Equivariant cohomology of the moduli space of genus three curves with symplectic level two structure via point counts
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Abstract
We determine the cohomology groups of the quartic and hyperelliptic loci inside the moduli space of genus three curves with symplectic level two structure as representations of the symmetric group \(S_7\) together with their mixed Hodge structures by means of making equivariant point counts over finite fields and via purity arguments. This determines the weighted Euler characteristic of the whole moduli space of genus three curves with level two structure.
Keywords
Moduli of curves Cohomology Point counts PurityMathematics Subject Classification
14H10 14H50 14F20 14F40 55R801 Introduction
Let n be a positive integer and let C be a curve. A level n structure on C is a basis for the ntorsion of the Jacobian of C. The purpose of this paper is to study the cohomology of the moduli space \(\mathscr {M}_{3}[2]\) of genus 3 curves with symplectic level 2 structure.
A genus 3 curve which is not hyperelliptic is embedded as a plane quartic via its canonical linear system. The corresponding locus in \(\mathscr {M}_{3}[2]\) is called the quartic locus and it is denoted \(\mathscr {Q}[2]\). A plane quartic with level 2 structure is specified, up to isomorphism, by an ordered septuple of points in general position in \(\mathbb {P}^{2}\), up to the action of Open image in new window (see Sect. 3, especially Theorem 3.2). This identification will be the basis for our investigation of \(\mathscr {Q}[2]\).
Our main focus will be on \(\mathscr {Q}[2]\) but we will also consider its complement in \(\mathscr {M}_{3}[2]\), i.e. the hyperelliptic locus \(\mathscr {H}_{3}[2]\). The spaces \(\mathscr {M}_{3}[2]\), \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) are all defined over Open image in new window , a fact which gives us the flexibility to consider them over the complex numbers as well as over finite fields of characteristic different from 2. In the present paper, the latter viewpoint will be the central one and the computations will be carried out via point counts over finite fields (see Sects. 6, 7). By virtue of Lefschetz trace formula (see Sect. 4), such point counts give cohomological information in the form of Euler characteristics. However, we shall see (in Proposition 5.3 resp. Sect. 7) that both \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) satisfy certain strong purity conditions which allow us to deduce information about the individual cohomology groups, in the form of Poincaré polynomials, from these Euler characteristics.
The group Open image in new window acts on \(\mathscr {M}_{3}[2]\) as well as on \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) by changing level structures. The cohomology groups thus become Open image in new window representations and our computations will therefore be equivariant. However, the action of Open image in new window is rather subtle on \(\mathscr {Q}[2]\) when \(\mathscr {Q}[2]\) is identified with the space of septuples of points in general position in \(\mathbb {P}^{2}\). On the other hand, the action of the symmetric group \(S_7\) on seven elements is very clear and we will therefore restrict our attention to this subgroup. The main results are presented in Tables 2 and 5 where we give the cohomology groups of \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) as representations of \(S_7\). In particular, we obtain the following.
Theorem 1.1
Theorem 1.2
The present paper may to a large extent be seen as a level 2 analogue of Looijenga [25] and many results, for instance Propositions 1.1 and 1.2, have counterparts in [25] and [19]. Our paper also builds heavily on the work of Dolgachev and Ortland [14], especially the description of \(\mathscr {Q}[2]\) in terms of configurations of points in the projective plane, and van Geemen, whose results are unpublished by himself but can be found in [14]. It may also be of interest to compare the present paper to the work of Bergström [3, 4], and Bergström and Tommasi [5] which also investigate cohomological questions about moduli spaces of low genus curves via point counts. Although there are many similarities, a key difference between the works mentioned above and the present paper is that in previous works the most refined information is obtained for compactifications of the various moduli spaces under investigation, and compactness is used in an essential way, whereas in the present paper we obtain the strongest information for “open” moduli spaces. There are also several other works which answer representation theoretic questions about the cohomology groups of various spaces, e.g. [10, 13, 22, 23]. It should also be mentioned that our results are an essential ingredient in the article [7] where further information about the action of Open image in new window on the cohomology \(\mathscr {M}_{3}[2]\) and \(\mathscr {Q}[2]\) is obtained via quite different methods.
2 Symplectic level structures
Let \(K\) be an algebraically closed field of characteristic not equal to 2 and let C be a smooth and irreducible curve of genus g over \(K\). The 2torsion part Open image in new window of the Jacobian of C is isomorphic to \(( \mathbb {Z}/2\mathbb {Z})^{2g}\) as an abelian group and the Weil pairing is a nondegenerate and alternating bilinear form on Open image in new window .
Definition 2.1
For more information about the Weil pairing and level structures, see e.g. [2] or [20]. Since we shall only consider symplectic level structures we shall refer to symplectic level structures simply as level structures.
A tuple \((C,D_1,\ldots , D_{2g})\) where C is a smooth irreducible curve and \((D_1, \ldots , D_{2g})\) is a level 2 structure on C is called a curve with level 2 structure. Let \((C',D'_1, \ldots ,D'_{2g})\) be another curve with level 2 structure. An isomorphism of curves with level 2 structures is an isomorphism of curves \(\phi :C \rightarrow C'\) such that \(\phi ^*(D'_i)=D_i\) for \(i=1, \ldots , 2g\). We denote the moduli space of genus g curves with level 2 structure by \(\mathscr {M}_{g}[2]\). We remark that we shall consider these moduli spaces as coarse spaces and not as stacks. The group Open image in new window acts on \(\mathscr {M}_{g}[2]\) by changing level structures.
A concept closely related to level 2 structures is that of theta characteristics.
Definition 2.2
Let C be a smooth and irreducible curve and let \(K_C\) be its canonical class. An element Open image in new window such that \(2 \theta = K_C\) is called a theta characteristic. We denote the set of theta characteristics of C by \(\Theta (C)\).
Let C be a curve of genus g. Given two theta characteristics \(\theta _1\) and \(\theta _2\) on C, we obtain an element Open image in new window by taking the difference \(\theta _1 \theta _2\). Conversely, given a theta characteristic \(\theta \) and a 2torsion element D, we obtain a new theta characteristic as \(\theta '=\theta +D\). More precisely we have that \(\Theta (C)\) is a Open image in new window torsor and the set Open image in new window is a vector space of dimension \(2g+1\) over the field \(\mathbb {Z}/2\mathbb {Z}\) of two elements.
Definition 2.3
Proposition 2.4
Let C be a smooth and irreducible curve. There is a bijection between the set of ordered Aronhold bases on C and the set of level 2 structures on C.
For a proof of Proposition 2.4 as well as a more thorough treatment of theta characteristics and Aronhold bases we refer to [21] and [26, Section 3.3].
Proposition 2.4 provides a more geometric way to think about level 2 structures. In the case of a plane quartic curve, which shall be the case of most importance to us, we point out that each theta characteristic occurring in an Aronhold basis is cut out by a bitangent line. Thus, in the case of plane quartics one can think of ordered Aronhold bases as ordered sets of bitangents (although not every ordered set of bitangents constitute an ordered Aronhold basis).
3 Plane quartics
Let \(K\) be an algebraically closed field of characteristic not equal to 2 and let C be a smooth and irreducible curve of genus g over \(K\). If C is not hyperelliptic it is embedded into \(\mathbb {P}^{\, g1}\) via its canonical linear system. Thus, a nonhyperelliptic curve of genus 3 is embedded into \(\mathbb {P}^{2}\) and by the genusdegree formula we see that the degree of the image is 4. We therefore refer to the corresponding locus in \(\mathscr {M}_{3}\), the moduli space of curves of genus 3, as the quartic locus and denote it by \(\mathscr {Q}\). It is the complement of the hyperelliptic locus, \(\mathscr {H}_{3}\). Similarly, we denote the complement of the hyperelliptic locus in \(\mathscr {M}_{3}[2]\) by \(\mathscr {Q}[2]\). Clearly, the action of Open image in new window on \(\mathscr {M}_{3}[2]\) restricts to an action on \(\mathscr {Q}[2]\).
The purpose of this section is to give an explicit, combinatorial description of \(\mathscr {Q}[2]\). This description will be in terms of points in general position. Intuitively, a set of n points in the projective plane is in general position if there is no “unexpected” curve passing through all of them. In our case, this is made precise by the following definition.
Definition 3.1
Let \((P_1, \ldots , P_7)\) be a septuple of points in \(\mathbb {P}^{2}\). We say that the septuple is in general position if there is no line passing through any three of the points and no conic passing through any six of them. We denote the moduli space of septuples of points in general position up to projective equivalence by \(\mathscr {P}^2_7\).
The following result can be found in [14] where it is attributed to van Geemen.
Theorem 3.2
Remark 3.3
Even though [14] only considers the case \(K= \mathbb {C}\), the proof only relies on the theory of geometrically marked Del Pezzo surfaces of degree 2. This theory is the same over any algebraically closed field of characteristic different from 2, see [24, Section III.3].
It should be pointed out that while the action of Open image in new window is clear on \(\mathscr {Q}[2]\) its action on \(\mathscr {P}^2_7\) is much more subtle. However, we can at least plainly see the symmetric group Open image in new window acting on \(\mathscr {P}^2_7\) by permuting points.
Remark 3.4
4 Lefschetz trace formula
We are interested in the spaces \(\mathscr {M}_{3}[2]\), \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) and in particular we want to know their cohomology. The Lefschetz trace formula provides a way to obtain cohomological information about a space via point counts over finite fields.
Let p be a prime number, let \(n\geqslant 1\) be an integer and let \(q=p^n\). Also, let \(\mathbb {F}_{q}\) denote a finite field with q elements, let \(\mathbb {F}_{q^m}\) denote a degree m extension of \(\mathbb {F}_{q}\) and let \(\overline{\mathbb {F}}_{q}\) denote an algebraic closure of \(\mathbb {F}_{q}\). Let X be a scheme defined over \(\mathbb {F}_{q}\) and let \(F\) denote its geometric Frobenius endomorphism. Finally, let l be a prime number different from p and let Open image in new window denote the kth compactly supported étale cohomology group of X with coefficients in \(\mathbb {Q}_l\).
Let \(\Gamma \) be a finite group acting on X by rational automorphisms (i.e. automorphisms over \(\mathbb {F}_{q}\)). Then each cohomology group Open image in new window is a \(\Gamma \)representation. The Lefschetz trace formula allows us to obtain information about these representations by counting the number of fixed points of \(F\sigma \) for different \(\sigma \in \Gamma \).
Theorem 4.1
For a proof, see [11, Rapport – Théorème 3.2].
Remark 4.2
This theorem is usually only stated in terms of \(F\). To get the above version one simply applies the “usual” theorem to the twist of X by \(\sigma \), i.e. by descending from \(\overline{X}\) via \(F\sigma \) instead of F. For more details, see e.g. [12, Section 3].
Remark 4.3
If \(\Gamma \) is a finite group acting on X by rational automorphisms and \(\sigma \in \Gamma \), then \(\overline{X}{}^{F\sigma }\) will only depend on the conjugacy class of \(\sigma \) in \(\Gamma \).
Definition 4.4
Let X be a separated scheme of finite type over \(\mathbb {F}_{q}\) with Frobenius endomorphism \(F\) and let \(\Gamma \) be a finite group acting on X by rational automorphisms. The determination of \(X^{F\sigma }\) for all \(\sigma \in \Gamma \) is then called a \(\Gamma \)equivariant point count of X over \(\mathbb {F}_{q}\).
5 Minimal purity
Definition 5.1
(Dimca and Lehrer [13]) Let X be an irreducible and separated scheme of finite type over \(\overline{\mathbb {F}}_{q}\) with Frobenius endomorphism \(F\) and let l be a prime not dividing q. The scheme X is called minimally pure if \(F\) acts on Open image in new window by multiplication by Open image in new window .
A pure dimensional and separated scheme X of finite type over \(\overline{\mathbb {F}}_{q}\) is minimally pure if for any collection \(\{X_1, \ldots , X_r\}\) of irreducible components of X, the irreducible scheme Open image in new window is minimally pure.
Examples of minimally pure varieties include complements of arrangements of hyperplanes and hypertori, toric varieties and quotients of reductive groups by maximal tori. For more examples, see [13, 22, 23].
5.1 Minimal purity of \(\mathscr {Q}[2]\)
Lemma 5.2
(Looijenga [25, Proposition 1.18, Lemma 3.6]) Open image in new window is minimally pure.
Proposition 5.3
\(\mathscr {Q}[2]\) is minimally pure.
Proof
Since \(\mathscr {Q}[2]\) is isomorphic to \(\mathscr {P}^2_7\), we may compute the cohomology of \(\mathscr {Q}[2]\) as a representation of \(S_7\) by making \(S_7\)equivariant point counts of \(\mathscr {P}^2_7\).
6 Equivariant point counts
In this section we shall perform an \(S_7\)equivariant point count of \(\mathscr {P}^2_7\). This amounts to the computation of \(( \mathscr {P}^2_7)^{F\sigma } \) for one representative \(\sigma \) of each of the fifteen conjugacy classes of \(S_7\). The computations will be rather different in the various cases but at least the underlying idea will be the same. Throughout this section we shall work over a finite field \(\mathbb {F}_{q}\) where q is odd.

the computation of \(\Delta _\mathsf{l}^{F\sigma }\),

the computation of \(\Delta _\mathsf{c}^{F\sigma }\),

the computation of Open image in new window .
Lemma 6.1
Let \(C \subset \mathbb {P}^{2}\) be a smooth conic over a field k and let \(P \in \mathbb {P}^{2}\) be a point such that n tangent lines of C pass through P. Then \(n \leqslant 2\) or Open image in new window .
Proof
The line \(P^{\vee }\) in the dual projective plane intersects the dual conic \(C^{\vee }\) in n points. The dual conic \(C^{\vee }\) is smooth if the characteristic of k is not 2 and we conclude that n can be at most 2. \(\square \)
One can, of course, also see this via a direct computation.
Lemma 6.2
Let \(C \in \mathbb {P}^2\) be a smooth, \(\mathbb {F}_{q}\)rational conic and let P be an \(\mathbb {F}_{q}\)rational point lying on precisely one tangent line L to C. Then P is a point on C.
Proof
We first observe that L must be \(\mathbb {F}_{q}\)rational since otherwise L and \(FL\) would be two distinct tangent lines passing through P. Let \(Q \in C\) be the point of tangency of L and assume \(Q \ne P\). If \(L'\) is another tangent to C, then \(L'\) cannot pass through Q since if that was the case the quadratic curve \(L \cup L'\) would intersect C with multiplicity at least 5, contradicting Bezout’s theorem.
Now consider the set S of \(\mathbb {F}_{q}\)rational tangents of C different from L. We have \(S=q\). By the above observation we have that none of the elements passes through Q and, by assumption, none of them passes through P. Since the number of \(\mathbb {F}_{q}\)rational points of L different from P and Q is \(q1\), the pigeon hole principle gives that there must be a point R on L such that two of the elements of S pass through R. But now R is a point with three tangents of C passing through it which is impossible by Lemma 6.1. We conclude that \(P=Q\) and that P is a point on C. \(\square \)
The above results justify the following definition, which will be useful in the analysis of Open image in new window . See also Fig. 1 for motivation of the terminology.
Definition 6.3

P is on the \(\mathbb {F}_{q}\)inside of C if there is no \(\mathbb {F}_{q}\)tangent to C passing through P,

P is on C if there is precisely one \(\mathbb {F}_{q}\)tangent to C passing through P,

P is on the \(\mathbb {F}_{q}\)outside of C if there are two \(\mathbb {F}_{q}\)tangents to C passing through P.
Definition 6.4
Let X be an \(\mathbb {F}_{q}\)scheme with Frobenius endomorphism \(F\) and let \(Z \subset X_{\overline{\mathbb {F}}_{q}}\) be a subscheme. We say that Z is a strict\(\mathbb {F}_{q^m}\)subscheme if Z is an \(\mathbb {F}_{q^m}\)subscheme which is not defined over \(\mathbb {F}_{q^n}\) for any \(n<m\).
If Z is a strict \(\mathbb {F}_{q^m}\)subscheme, the mtuple \((Z, \ldots , F^{m1}Z)\) is called a conjugatemtuple. Let r be a positive integer and let Open image in new window be a partition of r. An rtuple \((Z_1, \ldots , Z_r)\) of closed subschemes of X is called a conjugate\(\lambda \)tuple if it consists of \(\lambda _1\) conjugate 1tuples, \(\lambda _2\)conjugate 2tuples and so on. We denote the set of conjugate \(\lambda \)tuples of \(\overline{\mathbb {F}_{q}}\)points of X by \(X(\lambda )\).
We shall sometimes drop the adjective “conjugate” and simply write “\(\lambda \)tuple”. Since the conjugacy class of an element in \(S_7\) is given by its cycle type, we want to count the number of conjugate \(\lambda \)tuples in both U and \(\Delta \) for each partition of 7.
We are now ready for the task of counting the number of conjugate \(\lambda \)tuples for each element of \(S_7\).
Remark 6.5
Since \(\mathscr {P}^2_7\) is minimally pure, equation (6.1) gives that \(( \mathscr {P}^2_7)^{F\sigma } \) is a monic polynomial in q of degree six so it is in fact enough to make counts for six different finite fields and interpolate. This is however hard to carry out in practice, even with a computer, as soon as \(\lambda \) contains parts of large enough size (where “large enough” means 3 or 4). However, one can always obtain partial information which provides important checks for our computations and for partitions entirely with parts at most 2 we have been able to obtain the entire polynomials also via computer counts. This fact might help to convince the reader of the validity of our results since these cases are by far the hardest to do by hand.
6.1 The case \(\lambda = [7]\)
Lemma 6.6
If \((P_1, \ldots , P_7)\) is a \(\lambda \)tuple with three of its points on a line, then all seven points lie on a line defined over \(\mathbb {F}_{q}\).
Proof
 (i)
if S is of the form \(S=\{P_i,P_{i+1},P_{i+2}\}\), then Open image in new window ,
 (ii)
if S is of the form \(S=\{P_i,P_{i+2},P_{i+4}\}\), then Open image in new window ,
 (iii)
if S is of the form \(S=\{P_i,P_{i+1},P_{i+4}\}\), then Open image in new window .
Since \(P_1,P_2\) and \(P_4\) lie on a line we have \(P_4 \in L_{1,2}\). We have Open image in new window and Open image in new window so Open image in new window . But since \(P_2 \in L_{1,2}\) we must have Open image in new window . We thus have \(P_4 \in L_{1,2}\) and \(P_4 \in L_{3,6}\) so \(P_4=Q_1\). By analogous arguments one shows that \(P_5 =Q_3\) and \(P_7 =Q_2\). But we now have that \(\{Q_1,Q_2,Q_3\}=\{P_4,P_5,P_7\}=F^3 S\) so the points \(Q_1,Q_2\) and \(Q_3\) lie on the line \(F^3 L\). This contradiction establishes the claim. \(\square \)
Lemma 6.7
If \((P_1, \ldots , P_7)\) is a \(\lambda \)tuple with six of its points on a smooth conic, then all seven points lie on a smooth conic defined over \(\mathbb {F}_{q}\).
Proof
Suppose that the set \(S=\{P_{i_1}, \ldots , P_{i_6}\}\) lies on a smooth conic C. We have Open image in new window and since a conic is defined by any five points on it we have \(FC=C\). Hence, we have that C is defined over \(\mathbb {F}_{q}\) and that all seven points lie on C. \(\square \)
6.2 The case \(\lambda = [1,6]\)
Lemma 6.8
 (1)
the first six points of the \(\lambda \)tuple lie on an \(\mathbb {F}_{q}\)line or,
 (2)
the first six points lie on two conjugate \(\mathbb {F}_{q^2}\)lines, the \(\mathbb {F}_{q^2}\)lines contain three \(\mathbb {F}_{q^6}\)points each and these triples are interchanged by \(F\), or,
 (3)
the first six points lie pairwise on three conjugate \(\mathbb {F}_{q^3}\)lines which intersect in \(P_7\).
Proof
Suppose that Open image in new window lie on a line L. Then L is either defined over \(\mathbb {F}_{q},\mathbb {F}_{q^2},\mathbb {F}_{q^3}\) or \(\mathbb {F}_{q^6}\). One easily checks that for each of the \(\left( {\begin{array}{c}7\\ 3\end{array}}\right) =35\) possible choices of S there is an integer \(1 \leqslant r \leqslant 3\) such that Open image in new window so L is defined over \(\mathbb {F}_{q},\mathbb {F}_{q^2}\) or \(\mathbb {F}_{q^3}\), i.e. we are in one of the three cases above. \(\square \)
Let \(\Delta _{\mathsf{l},i}\) be the subset of \(\Delta _\mathsf{l}\) corresponding to case (i) in Lemma 6.8. The set \(\Delta _{\mathsf{l},1}^{F\sigma }\) is clearly disjoint from \(\Delta _{\mathsf{l},2}^{F\sigma }\) and \(\Delta _{\mathsf{l},3}^{F\sigma }\).
Lemma 6.9
If six of the points of a \(\lambda \)tuple \((P_1, \ldots , P_7)\) lie on a smooth conic, then \(P_1, \ldots , P_6\) lie on the conic and the conic is defined over \(\mathbb {F}_{q}\).
Proof
Suppose \(S=\{P_{i_1}, \ldots , P_{i_6}\}\) lie on a smooth conic C. Then \(FS \cap S \geqslant 5\) so \(FC=C\). Let \(P \in S\) be an \(\mathbb {F}_{q^6}\)point. Then we have Open image in new window . \(\square \)
6.3 The case \(\lambda =[2,5]\)
Lemma 6.10
If \((P_1, \ldots , P_7)\) is a \(\lambda \)tuple with three of its points on a line, then all five \(\mathbb {F}_{q^5}\)points lie on a line defined over \(\mathbb {F}_{q}\). If six of the points lie on a smooth conic C, then all seven points lie on C and C is defined over \(\mathbb {F}_{q}\).
Proof
The proof is very similar to the proofs of Lemmas 6.6 and 6.7 and is therefore omitted. \(\square \)
6.4 The case Open image in new window
6.5 The case Open image in new window
Lemma 6.11
 (1)
the four \(\mathbb {F}_{q^4}\)points lie on an \(\mathbb {F}_{q}\)line, or
 (2)
the three \(\mathbb {F}_{q^3}\)points lie on an \(\mathbb {F}_{q}\)line.
Proof
It is easy to see that if three \(\mathbb {F}_{q^4}\)points lie on a line, then all four \(\mathbb {F}_{q^4}\)points lie on that line and even easier to see the corresponding result for three \(\mathbb {F}_{q^3}\)points.
Suppose that two \(\mathbb {F}_{q^4}\)points \(P_i\) and Open image in new window and an \(\mathbb {F}_{q^3}\)point P lie on a line L. Since \(F^4P_i=P_i\) and Open image in new window we see that \(F^4L=L\). Thus, \(F^4P=FP \ne P\) lies on L. Repeating this argument again, with \(FP\) in the place of P, shows that also \(F^2P\) lies on L. We are thus in case (2).
If we assume that two \(\mathbb {F}_{q^3}\)points and an \(\mathbb {F}_{q^4}\)point lie on a line, then a completely analogous argument shows that all four \(\mathbb {F}_{q^4}\)points lie on that line. \(\square \)
6.6 The case \(\lambda =[1,2,4]\)

\(\Delta _{\mathsf{l},1}\) consists of \(\lambda \)tuples with three strict \(\mathbb {F}_{q^4}\)points lying on a line,

\(\Delta _{\mathsf{l},2}\) consists of \(\lambda \)tuples with two strict \(\mathbb {F}_{q^4}\)points and a strict \(\mathbb {F}_{q^2}\)point lying on a line,

\(\Delta _{\mathsf{l},3}\) consists of \(\lambda \)tuples with two strict \(\mathbb {F}_{q^4}\)points and the \(\mathbb {F}_{q}\)point lying on a line,

\(\Delta _{\mathsf{l},4}\) consists of \(\lambda \)tuples with a strict \(\mathbb {F}_{q^4}\)point and two strict \(\mathbb {F}_{q^2}\)points lying on a line,

\(\Delta _{\mathsf{l},5}\) consists of \(\lambda \)tuples with a strict \(\mathbb {F}_{q^4}\)point, a strict \(\mathbb {F}_{q^2}\)point and an \(\mathbb {F}_{q}\)point lying on a line, and,

\(\Delta _{\mathsf{l},6}\) consists of \(\lambda \)tuples with the two strict \(\mathbb {F}_{q^2}\)points and the \(\mathbb {F}_{q}\)point lying on a line.
The two slightly more complicated sets in the above list are \(\Delta _{\mathsf{l},2}\) and \(\Delta _{\mathsf{l},3}\). We shall therefore comment a bit about the computations involving them.

\(\Delta _{\mathsf{l},2}^1\) consists of \(\lambda \)tuples such that the four strict \(\mathbb {F}_{q^4}\)points and the two strict \(\mathbb {F}_{q^2}\)points lie on an \(\mathbb {F}_{q}\)line, or,

\(\Delta _{\mathsf{l},2}^2\) consists of \(\lambda \)tuples such that the two strict \(\mathbb {F}_{q^4}\)points and the strict \(\mathbb {F}_{q^2}\)point lie on a strict \(\mathbb {F}_{q^2}\)line L (and the other two strict \(\mathbb {F}_{q^4}\)points and the second strict \(\mathbb {F}_{q^2}\)point lie on \(FL\)), or,

\(\Delta _{\mathsf{l},2}^3\) consists of \(\lambda \)tuples such that the four strict \(\mathbb {F}_{q^4}\)points and the two strict \(\mathbb {F}_{q^2}\)points are intersection points of four conjugate \(\mathbb {F}_{q^4}\)lines.

\(\Delta _{\mathsf{l},3}^1\) consists of \(\lambda \)tuples such that the four strict \(\mathbb {F}_{q^4}\)points and the \(\mathbb {F}_{q}\)point lie on an \(\mathbb {F}_{q}\)line, and,

\(\Delta _{\mathsf{l},3}^2\) consists of \(\lambda \)tuples such that there are two conjugate \(\mathbb {F}_{q^2}\)lines intersecting in the \(\mathbb {F}_{q}\)point, each \(\mathbb {F}_{q^2}\)line containing two of the strict \(\mathbb {F}_{q^4}\)points.
We now want to choose an \(\mathbb {F}_{q}\)line through \(P_7\) intersecting C in two \(\mathbb {F}_{q^2}\)points. There are \(q+1\)\(\mathbb {F}_{q}\)lines through \(P_7\) of which two are tangent to C. These tangent lines contain an \(\mathbb {F}_{q}\)point of C each so there are \(q1\) remaining \(\mathbb {F}_{q}\)points on C. Picking such a point gives a line through this point, \(P_7\) and one further point on C. We thus see that exactly Open image in new window of the \(\mathbb {F}_{q}\)lines through \(P_7\) intersect C in two conjugate \(\mathbb {F}_{q^2}\)points. We label one of them as \(P_5\).
Since \(P_7\) now lies on the \(\mathbb {F}_{q}\)inside of C, every \(\mathbb {F}_{q}\)line through \(P_7\) will intersect C in two points. Exactly Open image in new window will intersect C in two \(\mathbb {F}_{q}\)points so the remaining Open image in new window will intersect C in two conjugate \(\mathbb {F}_{q^2}\)points. We pick such a pair of points and label one of them \(P_5\).
6.7 The case Open image in new window
 (i)
the point P may lie on the \(\mathbb {F}_{q}\)outside of C with one of the tangents through P also passing through \(P_i\),
 (ii)
the point P may lie on the \(\mathbb {F}_{q}\)outside of C with none of the tangents through P passing through \(P_i\),
 (iii)
the point P may lie on the \(\mathbb {F}_{q}\)inside of C.
6.8 The case \(\lambda =[1,3^2]\)

\(\Delta _{\mathsf{l},1}\) consists of \(\lambda \)tuples such that the points \(P_1,P_2\) and \(P_3\) lie on an \(\mathbb {F}_{q}\)line,

\(\Delta _{\mathsf{l},2}\) consists of \(\lambda \)tuples such that the points \(P_1,P_2\) and \(P_3\) are the intersection points of a conjugate triple of \(\mathbb {F}_{q^3}\)lines with each of the lines containing one of the points \(Q_1,Q_2\) and \(Q_3\),

\(\Delta _{\mathsf{l},3}\) consists of \(\lambda \)tuples such that the points \(Q_1,Q_2\) and \(Q_3\) are the intersection points of a conjugate triple of \(\mathbb {F}_{q^3}\)lines with each of the lines containing one of the points \(P_1,P_2\) and \(P_3\),

\(\Delta _{\mathsf{l},4}\) consists of \(\lambda \)tuples such that the points \(Q_1,Q_2\) and \(Q_3\) lie on an \(\mathbb {F}_{q}\)line, and

\(\Delta _{\mathsf{l},5}\) consists of \(\lambda \)tuples such that the point R is the intersection of three conjugate \(\mathbb {F}_{q^3}\)lines with each of the \(\mathbb {F}_{q^3}\)lines containing one of the points \(P_1,P_2\) and \(P_3\) and one of the points \(Q_1,Q_2\) and \(Q_3\) (Fig. 4).
We investigate each of the cases separately.
6.8.1 \(\Delta _{{l},1}\) and \(\Delta _{{l},4}\)
6.8.2 \(\Delta _{{l},2}\) and \(\Delta _{{l},3}\)
6.8.3 \(\Delta _{{l},5}\)
6.9 The case Open image in new window
6.10 The case Open image in new window
6.11 The case Open image in new window
6.12 The case \(\lambda =[1,2^3]\)

the set \(\Delta _{\mathsf{l},1}^a\) where \(P_1,P_2\) and O lie on an \(\mathbb {F}_{q}\)line,

the set \(\Delta _{\mathsf{l},1}^b\) where \(Q_1,Q_2\) and O lie on an \(\mathbb {F}_{q}\)line, and

the set \(\Delta _{\mathsf{l},1}^c\) where \(R_1,R_2\) and O lie on an \(\mathbb {F}_{q}\)line.

the two sets \(\Delta _{\mathsf{l},2}^{P_1,Q_i}\) where the line through the points \(P_1\) and \(Q_i\) also passes through the point O,

the two sets \(\Delta _{\mathsf{l},2}^{P_1,R_i}\) where the line through the points \(P_1\) and \(R_i\) also passes through the point O, and

the two sets \(\Delta _{\mathsf{l},2}^{Q_1,R_i}\) where the line through the points \(Q_1\) and \(R_i\) also passes through the point O.
We first consider the case when O is on the \(\mathbb {F}_{q}\)outside of C. There are \(q+1\) lines through O. Of these, precisely two are tangents and Open image in new window intersect C in \(\mathbb {F}_{q}\)points. Thus, the remaining Open image in new window lines will intersect C in two conjugate \(\mathbb {F}_{q^2}\)points. We thus pick one of these lines and label one of the intersection points by \(P_1\).
6.13 The case Open image in new window

\(\Delta _{\mathsf{l},1}\) consists of tuples such that the line through \(R_1\) and \(R_2\) also passes through \(P_1,P_2\) or \(P_3\),

\(\Delta _{\mathsf{l},2}\) consists of tuples such that the points \(R_1\) and \(R_2\) lie on the line through \(Q_1\) and \(Q_2\), and

\(\Delta _{\mathsf{l},3}\) consists of tuples such that a line through \(Q_1\) and one of the points \(P_1,P_2\) and \(P_3\) also contains \(R_1\) or \(R_2\).
Since \(P_1,P_2\) and \(P_3\) do not lie on a line we have that the intersection of \(\Delta _{\mathsf{l},1}^1,\Delta _{\mathsf{l},1}^2\) and \(\Delta _{\mathsf{l},1}^3\) is empty. We thus only have two types of triple intersections, namely Open image in new window and Open image in new window where, of course, i, j and k are assumed to be distinct.
6.14 The case Open image in new window

the set \(A_{i,0}^{\mathrm {out}}\) consisting of \(\lambda \)tuples such that the tangent lines to C passing through \(P_i\) do not pass through any of the other points of the \(\lambda \)tuple,

the set \(A_{i,1}^{\mathrm {out}}\) consisting of \(\lambda \)tuples such that exactly one of the tangent lines to C passing through \(P_i\) pass through one of the other points of the \(\lambda \)tuple,

the set \(A_{i,2}^{\mathrm {out}}\) consisting of \(\lambda \)tuples such that both the tangent lines to C passing through \(P_i\) passes through another point of the \(\lambda \)tuple.
6.15 The case \(\lambda =[1^7]\)
Definition 6.12
If P and Q are two points in \(\mathbb {P}^2\), then the line through P and Q shall be denoted \(P Q\).
6.15.1 The set \(\Delta _{{l}}\)

the points of \(\Delta _{\mathsf{l},1}\) are such that at least one of the points \(P_5,P_6\) or \(P_7\) lies in \(\mathscr {S}\),

the points of \(\Delta _{\mathsf{l},2}\) are such that one of the lines Open image in new window , \(5 \leqslant i < j \leqslant 7\), contains one of the points \(P_1,P_2,P_3\) and \(P_4\), but \(\{P_5,P_6,P_7\} \cap \mathscr {S} = \varnothing \), and

the points of \(\Delta _{\mathsf{l},3}\) are such that the three points \(P_5,P_6\) and \(P_7\) lie on a line which does not pass through \(P_1,P_2,P_3\) or \(P_4\).

the line \(P_5 P_6\) passes through \(P_r\), \(P_5 P_7\) passes through \(P_s\) and \(P_6 P_7\) passes through \(P_t\) but,

we allow \(P_5,P_6\) and \(P_7\) to lie in \(\mathscr {S}\), but,

we do not allow the lines Open image in new window , \(5 \leqslant i < j \leqslant 7\) to be contained in \(\mathscr {S}\).

\(\Delta _{\mathsf{l},3}(\{Q_r,Q_s\})\) consisting of those tuples of \(\Delta _{\mathsf{l},3}\) where \(P_5,P_6\) and \(P_7\) lie on the line \(Q_r Q_s\), \(1 \leqslant r < s \leqslant 3\), and,

\(\Delta _{\mathsf{l},3}(\{Q_r\})\) consisting of those tuples of \(\Delta _{\mathsf{l},3}\) with \(P_5,P_6\) and \(P_7\) on a line through \(Q_r\), \(1 \leqslant r \leqslant 3\), which does not pass through any of the other \(Q_i\), and

\(\Delta _{\mathsf{l},3}(\varnothing )\) consisting of those tuples of \(\Delta _{\mathsf{l},3}\) with \(P_5,P_6\) and \(P_7\) on a line which does not pass through \(Q_1,Q_2\) or \(Q_3\).
6.15.2 The set \(\Delta _{c}\)
6.15.3 The set \(\Delta _{l} \cap \Delta _{c}\)

the set \(\mathscr {F}_1\) consists of tuples such that at least one line contains three points of the tuple,

the set \(\mathscr {F}_2\) consists of tuples such that at least two lines contain three points of the tuple,

the set \(\mathscr {F}_3\) consists of tuples such that at least three lines contain three points of the tuple.
Since the points \(P_1,P_2,P_3\) and \(P_4\) are assumed to constitute a frame, we must do things a little bit differently depending on whether the point not on the conic is one of these four or not. We therefore make further subdivisions.
Here, we only have the possibility that P lies on three lines \(R_1 O_1,R_2 O_2\) and \(R_3 O_3\) where \(\{R_1,R_2,R_3\} = \{P_5,P_6,P_7\}\) and \(\{O_1,O_2,O_3\}=\{P_r,P_s,P_t\}\). However, we must take care of the case that P is on the outside of C and the case that P is on the inside of C separately. We call the corresponding numbers \(N_{3,\mathrm {out}}^{1,2,3,4}\) and \(N_{3,\mathrm {in}}^{1,2,3,4}\).
7 The hyperelliptic locus
Up to this point we have almost exclusively discussed plane quartics. We shall now briefly turn our attention to the other type of genus 3 curves — the hyperelliptic curves. There are many possible ways to approach the computation of the cohomology of \(\mathscr {H}_{3}[2]\). Our choice is by means of equivariant point counts as in the previous section.
Recall that a hyperelliptic curve C of genus g is determined, up to isomorphism, by \(2g+2\) distinct points on \(\mathbb {P}^1\), up to projective equivalence and that any such collection S of \(2g+2\) points determines a double cover \(\pi :C \rightarrow \mathbb {P}^{1}\) branched precisely over S (and C is thus a hyperelliptic curve). Moreover, if we pick \(2g+2\) ordered points \(P_1, \ldots , P_{2g+2}\) on \(\mathbb {P}^{1}\), the curve C also attains a level 2 structure. In the genus 3 case, we get eight points \(Q_i=\pi ^{1}(P_i)\) which determine \(\left( {\begin{array}{c}8\\ 2\end{array}}\right) =28\) odd theta characteristics Open image in new window , \(i <j\) and \(\{Q_1+Q_8, \ldots ,Q_7+Q_8\}\) is an ordered Aronhold basis, see [2, Appendix B.32–33] and [21], and an ordered Aronhold basis determines a level 2 structure.
However, not all level 2 structures on the hyperelliptic curve C arise from different orderings of the points. Nevertheless, there is an intimate relationship between the moduli space \(\mathscr {H}_{g}[2]\) of hyperelliptic curves with level 2 structure and the moduli space \(\mathscr {M}_{0,2g+2}\) of \(2g+2\) ordered points on \(\mathbb {P}^{1}\) given by the following theorem which can be found in [14, Theorem VIII.1].
Theorem 7.1
Each irreducible component of \(\mathscr {H}_{g}[2]\) is isomorphic to the moduli space \(\mathscr {M}_{0,2g+2}\) of \(2g+2\) ordered points on the projective line.
Dolgachev and Ortland [14] pose the question whether the irreducible components of \(\mathscr {H}_{g}[2]\) also are the connected components or, in other words, if \(\mathscr {H}_{g}[2]\) is smooth. In the complex case, the question was answered positively by Tsuyumine in [29] and later, by a shorter argument, by Runge in [28]. Using the results of [1], the argument of Runge carries over word for word to an algebraically closed field of positive characteristic different from 2.
Theorem 7.2
If \(g \geqslant 2\), then each irreducible component of \(\mathscr {H}_{g}[2]\) is also a connected component.
We have a natural action of \(S_{2g+2}\) on the space \(\mathscr {M}_{0,2g+2}\). Since different orderings of the points correspond to different symplectic level 2 structures, \(S_{2g+2}\) sits naturally inside Open image in new window and, in fact, for \(g=3\) and for even g it is a maximal subgroup, see [15]. With Theorems 7.1 and 7.2 at hand, the following slight generalization of a corollary in [14, p. 145] is clear.
Corollary 7.3
Remark 7.4
As pointed out in [28], the argument to prove the corollary stated in [14] is not quite correct in full generality as it is given there. However, it is enough to prove the result for \(g=3\) and for even g, and in [28] it is explained how to obtain the full result.
Compared to Q[2], the \(S_8\)equivariant point count of \(\mathscr {H}_{3}[2]\) is very easy. We first compute the number of \(\lambda \)tuples of \(\mathbb {P}^1\) for each partition of \(\lambda \) of 8 and then divide by Open image in new window in order to obtain \(\mathscr {M}_{0,8}^{F\sigma }\), where \(\sigma \) is a permutation in \(S_8\) of cycle type \(\lambda \). The result is given in Table 3. Once this is done, we induce up to Open image in new window in order to obtain the Open image in new window equivariant cohomology of \(\mathscr {H}_{3}[2]\). The results are given in Table 4. Finally, we restrict to \(S_7\) to get the results of Tables 5 and 6. The computations present no difficulties whatsoever.
The \(S_7\)equivariant point count of \(\mathscr {Q}[2]\)
\(\lambda \)  \(\mathscr {Q}[2]^{F \cdot \sigma _{\lambda }}\) 

[7]  \(q^6+q^3\) 
[6,1]  \(q^62q^3+1\) 
[5,2]  \(q^6q^2\) 
[\(5,1^2\)]  \(q^6q^2\) 
[4,3]  \(q^6q^52q^4+q^3+q^2\) 
[4,2,1]  \(q^6q^52q^4+q^32q^2+3\) 
[\(4,1^3\)]  \(q^6q^52q^4+q^32q^2+3\) 
\(q^62q^52q^48q^3+16q^2+10q+21\)  
[\(3,2^2\)]  \(q^6q^52q^4+3q^3+q^22q\) 
[\(3,2,1^2\)]  \(q^63q^5+5q^3q^22q\) 
[\(3,1^4\)]  \(q^65q^5+10q^45q^311q^2+10q\) 
\(q^63q^56q^4+19q^3+6q^224q+7\)  
\(q^67q^5+10q^4+15q^326q^28q+15\)  
[\(2,1^5\)]  \(q^615q^5+90q^4265q^3+374q^2200q+15\) 
[\(1^7\)]  \(q^635q^5+490q^43485q^3+13174q^224920q+18375\) 
The cohomology of \(\mathscr {Q}[2]\) as a representation of \(S_7\)
\(s_{7}\)  \(s_{6,1}\)  \(s_{5,2}\)  \(s_{5,1^2}\)  \(s_{4,3}\)  \(s_{4,2,1}\)  \(s_{4,1^3}\)  \(s_{3^2,1}\)  \(s_{3,2^2}\)  \(s_{3,2,1^2}\)  

\(H^0\)  1  0  0  0  0  0  0  0  0  0 
\(H^1\)  1  1  1  0  1  0  0  0  0  0 
\(H^2\)  0  3  4  4  3  5  1  3  1  1 
\(H^3\)  1  8  14  18  14  30  16  16  12  18 
\(H^4\)  4  20  44  47  44  99  56  56  54  83 
\(H^5\)  6  33  76  76  72  178  97  104  105  169 
\(H^6\)  6  23  51  54  54  127  74  76  77  126 
\(s_{3,1^4}\)  \(s_{2^3,1}\)  \(s_{2^2,1^3}\)  \(s_{2,1^5}\)  \(s_{1^7}\)  

\(H^0\)  0  0  0  0  0  
\(H^1\)  0  0  0  0  0  
\(H^2\)  0  0  0  0  0  
\(H^3\)  4  6  3  0  0  
\(H^4\)  32  31  25  6  1  
\(H^5\)  71  65  64  26  3  
\(H^6\)  54  54  50  22  5 
The \(S_8\)equivariant point count of \(\mathscr {M}_{0,8}\)
\(\lambda \)  \(\mathscr {M}_{0,8}^{F \cdot \sigma _{\lambda }}\) 

[8]  
[7,1]  \(( q+1 ) ( {q}^{2}+q+1 ) ( {q}^{2}q+1 )\) 
[6,2]  \(q ( q1 ) ( {q}^{3}+q1 )\) 
\([6,1^2]\)  \(q ( q+1 ) ( {q}^{3}+q1 )\) 
[5,3]  \( q ( q1 ) ( q+1 ) ( {q}^{2}+1 )\) 
[5,2,1]  \(q ( q1 ) ( q+1 ) ( {q}^{2}+1 )\) 
\([5,1^3]\)  \(q ( q1 ) ( q+1 ) ( {q}^{2}+1 )\) 
\([4^2]\)  \(q ( {q}^{4}{q}^{2}4 )\) 
[4,3,1]  
\([4,2^2]\)  
\([4,2,1^2]\)  
\([4,1^4]\)  
\(q ( q1 ) ( {q}^{3}q3 )\)  
\(q ( q+1 ) ( {q}^{3}q3 )\)  
\([3,2^2,1]\)  \(q ( q1 ) ( q2 ) ( q+1 )^{2}\) 
\([3,2,1^3 ]\)  
\([3,1^5]\)  \(q ( q1 ) ( q2 ) ( q3 ) ( q+1 )\) 
\([2^4]\)  \(( q2 ) ( q3 ) ( q+2 ) ( {q}^{2}q4 )\) 
\(q ( q2 ) ( q+1 ) ( {q}^{2}q4 )\)  
\(q ( q1 ) ( q+1 ) ( q2 ) ^{2}\)  
\([2,1^6]\)  \(q ( q1 ) ( q2 ) ( q3 ) ( q4 )\) 
\([1^8]\)  \(( q2 ) ( q3 ) ( q4 ) ( q5 ) ( q6 )\) 
The \(S_8\)equivariant point count of \(\mathscr {H}_3[2]\)
\(\lambda \)  \(\mathscr {H}_3[2]^{F \cdot \sigma _{\lambda }}\) 

[8]  
[7,1]  \({q}^{5}+{q}^{4}+{q}^{3}+{q}^{2}+q+1\) 
[6,2]  
[\(6,1^2\)]  \({q}^{5}+{q}^{4}+{q}^{3}q\) 
[5,3]  \({q}^{5}q\) 
[5,2,1]  \({q}^{5}q\) 
[\(5,1^3\)]  \({q}^{5}q\) 
[\(4^2\)]  
[4,3,1]  
[\(4,2^2\)]  
[\(4,2,1^2\)]  
[\(4,1^4\)]  
[\(3,2^2,1\)]  
[\(3,2,1^3\)]  
[\(3,1^5\)]  
[\(2^4\)]  
[\(2,1^6\)]  
[\(1^8\)] 
The \(S_7\)equivariant point count of \(\mathscr {H}_3[2]\)
\(\lambda \)  \(\mathscr {H}_3[2]^{F \cdot \sigma _{\lambda }}\) 

[7]  \({q}^{5}+{q}^{4}+{q}^{3}+{q}^{2}+q+1\) 
[6,1]  \({q}^{5}+{q}^{4}+{q}^{3}q\) 
[5,2]  \({q}^{5}q\) 
[\(5,1^2\)]  \({q}^{5}q\) 
[4,3]  
[4,2,1]  
[\(4,1^3\)]  
[\(3,2^2\)]  
[\(3,2,1^2\)]  
[\(3,1^4\)]  
[\(2,1^5\)]  
[\(1^7\)] 
The cohomology of \(\mathscr {H}_{3}[2]\) as a representation of \(S_7\)
\(s_{7}\)  \(s_{6,1}\)  \(s_{5,2}\)  \(s_{5,1^2}\)  \(s_{4,3}\)  \(s_{4,2,1}\)  \(s_{4,1^3}\)  \(s_{3^2,1}\)  \(s_{3,2^2}\)  \(s_{3,2,1^2}\)  

\(H^0\)  2  1  1  0  1  0  0  0  0  0 
\(H^1\)  2  7  9  5  5  7  1  3  2  1 
\(H^2\)  3  18  30  31  25  50  20  26  19  26 
\(H^3\)  6  35  74  80  72  162  86  92  83  129 
\(H^4\)  8  48  114  117  109  271  150  157  158  254 
\(H^5\)  5  31  72  77  72  180  103  108  108  180 
\(s_{3,1^4}\)  \(s_{2^3,1}\)  \(s_{2^2,1^3}\)  \(s_{2,1^5}\)  \(s_{1^7}\)  

\(H^0\)  0  0  0  0  0  
\(H^1\)  0  0  0  0  0  
\(H^2\)  5  7  4  0  0  
\(H^3\)  43  45  36  10  1  
\(H^4\)  105  96  92  35  4  
\(H^5\)  77  72  72  31  5 
8 Summary of computations
We summarize the computations related to \(\mathscr {Q}[2]\) in Table 1. Also recall that the Poincaré polynomial of \(\mathscr {Q}[2]\) is given in Theorem 1.1. In Table 2 we give the cohomology of \(\mathscr {Q}[2]\) as a representation of \(S_7\). The rows correspond to the cohomology groups and the columns correspond to the irreducible representations of \(S_7\). The symbol \(s_{\lambda }\) denotes the irreducible representation of \(S_7\) corresponding to the partition \(\lambda \) and a number n in row \(H^k\) and column \(s_{\lambda }\) means that \(s_{\lambda }\) occurs in \(H^k\) with multiplicity n.
Tables 3, 4 and 5 give equivariant point counts for various spaces and groups related to the equivariant point count of \(\mathscr {H}_{3}[2]\) and in Table 6 we give the cohomology groups of \(\mathscr {H}_{3}[2]\) as representations of \(S_7\). Recall that the Poincaré polynomial of \(\mathscr {H}_{3}[2]\) is given in Theorem 1.2.
We mention that since both \(H^k(\mathscr {Q}[2],\mathbb {C})\) and \(H^k(\mathscr {H}_{3}[2],\mathbb {C})\) are of pure Tate type (k, k), see [7], we have that their full mixed Hodge structure can be determined from Theorems 1.1 and 1.2. More precisely, replacing t by \(uv^2\) in the polynomials gives the corresponding Poincaré–Serre polynomials. Here u encodes the cohomological degree and v encodes the Hodge weight.
9 The total moduli space
By adding the results of \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\) we find the \(S_7\)equivariant point count of \(\mathscr {M}_{3}[2]\).
Theorem 9.1
Note in particular that the representations in the coefficients of each power of q occur with the same sign, except in the constant term where \(s_7\) occurs with coefficient 1 while all others occur with negative coefficients. Thus, in contrast with \(\mathscr {Q}[2]\) and \(\mathscr {H}_{3}[2]\), we cannot have that \(H^k(\mathscr {M}_{3}[2],\mathbb {C})\) is pure of Tate type (k, k). Moreover, Looijenga [25] shows that \(H^k(\mathscr {M}_{3},\mathbb {C})\) is pure of Tate type Open image in new window for some i which is sometimes different from k. In our case, neither this weaker purity holds.
To go a bit further in the investigation of the cohomology of \(\mathscr {M}_{3}[2]\), we recall the following result.
Lemma 9.2
Proposition 9.3
The cohomology group \(H^{k} (\mathscr {M}_{3}[2],\mathbb {C})\) consists of one part of Tate type \((k1,k1)\) and one part of Tate type (k, k).
Moreover, let \(m^k_X(\lambda )\) denote the multiplicity of \(s_{\lambda }\) in \(H^{k} (X)\) and let \(n^k(\lambda )= m^k_{\mathscr {Q}[2]}(\lambda )  m^{k2}_{\mathscr {H}_{3}[2]}(\lambda )\). If \(n^k(\lambda )\geqslant 0\), then \(s_{\lambda }\) occurs with multiplicity at least \(n^k(\lambda )\) in \(W_kH^{k} (\mathscr {M}_{3}[2])\) and if \(n^k(\lambda ) \leqslant 0\), then \(s_{\lambda }\) occurs with multiplicity at least Open image in new window in \(W_kH^{k+1} (\mathscr {M}_{3}[2])\). Thus, Tables 2 and 6 provide explicit bounds for the cohomology groups of \(\mathscr {M}_{3}[2]\). We note that a sharper bound for the dimension of \(H^{7} (\mathscr {M}_{3}[2],\mathbb {C})\) than that obtained here can be found in recent work of Fullarton and Putman [16]. However, the observation above gives bounds for all cohomology groups.
Notes
Acknowledgements
This paper is based on parts of my thesis [6] written at Stockholms Universitet. I would like to thank my advisors Carel Faber and Jonas Bergström for their help as well as Dan Petersen for useful comments on an early version of this paper. I also want to thank the anonymous referees for many improvements.
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