\ell(F)$ . This shows that we have a well-defined product

\[\textrm{K}^{\bullet}_\ell (F) \times \textrm{M}^{\bullet}(F)_\ell\to \textrm{M}^{\bullet}(F)_{\ell}, \quad ([\alpha],x) \mapsto \alpha \cdot x,\]

where $\alpha \in \textrm{K}^{\bullet}(F)$ is any representative of $[ \alpha ] \in \textrm{K}^{\bullet}_{\ell}(F)$ .

2.2 Equivariant Chow groups with coefficients

Chow groups with coefficients were introduced by Rost [Reference RostRos96] and their equivariant counterpart was first used by Guillot [Reference GuillotGui07], adapting ideas by Edidin, Graham and Totaro [Reference Edidin and GrahamEG98, Reference TotaroTot99] to compute classical cohomological invariants.

Let X be a scheme of finite type over $\mathbf{k}$ , and define

\[C_i(X,\textrm{M})=\bigoplus_{V \subset X, \textrm{dim}(V)=i}\textrm{M}^{\bullet}(\mathbf{k}(V)), \]

where the sum is over all irreducible subvarieties of dimension i of X. We have a complex

\[0 \leftarrow C_0(X,\textrm{M}) \xleftarrow{\partial_1}C_1(X,\textrm{M}) \xleftarrow{\partial_2} \cdots \xleftarrow{\partial_{\textrm{dim}(X)}} C_{\textrm{dim}(X)}(X,\textrm{M}) \leftarrow 0, \]

where the differential is obtained from the ramification map $\partial_v$ we described earlier. We define the ith dimensional Chow group with coefficients of X as

\[\textrm{A}_i(X, \textrm{M})= \textrm{Ker}(\partial_i)/\textrm{Im}(\partial_{i+1}). \]

If X is equidimensional we can also order by codimension and define $\textrm{A}^i(X,\textrm{M})=\textrm{A}_{\textrm{dim}(X)-i}(X,\textrm{M})$ .

The groups $\textrm{A}_*(X,\textrm{M})$ and $\textrm{A}^*(X,\textrm{M})$ are bigraded, with the two degrees coming from dimension (respectively codimension) and the grading of $\textrm{M}$ . We will always refer to the first grading as dimension or codimension and to the second grading as degree. When $\textrm{M}=\textrm{K}_\textrm{Mil}$ the zero-degree part of $\textrm{A}_i(X,\textrm{K}_\textrm{Mil})$ retrieves the usual Chow group $\textrm{CH}_i(X)$ and when $\textrm{M}=\textrm{K}_{\ell}$ the zero-degree part of $\textrm{A}_i(X,\textrm{K}_\ell)$ is equal to $\textrm{CH}_i(X)/\ell$ .

Often we will need to keep track of a fixed degree component. To do that, we write

\[C_i(X,\textrm{M}^d)=\bigoplus_{\textrm{dim}(V)=i} \textrm{M}^{d}(\mathbf{k}(V)),\quad \partial_i^d: C_i(X,\textrm{M}^d) \to C_{i-1}(X,\textrm{M}^{d-1})\]

so that we can define the quotient

\[\textrm{A}_i(X, \textrm{M}^d)= \textrm{Ker}(\partial^d_i)/\textrm{Im}(\partial_{i+1}^{d+1}) \]

as the component of dimension i and degree d of $\textrm{A}_\bullet(X,\textrm{M})$ . We define $\textrm{A}^i(X,\textrm{M}^d)=\textrm{A}_{\textrm{dim}(X)-i}(X,\textrm{M}^d)$ correspondingly. We will also often use the notation

\[ \textrm{A}^i(X,\textrm{M}^{\leq d}) = \bigoplus_{d' \leq d} \textrm{A}^i(X,\textrm{M}^{d'}).\]

Chow groups with coefficients enjoy all the properties of Chow groups and more. In the following list we will often use codimensional notation because it is the notation that will most often appear in the rest of the paper.

1. (A) Given a morphism $f:Y \to X$ there is a functorial pullback map $f^i:\textrm{A}^i(X,\textrm{M}) \to \textrm{A}^i(Y,\textrm{M})$ if f is flat equidimensional or X is smooth.

2. (B) Given a proper morphism $f:Y \to X$ then $f_*:\textrm{A}_i(Y,\textrm{M}) \to \textrm{A}_i(X,\textrm{M})$ is a functorial pushforward map.

3. (C) The pullback and pushforward maps are compatible along Cartesian diagrams.

4. (D) Given a closed immersion $i:V \to X$ of codimension c with complement U there is a ramification map $\partial_V:\textrm{A}^*(U, \textrm{M}) \to \textrm{A}^*(V,\textrm{M})$ and a long exact sequence

   \[\cdots \to \textrm{A}^{i}(X,\textrm{M}) \to \textrm{A}^{i}(U,\textrm{M}) \xrightarrow{\partial_V} \textrm{A}^{i-c+1}(V,\textrm{M}) \xrightarrow{i_*} \textrm{A}^{i+1}(X,\textrm{M}) \to \cdots\]

5. (E) Given a vector bundle $E \to X$ there are Chern classes

   \[c_i(E):\textrm{A}^*(X, \textrm{M}) \to \textrm{A}^{*-i}(X,\textrm{M})\]
   satisfying the usual properties of Chern classes.

6. (F) Given a vector bundle $E \to X$ we have the projective bundle formula

   \[\textrm{A}^{i}(\mathbb{P}(E),\textrm{M})=\bigoplus_{r+s=i}c_1(\mathcal{O}_{\mathbb{P}(E)}(-1))^r \cdot \textrm{A}^{s}(X,\textrm{M}).\]

7. (G) If $\textrm{M} =\textrm{K}_\textrm{Mil}$ or $\textrm{M}=\textrm{K}_{\ell}$ and X is smooth $\textrm{A}^*(X,\textrm{M})$ forms a ring which is graded commutative with respect to degree.

8. (H) If X is smooth, then $\textrm{A}^*(X, \textrm{M})$ is a $\textrm{A}^*(X,\textrm{K}_\textrm{Mil})$ -module. If, moreover, $\textrm{M}$ is $\ell$ -torsion then $\textrm{A}^*(X, \textrm{M})$ is a $\textrm{A}^*(X,\textrm{K}_{\ell})$ -module.

9. (I) The module structure above is compatible with pullbacks, pushforwards and ramification maps. Moreover, there is a projection formula

   \[f_*(f^*(x)\cdot y)=x \cdot f_*(y).\]

If our scheme is being acted upon by a smooth algebraic group $G/\mathbf{k}$ then we can form the equivariant Chow groups with coefficients by equivariant approximation, that is, we take a representation V of G such that the action is free on an open subset U whose complement has codimension greater than $\textrm{dim}(X)-i$ and define

\[\textrm{A}_{i}^G(X,\textrm{M})=\textrm{A}_{i+\textrm{dim}(V)-\textrm{dim}(G)}((X\times U)/G,\textrm{M}).\]

Similarly, if X is equidimensional and the codimension of the complement of U is greater than i we define

\[\textrm{A}_{G}^i(X,\textrm{M})=\textrm{A}^{i}((X\times U)/G,\textrm{M}).\]

These groups do not depend on the choice of representation, and in fact do not even depend on the particular presentation we chose for $\mathcal{X}=[X/G]$ : if $\mathcal{X}=[X/G]=[Y/G']$ are two different presentations of the same stack as a quotient by a smooth algebraic group over $\mathbf{k}$ then we have

\[\textrm{A}_i^{G}(X,\textrm{M})=\textrm{A}^{G'}_i(Y,\textrm{M})\]

for all i. Thus, we will sometimes write $\textrm{A}_i(\mathcal{X},\textrm{M})$ or $\textrm{A}^i(\mathcal{X},\textrm{M})$ when the chosen presentation is irrelevant.

Equivariant Chow groups with coefficients enjoy all the properties that ordinary Chow groups with coefficients do, as long as everything is G-equivariant.

2.3 Cohomological invariants

In this subsection we will briefly recall the properties of cohomological invariants which will be needed throughout the paper. For an in-depth treatment of the subject, the reader can refer to [Reference Di Lorenzo and PirisiDLP21a, § 2].

As mentioned in the introduction, given an algebraic stack $\mathcal{X}/\mathbf{k}$ , we consider [Reference Di Lorenzo and PirisiDLP21a, Definition 2.3] the functor

\[\textrm{Pt}_{\mathcal{X}}: (\mathcal{F}/\mathbf{k}) \to (\textrm{Set})\]

assigning to a finitely generated extension $F/\mathbf{k}$ the set $\mathcal{X}(F)/\!\sim$ of F-valued points of $\mathcal{X}$ modulo isomorphism. A cohomological invariant of $\mathcal{X}$ will be a natural transformation

\[\alpha: \textrm{Pt}_{\mathcal{X}} \to \textrm{M}\]

satisfying a continuity condition which basically says that the natural transformation is well behaved with regards to specializations. The cohomological invariants of $\mathcal{X}$ with coefficients in $\textrm{M}$ form a graded group $\textrm{Inv}^{\bullet}(\mathcal{X},\textrm{M})$ . Cohomological invariants enjoy the following properties.

1. (i) A morphism of algebraic stacks $f:\mathcal{Y} \to \mathcal{X}$ induces a pullback map $f^*$ on cohomological invariants defined by $f^*(\alpha)(P) = \alpha(f(P))$ .

2. (ii) Cohomological invariants form a sheaf in the smooth-Nisnevich topology, where coverings are smooth representable maps $\mathcal{X} \to \mathcal{Y}$ such that any F-valued point of $\mathcal{Y}$ lifts to a F-valued point of $\mathcal{X}$ .

3. (iii) In particular, if $\textrm{M}=\textrm{H}_D$ and $\mathcal{X}$ is smooth, then cohomological invariants with coefficients in $\textrm{H}_D$ are the sheafification of cohomology with coefficients in D on the smooth-Nisnevich site of $\mathcal{X}$ .

4. (iv) If $\mathcal{X}$ is smooth over $\mathbf{k}$ a dominant open immersion $\mathcal{U} \to \mathcal{X}$ induces an injective map on cohomological invariants.

5. (v) If $\mathcal{X}$ is smooth over $\mathbf{k}$ and $f:\mathcal{Y} \to \mathcal{X}$ is either an affine bundle, the projectivization of a vector bundle or an open immersion whose complement has codimension $\geq 2$ , then $f^*:\textrm{Inv}^{\bullet}(\mathcal{X}, \textrm{M}) \to \textrm{Inv}^{\bullet}(\mathcal{Y}, \textrm{M})$ is an isomorphism.

6. (vi) If $\mathcal{X}$ is smooth over $\mathbf{k}$ and $\mathcal{X}_{\mathcal{D},\ell}$ is the order- $\ell$ root stack of $\mathcal{X}$ at an irreducible Cartier divisor $\mathcal{D}$ , then the cohomological invariants of $\mathcal{X}_{D,\ell}$ are the invariants of $\mathcal{X} \smallsetminus \mathcal{D}$ whose ramification at $\mathcal{D}$ is of $\ell$ -torsion.

7. (vii) If $\mathcal{X}=[ X/G]$ is the quotient of a smooth scheme by a smooth algebraic group over $\mathbf{k}$ then

   $\textrm{Inv}^1(\mathcal{X},\textrm{K}_\ell)=\textrm{H}^1_{\textrm{lis-}\unicode{x00E9}\textrm{t}}(\mathcal{X},\mu_\ell)\quad \textrm{ and }\quad \textrm{Inv}^2(\mathcal{X}, \textrm{H}_{\mu_\ell^{\vee}})=\textrm{Br}'(\mathcal{X})_{\ell}.$

8. (viii) If $\mathcal{X}=[ X/G]$ is the quotient of a smooth scheme by a smooth algebraic group over $\mathbf{k}$ then

   \[\textrm{Inv}^{\bullet}(\mathcal{X},\textrm{M})=\textrm{A}^0_{G}(X,\textrm{M}).\]

Therefore, computing cohomological invariants with values in a cycle module $\textrm{M}$ of a smooth quotient stack $[X/G]$ with G smooth over $\mathbf{k}$ is equivalent to computing the zero-codimension G-equivariant Chow group $A^0_G(X,\textrm{M})$ . Cohomological invariants of a stack in turn give access to the Brauer group: this will be our main technical tool for the computation of the Brauer group of the moduli stacks of interest.

3.1 Moduli stacks of smooth plane curves

The moduli stack $\mathcal{X}_d$ of smooth plane curves of degree d is the stack whose objects are given by the data

\[(C\overset{\iota}{\hookrightarrow} P \to S),\]

where $P\to S$ is a Severi–Brauer variety whose fibers have dimension two (sometimes these varieties are called $\mathbb{P}^2$ -bundles), the scheme $C\to S$ is a smooth and proper morphism whose fibers have dimension one and $\iota$ is a closed embedding whose restriction over any geometric point s of S realizes $C_s$ as a smooth plane curve of degree d in $P_s\simeq \mathbb{P}^2_s$ . Recall from our Notation section that we will always assume that the characteristic of our base field $\mathbf{k}$ does not divide d.

Let $W_d$ be the vector space of trinary forms of degree d, so that $\mathbb{P}(W_d)$ can be identified with the Hilbert scheme of plane curves of degree d in $\mathbb{P}^2$ . Let $Z\subset\mathbb{P}(W_d)$ be the divisor of singular forms. Then the moduli stack $\mathcal{X}_d$ admits the following presentation as a quotient stack:

\[\mathcal{X}_d \simeq [ \mathbb{P}(W_d)\smallsetminus Z / \textrm{PGL}_3 ],\]

where the action of $\textrm{PGL}_3$ on $\mathbb{P}(W_d)$ is inherited from the natural action of $\textrm{GL}_3$ on $W_d$ .

A related stack, useful for our computations, is the stack $\mathcal{X}_d^\textrm{fr}$ of framed smooth plane curves of degree d. Its objects are pairs

\[(E\to S,C\overset{\iota}{\hookrightarrow} \mathbb{P}(E) \to S),\]

where $E\to S$ is a vector bundle of rank three, and $\iota$ is a closed embedding that over every geometric point s of S embeds $C_s$ in $\mathbb{P}(E)_s\simeq\mathbb{P}^2_s$ as a smooth plane curve of degree d. This stacks admits the following presentation:

\[ \mathcal{X}_d^\textrm{fr} := [(\mathbb{P}(W_d)\smallsetminus Z)/\textrm{GL}_3], \]

where $\mathbb{P}(W_d)$ inherits the $\textrm{GL}_3$ -action from $W_d$ .

The $\textrm{GL}_3$ -invariant divisor $Z\subset \mathbb{P}(W_d)$ can be further stratified. For our study, we will only deal with the following two $\textrm{GL}_3$ -invariant, locally closed subschemes:

\begin{align*}Z_{\textrm{irr}} &= \textrm{subscheme of irreducible, singular curves of degree} \, d; \\Z_{\textrm{bin}} &= \textrm{subscheme of irreducible curves of degree} \, d \, \textrm{with exactly two nodes.}\end{align*}

The subscheme $Z_{\textrm{irr}}$ is open in Z, and $\overline{Z_{\textrm{bin}}}$ has codimension one in Z. The divisor $Z\subset \mathbb{P}(W_d)$ admits the following desingularization: let $\widetilde{Z}\subset \mathbb{P}(W_d)\times \mathbb{P}^2$ be the singular locus of the universal curve over $\mathbb{P}(W_d)$ . More explicitly, we have

\[ \widetilde{Z} = \left\{ ([f],p)\textrm{ such that }\partial_{x_i} f (p) = 0,\textrm{ for }i=0,1,2 \right\},\]

where [f] is a point of $\mathbb{P}(W_d)$ and p is a point of $\mathbb{P}^2$ .

Proposition 3.2. The two projections

\[ \widetilde{Z} \longrightarrow \mathbb{P}^2, \quad \widetilde{Z} \longrightarrow \mathbb{P}(W_d) \]

realize $\widetilde{Z}$ as a projective bundle over $\mathbb{P}^2$ and as a birational model of Z, respectively. In particular, the induced morphism $\widetilde{Z} \to Z$ is an isomorphism over $Z_{\textrm{irr}}\smallsetminus Z_{\textrm{bin}}$ and is étale of degree two over $Z_{\textrm{bin}}$ .

We will denote $\widetilde{Z}_\textrm{irr}$ (respectively $\widetilde{Z}_\textrm{bin}$ ) as the preimage of $Z_\textrm{irr}$ (respectively $Z_\textrm{bin}$ ) along $\widetilde{Z} \to Z$ . The closure of $\widetilde{Z}_\textrm{bin}$ in $\widetilde{Z}$ will be denoted $\widetilde{Z}_\textrm{bin}^*$ .

3.2 Computation of equivariant fundamental classes

To compute the (low-degree) cohomological invariants of $\mathcal{X}_d^\textrm{fr}$ , we will need explicit expressions of $[Z]_{\textrm{GL}_3}$ in $\textrm{CH}^*_{\textrm{GL}_3}(\mathbb{P}(W_d)\times\mathbb{P}^2)$ and of $[\widetilde{Z}_\textrm{bin}^*]_{\textrm{GL}_3}$ in $\textrm{CH}^*_{\textrm{GL}_3}(\widetilde{Z})$ . For a brief review of the basic tools of equivariant intersection theory that are used below, the reader can consult [Reference Di Lorenzo, Fulghesu and VistoliDLFV21, § 2]. Recall that the $\textrm{GL}_3$ -equivariant Chow ring of a point is

\[\textrm{CH}_{\textrm{GL}_3}(\textrm{Spec}(\mathbf{k})) \simeq \mathbb{Z}[c_1,c_2,c_3], \]

where $c_i$ is the ith equivariant Chern class of the standard $\textrm{GL}_3$ -representation.

Let $T\subset\textrm{GL}_3$ the torus of diagonal matrices. Then the T-equivariant Chow ring of a point is

\[ \textrm{CH}_T(\textrm{Spec}(\mathbf{k})) \simeq\mathbb{Z}[l_1,l_2,l_3],\]

where $l_i$ is the T-equivariant first Chern class of the rank one representation determined by the formula $(u_1,u_2,u_3)\cdot x := u_i^{-1}x$ . With this choice of the generators, the embedding of equivariant Chow rings

$$\textrm{CH}_{\textrm{GL}_3}(\textrm{Spec}(\mathbf{k})) \simeq \mathbb{Z}[c_1,c_2,c_3]\longrightarrow \textrm{CH}_T(\textrm{Spec}(\mathbf{k}))$$

is given by

$$c_1\mapsto -(l_1+l_2+l_3), \quad c_2\mapsto l_1l_2+l_1l_3+l_2l_3, \quad c_3\mapsto-l_1l_2l_3.$$

Therefore, when working with T-equivariant Chow rings, the classes $c_i$ denote the cycles above.

3.2.1 The fundamental class of Z

The locus $\widetilde{Z}\subset \mathbb{P}(W_d)\times \mathbb{P}^2$ is a complete intersection of the three T-invariant hypersurfaces

\[ H_i := \{([f],p) \text{ such that }\partial_{x_i}f(p)=0\}, \quad i=1,2,3. \]

The T-equivariant fundamental class of $H_i$ is then [Reference Edidin and FulghesuEF09, Lemma 2.4]

\[ [H_i]_T = (h+(d-1)t+l_i) \]

where h (respectively t) is the hyperplane class of $\mathbb{P}(W_d)$ (respectively $\mathbb{P}^2$ ). It follows then that

\begin{align*} [\widetilde{Z}]_T&= \prod_{i=1}^{3}(h+(d-1)t+l_i) \\ &= (d-1)^3t^3 + (d-1)^2t^2 (3h + l_1 + l_2 + l_3) + t\cdot p + q,\end{align*}

where p and q are polynomials in h and $l_i$ . Using the fact that $\textrm{pr}_1|_{\widetilde{Z}}\colon\widetilde{Z}\to Z$ is generically an isomorphism, together with the formulas

$$\textrm{pr}_{1*}(1)=\textrm{pr}_{1*}(t)=0, \quad \textrm{pr}_{1*}(t^2)=1,\quad \textrm{pr}_{1*}(t^3)=-c_1, $$

we obtain

\begin{align*} [Z]_T &= \textrm{pr}_{1*}[\widetilde{Z}]_T \\ &= -(d-1)^3c_1 + (d-1)^2(3h-c_1) \\ &= (d-1)^2(3h-dc_1).\end{align*}

We have proven the following.

Lemma 3.3. The $\textrm{GL}_3$ -equivariant fundamental class of Z in the equivariant Chow ring of $\mathbb{P}(W_4)$ is equal to

\[ [Z]_{\textrm{GL}_3} = (d-1)^2(3h-dc_1). \]

3.2.2 The fundamental class of $\widetilde{Z}_\textrm{bin}^*$

This section is devoted to the computation of the $\textrm{GL}_3$ -equivariant fundamental class of $\widetilde{Z}_\textrm{bin}^*$ in $\textrm{CH}_{\textrm{GL}_3}(\widetilde{Z})$ .

The $\textrm{GL}_3$ -action on $\mathbb{P}(W_d)\times\mathbb{P}^2\times\mathbb{P}^2$ can be used to define a T-action on the same scheme. We have

\[\textrm{CH}_{T}(\mathbb{P}(W_4)\times\mathbb{P}^2\times\mathbb{P}^2) \simeq \mathbb{Z}[l_1,l_2,l_3,h,s,t]/I,\]

where h (respectively s, t) is the hyperplane section of $\mathbb{P}(W_d)$ (respectively of the first $\mathbb{P}^2$ , of the second $\mathbb{P}^2)$ .

For $i=1,2,3$ , let $H_i\subset \mathbb{P}(W_d)\times\mathbb{P}^2\times\mathbb{P}^2$ be the hypersurface of the equation

\[ H_i:=\{([f],p,q) \text{ such that }\partial_{x_i}f(p)=0\},\]

where $x_1,x_2$ and $x_3$ are the variables of the trinary form f. Similarly, we define the hypersurface $F_i \subset \mathbb{P}(W_4) \times \mathbb{P}^2 \times \mathbb{P}^2$ as

\[ F_i:=\{([f],p,q) \text{ such that }\partial_{x_i}f(q)=0\}.\]

We have [Reference Edidin and FulghesuEF09, Lemma 2.4]

\[ [H_i]_T = (h+(d-1)s+l_i),\quad [F_i]_T=(h+(d-1)t+l_i).\]

Set $Y=(\bigcap_{i=1}^{3} H_i)\cap(\bigcap_{i=1}^{3} F_i)$ to be the set-theoretic intersection, and let

\[ \xi := \prod_{i=1}^{3} (h+(d-1)s+l_i) \cdot \prod_{i=1}^{3}(h+(d-1)t+l_i) \]

be the intersection-theoretic product. Observe that Y is supported on the union $B\cup R$ , where

• – the scheme B is the locus of triples ([f],p,q) such that [f] belongs to $\overline{Z}_{\textrm{bin}}$ and p, q are singular points;

• – the scheme R is the locus of triples ([f],p,q) where $p=q$ , or equivalently is the image of $\widetilde{Z}$ through the $\textrm{id}\times\delta:\mathbb{P}(W_4)\times\mathbb{P}^2\to \mathbb{P}(W_4)\times\mathbb{P}^2\times\mathbb{P}^2$ morphism.

Therefore, as the intersection is transversal on B, we have $\xi=[B]_T+\rho$ , where $\rho$ is a cycle supported on R. The cycle $\rho$ can be computed via a simple version of the residual intersection formula [Reference FultonFul98, Proposition 9.11]. Consider the closed embedding

\[ \iota: \widetilde{Z} \overset{j}{\longrightarrow} \mathbb{P}(W_d) \times \mathbb{P}^2 \overset{\textrm{id}\times \delta}{\longrightarrow} \mathbb{P}(W_d) \times \mathbb{P}^2 \times \mathbb{P}^2. \]

For what follows, we set the following notation:

• – the hyperplane class of $\widetilde{Z}$ , regarded as a projective bundle over $\mathbb{P}^2$ , is denoted as $h_{\widetilde{Z}}$ ;

• – the hyperplane class of $\mathbb{P}^2$ , pulled back along $\widetilde{Z}\to\mathbb{P}^2$ , is denoted as u.

With this notation, we see that $\iota^*h=h_{\widetilde{Z}}$ and $\iota^*s=\iota^*t=u$ . Then the residual intersection formula tells us that

\begin{align*} \rho &= \iota_*\bigg\{ \prod_{i=1}^{3} c^T(N_{H_i}|_{\widetilde{Z}})\cdot \prod_{i=1}^{3} c^T(N_{F_i}|_{\widetilde{Z}}) \cdot c^T(N_{\iota})^{-1}\bigg\}_{\!\!1} \\ &= \iota_*\bigg(\sum_{i=1}^{3}( c^T_1(N_{H_i}|_{\widetilde{Z}})+c^T_1(N_{F_i}|_{\widetilde{Z}}))-c^T_1(N_{\iota})\bigg) \\ &= \iota_*\bigg(2\bigg(\sum_{i=1}^{3} h_{\widetilde{Z}} + (d-1)u + l_i)-c_1^T(N_{\iota})\bigg)\bigg),\end{align*}

where $N_{\iota}$ is the normal bundle relative to the closed embedding $\iota:\widetilde{Z} \hookrightarrow \mathbb{P}(W_4)\times\mathbb{P}^2\times\mathbb{P}^2$ . Using the exact sequence of normal bundles

\[ 0 \longrightarrow N_{j} \longrightarrow N_{\iota} \longrightarrow j^*N_{\textrm{id}\times\delta} \longrightarrow 0 \]

we obtain

\begin{align*} c_1^T(N_{\iota}) &= c_1^T(N_{j}) + j^*(c_1^T(N_{\textrm{id}\times\delta}))\\ &=\bigg(\sum_{i=1}^{3} h_{\widetilde{Z}} + (d-1)u + l_i\bigg) + 3u + l_1+l_2+l_3.\end{align*}

In the last equality above we used the fact that, for any smooth variety X, the diagonal embedding $\delta\colon X \hookrightarrow X\times X$ is a regular embedding and we have $N_{\delta}\simeq \Omega_X^\vee$ , and $c_1^G(\Omega_X^\vee)= -c_1^G(\det(\Omega_X))$ . This, applied to $X=\mathbb{P}^2$ , gives us $c_1^T(N_\delta)=-c_1^T(\omega_{\mathbb{P}^2})=3u+l_1+l_2+l_3$ .

From this we deduce that

\begin{align*}\rho = \iota_*(3h_{\widetilde{Z}} + 3(d-2)u) = \iota_*(\iota^*(3h + 3(d-2)s)) = (3h+3(d-2)s)\cdot [R]_T.\end{align*}

This implies that

\begin{align*}[B]_T &= \xi - \rho = \prod_{i=1}^{3} (h+(d-1)s+l_i) \cdot \prod_{i=1}^{3}(h+(d-1)t+l_i) - (3h+3(d-2)s)\cdot [R]_T \\&= \textrm{pr}_{12}^*[\widetilde{Z}]_T\cdot \bigg( \prod_{i=1}^{3}(h+(d-1)t+l_i)\bigg) - (3h+3(d-2)s)\cdot [R]_T,\end{align*}

where $\textrm{pr}_{12}:\mathbb{P}(W_d)\times\mathbb{P}^2\times\mathbb{P}^2\to \mathbb{P}(W_d)\times\mathbb{P}^2$ is the projection on the first two factors. Observe that $\textrm{pr}_{12*}([B]_T)=[\widetilde{Z}_\textrm{bin}^* \subset \mathbb{P}(W_d)\times \mathbb{P}^2]_T$ , i.e. the equivariant fundamental class of $\widetilde{Z}_\textrm{bin}^*$ in the Chow ring of $\mathbb{P}(W_d)\times\mathbb{P}^2$ . Using the previous computation of $[B]_T$ , and applying the projection formula, we get

\begin{align*} \textrm{pr}_{12*}[B]_T &= [\widetilde{Z}]_T\cdot \textrm{pr}_{12*}\bigg(\prod_{i=1}^{3}(h+(d-1)t+l_i)\bigg) - (3h+3(d-2)s)\textrm{pr}_{12*}(\iota_*[\widetilde{Z}]_T)\\ &= [\widetilde{Z}]_T\cdot \bigg(\textrm{pr}_{12*}\bigg(\prod_{i=1}^{3}(h+(d-1)t+l_i)\bigg) - (3h+3(d-2)s)\bigg).\end{align*}

Observe now that we have $\textrm{pr}_{12*}(1)=\textrm{pr}_{12*}(t)=0$ , $\textrm{pr}_{12*}(t^2)=1$ . Moreover, using the relation $t^3+c_1t^2+c_2t+c_3=0$ , we deduce that $\textrm{pr}_{12*}(t^3)=-c_1$ . Therefore, we get

\begin{align*} \textrm{pr}_{12*}\bigg(\prod_{i=1}^{3}(h+(d-1)t+l_i)\bigg) - (3h+3(d-2)s) &=3((d-1)^2-1)h - ((d-1)^3+(d-1)^2)c_1\nonumber\\ & \quad- 3(d-2)s.\end{align*}

Let $[\widetilde{Z}_{\textrm{bin}}^*]_T$ be the fundamental class of $\widetilde{Z}_\textrm{bin}^*$ in the Chow ring of $\widetilde{Z}$ , so that we have $[\widetilde{Z}_\textrm{bin}^*]_T=p(h_{\widetilde{Z}},u,c_1)$ . As the pullback homomorphism

\[j^*: \textrm{CH}_T^*(\mathbb{P}(W_d)\times\mathbb{P}^2) \longrightarrow \textrm{CH}_T^*(\widetilde{Z}) \]

is surjective, it follows that

\[[\widetilde{Z}_\textrm{bin}^* \subset \mathbb{P}(W_d)\times\mathbb{P}^2 ]_T = j_*[\widetilde{Z}_\textrm{bin}^*]_T = p(h,s,c_1)\cdot [\widetilde{Z}]_T. \]

It is easy to check that there is no non-zero polynomial $q(h,s,c_1)$ of degree one such that $q\cdot [\widetilde{Z}]_T =0$ : indeed, we have $[\widetilde{Z}]_T=h^3+f(h,s,c_1,c_2,c_3)$ (see § 3.2.1), where f has degree two with respect to the variable h. The subgroup of relations in $\textrm{CH}_T^4(\mathbb{P}(W_d)\times\mathbb{P}^2)$ is generated by the cycles

\[ h(s^3+c_1s^2+c_2s+c_3),\quad s(s^3+c_1s^2+c_2s+c_3), \quad l_i(s^3+c_1s^2+c_2s+c_3) \text{ for }i=1,2,3. \]

In particular, there is no relation in degree four containing a monomial divisible by $h^3$ , from which follows that $q\cdot [\widetilde{Z}]_T=0$ if and only if $q=0$ .

As we have proved that

$$p(h,s,c_1)\cdot [\widetilde{Z}]_T = (3((d-1)^2-1)h - ((d-1)^3+(d-1)^2)c_1 - 3(d-2)s) \cdot [\widetilde{Z}]_T$$

where $p(h_{\widetilde{Z}},u,c_1)=[\widetilde{Z}_\textrm{bin}^*]_T$ , we finally deduce the following.

Lemma 3.4. The $\textrm{GL}_3$ -equivariant fundamental class of $\widetilde{Z}_\textrm{bin}^*$ in the equivariant Chow ring of $\widetilde{Z}$ is equal to

\[ [\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3} = (3d(d-2)h_{\widetilde{Z}} - d(d-1)^2c_1 - 3(d-2)u). \]

3.3 Cohomological invariants of $\mathcal{X}_d^\textrm{fr}$ in low degree

In this section, we study the low-degree cohomological invariants of $\mathcal{X}_d^{\textrm{fr}}$ with coefficients in a cycle module $\textrm{M}$ , leveraging their interpretation as the codimension-zero $\textrm{GL}_3$ -equivariant Chow group of $\mathbb{P}(W_d)\smallsetminus Z$ with coefficients in $\textrm{M}$ . We start with a simple lemma.

Lemma 3.5 [Reference Di Lorenzo and PirisiDLP21a, Proposition 4.2]. Let $\mathcal{E}=[ \mathbb{A}^3/\textrm{GL}_3] \to \textrm{BGL}_3$ be the standard representation of $\textrm{GL}_3$ and let $c_i = c_i(\mathcal{E})$ for $i = 1,2,3$ . Then,

\[\textrm{A}^{\bullet}_{\textrm{GL}_3}(\textrm{Spec}(\mathbf{k}),\textrm{M})=\mathbb{Z}[c_1, c_2, c_3 ] \otimes \textrm{M}^{\bullet}(\mathbf{k}).\]

In particular, if X is a tower of $\textrm{GL}_3$ -equivariant affine bundles and $\textrm{GL}_3$ -equivariant projective bundles, we can apply the lemma and the projective bundle formula to completely reduce its equivariant Chow groups with coefficients to the ordinary (equivariant) Chow groups and the chosen cycle module.

Lemma 3.6. Let X be a $\textrm{GL}_3$ -equivariant scheme that is obtained from $\textrm{Spec}(\mathbf{k})$ as a sequence of $\textrm{GL}_3$ -equivariant affine and projective bundles. Then

\[\textrm{A}^{\bullet}_{\textrm{GL}_3}(X)=\textrm{CH}^{\bullet}_{\textrm{GL}_3}(X) \otimes \textrm{M}^{\bullet}(\mathbf{k}).\]

This applies to $\widetilde{Z}$ , which by [Reference Fulghesu and VistoliFV18, Proposition 4.2] is an equivariant projective bundle over $\mathbb{P}^2$ . Using this we can deal with the divisor Z of singular plane curves.

1. (i) Consider a smooth algebraic group G acting on a scheme X, and let $U \subseteq X$ be its smooth locus. If U has a $\mathbf{k}$ -rational point, then the composition

   \[\textrm{A}^0_G(X,\textrm{M}) \to \textrm{A}^0(U,\textrm{M}) \to A^0(\textrm{Spec}(\mathbf{k}),\textrm{M})=\textrm{M}^{\bullet}(\mathbf{k})\]
   is a retraction of the pullback $\textrm{M}^{\bullet}(\mathbf{k})=A^0(\textrm{Spec}(\mathbf{k}),\textrm{M}) \to A^0_G(X,\textrm{M})$ so that $\textrm{M}^{\bullet}(\mathbf{k})$ is a direct summand in $A^0_G(X,\textrm{M})$ .

Note that if the field $\mathbf{k}$ is finite (which is often a problem when looking for points on an open subset) we can replace it with $\mathbf{k}(t)$ as the map

\[x \mapsto \partial_{t = 0}(\lbrace t \rbrace \cdot x)\]

is a retraction from $\textrm{M}^{\bullet}(\mathbf{k}(t))$ to $\textrm{M}^{\bullet}(\mathbf{k})$ .

1. (ii) Consider a G-equivariant closed immersion $D \to X$ of codimension one. Assume that there exist positive integers $\ell_i \mid \ell$ and elements $\alpha_i\in \textrm{A}^0_G(D,\textrm{K}_{\ell_i})$ such that for all $\ell$ -torsion modules we have:

   1. (a) the map $\textrm{M}^{\bullet}(\mathbf{k})_{\ell_i} \to \textrm{A}^{\bullet}(D, \textrm{M})$ given by $x \mapsto \alpha_{i}\cdot x$ is injective for all i;

   2. (b) an equality

      \[\textrm{Im}(\partial_D: \textrm{A}^{0}_G(X\smallsetminus D,\textrm{M})\to \textrm{A}^{0}_G(D,\textrm{M})) = \bigoplus_i \alpha_i\cdot\textrm{M}^{\bullet}(\mathbf{k})_{\ell_i}. \]

Then picking $\textrm{M}=\textrm{K}_{\ell_i}$ we can find elements $\beta_i \in \textrm{A}^0_G(X\smallsetminus D,\textrm{K}_{\ell_i})$ such that $\partial_D(\beta_i)=\alpha_i$ and the map

\[\bigoplus_i \alpha_i\textrm{M}^{\bullet}(\mathbf{k})_{\ell_i} \to A^{0}_G(X \smallsetminus D, \textrm{M}), \quad \alpha_i \cdot x \mapsto \beta_i \cdot x \]

is a splitting of $\partial_D$ as by the compatibility of product and boundary maps [Reference RostRos96, pp. 229–330, R3d and R3f] we have $\partial_D(\beta_i \cdot x)=\alpha_i \cdot x$ .

Proposition 3.8. We have

\[ \textrm{A}^0_{\textrm{GL}_3}(Z, \textrm{M}^{\leq 1}) =\textrm{A}^0_{\textrm{GL}_3}(Z_{\textrm{irr}}, \textrm{M}^{\leq 1}) =\begin{cases} \textrm{M}^{\leq 1}(\mathbf{k}) \oplus \textrm{M}^{0}(\mathbf{k})_2[1], & \textrm{if } d \textrm{ is even,}\\\textrm{M}^{\leq 1}(\mathbf{k}), & \textrm{if } d \textrm{ is odd.}\end{cases}\]

Here, $\textrm{M}^{0}(\mathbf{k})_2[1]= \alpha\cdot\textrm{M}^{0}(\mathbf{k})_2$ for an element $\alpha\in \textrm{A}^0_{\textrm{GL}_3}(Z_{\textrm{irr}},\textrm{K}_2)$ of degree one.

Recall that, as $\widetilde{Z}$ is obtained as a sequence of projective bundles over $\textrm{Spec}(\mathbf{k})$ , we have

\[\textrm{A}^{\bullet}_{\textrm{GL}_3}(\widetilde{Z},\textrm{M})=\textrm{CH}_{\textrm{GL}_3}^{\bullet}(\widetilde{Z})\otimes \textrm{M}(\mathbf{k}),\]

so in particular $\textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}, \textrm{M})=\textrm{M}^{\bullet}(\mathbf{k})$ . As the complement of $\widetilde{Z}_\textrm{irr}$ has codimension 2 in $\widetilde{Z}$ we must have that $\textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_\textrm{irr}, \textrm{M})$ is trivial as well.

Set $\overline{Z}_{\textrm{bin},\textrm{irr}}:=\overline{Z}_\textrm{bin}\cap Z_\textrm{irr}$ and $\widetilde{Z}^*_{\textrm{bin},\textrm{irr}}:=\widetilde{Z}_{\textrm{bin}}^*\cap \widetilde{Z}_\textrm{irr}$ . The projection $\widetilde{Z}_\textrm{irr} \to Z_\textrm{irr}$ induces the following commutative diagram of equivariant Chow groups with coefficients.

We are only interested in the cohomological invariants of $Z_\textrm{irr}$ of cohomological degree $\leq 1$ .

Observe that $\widetilde{Z}_{\textrm{bin}}^*$ is irreducible: this can be seen for instance by first observing that $\widetilde{Z}_\textrm{bin}$ is isomorphic to an open, dense subset of the subscheme $B^\circ$ of $\mathbb{P}(W_4)\times(\mathbb{P}^2\times\mathbb{P}^2\smallsetminus\Delta_{\mathbb{P}^2})$ parametrizing triples ([f],p,q) such that $\partial_{x_i}f(p)=\partial_{x_i}f(q)=0$ for $i=0,1,2$ .

The fiber of the morphism $\textrm{pr}:B^\circ\to \mathbb{P}^2\times \mathbb{P}^2\smallsetminus\Delta_{\mathbb{P}^2}$ over a pair (p,q) is given by the vanishing of the six linear polynomials $\partial_{x_i}f(p)$ , $\partial_{x_i}f(q)$ , for $i=0,1,2$ .

From this we deduce that the projection realizes $B^\circ$ as a projective bundle over $\mathbb{P}^2\times \mathbb{P}^2\smallsetminus\Delta_{\mathbb{P}^2}$ , from which the irreducibility of $B^\circ$ (hence of $\widetilde{Z}_\textrm{bin}$ and $\widetilde{Z}^*_\textrm{bin}$ ) follows. We therefore have $\textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_{\textrm{bin}}^*, \textrm{M}^0) \simeq \textrm{M}^0(\mathbf{k})$ .

The pushforward $i_*$ sends any $x \in \textrm{M}^{\bullet}(\mathbf{k})$ to $[\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3}\cdot x \in \textrm{A}^1_{\textrm{GL}_3}(\widetilde{Z}_\textrm{irr}, \textrm{M})$ . The complement of $\widetilde{Z}_\textrm{irr}$ in $\widetilde{Z}$ has codimension 2, and thus the pullback

\[ A^{1}_{\textrm{GL}_3}(\widetilde{Z},\textrm{M}) \to A^{1}_{\textrm{GL}_3}(\widetilde{Z}_\textrm{irr},\textrm{M})\]

is injective. As $[\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3}\cdot x$ comes from the left-hand side, we can check whether it is zero there. Then the element $[\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3}\cdot x$ is zero if and only if $[\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3}\otimes x$ is zero in $\textrm{CH}^1_{\textrm{GL}_3}(\widetilde{Z}_\textrm{irr})\otimes\textrm{M}^{\bullet}(\mathbf{k})$ . As the group on the left of the tensor product is finitely generated and free, all that matters is how divisible $[\widetilde{Z}_{\textrm{bin}}^*]_{\textrm{GL}_3}$ is (because the subgroup generated by a primitive element is a direct summand in $\textrm{CH}^1_{\textrm{GL}_3}(\widetilde{Z}_\textrm{irr})$ ). By Lemma 3.4 we conclude that

\begin{align*} \textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_\textrm{irr} \smallsetminus \widetilde{Z}_{\textrm{bin}}^*, \textrm{M}^{\leq 1})=\textrm{M}^{\leq 1}(\mathbf{k}) \oplus \textrm{M}^{0}(\mathbf{k})_{r}[1].\end{align*}

Here, $r=\textrm{gcd}(d(d-1)^2,3(d-2))$ , and the $\textrm{M}^0(\mathbf{k})_r[1]$ component is equal to $\beta \cdot \textrm{M}^{0}(\mathbf{k})_r$ , where $\beta \in \textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_\textrm{irr} \smallsetminus \widetilde{Z}_{\textrm{bin}}^*, \textrm{K}_r)$ is an inverse image of $1 \in \textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_{\textrm{bin}}^*, \textrm{K}_r)$ . We have a direct sum as explained in Remark 3.7, case (ii).

Now we need to understand

\[\textrm{A}_{\textrm{GL}_3}^0(Z_{\textrm{irr}}, \textrm{M}^{\leq 1})=\textrm{Ker}(\textrm{A}_{\textrm{GL}_3}^0(Z_{\textrm{irr}} \smallsetminus \overline{Z}_{\textrm{bin}}, \textrm{M}^{\leq 1}) \longrightarrow \textrm{A}_{\textrm{GL}_3}^0(\overline{Z}_{\textrm{bin}}, \textrm{M}^{\leq 1})).\]

We know that $\textrm{A}_{\textrm{GL}_3}^0(Z_{\textrm{irr}} \smallsetminus \overline{Z}_{\textrm{bin}}, \textrm{M})=\textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z} \smallsetminus \widetilde{Z}_{\textrm{bin}}^*, \textrm{M})$ and that the kernel of

\[\textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_\textrm{irr} \smallsetminus \widetilde{Z}_{\textrm{bin}}^*, \textrm{M}) \longrightarrow \textrm{A}_{\textrm{GL}_3}^0(\widetilde{Z}_{\textrm{bin}}^*, \textrm{M})\]

is exactly $\textrm{M}^{\bullet}(\mathbf{k})$ , so by the compatibility of pushforward and ramification maps we conclude that

\[\textrm{A}_{\textrm{GL}_3}^0(Z_{\textrm{irr}}, \textrm{M}^{\leq 1}) = \textrm{M}^{\leq 1}(\mathbf{k}) \oplus \textrm{Ker}(\textrm{M}^{0}(\mathbf{k})_r \longrightarrow \textrm{A}_{\textrm{GL}_3}^0(\overline{Z}_{\textrm{bin}}, \textrm{M}^{0}))[1].\]

The map $\textrm{M}^{0}(\mathbf{k})_r \to \textrm{A}_{\textrm{GL}_3}^0(\overline{Z}_{\textrm{bin}}, \textrm{M}^{0})$ is just the restriction to $\textrm{M}^{0}(\mathbf{k})_r$ of multiplication by 2 on $\textrm{M}^{\bullet}(\mathbf{k})$ , so

\[\textrm{Ker}(\textrm{M}^{0}(\mathbf{k})_r \longrightarrow \textrm{A}_{\textrm{GL}_3}^0(\overline{Z}_{\textrm{bin}}, \textrm{M}^{0}))=\begin{cases} \textrm{M}^{0}(\mathbf{k})_2, & \textrm{if } d \textrm{ is even,}\\ 0, & \textrm{if } d \textrm{ is odd,}\end{cases} \]

where in the first case the kernel is generated by $\alpha = (r/2)\beta$ .

Now we are ready to compute the low-degree cohomological invariants of $\mathcal{X}^{\textrm{fr}}_d$ .

Proposition 3.9. Set $i_d=1$ when $3|d$ , and zero otherwise. Then for d even we have

\[\textrm{A}^0_{\textrm{GL}_3}(\mathbb{P}(W_d)\smallsetminus Z,\textrm{M}^{\leq 2})=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \beta_1 \cdot \textrm{M}^{\leq 1}(\mathbf{k})_{3^{i_d}(d-1)^2} \oplus N[2]\]

where $N \subseteq \textrm{M}^{0}(\mathbf{k})_2$ is the kernel of

\[i_*:\alpha\cdot \textrm{M}^{0}(\mathbf{k})_2 \subset\textrm{A}^0_{\textrm{GL}_3}(Z,\textrm{M}^1) \longrightarrow \textrm{A}^1_{\textrm{GL}_3}(\mathbb{P}(W_d),\textrm{M}^1).\]

For d odd, we have

\[\textrm{A}^0_{\textrm{GL}_3}(\mathbb{P}(W_d)\smallsetminus Z,\textrm{M}^{\leq 2})=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \beta_1 \cdot \textrm{M}^{\leq 1}(\mathbf{k})_{3^{i_d}(d-1)^2}. \]

Proof. The claim is an immediate consequence of the exact sequence

\[0 \to \textrm{A}_{\textrm{GL}_3}^0(\mathbb{P}(W_d), \textrm{M}) \to \textrm{A}_{\textrm{GL}_3}^0(\mathbb{P}(W_d)\smallsetminus Z, \textrm{M}) \to \textrm{A}_{\textrm{GL}_3}^0(Z, \textrm{M}) \to \textrm{A}_{\textrm{GL}_3}^1(\mathbb{P}(W_d), \textrm{M})\]

and the fact that, by Lemma 3.3, the class $\left[ Z \right]_{\textrm{GL}_3}$ is divisible by $3^{i_d}(d-1)^2$ in $\textrm{A}_{\textrm{GL}_3}^1(\mathbb{P}(W_d), \textrm{M})\simeq \textrm{CH}^1_{\textrm{GL}_3}(\mathbb{P}(W_d))\otimes \textrm{M}^{\bullet}(\mathbf{k})$ . The splitting of the exact sequence is as in Remark 3.7, case (i). We know there is always a point in $\mathbb{P}(W_d)\smallsetminus Z$ thanks to [Reference PoonenPoo05].

Recall that, for a smooth quotient stack $[X/G]$ , we have an identification

\[\textrm{Inv}^{\bullet}([X/G], \textrm{M}) \simeq \textrm{A}^0_G(X,\textrm{M}). \]

Moreover, note that if $\mathbf{k}$ is algebraically closed and $\textrm{M}$ is equal to $\textrm{K}_\ell$ or $\textrm{H}_D$ for some $\ell$ -torsion Galois module then there are no elements of positive degree in $\textrm{M}^{\bullet}(\mathbf{k})$ and consequently by the projective bundle formula in $\textrm{A}^1_{\textrm{GL}_3}(\mathbb{P}(W_4),\textrm{M})$ , so we can conclude that $N=\textrm{M}^{0}(\mathbf{k})_2$ . Then using Proposition 3.9 we can easily deduce the following.

Theorem 3.10. Assume that $d\geq 4$ and that the ground field $\mathbf{k}$ is an algebraically closed field whose characteristic is not 2 and does not divide d, and that $\textrm{M}$ is equal to $\textrm{K}_\ell$ or $\textrm{H}_D$ for some $\ell$ -torsion Galois module. Set $i_d=1$ if $3|d$ , and zero otherwise. Then we have

\[\textrm{Inv}^{\leq 2}(\mathcal{X}_d^\textrm{fr},\textrm{M}) \simeq \begin{cases} \textrm{M}^{\leq 2}(\mathbf{k}) \oplus \textrm{M}^{\leq1}(\mathbf{k})_{3^{i_d}(d-1)^2}[1] \oplus \textrm{M}^0(\mathbf{k})_2[2], & \textrm{if } d \textrm{ is even,}\\\\ \textrm{M}^{\leq 2}(\mathbf{k}) \oplus \textrm{M}^{\leq1}(\mathbf{k})_{3^{i_d}(d-1)^2}[1], & \textrm{if } d \textrm{ is odd.}\\ \end{cases} \]

Moreover, when d is odd the presentation above holds for any field without the assumption of $\mathbf{k}$ being algebraically closed and for any cycle module $\textrm{M}$ .

3.4 Brauer group of moduli of smooth plane curves

As an immediate application of Theorem 3.10 we obtain the following result on the Brauer group of $\mathcal{X}_d^\textrm{fr}$ .

1. (i) if ${char}(\mathbf{k})=0$ , we have ${Br}(\mathcal{X}_d^\textrm{fr}) \simeq \mathbb{Z}/\langle 2,d\rangle$ ;

2. (ii) if ${char}(\mathbf{k})=p>0$ , we have ${Br}(\mathcal{X}_d^\textrm{fr}) \simeq \mathbb{Z}/\langle 2, d\rangle\oplus B_{d,p}$ , where $B_{d,p}$ is of p-primary torsion.

1. (i) if $\textrm{char}(\mathbf{k})=0$ , we have $\textrm{Br}(\mathcal{X}_d^\textrm{fr}) \simeq \textrm{Br}(\mathbf{k}) \oplus \textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z})$ ;

2. (ii) if $\textrm{char}(\mathbf{k})=p>0$ , we have $\textrm{Br}(\mathcal{X}_d^\textrm{fr})_\ell \simeq \textrm{Br}(\mathbf{k})_\ell \oplus \textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z})_\ell$ .

The $\textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z})$ component comes from the cup products of

$$\beta_1 \in \textrm{Inv}^1(\mathcal{X}_d^\textrm{fr},\textrm{K}_{3^{i_d}(d-1)^2})=\textrm{H}^1(\mathcal{X}_d^\textrm{fr},\mu_{3^{i_d}(d-1)^2})$$

and elements of $\textrm{H}^1(\mathbf{k},\mathbb{Z}/{3^{i_d}(d-1)^2}\mathbb{Z})$ . The splitting is because $\mathcal{X}_d^{\textrm{fr}}$ always has a $\mathbf{k}$ -rational point.

Recall from § 3.1 that $\mathcal{X}_d^{\textrm{fr}}$ is a $\mathbb{G}_\textrm{m}$ -gerbe over the moduli stack of smooth plane curves $\mathcal{X}_d$ . We want to leverage this result to get to the Brauer group of $\mathcal{X}_d$ . First, let us recall the following.

Theorem 3.13. [Reference ShinShi21, Theorem 1.2]. Let S be a scheme, and let $\pi:\mathcal{G} \to S$ be a $\mathbb{G}_\textrm{m}$ -gerbe, corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${Br}'(S)$ . Then we have the exact sequence

\[ \textrm{H}^0(S,\underline{\mathbb{Z}}) \longrightarrow {Br}'(S) \longrightarrow {Br}'(\mathcal{G}) \longrightarrow 0\]

where the first map maps 1 to $[\mathcal{G}]$ .

Assuming that the gerbe $\mathcal{G}$ is q-torsion for q coprime with $\textrm{char}(\mathbf{k})$ , the result above easily generalizes to the case when $S=[ X/G]$ is a quotient stack with X a smooth quasi-projective scheme and the action of G is linearized.

Indeed, with these hypotheses there exist a G-representation V and a G-invariant open subset $U\subset V$ whose complement has codimension ${>}2$ on which G acts freely, such that $X\times^G U:=X\times U/G$ is a smooth quasi-projective scheme [Reference Edidin and GrahamEG98, Proposition 23]. Then for any $\ell$ we have the equality $\textrm{Br}'(X\times^G U)_{\ell}=\textrm{Br}'([ X/G])_{\ell}$ .

We have a natural morphism $f:X\times^G U \to [X/G]$ which realizes $X\times^G U$ as an open subset of a vector bundle over $[X/G]$ . Given a $\mathbb{G}_\textrm{m}$ -gerbe $\mathcal{G}\to [X/G]$ we obtain a $\mathbb{G}_\textrm{m}$ -gerbe $f^*\mathcal{G} \to X\times^G U$ . Applying Shin’s theorem, we obtain an exact sequence

\[ \textrm{H}^0(X\times^G U,\underline{\mathbb{Z}}) \longrightarrow \textrm{Br}'(X\times^G U) \longrightarrow \textrm{Br}'(f^*\mathcal{G}) \longrightarrow 0\]

where the left arrow maps 1 to $f^*[\mathcal{G}]$ . Now if $\mathcal{G}$ is q-torsion the sequence above can be extended to a commutative diagram

where the bottom left arrow sends 1 to $[\mathcal{G}]$ , and the vertical arrows are the pullbacks along f, which by construction are all isomorphisms. Then from the exactness of the top sequence we immediately deduce the exactness of the bottom sequence.

Before getting to the proof of the main theorem, we need an intermediate result. Recall that given a $\textrm{PGL}_n$ -torsor $P \to S$ we have

\[\textrm{H}^1(S,\textrm{PGL}_n)\ni (P\to S) \mapsto ([P/\textrm{GL}_n]\to S) \in \textrm{H}^2(S,\mathbb{G}_\textrm{m}).\]

On the other hand, the $\textrm{PGL}_n$ -torsor P corresponds to the Severi–Brauer variety $(P\times_S \mathbb{P}^n)/\textrm{PGL}_n$ . In particular, the universal Severi–Brauer variety $[\mathbb{P}^n/\textrm{PGL}_n]$ corresponds to the $\textrm{PGL}_n$ -torsor $\textrm{Spec}(\mathbf{k}) \to \textrm{BPGL}_n$ which in turn maps to the $\mathbb{G}_\textrm{m}$ -gerbe $\textrm{BGL}_n \to \textrm{BPGL}_n$ .

Lemma 3.14. Suppose that ${char}(\mathbf{k})\neq 3$ and that $3 \mid \ell$ . Then

\[{Br}'(\textrm{B}\textrm{PGL}_3)_{\ell} \simeq {Br}(\mathbf{k})_{\ell}\oplus \mathbb{Z}/3\mathbb{Z}\cdot \gamma , \]

where $\gamma$ is the class of the $\mathbb{G}_\textrm{m}$ -gerbe $\textrm{B}\textrm{GL}_3 \to \textrm{B}\textrm{PGL}_3$ or, equivalently, the class of the universal Severi–Brauer variety over $\textrm{B}\textrm{PGL}_3$ .

Proof. First observe that the class of the $\mathbb{G}_\textrm{m}$ -gerbe $\textrm{B}\textrm{GL}_3 \to \textrm{B}\textrm{PGL}_3$ is of 3-torsion as it corresponds to the universal $\textrm{PGL}_3$ -torsor $\textrm{Spec}(\mathbf{k}) \to \textrm{B}\textrm{PGL}_3$ which has index 3. Therefore, applying Shin’s theorem we get the exact sequence

\[ \mathbb{Z}/3\mathbb{Z} \longrightarrow \textrm{Br}'(\textrm{B}\textrm{PGL}_3)_{\ell} \longrightarrow \textrm{Br}'(\textrm{B}\textrm{GL}_3)_{\ell}=\textrm{Br}'(\mathbf{k})_{\ell}. \]

We observe that the first map is injective, as $\textrm{B}\textrm{GL}_3\to\textrm{B}\textrm{PGL}_3$ is a non-trivial gerbe, and the last map has a retraction as $\textrm{BPGL}_3$ has a $\mathbf{k}$ -rational point. This concludes the proof.

We are now ready to prove the main theorem of this section. Recall (see [Reference Edidin, Hasset, Kresch and VistoliEHKV01, Reference Kresch and VistoliKV04]) that for a smooth, generically tame and separated Deligne–Mumford stack whose coarse moduli space is quasi-projective the Brauer group and cohomological Brauer group coincide. As these conditions are always verified for $\mathcal{X}_d$ we have $\textrm{Br}'(\mathcal{X}_d)=\textrm{Br}(\mathcal{X}_d)$ .

• – if ${char}(\mathbf{k})=0$ then ${Br}(\mathcal{X}_d)\simeq \mathbb{Z}/\langle d,6 \rangle$ ;

• – if ${char}(\mathbf{k})=p>0$ then ${Br}(\mathcal{X}_d)\simeq \mathbb{Z}/\langle d,6 \rangle \oplus B'_{p,d}$ , where $B'_{p,d}$ is of p-primary torsion.

where the horizontal maps come from the presentations of $\mathcal{X}_d$ and $\mathcal{X}_d^{\textrm{fr}}$ as quotient stacks. From this we deduce that $\mathcal{X}_d^{\textrm{fr}} \to \mathcal{X}_d$ is a $\mathbb{G}_\textrm{m}$ -gerbe whose class in $\textrm{Br}'(\mathcal{X}_d)$ is of 3-torsion. By applying Shin’s theorem and Lemma 3.14, we obtain a commutative diagram.

(1)

Suppose that $3\mid d$ . Then $W_d$ is a $\textrm{PGL}_3$ -representation, where the action is defined as

\[ [A] \cdot F(x_1,x_2,x_3) = \det(A)^{{d}/{3}}F(A^{-1}(x_1,x_2,x_3)).\]

This implies that $g\colon\mathcal{X}_d\to\textrm{B}\textrm{PGL}_3$ is a composition of an open embedding and a projective bundle, and hence the induced pullback homomorphism $g^*\colon\textrm{Br}'(\textrm{B}\textrm{PGL}_3) \to \textrm{Br}'(\mathcal{X}_d)$ is injective. From this it follows that the top left morphism in the sequence above is injective, proving $\textrm{Br}'(\mathcal{X}_d)_\ell \simeq \textrm{Br}'(\mathcal{X}_d^\textrm{fr})_\ell \times \mathbb{Z}/3\mathbb{Z}$ .

Suppose instead that $3 \nmid d$ . Then the pullback of the universal Severi–Brauer variety to $\mathcal{X}_d$ is a projective bundle (hence its class is trivial in the Brauer group): indeed, denoting this Severi–Brauer variety as $\pi\colon\mathcal{P}\to\mathcal{X}_d$ , we have that over $\mathcal{P}$ there are two well-defined line bundles, namely the relative dualizing line bundle $\omega_{\mathcal{P}/\mathcal{X}_d}$ and $\mathcal{O}(\mathcal{C})$ , where $\mathcal{C}\subset \mathcal{P}$ is the universal smooth plane curve.

Then there exist integers a and b such that $\mathcal{L}:=\mathcal{O}(a\mathcal{C})\otimes(\omega_{\mathcal{P}/\mathcal{X}_d})^{\otimes b}$ has degree 1 on every fiber of $\pi\colon\mathcal{P}\to \mathcal{X}_d$ which implies that $\mathcal{P}\simeq \mathbb{P}(\pi_*\mathcal{L})$ . As the class in the Brauer group of $\mathcal{P}$ coincides with the one of the gerbe $\mathcal{X}_d^{\textrm{fr}}\to \mathcal{X}_d$ , we conclude that the top left arrow in (1) is zero, thus proving that $\textrm{Br}'(\mathcal{X}_d)\simeq \textrm{Br}'(\mathcal{X}_d^\textrm{fr})$ .

1. (i) if $\textrm{char}(\mathbf{k})=0$ we have

   \[\textrm{Br}(\mathcal{X}_d) \simeq \textrm{Br}(\mathbf{k}) \oplus \textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z}) \oplus \mathbb{Z}/\langle 3, d\rangle;\]

2. (ii) if $\textrm{char}(\mathbf{k})=p>0$ we have

   \[\textrm{Br}(\mathcal{X}_d)_\ell \simeq \textrm{Br}(\mathbf{k})_\ell \oplus \textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z})_\ell\oplus \mathbb{Z}/\langle 3, d\rangle_\ell. \]

To see the splitting of the exact sequence note that the torsion in the Picard group of $\mathcal{X}_d$ is the same as the torsion in the Picard group of $\mathcal{X}_d^{\textrm{fr}}$ , which can be seen using [Reference ShinShi19, Equation 2.9.2] and the fact that the character group of $\mathbb{G}_\textrm{m}$ is $\mathbb{Z}$ . This shows that $\textrm{H}^1(\mathcal{X}_d,\mu_{3^{i_d}(d-1)^2})$ contains an element $\tilde{\beta}_1$ which maps to the element $\beta_1 \in \textrm{H}^1(\mathcal{X}_d^{\textrm{fr}},\mu_{3^{i_d}(d-1)^2})$ generating the corresponding component in $\textrm{Br}(\mathcal{X}_d^{\textrm{fr}})$ . Then the map

\[\textrm{Br}(\mathcal{X}_d^{\textrm{fr}})=\textrm{Br}(\mathbf{k})\oplus\textrm{H}^1(\mathbf{k},\mathbb{Z}/3^{i_d}(d-1)^2\mathbb{Z})\to \textrm{Br}(\mathcal{X}_d), \quad (a,x) \mapsto a + \tilde{\beta}_1 \cdot x\]

is a splitting of $\textrm{Br}(\mathcal{X}_d) \to \textrm{Br}(\mathcal{X}_d^{\textrm{fr}})$ .

4.1 Genus three curves, their canonical model and theta characteristics

Recall that the canonical model of a smooth, non-hyperelliptic curve of genus three over a field is a plane quartic curve in $\mathbb{P}^2_{\mathbf{k}}$ . Therefore, we have the following guiding principle: every intrinsic geometric property of a non-hyperelliptic curve of genus three has an interpretation in terms of plane quartic curves, and vice versa.

Let $\mathcal{M}_3$ be the moduli stack of smooth genus three curves, which is a smooth, connected Deligne–Mumford stack, and let $\mathcal{H}_3$ be the divisor of hyperelliptic curves. A first instance of the principle above is then the following.

Vice versa, given a smooth, non-hyperelliptic, genus three curve $\pi\colon C\to S$ , we have the canonical, surjective morphism $\pi^*\pi_*\omega_{\pi}\to \omega_\pi$ , where $\omega_\pi$ is the relative dualizing sheaf. This induces a morphism $C\hookrightarrow \mathbb{P}(\pi_*\omega_\pi)\to S$ , which is a closed embedding and realizes each geometric fiber as a smooth plane curve of degree four.

In this way we have a morphism $g\colon\mathcal{M}_3\smallsetminus\mathcal{H}_3\to \mathcal{X}_4$ which constitutes an inverse to the previous map. Indeed, it is straightforward to check that $g\circ f=\textrm{id}$ . On the other hand, given an object $C\overset{\iota}{\hookrightarrow} P \overset{\rho}{\to} S$ , observe that $L:=\omega_{P/S}\otimes \mathcal{O}(C)$ has degree one on every fiber of $P\to S$ , and hence we have a canonical isomorphism $P\simeq \mathbb{P}(\rho_*L)$ .

Observe that there is an isomorphism $L|_C\simeq \omega_{C/S}\otimes \pi^*M$ for some line bundle M on the base. This implies that $\mathbb{P}(\pi_*\omega_{C/S})$ is canonically isomorphic to $\mathbb{P}(\pi_*(L|_C))$ . The morphism $\rho_*L \to \rho_*(\iota_*(L|_C))\simeq \pi_*(L|_C)$ is an isomorphism, because both $\rho_*L(-C)$ and $R^1\rho_*(L(-C))$ are zero; this implies that $\mathbb{P}(\pi_*(L|_C))\simeq \mathbb{P}(\rho_*L)\simeq P$ , and hence $f\circ g=\textrm{id}$ .

We will also need the following.

Proposition 4.2 [Reference Di LorenzoDL21, Proposition 3.1.3]. Let $W_4$ be the vector space of quartic forms in three variables, endowed with the $\textrm{GL}_3$ -action $A\cdot f(x_0,x_1,x_2):=\det(A)f(A^{-1}(x_0,x_1,x_2))$ , and let $\Delta \subset W_4$ be the divisor of singular quartic forms. Then we have an isomorphism of Deligne–Mumford stacks

\[ \mathcal{M}_3\smallsetminus\mathcal{H}_3 \simeq [ (W_4\smallsetminus \Delta)/\textrm{GL}_3]. \]

A second instance of the principle above is provided by theta characteristics. Recall that a theta characteristic on a smooth curve C over a field $\mathbf{k}$ is an isomorphism class of a line bundle $\theta$ such that $\theta^{\otimes 2}\simeq \omega_C$ , where $\omega_C$ denotes the canonical bundle of the curve. The set of theta characteristics in the Picard variety $\textrm{Pic}^{g-1}(C)$ forms a torsor under the additive group $\textrm{Jac}(C)_2$ of 2-torsion points in the Jacobian of C, i.e. the connected component of the identity of $\textrm{Pic}(C)$ . In particular, a smooth curve of genus g has exactly $2^{2g}$ theta characteristics.

Theta characteristics can be further subdivided into even or odd, depending on the parity of $h^0(C,\theta)$ . A smooth curve C of genus g has $2^{g-1}(2^{g}-1)$ odd theta characteristics and $2^{g-1}(2^{g}+1)$ even theta characteristics.

If C is a smooth curve of genus three, we deduce that there are exactly 28 odd theta characteristics, and for each of them we must have $h^0(C,\theta)=1$ , because $h^0(C,\theta)\leq \deg(\theta)=2$ . On the other hand, global sections of $\theta$ correspond to effective divisor of degree two on C, and hence there are exactly 28 distinct pairs of (not necessarily distinct) points $p+q$ such that $\mathcal{O}(2p+2q)\simeq \omega_{C}$ . Given a canonical embedding $C\subset \mathbb{P}^2$ , we have that $\omega_{C}\simeq \mathcal{O}_{\mathbb{P}^2}(1)|_C$ , and hence the previous statement can be reformulated as follows.

The notion of theta characteristics can be extended to families of curves: given a smooth curve of genus three $\pi\colon C\to S$ , a theta characteristic consists of a line bundle $\theta$ on C and an isomorphism $\theta^{\otimes 2} \overset{\sim}{\to} \omega_{\pi}$ . It is known (see for instance [Reference MumfordMum71]) that the parity of the rank function of the sheaf $\pi_*\theta$ is constant, and thus the definition of odd or even theta function extends to families of curves as well. The datum

$$(\pi\colon C\to S, \theta, \alpha\colon \theta^{\otimes 2}\overset{\sim}{\to}\omega_\pi)$$

with $\theta$ an odd (respectively even) theta characteristic is called an odd spin curve (respectively an even spin curve).

4.2 Extending invariants from moduli of plane curves

In this subsection we will build on the results of the previous section to obtain a partial computation of the cohomological invariants of $\mathcal{M}_3$ and $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ . First, we show that the low-degree invariants of $\mathcal{X}_{d}^\textrm{fr}$ and $\mathcal{M}_3\smallsetminus\mathcal{H}_3$ coincide.

Proof. First note that the pullback $\pi^*$ is injective because a $\mathbb{G}_\textrm{m}$ -torsor is always a smooth-Nisnevich covering. We know by Proposition 4.1 that the composition $\mathcal{M}_3 \smallsetminus\mathcal{H}_3 \to \mathcal{X}_4$ is an isomorphism, which implies that the composition

\[\textrm{Inv}^{\bullet}(\mathcal{X}_4,\textrm{M}) \to \textrm{Inv}^{\bullet}(\mathcal{X}_4^\textrm{fr},\textrm{M}) \to \textrm{Inv}^{\bullet}(\mathcal{M}_3 \smallsetminus \mathcal{H}_3, \textrm{M})\]

is an isomorphism (note that we can pull back cohomological invariants through non-representable maps), showing that $\pi^*$ must be surjective as well.

The stack $\mathcal{H}_3$ of hyperelliptic curves of genus three is a smooth Deligne–Mumford stack, whose generic stabilizer is isomorphic to $\mu_2$ , and it forms a Cartier divisor in $\mathcal{M}_3$ . The open embedding $\mathcal{M}_3\smallsetminus\mathcal{H}_3 \hookrightarrow \mathcal{M}_3$ induces an inclusion at the level of cohomological invariants. We will show that this inclusion is surjective on 2-torsion invariants.

1. (i) $M_3^{_\textrm{sm}}$ is the locus of curves which either have no non-trivial automorphisms or whose only non-trivial automorphism is the hyperelliptic involution;

2. (ii) $H_3^{_\textrm{sm}}$ is the Cartier divisor in $M_3^{sm}$ of hyperelliptic curves whose automorphism group is exactly $\mathbb{Z}/2\mathbb{Z}$ ;

3. (iii) $M_3 \smallsetminus M_3^{_\textrm{sm}}$ has codimension $\geq$ 2 in $M_3$ .

Set $\mathcal{M}_3^{_\textrm{sm}}:=\mathcal{M}_3\times_{M_3}M_3^{_\textrm{sm}}$ . Observe that, as the complement of $\mathcal{M}_3^{_\textrm{sm}}$ in $\mathcal{M}_3$ has codimension $\geq 2$ , we have

\[ \textrm{Inv}^{\bullet}(\mathcal{M}_3, \textrm{M}) \simeq\textrm{Inv}^{\bullet}(\mathcal{M}_3^{_\textrm{sm}}, \textrm{M}). \]

Proposition 4.8. Let $M^{_\textrm{sm}}_{3, \textrm{root}}$ be the order-two root stack of $M_3^{_\textrm{sm}}$ along the Cartier divisor $H_3^{_\textrm{sm}}$ . We have an isomorphism:

\[\mathcal{M}_3^{_\textrm{sm}} \simeq M^{_\textrm{sm}}_{3,\textrm{root}}.\]

Proof. The coarse moduli space morphism $\mathcal{M}_3^{_\textrm{sm}} \to M_3^{_\textrm{sm}}$ ramifies with order two over $H_3^{_\textrm{sm}}$ , and thus [Reference Geraschenko and SatrianoGS16, Theorem 1] we have a factorization

\[ \mathcal{M}_3^{_\textrm{sm}} \longrightarrow M^{_\textrm{sm}}_{3, \textrm{root}} \longrightarrow M_3^{_\textrm{sm}}. \]

The first morphism in the factorization above is representable by schemes: it is representable by algebraic spaces because it induces an isomorphism on automorphism groups, and it is quasi-finite, so by [Reference Laumon and Moret-BaillyLMB00, Théoréme A.2] it is representable by schemes. Moreover, it is by construction universally closed, generically an isomorphism and surjective. Applying Zariski’s main theorem, we deduce that $\mathcal{M}_3^{_\textrm{sm}} \to M^{_\textrm{sm}}_{3, \textrm{root}}$ is an isomorphism.

As a consequence, we have an isomorphism of cohomological invariants

\[\textrm{Inv}^{\bullet}(\mathcal{M}_3, \textrm{M}) \simeq \textrm{Inv}^{\bullet}(\mathcal{M}_3^{_\textrm{sm}}, \textrm{M}) \simeq \textrm{Inv}^{\bullet}(M^{_\textrm{sm}}_{3, \textrm{root}}, \textrm{M}) \]

and we get the following proposition.

Proposition 4.9. We have an equality

\[\textrm{Inv}^{\bullet}(M^{_\textrm{sm}}_3\smallsetminus H_3^{_\textrm{sm}}, \textrm{M}) = \textrm{Inv}^{\bullet}(\mathcal{M}_3\smallsetminus\mathcal{H}_3, \textrm{M}),\]

and the subgroup

\[\textrm{Inv}^{\bullet}(\mathcal{M}_3,\textrm{M}) \subseteq \textrm{Inv}^{\bullet}(\mathcal{M}_3 \smallsetminus \mathcal{H}_3,\textrm{M})\]

is composed of those invariants whose ramification at $H_3^{_\textrm{sm}}$ is of 2-torsion.

In particular, this implies that

\[\textrm{Inv}^{\bullet}(\mathcal{M}_3\smallsetminus\mathcal{H}_3, \textrm{M})_2 \simeq \textrm{Inv}^{\bullet}(\mathcal{M}_3, \textrm{M})_2. \]

On the other hand, we have an isomorphism $M_3^{_\textrm{sm}}\smallsetminus H_3^{_\textrm{sm}} \simeq \mathcal{M}_3^{_\textrm{sm}}\smallsetminus\mathcal{H}_3^{_\textrm{sm}}$ , and the complement of the latter in $\mathcal{M}_3\smallsetminus \mathcal{H}_3$ has codimension $\geq 2$ . We deduce that

\[ \textrm{Inv}^{\bullet}(M^{_\textrm{sm}}_3\smallsetminus H_3^{_\textrm{sm}}, \textrm{M}) = \textrm{Inv}^{\bullet}(\mathcal{M}_3\smallsetminus\mathcal{H}_3, \textrm{M}) .\]

The last statement follows trivially as the ramification of a 2-torsion invariant is always of 2-torsion.

This shows that the subgroup $N[2] \subset \textrm{Inv}^{\bullet}(\mathcal{M}_3 \smallsetminus \mathcal{H}_3, \textrm{M})$ , which is 2-torsion, belongs to the cohomological invariants of $\mathcal{M}_3$ . All that is left to do is to understand if the same happens for the 9-torsion subgroup $\beta_1 \cdot \textrm{M}^{\bullet}(\mathbf{k})_9$ .

Corollary 4.10. We have

\[\textrm{Inv}^{\bullet}(\mathcal{M}_3,\textrm{M}^{\leq 2}) = \textrm{Inv}^{\bullet}(\mathbf{k}, \textrm{M}^{\leq 2}) \oplus N[2],\]

where N[2] restricts on $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ to the pullback of the corresponding summand in $\textrm{Inv}^{\bullet}(\mathcal{X}_{4}^{\textrm{fr}},\textrm{M})$ .

Proof. We want to see which elements of

\[\textrm{Inv}^{\bullet}(\mathcal{M}_3 \smallsetminus \mathcal{H}_3, \textrm{M}^{\leq 2})=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \beta_1 \cdot \textrm{M}^{\leq 1}(\mathbf{k})_9 \oplus N[2]\]

extend to $\mathcal{M}_3$ . First note that obviously all invariants coming from the base field extend, and by Proposition 4.9 the 2-torsion component, which includes N[2], extends to $\mathcal{M}_3$ . Now, by [Reference Di LorenzoDL21, Theorem 3.1.5] we know that $\textrm{Pic}(\mathcal{M}_3)=\mathbb{Z}$ , and it is well known that $\mathcal{O}^*(\mathcal{M}_3)=\mathbf{k}^*$ . Finally, recall that by [Reference Di Lorenzo and PirisiDLP21a, Lemma 2.18] we have $\textrm{Inv}^1(\mathcal{M}_3,\textrm{K}_\ell)=\textrm{H}^1(\mathcal{M},\mu_\ell)$ . Then the Kummer exact sequence for $\ell = 9$ reads

\[\mathbf{k}^*/(\mathbf{k}^*)^9 \to \textrm{Inv}^1(\mathcal{M}_3,\textrm{K}_9) \to \textrm{Pic}(\mathcal{M}_3)_9 = 0;\]

the component on the left is just $\textrm{K}^1_9(\mathbf{k})$ so we conclude that there are no elements in $\textrm{Inv}^1(\mathcal{M}_3,\textrm{K}_9)$ outside those coming from the base, which implies that the ramification of $\beta_1$ on $\mathcal{H}_3$ , which belongs to $\textrm{A}^0(\mathcal{H}_3,\textrm{H}^0_9)=\mathbb{Z}/9\mathbb{Z}$ , must be coprime to 3. But then the ramification of $\beta_1 \cdot x$ must be non-zero for any $x \in \textrm{M}^{\bullet}(\mathbf{k})_9$ , proving our claim.

4.3 Ètale algebras and Stiefel–Whitney classes

To explicitly construct a generator for the factor N[2] of the ring of cohomological invariants of $\mathcal{M}_3$ we will make use of some results in the theory of classical cohomological invariants, to the authors’ knowledge due to Serre.

An étale algebra E over F is a finite F-algebra that is isomorphic to $(F')^n$ for some n after passing to some finite extension $F'/F$ , i.e. $\textrm{Spec}(E) \to \textrm{Spec}(F)$ is a finite étale morphism. Passing to the group scheme $T_E(R) = \textrm{Aut}_R(E \otimes R)$ we obtain a $\textrm{S}_n$ -torsor over F. The category of finite étale algebras of rank n over $\mathbf{k}$ forms a Deligne–Mumford stack $\unicode{x00C9}\textrm{t}_n$ isomorphic to $\textrm{BS}_n$ , and any finite étale morphism $\mathcal{S} \to \mathcal{X}$ of constant degree n induces a morphism $\mathcal{X} \to \unicode{x00C9}\textrm{t}_n$ $\mathcal{X} \to \unicode{x00C9}\textrm{t}_n$ , meaning we can pull the cohomological invariants of $\unicode{x00C9}\textrm{t}_n$ back to $\mathcal{X}$ .

The cohomological invariants of $\unicode{x00C9}\textrm{t}_n$ are fully understood [Reference Garibaldi, Merkurjev and SerreGMS03, Chapter 7], and can be described as follows.

Given $E/F \in \unicode{x00C9}\textrm{t}_n(F)$ , we can define a scalar product on E, seen as a F-vector space, by $\langle x, y \rangle_\textrm{tr} = \textrm{tr}(\cdot xy)$ , that is the couple x,y is sent to the trace of the multiplication by xy seen as an endomorphism of E. Diagonalizing this product yields an n-uple of coefficients $(d_1, \ldots, d_n) \in F^n$ ; obviously picking two different diagonal representations will yield different coefficients: it is immediate that an invariant of E must be left unchanged when multiplying coefficients by a square or permuting them. We define the ith Stiefel–Whitney class of E as

\[\alpha^\textrm{SW}_i(E) = \sigma_i(d_1,\ldots, d_n) \in \textrm{K}^i_{2}(F),\]

where $\sigma_0=1 \in \textrm{K}^0_2(F)$ and $\sigma_i$ is the ith elementary symmetric polynomial in $d_1, \ldots, d_n$ seen as element in $\textrm{K}_2^i(F)=\textrm{H}^i(F,\mathbb{Z}/2\mathbb{Z}(i))$ , e.g.

\[\sigma_2(a,b,c)=\lbrace a , b \rbrace + \lbrace a , c \rbrace + \lbrace b , c \rbrace \in \textrm{K}^2_2(\mathbf{k}(a,b,c)).\]

We have

\[\textrm{Inv}^{\bullet}(\unicode{x00C9}\textrm{t}_n,\textrm{M})=\textrm{M}^{\bullet}(\mathbf{k}) \oplus \alpha^\textrm{SW}_1 \cdot \textrm{M}^{\bullet}(\mathbf{k}) \oplus \cdots \oplus \alpha^\textrm{SW}_{\lfloor n/2\rfloor} \cdot \textrm{M}^{\bullet}(\mathbf{k}).\]

Now we will define new generators for the cohomological invariants of $\unicode{x00C9}\textrm{t}_n$ , the Galois–Stiefel–Whitney classes, which have better properties. Set [Reference Garibaldi, Merkurjev and SerreGMS03, Theorem 25.10]

\[\alpha_i=\begin{cases}\alpha_i^\textrm{SW}, & i \;\textrm{is odd,}\\\alpha_i^\textrm{SW} + \lbrace 2 \rbrace \cdot\alpha^\textrm{SW}_{i-1}, & i\; \textrm{is even.}\end{cases}\]

Moreover, define the total Galois–Stiefel–Whitney class of E as $\alpha_\textrm{tot}(E)=\sum_i \alpha_i(E)$ . Then the following properties hold:

1. (i) $\textrm{Inv}^{\bullet}(\unicode{x00C9}\textrm{t}_n,\textrm{M})=\textrm{M}^{\bullet}(\mathbf{k}) \oplus \alpha_1 \cdot \textrm{M}^{\bullet}(\mathbf{k}) \oplus \ldots \oplus \alpha_{\lfloor n/2\rfloor} \cdot \textrm{M}^{\bullet}(\mathbf{k}) $ ;

2. (ii) $\alpha_i(E)=0$ if i is greater than $\lfloor n/2 \rfloor$ ;

3. (iii) $\alpha_\textrm{tot}(F^n/F)=1$ ;

4. (iv) $\alpha_\textrm{tot}(F(\sqrt{a})/F)=1+\lbrace a \rbrace$ ;

5. (v) $\alpha_\textrm{tot}(E \times E')=\alpha_\textrm{tot}(E)\cdot \alpha_\textrm{tot}(E').$

The following examples will be helpful later on.

Example 4.11. Consider the field $F=\mathbf{k}(a,b)$ , and the étale algebra over F:

\[E= F(\sqrt{a}) \times F(\sqrt{b}) \times F(\sqrt{ab}).\]

Then we have

\begin{align*} \alpha_\textrm{tot}(E) &= \alpha_\textrm{tot}(F(\sqrt{a})) \cdot \alpha_\textrm{tot}(F(\sqrt{b})) \cdot \alpha_\textrm{tot}(F(\sqrt{c}))=(1+ \lbrace a \rbrace)(1+ \lbrace b \rbrace)(1+ \lbrace ab \rbrace)\\&=1+\lbrace a \rbrace + \lbrace b \rbrace + \lbrace ab \rbrace + \lbrace a, b \rbrace + \lbrace a, ab \rbrace + \lbrace b, ab \rbrace + \lbrace a, b, ab \rbrace \\&=1 + 2\lbrace ab \rbrace + 3\lbrace a, b \rbrace + \lbrace - 1, a \rbrace + \lbrace -1, b \rbrace - 2\lbrace -1, a, b \rbrace \\&=1 + \lbrace a, b \rbrace + \lbrace -1, a \rbrace +\lbrace -1, b \rbrace = 1 +\lbrace a , b \rbrace + \lbrace -1, ab \rbrace.\end{align*}

Example 4.12. Consider again $F=\mathbf{k}(a,b)$ and let $K=F(\sqrt{a},\sqrt{b})$ , seen as an étale algebra over F. We know that $\alpha_i(K)=0$ for $i >2$ , so we are interested in computing $\alpha_1(K)$ and $\alpha_2(K)$ . Consider the trace form on K. By direct computation we obtain

\[\textrm{tr}(\cdot \sqrt{a})=\textrm{tr}(\cdot \sqrt{b})=\textrm{tr}(\cdot \sqrt{ab})=0,\quad \textrm{tr}(\cdot 1)=4,\quad \textrm{tr}(\cdot a)=4a,\quad \textrm{tr}(\cdot b)=4b,\quad \textrm{tr}(\cdot ab)=4ab,\]

which shows that the matrix representing the trace form is

\[\begin{pmatrix} 4 & 0 & 0 & 0 \\ 0 & 4a & 0 & 0\\ 0 & 0 & 4b & 0 \\ 0 & 0 & 0 & 4ab \end{pmatrix},\]

which shows that

\[\alpha_1(K)=\alpha^\textrm{SW}_1(K)=\lbrace a \rbrace+\lbrace b \rbrace+\lbrace ab \rbrace=0,\]

and in degree two we have

\[\alpha_2(K)=\alpha^\textrm{SW}_2(K)+\lbrace 2\rbrace\alpha^\textrm{SW}_1(K)=\lbrace a , b \rbrace+\lbrace a , ab \rbrace+\lbrace b , ab \rbrace= \lbrace a , b \rbrace + \lbrace -1, ab \rbrace.\]

4.4 An explicit generator: bitangents and lines on cubic surfaces

In this section we will produce an explicit étale algebra S over $\mathcal{M}_3$ and $\mathcal{X}_{4}^{\textrm{fr}}$ whose second Galois–Stiefel–Whitney class is non-trivial, or more specifically such that the map $\textrm{M}^{\bullet}(\mathbf{k})_2 \to \textrm{Inv}^{\bullet}(\mathcal{X}_{4}^{\textrm{fr}}, \textrm{M})$ given by $x \mapsto \alpha_2(S) \cdot x$ is injective for any cycle module M.

Recall from § 4.1 that there is a finite étale morphism $\mathcal{S}_3^- \to \mathcal{M}_3$ of degree 28, which is given by the functor $(C,\theta)\mapsto C$ that forgets the odd theta characteristic $\theta$ .

There are two maps

\[\mathcal{M}_3 \smallsetminus \mathcal{H}_3 \to \mathcal{X}_4, \quad \mathcal{X}_4 \to \mathcal{M}_3 \smallsetminus \mathcal{H}_3\]

the first being the composition of $\mathcal{M}_3 \smallsetminus \mathcal{H}_3 \to \mathcal{X}_4^{\textrm{fr}}$ and $\mathcal{X}_4^{\textrm{fr}} \to \mathcal{X}_4$ and the second given by the $(C\to P \to S) \mapsto (C \to S)$ forgetful map. It is straightforward to check that their composition is the identity on $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ . This shows that the restriction of $\mathcal{S}_3^-$ to $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ induces a corresponding étale covering of $\mathcal{X}_4^{\textrm{fr}}$ and $\mathcal{X}_4$ , and everything we do can be applied to either of these. By Proposition 4.3, given a non-hyperelliptic curve $C/F$ and an embedding $C \to \mathbb{P}^2_F$ the fiber $\mathcal{S}^-_3(C)$ parametrizes bitangent lines to C.

Name S the corresponding étale algebra on $\mathcal{M}_3$ and $\mathcal{X}^{\textrm{fr}}_4$ . We will show the following.

Proposition 4.13. Let N[2] be the subgroup of ${Inv}^{\bullet}(\mathcal{X}^{\textrm{fr}}_4,\textrm{M})$ and ${Inv}^{\bullet}(\mathcal{M}_3, \textrm{M})$ in Proposition 3.9 and Corollary 4.10. We have

\[ N[2] = \textrm{M}^{0}(\mathbf{k})_2[2] = \alpha_2(S)\cdot \textrm{M}^{0}(\mathbf{k})_2,\]

where S is the étale algebra coming from the finite étale morphism of degree 28,

\[\mathcal{S}_3^- \to \mathcal{M}_3,\]

and the corresponding morphism on $\mathcal{X}^{\textrm{fr}}_4$ .

This will complete the computation of low-degree cohomological invariants of $\mathcal{M}_3$ as well as $\mathcal{X}^{\textrm{fr}}_4$ and $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ .

We recall now a very well-known construction of a smooth cubic surface as a two-sheeted covering of $\mathbb{P}^2$ , branched along a smooth quartic (see, for example, [Reference HarrisHar79a]).

Proposition 4.14. Assume ${char}(F) \neq 2$ , and consider a smooth cubic surface $X \subset \mathbb{P}^3_F$ , an F-valued point $P \in X$ which does not lie on any line of X. Let $\widetilde{X}=\textrm{Bl}_P X$ be the blow-up of X at P. Then there is a finite morphism of degree two

\[f:\widetilde{X} \to \mathbb{P}^2_F\]

whose branching locus is a smooth quartic $B_X \subset \mathbb{P}^2_F$ . Moreover, the bitangent lines to $B_X$ correspond to the lines on X plus the exceptional divisor.

Proof. Let H be a hyperplane in $\mathbb{P}^3_F$ , not containing P. We can define a map $X\smallsetminus P \to H$ by sending $Q\in X$ to the intersection of H and the line through P and Q. By taking the blow-up of X at P we can extend this map to a projection

\[f:\widetilde{X}=\textrm{Bl}_P X \to H.\]

This is a finite morphism of degree two as every line in $\mathbb{P}^3_F$ which passes through $P \in X$ intersects X in two more points, counted with multiplicity.

Observe that the canonical divisor $K_{\widetilde{X}}$ is equal to $f^*K_{H}+R_X$ , where $R_X$ is the ramification divisor. From this we deduce that

\begin{align*} f_*(K_{\widetilde{X}})&= \deg(f)\cdot K_H + B_X = -6L + B_X,\end{align*}

where $B_X$ is the branching divisor and L is the class of a line in H. On the other hand, from the formula for the canonical divisor of a blow-up, we have that $K_{\widetilde{X}} = b^*K_X + E$ , where $b\colon \widetilde{X}\to X$ is the blow-down morphism and E is the exceptional divisor.

As $X\subset\mathbb{P}^3_F$ is a cubic hypersurface, by the adjunction formula $K_X = -S|_X$ , where $S\subset \mathbb{P}^3_F$ is a hyperplane. Moreover, from the formula for the fundamental class of a strict transform, we have $b^*(S|_X)=C + E$ , where C is the strict transform of $S|_X$ . We deduce that $K_{\widetilde{X}} = -C$ .

Observe that, for a generic hyperplane $S\subset\mathbb{P}^3_F$ passing through P, the curve $S|_X$ is a plane curve of degree three in $S\simeq\mathbb{P}^2_F$ , and hence a curve of genus one. On the other hand, the closure of the image of $S \subset \mathbb{P}^3_F$ through the projection $\mathbb{P}^3\dashrightarrow H$ is a line L, because the projection from a point maps generic hyperplanes to lines. These two facts together imply that the induced morphism $f|_C\colon C \to f(C)=L$ is a 2:1 covering, and hence $f_*C= 2L$ .

Putting everything together, we deduce that $B_X=4L$ , and hence $f\colon\widetilde{X} \to H$ is a two-sheeted covering whose ramification locus is a smooth plane quartic $B_X$ .

Let $D\subset X$ be a line, which by hypothesis does not contain the point P. Then its image through $X\smallsetminus P \to \mathbb{P}^2_F$ is a curve; on the other hand, the projection of D through $\mathbb{P}^3_F\smallsetminus P \to \mathbb{P}^2_F$ is either a line or a point, so hence it must be a line. If $L\subset \mathbb{P}^2_F$ is this line, then the restriction of $f|_D\colon D\to L$ has degree one, where with a little abuse of notation we are denoting both D and its strict transform in $\widetilde{X}$ with the same letter.

As $f^{-1}(L)\to L$ is finite of degree two, and $D\subset f^{-1}(L)$ is not contained in the branching locus, we must have $f^{-1}(L)=D\cup C$ for some other curve $C\subset \widetilde{X}$ which is mapped one-to-one onto L.

Let $S\subset \mathbb{P}^3_F$ be the hyperplane obtained as the closure of the preimage of L through the projection $\mathbb{P}^3_F\smallsetminus P \to \mathbb{P}^2_F$ ; then $S\cap X= D \cup \overline{C}$ , where $\overline{C}$ is the image of C through the blow-down.

As $D \cup \overline{C}\subset S$ is a plane curve of degree three, we must have that $\overline{C}$ has degree two, and hence it meets D in two points, counted with multiplicity (which must be different from P). By construction, the ramification points of $D\cup C \to L$ coincide with $D\cap C$ , and hence L is a line meeting the branching divisor in two points. We deduce that L is bitangent to $B_X$ .

Finally, observe that the exceptional divisor E is also mapped to a line L: this follows from the fact that the morphism $\textrm{Bl}_P \mathbb{P}^3_F \to \mathbb{P}^2_F$ maps the exceptional divisor one-to-one onto $\mathbb{P}^2_F$ , and the map $E\to f(E)$ factors through the embedding of E in the exceptional divisor of $\textrm{Bl}_P \mathbb{P}^3_F$ .

Then, as before, we must have $f^{-1}(L)=E\cup C$ for some curve $C\subset \widetilde{X}$ that maps one-to-one onto L. Let $\overline{C}$ be its image in X: then $\overline{C}$ is equal to the intersection of X with some hyperplane S, and hence must have genus one. The only option left is that $\overline{C}$ is singular at P, from which we deduce that C meets E in two points (counted with multiplicity).

As the branching locus of $f^{-1}(L) \to L$ coincides with $f(E\cap C)$ , we deduce that L is also a bitangent to the ramification locus.

As the number of bitangents of $B_X$ is 28, we deduce that the 27 lines on X are sent one-to-one to the 27 bitangents of $B_X$ , with the last bitangent being the image of the exceptional divisor.

Thus if we pick the point $B_X \in \mathcal{X}_4(F)$ obtained in this way, then the fiber $\mathcal{S}^{-}_3(B_X)$ is equal to $\textrm{Spec}(F) \cup L_X$ , where $L_X$ is the Fano scheme of lines on X.

Our goal is then to produce a cubic surface $X \subset \mathbb{P}^3_{F}$ such that the scheme $L_X$ representing lines on X, or more precisely the associated étale algebra $E_X/F$ , has a second Galois–Stiefel–Whitney class which satisfies $x \mapsto \alpha_2(E_X) \cdot x \neq 0$ for any cycle module $\textrm{M}$ and non-zero element $x \in \textrm{M}^{\bullet}(\mathbf{k})_2$ .

Let F be the field $\mathbf{k}(a,b)$ . A well-known way to produce a cubic surface in $\mathbb{P}^3$ (see, for example, [Reference HartshorneHar79b, Chapter V, § 4]), is to blow up six points in general position on $\mathbb{P}^2$ ; we want to make a more arithmetic version of this construction, taking inspiration from Example 4.11. Consider three length two points $P_1, P_2, P_3$ on $\mathbb{P}^2_F$ , defined respectively over $E_1=F(\sqrt{a})$ , $E_2 = F(\sqrt{b})$ and $E_3 = F(\sqrt{ab})$ . If we write $K=F(\sqrt{a}, \sqrt{b})$ and $\textrm{Gal}(K) = \langle \sigma, \tau \rangle$ where

\[\sigma(\sqrt{a})=-\sqrt{a}, \quad \sigma(\sqrt{b})=\sqrt{b}, \quad \tau(\sqrt{a})=\sqrt{a}, \quad \sigma(\sqrt{b})=-\sqrt{b},\]

this is equivalent to picking six points $Q_1, Q'_1, Q_2, Q'_2, Q_3, Q'_3$ on $\mathbb{P}^2_{K}$ where $Q_1,Q'_1, Q_2, Q'_2$ and $Q_3, Q'_3$ are exchanged by the Galois group and are fixed by, respectively, $\tau, \sigma$ and $\tau \cdot \sigma$ . We, moreover, ask for these points to be in general position, i.e. no three points belong to the same line and the six points are not all on a conic. An example of six such points that works for every base field is

\[Q_1, Q'_1 = ( \pm\sqrt{a}: 1 : 0),\quad Q_2, Q'_2=(\pm\sqrt{b}: 0: 1),\quad Q_3, Q'_3 = (0 : \pm\sqrt{ab} : 1).\]

Blowing up these three points yields a cubic surface $X/F$ , as can be seen by passing to K.

The 27 lines on a cubic surface obtained this way can be easily described. Over K, they are the six exceptional divisors corresponding to the points, the 15 lines passing through two points each, and the six conics passing through five of the six points. Then to describe the étale algebra $E_X/F$ it suffices to look at the action of the Galois group on $E_X \otimes_F K= K^{27}$ . We immediately see the following.

1. (i) The six lines given by the exceptional divisors are fixed by the stabilizer of the corresponding point and thus are defined over the same field.

2. (ii) The three lines passing by conjugate points are fixed by the whole group and thus defined over F.

3. (iii) The 12 lines passing by two non-conjugate points are only fixed by the identity and thus are defined over K.

4. (iv) The six conics passing through five of six points are stabilized by the stabilizer of the sixth point and thus are defined over the same field.

This shows that

\[E_X = F^3 \times E_1^2 \times E_2^2 \times E_3^2 \times K^3.\]

We know that $\alpha_\textrm{tot}(\textrm{Spec}(F))=1$ and $\alpha_\textrm{tot}(\textrm{Spec}(E_1))$ , $\alpha_\textrm{tot}(\textrm{Spec}(E_2))$ and $\alpha_\textrm{tot}(\textrm{Spec}(E_3))$ is equal to $1+\lbrace a\rbrace,1+\lbrace b\rbrace$ and $1+\lbrace ab\rbrace$ , respectively, which shows that

\[\alpha_\textrm{tot}(E_X)=(1+\lbrace a \rbrace)^2(1+\lbrace b \rbrace)^2(1+\lbrace ab \rbrace)^2\alpha_\textrm{tot}(\textrm{Spec}(K))^3,\]

and expanding the first three components we get

\begin{align*}(1+\lbrace a \rbrace)^2(1+\lbrace b \rbrace)^2(1+\lbrace ab \rbrace)^2 & =(1+\lbrace -1 , a \rbrace)(1+\lbrace -1 , b \rbrace)(1+\lbrace -1 , ab \rbrace)\\& =1 + \lbrace -1 , a \rbrace + \lbrace -1 , b \rbrace + \lbrace -1 , a \rbrace + \lbrace -1 , b \rbrace + r = 1 + r,\end{align*}

where every term in r has degree higher than 2. This shows that the degree-two term of $\alpha_\textrm{tot}(E_X)$ is the degree-two term of $\alpha_\textrm{tot}(K)^3$ . By Example 4.12 we know that

\[\alpha_1(K)=0, \quad \alpha_2(K)=\lbrace a, b \rbrace + \lbrace -1, ab \rbrace, \]

so

\[\alpha_\textrm{tot}(E_X)=(1+r)\alpha_\textrm{tot}(K)^3=(1+r)(1+\lbrace a , b \rbrace + r')^3=1+\lbrace a , b \rbrace + \lbrace -1, ab \rbrace + r'',\]

where again every term in r” has degree higher than 2, showing that

\[\alpha_2(E_X)=\lbrace a , b \rbrace + \lbrace -1, ab \rbrace.\]

Observe that for any $x \in \textrm{M}^{\bullet}(\mathbf{k})_2$ by the compatibility of the $\textrm{K}_{2}$ -module structure and the differential we have

\[\partial_{a=0} (\partial_{b=0}(\lbrace a , b \rbrace + \lbrace -1, ab \rbrace)\cdot x) = \partial_{a=0}(\lbrace a\rbrace \cdot x) = x,\]

showing that the map $\textrm{M}^{\bullet}(\mathbf{k}) \to \textrm{Inv}^{\bullet}(\mathcal{X}^{\textrm{fr}}_4, \textrm{M})$ given by $x \mapsto \alpha_2(S)\cdot x$ is injective. Note, moreover, that the only constant invariant in the image of this map is necessarily zero, as a constant invariant would be unramified at $b=0$ , as would come from the $\textrm{M}^{\bullet}(\mathbf{k}) \to \textrm{M}^{\bullet}(\mathbf{k}(a,b))$ pullback.

Remark 4.15. Another way to see that the subgroup $\alpha_2(S) \cdot \textrm{M}_2^{\bullet}(\mathbf{k}) \subset \textrm{Inv}^2(\mathcal{X}_4^\textrm{fr})$ does not contain any non-zero constant invariant is the following: repeat the construction above but this time blowing up six rational points in $\mathbb{P}^2_\mathbf{k}$ . Then we get a smooth quartic C over $\mathbf{k}$ whose theta characteristics are all rational, so that $\alpha_2(S_C)=0$ . This clearly applies to any element in $\alpha_2(S)\cdot \textrm{M}^\bullet(\mathbf{k})_2$ . But the only constant invariant which restricts to zero on a curve is the zero invariant, showing that

\[\alpha_2(S) \cdot \textrm{M}_2^{\bullet}(\mathbf{k}) \cap \textrm{M}^{\bullet}(\mathbf{k})=\{0\}.\]

Proof of Proposition 13. We constructed an injective map of $\textrm{K}_2^0(\mathbf{k})$ -modules

\[\textrm{M}^0(\mathbf{k})_2 \xrightarrow{\alpha_2(S)} \textrm{Inv}^2(\mathcal{X}_4^\textrm{fr},\textrm{M}^2)\]

whose image does not contain any non-zero constant invariant. Consider the case when $\textrm{M}=\textrm{K}_2$ : as $\textrm{K}_2^0(\mathbf{k})=\mathbb{Z}/2\mathbb{Z}$ we conclude that $N=\textrm{K}_2^0(\mathbf{k})=\mathbb{Z}/2\mathbb{Z}$ , as it must be non-trivial, and moreover that the image of $\alpha_2(S)$ through the ramification map

\[\textrm{A}^0_{\textrm{GL}_3}(\mathbb{P}(W_4)\smallsetminus Z, \textrm{M}^2) \xrightarrow{\partial} \textrm{A}^0_{\textrm{GL}_3}(Z, \textrm{M}^1)\]

is the element $\alpha$ appearing in Propositions 3.8 and 3.9, because otherwise $\alpha_2(S)$ would come from the base field. Now, by the compatibility of the residue map and the module structure [Reference RostRos96, pp. 229–230, R3d and R3f] we see that

\[\partial (\alpha_2(S) \cdot x) = \alpha \cdot x\]

for all $x \in \textrm{M}^0(\mathbf{k})_2$ , which shows that every element in the form $\alpha \cdot x$ belongs to the kernel of $A^0(Z,\textrm{M}) \xrightarrow{i_*} A^1(\mathbb{P}(W_4),\textrm{M})$ , and thus

\[N=\textrm{M}^0(\mathbf{k})_2\]

as claimed.

If we repeat the construction above using $K'=\mathbf{k}(\sqrt{a},\sqrt{b},\sqrt{c})$ and picking points $Q_1, \ldots, Q'_3$ defined over $F'(\sqrt{a}),\ldots, F'(\sqrt{c})$ we obtain

\begin{align*}\alpha_\textrm{tot}(E_{X'}) & =1 + \lbrace -1, a b c\rbrace + \lbrace a,b \rbrace + \lbrace a, c \rbrace + \lbrace b, c \rbrace + \lbrace -1, -1, -1, abc \rbrace \\& \quad + \lbrace -1\rbrace^3 \lbrace a, b, c \rbrace + \lbrace - 1 \rbrace^5 \lbrace abc \rbrace + r,\end{align*}

where every term in r has degree at least 8. Note that the classes

\[1,\quad \alpha_2(E_{X'}),\quad \alpha_4(E_{X'}),\quad \alpha_6(E_{X'})\]

are all $\textrm{K}_2(\mathbf{k})$ -linearly independent, which shows that there must be at least two $\textrm{K}_2(\mathbf{k})$ -linearly independent cohomological invariants of $\mathcal{M}_3$ of degree three or more that do not belong to the submodule generated by invariants of degree two or less.

4.5 Main results

We are ready to assemble our main results. Combining Corollary 4.10 and Proposition 4.13 we immediately obtain a full computation of the low-degree cohomological invariants of $\mathcal{M}_3$ . Moreover, we can combine Theorem 3.10 and Proposition 4.5 with Corollary 4.13 to additionally compute the low-degree cohomological invariants of $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ and $\mathcal{X}_4^{\textrm{fr}}$ over a non-algebraically closed field.

Theorem 4.17. Assume our base field $\mathbf{k}$ has characteristic different from 2, and let $\textrm{M}$ be an $\ell$ -torsion cycle module. We have

\begin{align*} {Inv}^{\leq 2}(\mathcal{M}_3,\textrm{M})&=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \alpha_2 \cdot \textrm{M}^{0}(\mathbf{k})_2,\\ {Inv}^{\leq 2}(\mathcal{M}_3\smallsetminus \mathcal{H}_3,\textrm{M})&=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \beta_1 \cdot\textrm{M}^{\leq 1}(\mathbf{k})_9 \oplus \alpha_2 \cdot \textrm{M}^{0}(\mathbf{k})_2,\\ {Inv}^{\leq 2}(\mathcal{X}_4^{\textrm{fr}},\textrm{M})&=\textrm{M}^{\leq 2}(\mathbf{k}) \oplus \beta_1 \cdot\textrm{M}^{\leq 1}(\mathbf{k})_9 \oplus \alpha_2 \cdot \textrm{M}^{0}(\mathbf{k})_2,\end{align*}

where $\alpha_2$ is the second Galois–Stiefel–Whitney class of the étale covering $\mathcal{S}^-_3 \to \mathcal{M}_3$ and $\beta_1 \in {Inv}^1(\mathcal{M}_3\smallsetminus \mathcal{H}_3,\textrm{K}_9)$ .

Restricting the result above to $\textrm{M} = \textrm{H}_{\mu_{\ell}^{\vee}}$ , and using the fact (see [Reference Edidin, Hasset, Kresch and VistoliEHKV01, Reference Kresch and VistoliKV04]) that for a smooth, generically tame and separated Deligne–Mumford stack whose coarse moduli space is quasi-projective the Brauer group and cohomological Brauer group coincide, we obtain the computation of the Brauer groups of $\mathcal{M}_3$ and $\mathcal{M}_3 \smallsetminus \mathcal{H}_3$ .

Corollary 4.2. Assume our base field $\mathbf{k}$ has characteristic different from 2, and let $\ell$ be a positive integer divisible by 2 and coprime to ${char}(\mathbf{k})$ . Then

\[\textrm{Br}(\mathcal{M}_3)_\ell=\textrm{Br}(\mathbf{k})_\ell \oplus \mathbb{Z}/2\mathbb{Z},\]

where the $\mathbb{Z}/2\mathbb{Z}$ is the pullback of a generator of $\textrm{Br}(\textrm{B}\textrm{S}_{28})/\textrm{Br}(\mathbf{k})$ through the map induced by $\mathcal{S}^-_3 \to \mathcal{M}_3$ . Moreover, we have

\[{Br}(\mathcal{M}_3 \smallsetminus \mathcal{H}_3)_{\ell}={Br}(\mathcal{X}_4^{\textrm{fr}})_{\ell}={Br}(\mathcal{X}_4)_{\ell}=\textrm{Br}(\mathbf{k})_\ell \oplus \textrm{H}^1(\mathbf{k},\mathbb{Z}/9\mathbb{Z})_{\ell} \oplus \mathbb{Z}/2\mathbb{Z},\]

where when ${char}(\mathbf{k})$ is coprime to 3 the $\textrm{H}^1(\mathbf{k},\mathbb{Z}/9\mathbb{Z})$ component comes from the cup products of $\beta_1 \in{Inv}^1(\mathcal{M}_3\smallsetminus \mathcal{H}_3,\textrm{K}_9)=\textrm{H}^1(\mathcal{M}_3 \smallsetminus \mathcal{H}_3,\mu_9)$ and elements of $\textrm{H}^1(\mathbf{k},\mathbb{Z}/9\mathbb{Z})$ .

We would like to highlight three remaining questions.

1. (i) What is the image of $\textrm{Br}(\mathcal{M}_3)$ inside $\textrm{Br}(\mathcal{H}_3)$ ?

2. (ii) Is it possible to compute the full cohomological invariants of $\mathcal{M}_3$ , at least over an algebraically closed field?

3. (iii) Do the étale algebras obtained by odd and even theta characteristics induce any non-trivial invariants on $\mathcal{M}_g$ for $g \geq 4$ ?