1 Introduction
Cohomological invariants of algebraic groups are a classical topic, whose first examples date back to the beginning of the twentieth century [Reference Witt25]. The modern definition was developed by Serre in the late ’90s, and the general theory can be found in Garibaldi, Merkurjev, and Serre’s book [Reference Garibaldi, Merkurjev and Serre12]. From a more geometric point of view, they can be seen as invariants of the classifying stack
${\mathrm{B}}G$
rather than the group G. Following this idea, in [Reference Pirisi21], the second author defined an extension of the notion of cohomological invariants to general algebraic stacks.
Let
$\mathcal {X}$
be an algebraic stack over a field k and
${\mathrm{M}}$
a cycle module over k, as defined in [Reference Rost22]. A cohomological invariant for
$\mathcal {X}$
with coefficients in
${\mathrm{M}}$
is a functorial way to associate to any point
$\operatorname {\mathrm {Spec}}(K) \rightarrow \mathcal {X}$
an element of
${\mathrm{M}}^{\bullet }(K)$
, satisfying a certain continuity condition. Originally, in [Reference Garibaldi, Merkurjev and Serre12], cohomological invariants were defined as taking values in the cycle module
${\mathrm{H}}_D$
, where D is an
$\ell $
-torsion Galois module for some positive number
$\ell $
not divisible by
${\mathrm{char}}(k)$
, and
${\mathrm{H}}_D^{\bullet }(F):=\oplus _i {\mathrm{H}}^i_{\mathrm{Gal}}(F,D(i))$
. The theory for general cycle modules was recently developed by Gille and Hirsch [Reference Gille and Hirsch13], and adapted to the setting of cohomological invariants of algebraic stacks by the authors in [Reference Di Lorenzo and Pirisi9].
Our definition retrieves the classical theory of cohomological invariants of algebraic groups when
$\mathcal {X}={\mathrm{B}}G$
, and if
$\mathcal {X}=X$
is a scheme, its cohomological invariants with coefficients in
${\mathrm{H}}_D$
are equal to its unramified cohomology
${\mathrm{H}}^{\bullet }_{\mathrm{nr}}(X,D)$
. Moreover, in [Reference Di Lorenzo and Pirisi9], the authors proved that the cohomological invariants compute the Brauer group, and used them to compute the Brauer groups of the moduli stacks
$\mathcal {H}_g$
of hyperelliptic curves of genus g.
In [Reference Di Lorenzo and Pirisi8, Appendix A], the authors showed that the cohomological invariants (and consequently, by [Reference Di Lorenzo and Pirisi9, Remark 6.11], the Brauer group) of the standard compactification
$\overline {\mathcal {H}_g}$
of
$\mathcal {H}_g$
are trivial, while claiming this not to occur for the compactification
$\widetilde {\mathcal {H}}_g$
obtained via the theory of admissible double coverings. In this paper, we prove the aforementioned claim.
Theorem (5.7, 5.10)
Let
${\mathrm{M}}$
be an
$\ell $
-torsion cycle module. Then:
-
• If
$\ell $
is odd, we have
$\operatorname {\mathrm {Inv^{\bullet }}}(\widetilde {\mathcal {H}}_g,{\mathrm{M}})={\mathrm{M}}^{\bullet }(k)$
. -
• If
$\ell =2^m$
, then we have an exact sequence
$$ \begin{align*}0 \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}({\mathrm{BS}}_{2g+2},{\mathrm{M}}) \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}(\widetilde{\mathcal{H}}_g,{\mathrm{M}}) \rightarrow {\mathrm{M}}^{\bullet}(k)_2 ,\end{align*} $$
where the last map lowers degree by
$g+2$
. Moreover, if g is even, the last map is surjective and split.
Thanks to the result above, we are able compute the prime-to-c part of the Brauer group of
$\widetilde {\mathcal {H}}_g$
, which coincides with the whole Brauer group of the stack when the base field has characteristic zero.
Corollary (5.11)
We have
$$ \begin{align*}^c{\mathrm{Br}}(\widetilde{\mathcal{H}}_g)= {^c{\mathrm{Br}}}(k) \oplus {\mathrm{H}}^{1}(k,\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}/2\mathbb{Z}.\end{align*} $$
If one considers the analogy with unramified cohomology, this appears to be in stark contrast to what happens for proper smooth varieties, where unramified cohomology is a birational invariant. The stacks
$\overline {\mathcal {H}_g}$
and
$\widetilde {\mathcal {H}}_g$
are smooth, proper (in fact, projective, according to Kresch’s definition of a projective Deligne–Mumford stack [Reference Kresch19]), and birational to each other, but their cohomological invariants are very different. The reason why this happens is related to the so-called root stack construction.
1.1 Weak factorization, root stacks, and cohomological invariants
Given smooth projective varieties
$X,Y$
over a base field k of characteristic zero, the weak factorization theorem states that a birational map
$X \dashrightarrow Y$
, which is an isomorphism on an open subset U, always factors as a composition
$$ \begin{align*}X=X_0 \overset{f_0}{\dashrightarrow}X_1 \overset{f_1}{\dashrightarrow} \cdots \overset{f_{n-1}}{\dashrightarrow} X_n = Y,\end{align*} $$
where for each i either
$f_i$
or
$f_i^{-1}$
is a blowup in a smooth center and each
$f_i$
is an isomorphism on U.
Recently, Harper [Reference Harper16] proved a similar result for Deligne–Mumford stacks over a field of characteristic zero. Given a proper birational map
$\mathcal {X} \rightarrow \mathcal {Y}$
, we always have a factorization
$$ \begin{align*}\mathcal{X}=\mathcal{X}_0 \overset{f_0}{\dashrightarrow}\mathcal{X}_1 \overset{f_1}{\dashrightarrow} \cdots \overset{f_{n-1}}{\dashrightarrow} \mathcal{X}_n = \mathcal{Y},\end{align*} $$
where for each i either
$f_i$
or
$f_i^{-1}$
is a stacky blowup, that is, either a regular blowup in a smooth center or a root stack along a smooth divisor. Given a Cartier divisor V, the order r root stack along V, denoted
$\mathcal {X}_{(V,r)} \rightarrow \mathcal {X}$
, is a canonical way to add a
$\mu _r$
stabilizer along the points of V.
Unramified cohomology and cohomological invariants are left unchanged when taking a blowup, but can change drastically when taking a root stack. If one wants to be able to compare the cohomological invariants of birational proper Deligne–Mumford stacks, it is critical to be able to understand what happens when taking a root stack. This is described in the following theorem.
Theorem (2.5)
Let
$\mathcal {X}$
be a smooth algebraic stack over a field k, and let
$r\geq 2$
be an integer that is not divisible by
${\mathrm{char}}(k)$
. Let
$V \subset \mathcal {X}$
be a smooth divisor.
Then,
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X}_{(V,r)},{\mathrm{M}})$
is equal to the subgroup of
$\operatorname {\,\mathrm {Inv^{\bullet }}}(\mathcal {X}\backslash V,{\mathrm{M}})$
of invariants
$\alpha $
such that the ramification of
$\alpha $
at V is of r-torsion.
The stack
$\widetilde {\mathcal {H}}_g$
is, up to codimension
$2$
closed substacks, a root stack over
$\overline {\mathcal {H}}_g$
. More precisely, after excising some loci, it can be obtained from an open substack of
$\overline {\mathcal {H}}_g$
by adding a
$\mu _2$
stabilizer along the divisor of irreducible singular curves.
As cohomological invariants are unchanged when removing a closed subset of codimension
$\geq 2$
, we can (and we do) apply the theorem above to compute the cohomological invariants of
$\widetilde {\mathcal {H}}_g$
, leveraging our previous knowledge of the cohomological invariants of
$\mathcal {H}_g$
and
$\overline {\mathcal {H}}_g$
.
1.2 Cohomological invariants and Chow groups with coefficients
We briefly recall some properties of cohomological invariants which will be used in the paper. For a detailed introduction to cohomological invariants and Chow groups with coefficients, the reader can refer to [Reference Di Lorenzo and Pirisi9, Section 1].
Given a field F, let
${\mathrm{T}}^{\bullet }(F)$
be the tensor algebra
$\oplus _{i\geq 0}(F^*)^{\otimes _{\mathbb {Z}} i}$
. Milnor’s
${\mathrm{K}}$
-theory
${\mathrm{K}}_{\mathrm{M}}$
is the functor from field extensions of k to graded-commutative rings which sends a field F to
$$ \begin{align*} {\mathrm{K}}_{{\mathrm{M}}}^{\bullet}(F) = {\mathrm{T}}^{\bullet}(F)/\lbrace a\otimes b \mid a,b \in F^*, \, a + b =1 \rbrace. \end{align*} $$
Milnor’s
${\mathrm{K}}$
-theory is the basic, and most important, cycle module. A general cycle module
${\mathrm{M}}$
is a functor from fields over k to abelian groups which is functorially a
${\mathrm{K}}_{\mathrm{M}}$
-module, and shares the main properties of Milnor’s
${\mathrm{K}}$
-theory. Given a torsion Galois module D over k, the functor
$$ \begin{align*} {\mathrm{H}}_D^{\bullet}(F)=\oplus_{i\geq 0}{\mathrm{H}}^i_{\mathrm{Gal}}(F,D(i)) \end{align*} $$
is a cycle module. In particular, if we take
$D=\mathbb {Z}/\ell \mathbb {Z}$
by Voevodsky’s norm-residue isomorphism [Reference Voevodsky24, Theorem 6.1], we have
$$ \begin{align*} {\mathrm{H}}_{\mathbb{Z}/\ell \mathbb{Z}}={\mathrm{K}}_M/(\ell)\stackrel{{\mathrm{def}}}{=} {\mathrm{K}}_\ell. \end{align*} $$
If
${\mathrm{M}}$
is of
$\ell $
-torsion, it is a
${\mathrm{K}}_\ell $
-module.
Cohomological invariants with coefficients in a cycle module
${\mathrm{M}}$
form a graded abelian group. When
${\mathrm{M}}$
is given by
${\mathrm{K}}_{{\mathrm{M}}}$
or
${\mathrm{K}}_{\ell }$
, they form a graded-commutative ring. In general,
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X},{\mathrm{M}})$
is an
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X},{\mathrm{K}}_{{\mathrm{M}}})$
-module, and if
${\mathrm{M}}$
is of
$\ell $
-torsion, it is an
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X},{\mathrm{K}}_{\ell })$
-module.
The functor
$\operatorname {\mathrm {Inv^{\bullet }}}(-,{\mathrm{M}})$
forms a sheaf with respect to the smooth-Nisnevich topology, where the coverings are smooth representable maps
$\mathcal {Y} \to \mathcal {X}$
such that any point
$\operatorname {\mathrm {Spec}}(K) \to \mathcal {X}$
has a lifting
$\operatorname {\mathrm {Spec}}(K) \to \mathcal {Y}$
. Moreover, cohomological invariants are unchanged when passing to an affine bundle or an open subset whose complement has codimension
$\geq 2$
, and in general, restriction to a dense open subset is injective.
Our main tool for computations will be Rost’s Chow groups with coefficients [Reference Rost22]. These are an extension of ordinary Chow groups: instead of having
$\mathbb {Z}$
-linear sums of cycles, they are generated by sums in the form
$\sum _i \alpha _i P_i$
, where
$\alpha _i$
belongs to
${\mathrm{M}}^{\bullet }(k(P))$
. The i-codimensional Chow group of X with coefficients in
${\mathrm{M}}$
is denoted by
$A^{i}(X,{\mathrm{M}})$
.
Chow groups with coefficients share the same properties of ordinary Chow groups, and moreover, given a closed subset V of X of pure codimension d, they have a long exact sequence
$$ \begin{align*} \cdots \to A^i(X,D) \to A^i(X\backslash V,D) \to A^{i+1-d}(V,D) \to A^{i+1}(X,D) \to \cdots. \end{align*} $$
When X is a smooth scheme, we have an isomorphism
$A^0(X,{\mathrm{M}}) \simeq \operatorname {\mathrm {Inv^{\bullet }}}(X,{\mathrm{M}})$
.
More generally, if X is a smooth scheme being acted upon by a smooth affine algebraic group G, we can use the Edidin–Graham–Totaro approximation process [Reference Edidin and Graham11] to define equivariant Chow groups with coefficients
$A^i_{G}(X,{\mathrm{M}})$
. This was first done by Guillot [Reference Guillot15]. Again, we have
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\left[X/G\right],{\mathrm{M}}) \simeq A^{0}_{G}(X,{\mathrm{M}}).\end{align*} $$
1.3 Outline of the paper
In Section 2, we discuss the root stack construction in greater detail. In particular, we give a presentation of the root stack
$X_{(V,r)}$
as a quotient (Proposition 2.2), which we use to deduce our formula for cohomological invariants (Theorem 2.4).
In Section 3, we recall some basic facts on the stack of stable hyperelliptic curves, and we provide a new description of an open substack of
$\overline {\mathcal {H}}_g$
, which will be useful for our purposes.
In Section 4, we describe the theory of admissible double coverings and its connection to hyperelliptic curves.
Finally, in Section 5, we compute the cohomological invariants of
$\widetilde {\mathcal {H}}_g$
(Theorem 5.7 for the even genus case and Theorem 5.10 for the odd genus case), as well as its Brauer group (Theorem 5.11).
1.4 Notation and conventions
Throughout the paper, we will fix a base field k of characteristic
$c\neq 2$
. All stacks and schemes are assumed to be of finite type over k unless otherwise specified. By
$\ell $
or r, we will always mean a positive integer that is not divisible by the characteristic of k. If A is an abelian group, we denote
$A_{\ell }$
the
$\ell $
-torsion subgroup. If moreover A is torsion, we denote
$^cA$
the subgroup of elements whose order is not divisible by c.
2 Cohomological invariants of root stacks
Fix an integer
$r\geq 1$
, and let X be an integral scheme over some field k whose characteristic does not divide r. Choose an invertible sheaf
$\mathcal {L}$
over X and define
$X_{(\mathcal {L},r)}$
as the root stack of order r of X along the invertible sheaf
$\mathcal {L}$
: this is the stack over the big étale site of X whose objects are triples
$(S\to X,\mathcal {M},\phi )$
, where
$\mathcal {M}$
is an invertible sheaf over S together with an isomorphism
$\phi :\mathcal {L}_S\simeq \mathcal {M}^{\otimes r}$
. A morphism
$(S\to X,\mathcal {M},\phi )\to (S\to X,\mathcal {M}',\phi ')$
over S is given by an isomorphism of invertible sheaves
$\psi :\mathcal {M}\simeq \mathcal {M}'$
such that
$\phi '\circ \psi ^{\otimes r}=\phi $
.
The root stack
$X_{(\mathcal {L},r)}$
has a simple presentation as quotient stack: let L be the line bundle associated to the invertible sheaf
$\mathcal {L}$
, i.e.,
$L:=\operatorname {\mathrm {\underline {Spec}}}_{\mathcal {O}_X}(\operatorname {\mathrm {Sym}} \mathcal {L}^{\vee })$
, and denote
$L^*$
the complement of the zero section in L.
The natural
$\mathbb {G}_m$
-action on L turns
$L^*\to X$
into a
$\mathbb {G}_m$
-torsor. For every
$r\geq 1$
, we can also twist the
$\mathbb {G}_m$
-action by the character
$\chi _r:\mathbb {G}_m\overset {x^r}{\mapsto }\mathbb {G}_m$
. In what follows, for every scheme S endowed with a
$\mathbb {G}_m$
-action, we denote
$[S/\mathbb {G}_{m,r}]$
the quotient stack where the
$\mathbb {G}_m$
-action is twisted by
$\chi _r$
. The next result is [Reference Cadman5, Proposition 2.3.5].
Proposition 2.1 We have
$X_{(\mathcal {L},r)}\simeq \left [ L^*/\mathbb {G}_{m,r} \right ]$
.
Consider a nonzero global section
$\sigma $
of
$\mathcal {L}$
, and let V be the divisor where the section
$\sigma $
vanishes. The stack
$X_{(V,r)}$
is the root stack of order r of X along V: its objects are quadruples
$(S\to X,\mathcal {M},\tau ,\phi )$
where
$\mathcal {M}$
is an invertible sheaf over S and
$\tau $
is a global section of
$\mathcal {M}$
, together with an isomorphism
$\phi :\mathcal {L}_S\simeq \mathcal {M}^{\otimes r}$
such that
$\phi (\sigma )=\tau ^{\otimes r}$
.
The root stack
$X_{(V,r)}$
also admits a simple description as a quotient stack: regard
$\sigma $
as a morphism from L to
$\mathbb {A}^1_X$
, and let
$(\cdot )^r:\mathbb {A}^1_X\to \mathbb {A}^1_X$
be the
$r^{\mathrm{}}$
th-power morphism, so that we can form the Cartesian diagram:
Roughly, the scheme Y can also be thought as the closed subscheme of
$L^*\times \mathbb {A}^1$
locally described by the equation
$u^r=\sigma (x,t)$
, where x and t are local coordinates for
$L^*$
and u is a coordinate for
$\mathbb {A}^1$
.
We can define an action of
$\mathbb {G}_m$
on
$L^*$
and
$\mathbb {A}^1_X$
, so that the morphisms
$\sigma $
and
$(\cdot )^r$
become equivariant: namely, we let
$\mathbb {G}_m$
acts on both
$L^*$
and the
$\mathbb {A}^1_X$
on the bottom-right corner via its
$r^{\mathrm{}}$
th power, and on the
$\mathbb {A}^1_X$
in the top-right corner via the standard action. In this way,
$Y^*$
inherits a
$\mathbb {G}_m$
-action too, and we get a Cartesian diagram of quotient stacks as follows:
Then, [Reference Cadman5, Proposition 2.3.5] says the following.
Proposition 2.2 We have
$X_{(V,r)}\simeq [Y^*/\mathbb {G}_m]$
.
We can use Proposition 2.2 to investigate the cohomological invariants of
$X_{(V,r)}$
, for X smooth. We start with the case of a root stack along an invertible sheaf.
Proposition 2.3 Let X be a smooth scheme over k. Consider an invertible sheaf
$\mathcal {L}$
over X, and let
$X_{(\mathcal {L},r)}$
be the root stack of order r of X along
$\mathcal {L}$
. Then, there is an exact sequence of
${\mathrm{M}}^{\bullet }(k)$
-modules:
$$ \begin{align*} 0 \to \operatorname{\mathrm{Inv^{\bullet}}}(X,{\mathrm{M}}) \to \operatorname{\mathrm{Inv^{\bullet}}}(X_{(\mathcal{L},r)},{\mathrm{M}}) \to \ker(c_1(\mathcal{L}))_r \to 0, \end{align*} $$
where the term on the right stands for the subgroup of r-torsion elements in the kernel of
$c_1(\mathcal {L}):A^0(X,{\mathrm{M}})\to A^1(X,{\mathrm{M}})$
.
Proof The open embedding of stacks
$[L^*/\mathbb {G}_{m,r}]\hookrightarrow [L/\mathbb {G}_{m,r}]$
induces the following long exact sequence of equivariant Chow groups with coefficients:
$$ \begin{align*} 0\to A^0_{\mathbb{G}_{m,r}}(L,{\mathrm{M}}) \to A^0_{\mathbb{G}_{m,r}}(L^*,{\mathrm{M}})\to A^0_{\mathbb{G}_{m,r}}(X,{\mathrm{M}}) \xrightarrow{\sigma_{0*}} A^1_{\mathbb{G}_{m,r}}(L,{\mathrm{M}}). \end{align*} $$
We have an isomorphism
$$ \begin{align*}A^*_{\mathbb{G}_{m,r}}(L,{\mathrm{M}})\simeq A^*_{\mathbb{G}_{m,r}}(X,{\mathrm{M}})\simeq A^*(X,{\mathrm{M}})[\xi]\,, \end{align*} $$
where
$\xi $
is the generator of
$A^*_{\mathbb {G}_m}(\operatorname {\mathrm {Spec}}(k))$
as
$\operatorname {\mathrm {H^{\bullet }}}(k)$
-module. It has codimension
$1$
and cohomological degree
$1$
. In particular,
$$ \begin{align*}A^1_{\mathbb{G}_{m,r}}(L,{\mathrm{M}})\simeq A^1(X,{\mathrm{M}})\oplus A^0(X,{\mathrm{M}})\cdot\xi. \end{align*} $$
With this identification, the morphism
$\sigma _{0*}$
is equal to
$$ \begin{align*} \alpha \mapsto c_1^{\mathbb{G}_{m,r}}(\mathcal{L})(\alpha)=c_1(\mathcal{L})(\alpha)+r\alpha\cdot\xi .\end{align*} $$
If
$\alpha $
is an element in the kernel of
$\sigma _{0*}$
, then it must necessarily satisfy
$$ \begin{align*} c_1(L)\cap \alpha=0,\quad r\alpha=0 .\end{align*} $$
We deduce that
$\ker (\sigma _{0*})$
is equal to subgroup of r-torsion elements of the kernel of
$c_1(\mathcal {L})$
.
Combining what we have just found with the long exact sequence above, the proposition easily follows.▪
Next, we study the case of a root stack along a divisor.
Theorem 2.4 Let X be a smooth scheme over k. Consider a smooth Cartier divisor
$V\subset X$
, and let
$X_{(V,r)}$
be the root stack of order r of X along V. Then,
$\operatorname {\mathrm {Inv^{\bullet }}}(X_{(V,r)},{\mathrm{M}})$
is equal to the subset of
$\operatorname {\mathrm {Inv^{\bullet }}}(X\backslash V,{\mathrm{M}})$
of invariants
$\alpha $
such that
$r\cdot \partial _V(\alpha )=0$
.
Equivalently, there is an exact sequence of
${\mathrm{M}}^{\bullet }(k)$
-modules:
$$ \begin{align*} 0 \to \operatorname{\mathrm{Inv^{\bullet}}}(X,{\mathrm{M}}) \to \operatorname{\mathrm{Inv^{\bullet}}}(X_{(V,r)},{\mathrm{M}}) \to \ker(i_*)_r \to 0\,, \end{align*} $$
where the term on the right stands for the subgroup of r-torsion elements in the kernel of
$i_*:A^0(V,{\mathrm{M}})\to A^1(X,{\mathrm{M}})$
.
Proof Recall from Proposition 2.2 that
$X_{(V,r)}\simeq [Y^*/\mathbb {G}_m]$
: here
$Y^*$
stands for the closed subscheme in
$L^*\times \mathbb {A}^1$
of equation
$\sigma (x,t)=u^r$
, where
$\sigma :L\to \mathbb {A}^1$
is determined by the section
$\sigma $
of
$\mathcal {L}$
defining V and u is a coordinate for
$\mathbb {A}^1$
. The group
$\mathbb {G}_m$
acts in the standard way on
$\mathbb {A}^1$
and as
$\mathbb {G}_{m,r}$
on L.
Let Y be the closure of
$Y^*$
in
$L\times \mathbb {A}^1$
. Observe that we have
$Y\backslash Y^*\simeq V$
, as it is equal to the intersection of the
$\sigma ^{-1}(0)$
with the image of the zero section
$t_0:X\to L$
(the coordinate u must necessarily be equal to zero). Moreover, the scheme Y is
$\mathbb {G}_m$
-invariant.
In the following, we denote the root stack
$V_{(\mathcal {O}_V(V),r))}$
by
$\widetilde {V}$
. Observe that there is a closed immersion of stacks
$\widetilde {V}\hookrightarrow X_{(V,r)}$
: the complement of
$\widetilde {V}$
in
$X_{(V,r)}$
is isomorphic to
$X\backslash V$
. Therefore, we have an exact sequence of Chow groups with coefficients as follows:
$$ \begin{align} 0\to A^0(X_{(V,r)},{\mathrm{M}}) \to A^0(X_{(V,r)}\backslash \widetilde{V},{\mathrm{M}}) \simeq A^0(X\backslash V,{\mathrm{M}}) \to A^0(\widetilde{V},{\mathrm{M}}). \end{align} $$
We need to understand the kernel of the last map in the sequence above.
We claim that there is a factorization
$$ \begin{align} A^0(X_{(V,r)} \backslash \widetilde{V},{\mathrm{M}}) \simeq A^0(X\backslash V,{\mathrm{M}}) \xrightarrow{r\partial_V} A^0(V,{\mathrm{M}}) \hookrightarrow A^0(\widetilde{V},{\mathrm{M}}), \end{align} $$
where the first map is r times the boundary map, and the second map is the pullback along
$\widetilde {V}\to V$
, whose injectivity follows from Proposition 2.3.
Let
$Z^*$
be the closed subscheme of
$Y^*$
where
$u=0$
, which is isomorphic to
$L|_V^*$
, and set
$U^*:=Y^*\backslash Z^*$
. Observe that we have
$[Z^*/\mathbb {G}_{m,r}]\simeq \widetilde {V}$
and
$[U^*/\mathbb {G}_m]\simeq X\backslash V$
.
Observe that
$f:Y^*\to L^*$
is a cyclic cover of degree r branched along
$Z^*$
and ramified over
$L^*|_V$
. Moreover,
$A^0_{\mathbb {G}_m}(U^*,{\mathrm{M}})\simeq A^0(X\backslash V,{\mathrm{M}})$
. Let
$g:L^*|_{X\backslash V}\to X\backslash V$
be the projection morphism. Then, it follows from what we just observed that every element of
$A^0_{\mathbb {G}_m}(U^*,{\mathrm{M}})$
is of the form
$f^*g^*\alpha $
. By [Reference Rost22, Definition 1.1(R3a)], we have:
$$ \begin{align*} \partial_{Z^*}(f^*g^*\alpha) = r \cdot f_V^* (\partial_{L^*|_V} (g^*\alpha)), \end{align*} $$
where
$$ \begin{align*} \partial_{L^*|_V}:A^0_{\mathbb{G}_m}(L^*|_{X\backslash V},{\mathrm{M}}) \longrightarrow A^0_{\mathbb{G}_m}(L^*|_V,{\mathrm{M}}) \end{align*} $$
is the boundary map induced by the closed embedding
$L^*|_V\hookrightarrow L^*$
, and
$f_V:Z^*\to L^*|_V$
is the restriction of f to
$Z^*$
. Consider the Cartesian diagram:
Then, the compatibility formula implies
$$ \begin{align*} r \cdot \partial_{L^*|_V} (g^*\alpha) = r \cdot g_V^* (\partial_V \alpha ), \end{align*} $$
where
$\partial _V:A^0(X\backslash V,{\mathrm{M}})\to A^0(V,{\mathrm{M}})$
is the boundary map induced by the closed embedding
$V\hookrightarrow X$
. Putting all together, we have shown that (2.2) holds true.
We deduce from (2.1) that
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(X_{(V,r)},{\mathrm{M}}) = \ker \left(r\cdot\partial_V:A^0(X\backslash V,{\mathrm{M}}) \longrightarrow A^0(V,{\mathrm{M}}) \right). \end{align*} $$
In other words, the cohomological invariants of
$X_{(V,r)}$
are the invariants of
$X \backslash V$
whose ramification along V is of r-torsion.
It is easy to see that
$A^0(X,{\mathrm{M}})$
injects into
$\ker (r\partial _V)$
. The quotient is then a subgroup of
$\ker (A^0(V,{\mathrm{M}})\to A^1(X,{\mathrm{M}}))$
, and it is precisely the subgroup of r-torsion elements. This concludes the proof.▪
Let us recall the definition of a smooth-Nisnevich morphism [Reference Pirisi21, Definition 3.2]. A morphism of algebraic stacks
$f:\mathcal {M}\to \mathcal {N}$
is said to be smooth-Nisnevich if the map f is representable, smooth, and for every field K and morphism
${\mathrm{{Spec}}}(K)\to \mathcal {N}$
, there exists a lifting
${\mathrm{{Spec}}}(K)\to \mathcal {M}$
such that
Cohomological invariants of a stack form a sheaf in the smooth-Nisnevich topology [Reference Pirisi21, Theorem 3.8]. When the base field k is infinite, any finite type algebraic stack over k with affine geometric stabilizers admits a smooth-Nisnevich covering by a scheme of finite type over k [Reference Pirisi21, Theorem 3.6].
Theorem 2.5 Let
$\mathcal {X}$
be a smooth algebraic stack over k, and let
$r\geq 2$
be an integer that is not divisible by
${\mathrm{char}}(k)$
. Let
$V \subset \mathcal {X}$
be a smooth divisor. Suppose moreover that
$\mathcal {X}$
admits a smooth-Nisnevich cover
$X\xrightarrow {\pi }\mathcal {X}$
with X a scheme, and let
$V'$
be the inverse image of V.
Then.
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X}_{(V,r)},{\mathrm{M}})$
is equal to the subset of
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X}\backslash V,{\mathrm{M}})$
of invariants
$\alpha $
such that
$r\cdot \partial _{V'}(\pi ^*\alpha )=0$
.
If moreover
$\mathcal {X} = \left [X/G\right ]$
with X a smooth scheme and G a smooth affine group scheme, we have an equivariant ramification map
$$ \begin{align*}\partial_V: \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{X},{\mathrm{M}})=A^0_G(X,{\mathrm{M}}) \to A^0_G(V,{\mathrm{M}}).\end{align*} $$
Then,
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X}_{(V,r)},{\mathrm{M}})$
is equal to the subset of
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {X}\backslash V,{\mathrm{M}})$
of invariants
$\alpha $
such that
$r\cdot \partial (\alpha )=0$
.
Proof Let
$\alpha $
be an invariant of
$\mathcal {X}_{(V,r)}$
, and let
$\pi :X\to \mathcal {X}$
be any smooth-Nisnevich cover of
$\mathcal {X}$
. Call
$V'$
the preimage of V in X. The fiber product
$X \times _{\mathcal {X}} \mathcal {X}_{(V,r)}$
is equal to
$X_{(V',r)}$
, the root stack of X along
$V'$
of order r, and
$X_{(V',r)}$
is a smooth-Nisnevich cover of
$\mathcal {X}_{(V,r)}$
. Then, by Theorem 2.4, the ramification
$\partial _{V'}(\pi ^*\alpha )$
must be of r-torsion.
On the other hand, let
$\alpha $
be an invariant of
$\mathcal {X} \backslash V$
such that
$r\cdot \partial _{V}(\alpha )=0$
. We claim that
$\alpha $
extends to
$\mathcal {X}_{(V,r)}$
.
Again by Theorem 2.4, the pullback of
$\alpha $
to
$X\backslash V'$
extends to a cohomological invariant of
$X_{(V',r)}$
, so we only need to show that the gluing conditions are satisfied. This is trivially true, because the gluing conditions can be checked on the generic point of
$X_{(V',r)}$
and we know that the invariant glues on the open subset isomorphic to
$X \backslash V'$
.
To prove the last point, it suffices to notice that
$\pi ^*(\partial _V(\alpha ))=\partial _{V'}(\pi ^*\alpha )$
.▪
Remark 2.6 By [Reference Pirisi21, Proposition 3.6], when k is infinite, the theorem above applies to all smooth algebraic stacks with affine stabilizers. In particular, it applies to the moduli stack
$\overline {\mathcal {H}}_g$
of stable hyperelliptic curves.
When k is finite, note that by [Reference Olsson20, p.276 13.2.23, p.187 8.4.13] the stack
$\overline {\mathcal {M}}_g$
has a presentation as
$\left [X/{\mathrm{GL}}_{5g-5}\right ]$
, where X is a smooth scheme. Consequently,
$\overline {\mathcal {H}}_g$
has a presentation as
$\left [V/{\mathrm{GL}}_{5g-5}\right ]$
, where
$V\subset X$
is some closed,
${\mathrm{GL}}_{5g-5}$
-invariant subscheme. The induced map
$V\to [V/{\mathrm{GL}}_{5g-5}]$
is smooth-Nisnevich, because
${\mathrm{GL}}_{5g-5}$
is special; hence,
$\overline {\mathcal {H}}_g$
has a smooth-Nisnevich cover of finite type.
The precise description of the smooth-Nisnevich covers will not be relevant to our computations, as we will use the equivariant part of the theorem which only requires the existence of such a cover.
Example 2.7 Consider the scheme
$\mathbb {P}^1$
with two marked points
$0, \infty $
. It is well known that
$\mathbb {P}^1$
has no nontrivial cohomological invariants. If we take the root stack
$\mathbb {P}^1_{(0,r)}$
, Theorem 2.4 shows us that the cohomological invariants are still trivial, because the complement
$\mathbb {P}^1\backslash 0$
is isomorphic to the affine line
$\mathbb {A}^1$
, which has trivial invariants.
On the other hand, if we take the divisor
$V=0 + \infty $
, we have
$\mathbb {P}^1 \backslash V \simeq \mathbb {G}_m$
, and for every r,
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\mathbb{P}^1_{(V,r)},{\mathrm{M}})={\mathrm{M}}^{\bullet}(k) \oplus \alpha \cdot {\mathrm{M}}^{\bullet}(k)_{r}. \end{align*} $$
Another way to see this is to note that
$\mathbb {P}^1_{(V,r)} \simeq \left [\mathbb {P}^1/\mu _r\right ]$
, where
$\mu _r$
acts by multiplication on either homogeneous coordinate.
3 The stack of stable hyperelliptic curves
A stable hyperelliptic curve
$(C,\iota )$
over
$\operatorname {\mathrm {Spec}}(k)$
is a stable curve C endowed with an involution
$\iota :C\to C$
such that the quotient
$f:C\to C/\langle \iota \rangle $
is a curve of genus
$0$
.
Let
$\overline {\mathcal {H}}_g$
be the stack of stable hyperelliptic curves, i.e., the closure of
$\mathcal {H}_g$
inside the stack
$\overline {\mathcal {M}}_g$
of stable curves. The irreducible components of the boundary divisor
$\partial \overline {\mathcal {H}}_g:=\overline {\mathcal {H}}_g\backslash \mathcal {H}_g$
can be described as follows (see [Reference Cornalba6]):
-
• The irreducible divisor
$\Delta _0$
is the stack of stable hyperelliptic curves having a nonseparating node that is fixed by the hyperelliptic involution. -
• The irreducible divisor
$\Delta _i$
, for
$i=1,\ldots ,\lfloor \frac {g}{2} \rfloor $
, is the stack of stable hyperelliptic curves with two components of arithmetic genus i and
$g-i$
meeting at a separating node that is fixed by the hyperelliptic involution. -
• The irreducible divisor
$\Xi _i$
, for
$i=1,\ldots ,\lfloor \frac {g}{2}\rfloor -1$
, is the stack of stable hyperelliptic curves with two components of genus i and
$g-i-1$
meeting at two points that are switched by the hyperelliptic involution.
Let
$\overline {\mathcal {H}}^{o}_g$
be the open substack of
$\overline {\mathcal {H}}_g$
formed by those curves that either have at most one node that is fixed by the involution or exactly two nodes that are switched by the involution. Observe that the complement of
$\overline {\mathcal {H}}_g^o$
in
$\overline {\mathcal {H}}_g$
has codimension
$2$
. We will call
$\Delta _i^o$
and
$\Xi _i^o$
, respectively, the restriction of
$\Delta _i$
and
$\Xi _i$
to
$\overline {\mathcal {H}}_g^o$
.
We give now an alternative description of the stack
$\overline {\mathcal {H}}_g^o$
that resembles the one in [Reference Arsie and Vistoli3, Corollary 4.7]. Let
$\mathcal {F}_g^o$
be the fibered category over the site of k-schemes
$$ \begin{align*} \mathcal{F}_g^o(S) := \left\{ (P\to S, \mathcal{L}, s) \right\}, \end{align*} $$
where:
-
•
$P\to S$
is a family of twisted conics of rank
${\geq}2$
, i.e., each geometric fiber is a stacky curve whose coarse moduli space is a conic of rank
${\geq}2$
, and the only stacky structure allowed is at most a
$B\mu _2$
at the node (see [Reference Abramovich, Corti and Vistoli1, Definition 2.1.4] for the notion of twisted curve). -
•
$\mathcal {L}$
is a line bundle on P of degree
$-g-1$
whose degree on each irreducible component is
${>}1$
. Moreover, if the fiber possesses a twisted node, the degree of
$\mathcal {L}$
restricted to an irreducible component must be an half-integer (more on this a few lines below). -
• s is a global section of
$\mathcal {L}^{\otimes (-2)}$
. For every geometric point z of S, if the conic
$P_{z}$
is smooth, then s can have at most one double root, otherwise no double roots are allowed.
More explicitly, if
$P\to \operatorname {\mathrm {Spec}}(\overline {k})$
is a rank
$2$
twisted conic, the restriction of
$\mathcal {L}$
to an irreducible component can have degree
$\frac {r}{2}$
with
$r=3,4,\ldots , 2\lfloor \frac {g}{2} \rfloor +1$
. Indeed, consider an auxiliary rank
$2$
conic
$P'$
, so that P is the rigidification of
$[P'/\mu _2]$
away from the node (here
$\mu _2$
acts by sending
$(x:y)\mapsto (x:-y)$
). A line bundle of bidegree
$(\frac {r}{2},\frac {l}{2})$
can then be thought of as a line bundle on P whose lifting to
$P'$
has bidegree
$(r,l)$
: by this we mean that, regarding
$P'$
as two copies of
$\mathbb {P}^1$
glued in one point, the restriction of the line bundle has degree r on one component and degree l on the other.
The category
$\mathcal {F}_g^o$
is actually a Deligne–Mumford stack. Observe that
$\mathcal {F}_g^o$
contains an open substack isomorphic to
$\mathcal {H}_g$
, namely the substack of objects
$(P\to S,L,s)$
where
$P\to S$
is smooth and the vanishing locus of s is étale over the base.
Proposition 3.1 There is an isomorphism
$\overline {\mathcal {H}}_g^o\simeq \mathcal {F}_g^o$
that extends the one of [Reference Arsie and Vistoli3, Corollary 4.7].
Proof Let
$(C\to S,\iota )$
be an object of
$\overline {\mathcal {H}}_g^o$
. We can form the stacky quotient
$[C/\iota ]$
, which is by construction a stacky conic of rank
${\geq}2$
. As the stacky structure is supported on a divisor, we can apply the rigidification process of [Reference Abramovich, Corti and Vistoli1]*5.1 to rigidify
$[C/\iota ]$
away from the singular locus: the resulting stack
$P\to S$
is a family of twisted conics of rank
${\geq}2$
.
Observe that the morphisms
$C\to [C/\iota ]$
and
$[C/\iota ]\to P$
are both flat, hence so must be their composition
$f:C\to P$
. In particular, this easily implies that
$f_*\mathcal {O}_C$
is a locally free sheaf of rank
$2$
. Moreover, the inclusion
$\mathcal {O}_P \hookrightarrow f_*\mathcal {O}_{C}$
of
$\iota $
-invariant elements has a splitting given by the Reynolds morphism, hence
$f_*\mathcal {O}_C\simeq \mathcal {O}_P\oplus \mathcal {L}$
for some line bundle
$\mathcal {L}$
on P.
The structure of
$f_*\mathcal {O}_C$
as
$\mathbb {Z} / 2\mathbb {Z}$
-graded algebra determines a morphism
$\mathcal {L}^{\otimes 2}\to \mathcal {O}_P$
or, equivalently, a global section
$s:\mathcal {O}_P\to \mathcal {L}^{\otimes (-2)}$
. The vanishing locus of s coincides with the branch locus of
$C\to P$
, which has degree
$2g+2$
by the Hurwitz formula: this implies that
$\mathcal {L}$
has total degree
$-g-1$
. As C is stable, the degree of
$\mathcal {L}$
restricted to every irreducible component must be
${\geq}2$
. Eventually, stability of C also implies that the vanishing locus of s satisfies the required assumptions.
In this way, we have constructed a morphism of stacks
$\varphi :\overline {\mathcal {H}}_g^o\to \mathcal {F}_g^o$
. To construct its quasi-inverse, we can do the following: given an object
$(P\to S,\mathcal {L},s)$
of
$\mathcal {F}_g^o$
, consider the
$\mathcal {O}_P$
-module
$\mathcal {O}_P\oplus \mathcal {L}$
. We can give to
$\mathcal {O}_P\oplus \mathcal {L}$
the structure of an
$\mathcal {O}_P$
-algebra as follows. First observe that the multiplication by the global section s of
$\mathcal {L}^{\otimes -2}$
defines a morphism
$\mathcal {O}_P\to \mathcal {L}^{\otimes -2}$
. We can dualize this morphism of
$\mathcal {O}_P$
-modules, obtaining in this way
$s^{\vee }:\mathcal {L}^{\otimes 2}\to \mathcal {O}_P$
.
Given
$U\to P$
and elements
$a\oplus b$
and
$a'\oplus b'$
of
$\mathcal {O}_P(U)\oplus \mathcal {L}(U)$
, we set
$$ \begin{align*} (a\oplus b)\cdot (a'\oplus b') = (aa'+s^{\vee}(b\otimes b'),ab'+a'b). \end{align*} $$
This gives a multiplicative structure to
$\mathcal {O}_P\oplus \mathcal {L}$
, turning it into an
$\mathcal {O}_P$
-algebra. Define
$C:=\operatorname {\mathrm {\underline {Spec}}}_{\mathcal {O}_P}(\mathcal {O}_P\oplus \mathcal {L})$
: the involution on
$\mathcal {O}_P\oplus \mathcal {L}$
defined as
$$ \begin{align*}a\oplus b \longmapsto a\oplus -b \end{align*} $$
is a morphism of algebras and determines an involution
$\iota :C\to C$
, whose quotient
$C/\iota $
is the relative spectrum of the
$\iota $
-invariant part of
$(\mathcal {O}_P\oplus \mathcal {L})$
, that is,
$\mathcal {O}_P$
. In other terms,
$[C/\iota ]$
has the same coarse moduli space of P, and the rigidification of
$[C/\iota ]$
at the vanishing locus of s coincides with P.
This shows that
$C\to P$
is a ramified cover of degree
$2$
. We claim that
$(C\to S,\iota )$
is an object of
$\overline {\mathcal {H}}_g^o$
. For this, it is enough to check that in the case
$S=\operatorname {\mathrm {Spec}}(k)$
, the curve C is a stable hyperelliptic curve belonging to
$\overline {\mathcal {H}}_g^o$
.
First assume that p is a point in the smooth locus of P. Then, we have
$\mathcal {O}_{P,p}\simeq k[x]_{(x)}$
, and there exists a trivialization
$\mathcal {L}_p\simeq \mathcal {O}_{P,p}$
. Then, we have that locally over p, the curve C is the localization at
$(x)$
of
$\operatorname {\mathrm {Spec}}(k[x,y]/(y^2=f))$
, where f is the image of
$1$
along the map
$\mathcal {O}_{P,p}\xrightarrow {s} \mathcal {L}_p^{\otimes -2} \simeq \mathcal {O}_{P,p}$
. This shows that if P is smooth, then C is a double cover ramified over the vanishing locus of s, which is a divisor of degree
$2g+2$
with at most one double point. This implies that C is an irreducible stable hyperelliptic curve of genus g with at most one node.
Now, suppose that p is an untwisted node of P: as the section s does not vanish on p, we have that
$C\to P$
is étale locally around p. In particular, this together with the hypotheses on the vanishing locus of s forces C to have two irreducible components of genus i and
$g-i-1$
meeting in the two nodes over p. The two irreducible components are double covers of the two components of p; hence, C is again a stable hyperelliptic curve in
$\overline {\mathcal {H}}_g^o$
.
Finally, suppose that p is a twisted node of P: then again
$C\to P$
is étale of degree two around p, but the presence of the twist implies that the map from C to the coarse moduli space of P ramifies at the node (see the proof of [Reference Di Lorenzo and Vistoli10, Theorem 2.5]) and that C is the union of two curves of genus i and
$g-i$
meeting in a point. Moreover, the two components are double covers of the two components of P; hence, C is a stable hyperelliptic curve belonging to
$\overline {\mathcal {H}}_g^o$
.
In this way, we have constructed a morphism of stacks
$\mathcal {F}_g^o\to \overline {\mathcal {H}}_g^o$
, and is straightforward to check that this gives a quasi-inverse to
$\varphi $
.▪
Remark 3.2 The necessity of dealing with twisted conics comes from the fact that the morphism
$C\to P$
, whenever it ramifies over the node, cannot be flat unless the node is twisted. This is explained in greater detail in the introduction of [Reference Di Lorenzo and Vistoli10] (see also [Reference Di Lorenzo and Vistoli10, Section 2]).
On the other hand, the line bundle
$\mathcal {L}^{\otimes (-2)}$
comes from the coarse moduli space of P, so the global section s does as well. This shows that we still have a morphism from
$\overline {\mathcal {H}}_g^o$
to the stack of conics of rank
${\geq}2$
with a line bundle
$\mathcal {M}$
of degree
$2g+2$
and a global section s of
$\mathcal {M}$
, but this is not an isomorphism.
The divisor
$\Delta _0^o$
is sent by the isomorphism of Proposition 3.1 to the closed substack in
$\mathcal {F}_g^o$
whose objects are triples
$(P\to S,\mathcal {L},s)$
with
$P\to S$
smooth. We denote this divisor by
$\mathcal {D}_0^o$
.
The divisors
$\Delta _i^o$
for
$i>0$
are isomorphic to the divisors in
$\mathcal {F}_g^o$
whose objects are
$(P\to S,\mathcal {L}, s)$
are such that P is a family of conics of rank
$2$
and
$\mathcal {L}$
has bidegree
$(-\frac {2i+1}{2},-\frac {2g-2i+1}{2})$
on each geometric fiber. In particular, every fiber of
$P\to S$
must have a twist in the node. We denote these divisors by
$\mathcal {D}_i^o$
.
On the other hand, the divisors
$\Xi _i^o$
are isomorphic to the stacks whose objects are triples
$(P\to S,\mathcal {L},s)$
such that
$P\to S$
is a family of rank
$2$
conics and
$\mathcal {L}$
has bidegree
$(-i,-g-i-1)$
on each fiber. These divisors will be denoted
$\mathcal {X}_i^o$
.
4 The stack of admissible double coverings
In the following, a nodal curve is a connected, reduced, and proper scheme C over
$\operatorname {\mathrm {Spec}}(k)$
of dimension
$1$
whose singularities are all nodal, i.e., the formal completion of the local ring of C at a singular point p is isomorphic to
$k[[x,y]]/(xy)$
, for every singular point of C.
A family of nodal curves is a proper and flat morphism of schemes
$C\to S$
such that every geometric fiber is a nodal curve.
Definition 4.1 [Reference Harris and Mumford17, Section 4]
An admissible double covering of genus g is a finite morphism
$C\to P$
between a genus g nodal curve C and a genus
$0$
nodal curve P such that:
-
• On the complement of the singular locus of P, the morphism
$C\to P$
has degree
$2$
. -
• The image of every node q of C is a node p of P, and we have that
sends
$$ \begin{align*} \widehat{\mathcal{O}_{V,p}}\simeq k[[x,y]]/(xy) \longrightarrow \widehat{\mathcal{O}_{C,q}}\simeq k[[u,v]]/(u,v) \end{align*} $$
$x\mapsto u^e$
and
$y\mapsto v^e$
with
$1\leq e \leq 2$
.
-
• Only a finite number of automorphisms of P fixes both the ramification divisor of
$C\to P$
and the singular locus.
Given a scheme S, a family of admissible double coverings of genus g over S is an S-morphism
$C\to P$
between families of nodal curves over S such that over every geometric point
$\overline {s}$
of S, the morphism
$C_{\overline {s}}\to P_{\overline {s}}$
is an admissible covering of degree
$2$
and genus g.
Remark 4.1 The definition above differs from the one of [Reference Harris and Mumford17], because we do not ask for
$2g+2$
sections
$\sigma _i:\operatorname {\mathrm {Spec}}(k)\to P$
such that the ramification divisor, away from the nodes, is equal to
$\sum \sigma _i$
.
We can form the fibered category
$\widetilde {\mathcal {H}}_g$
of admissible double coverings of genus g, that is,
$$ \begin{align*} \widetilde{\mathcal{H}}_g (S) := \left\{ (C \to P \to S) \right\}, \end{align*} $$
where a morphism
$(C\to P\to S)\to (C'\to P'\to S')$
consists of a morphism
$S\to S'$
and two isomorphisms
$C\simeq C^{\prime }_S$
and
$P\simeq P^{\prime }_S$
which commute with the covering maps.
The stack
$\widetilde {\mathcal {H}}_g$
is a smooth Deligne–Mumford stack. One way to see this is to consider the stack
$\mathcal {B}_{0,2g+2}(S_{2})$
of twisted admissible coverings (see [Reference Abramovich, Corti and Vistoli1, Section 2.2]), which is isomorphic to the stack of admissible coverings of degree
$2$
of Harris–Mumford by [Reference Abramovich, Corti and Vistoli1, Proposition 4.2.2]. The stack
$\mathcal {B}_{0,2g+2}(S_2)$
is a smooth Deligne–Mumford stack, and the automorphism group of every twisted
$S_2$
-admissible covering contains the symmetric group
$S_{2g+2}$
, which acts by permutation on the markings. By [Reference Abramovich, Corti and Vistoli1, Section 5.1], we can take the rigidification
$[\mathcal {B}_{0,2g+2}(S_2)/\mathcal {B} S_{2g+2}]$
, and it is easy to see that this last stack is isomorphic to
$\widetilde {\mathcal {H}}_g$
.
We can recast Definition 4.1 using the notion of admissible involution.
Definition 4.2 [Reference Beauville4, Section 3]
An admissible involution of a nodal curve C over
$\operatorname {\mathrm {Spec}}(k)$
is an involution
$\iota :C\to C$
such that:
-
• No irreducible component is fixed by
$\iota $
. -
• The quotient
$C/\langle \iota \rangle $
is a nodal curve of genus
$0$
. -
• If a node is fixed by
$\iota $
, then
$\iota $
acts on the local ring
$\mathcal {O}_{C,p}$
, hence on its formal completion
$\widehat {\mathcal {O}_{C,p}}$
, and it does not switch the two branches of the node.
The definition above extends in an obvious way to families of nodal curves.
Proposition 4.2 [Reference Scavia23, Proposition 3.3.7]
Let
$C\to P\to S$
be a family of admissible double coverings over S. Then, there exists a unique admissible involution
$\iota :C\to C$
over S such that
$P=C/\langle \iota \rangle $
.
Conversely, given an admissible involution
$\iota :C\to C$
of a family of nodal curves, the morphism
$C\to C/\langle \iota \rangle $
is an admissible double covering.
The following results give a somewhat more concrete description of how an admissible double covering looks like.
Proposition 4.3 [Reference Scavia23, Lemma 4.1.2 and Proposition 4.1.3]
Let
$C\to P$
be an admissible double covering, and let
$\iota :C\to C$
be the admissible involution of Proposition 4.2. Then:
-
(1) The nodal curve
$C\to S$
is semistable. -
(2) Every irreducible component of C is smooth. Moreover, if
$\iota (C_i)=C_i$
, then
$C_i\to C_i/\langle \iota \rangle $
is a hyperelliptic curve, otherwise
$C_i$
and
$\iota (C_i)$
are two disconnected projective lines. -
(3) Every nonstable component is a projective line intersecting the rest of the curve in two points, which get switched by
$\iota $
.
An immediate consequence of the proposition above is that the admissible involution
$\iota $
of an admissible double covering
$C\to P$
descends to a hyperelliptic involution
$\overline {\iota }$
of the stable model
$\overline {C}$
of C: in other terms, the stable curve
$\overline {C}$
is a stable hyperelliptic curve. This argument works for families too.
Corollary 4.4 Let
$C\to P\to S$
be a family of admissible double covering, and let
$\overline {C}\to S$
be its stable model. Then, the admissible involution
$\iota :C\to C$
descends to a hyperelliptic involution
$\overline {\iota }:\overline {C}\to \overline {C}$
, and the pair
$(\overline {C}\to S,\overline {\iota })$
is a family of stable hyperelliptic curves.
The stabilization morphism commutes with base change; hence, the corollary above determines a morphism of stacks
$$ \begin{align*} \varphi: \widetilde{\mathcal{H}}_g \longrightarrow \overline{\mathcal{H}}_g,\quad (C\to P\to S)\longmapsto (\overline{C}\to S,\overline{\iota}) \end{align*} $$
called the collapsing morphism [Reference Scavia23, Definition 4.1.13].
Observe that the collapsing morphism induces a bijection at the level of geometric points, and it is obviously an isomorphism over the open substack of smooth curves. This in turn implies that the irreducible components of the boundary
$\widetilde {\mathcal {H}}_g\backslash \mathcal {H}_g$
are
$\widetilde {\Delta }_i:=\varphi ^{-1}(\Delta _i)$
for
$i=0,1,\ldots ,\lfloor \frac {g}{2}\rfloor $
and
$\widetilde {\Xi }_j:=\varphi ^{-1}(\Xi _j)$
for
$j=1,2,\ldots ,\lfloor \frac {g}{2}\rfloor $
.
Recall that in Section 3, we defined the open substack
$\overline {\mathcal {H}}_g^o$
of
$\overline {\mathcal {H}}_g$
. Set then
$\widetilde {\mathcal {H}}_g^o:=\varphi ^{-1}(\overline {\mathcal {H}}_g^o)$
. As for the case of stable hyperelliptic curves, the complement of
$\widetilde {\mathcal {H}}_g^o$
in
$\widetilde {\mathcal {H}}_g$
has codimension
$2$
, and the irreducible components of
$\widetilde {\mathcal {H}}_g^o\backslash \mathcal {H}_g$
are
$\widetilde {\Delta }_i^o:=\varphi ^{-1}(\Delta _i^o)$
and
$\widetilde {\Xi }_j^o:=\varphi ^{-1}(\Xi _j^o)$
.
Observe that the collapsing map induces an isomorphism of stacks for almost all the irreducible components of the boundary of
$\widetilde {\mathcal {H}}_g^o$
and
$\overline {\mathcal {H}}_g^o$
, with the only exception of
$\widetilde {\Delta }_0^o\to \Delta _0^o$
: indeed, compared to its stable model, an admissible covering in
$\widetilde {\Delta }_0^o$
has an additional automorphism, namely the involution of the projective line that switches the two intersection points.
Proposition 4.5 The stack
$\widetilde {\mathcal {H}}_g^o$
is a root stack of
$\overline {\mathcal {H}}_g^o$
of order
$2$
along the divisor
$\Delta _0^o$
.
Proof Denote
$\widehat {\mathcal {H}}_g^o$
the root stack of
$\overline {\mathcal {H}}_g^o$
of order
$2$
along
$\Delta _0^o$
. We want to prove that
$\widetilde {\mathcal {H}}_g^o\simeq \widehat {\mathcal {H}}_g^o$
. First, we are going to lift the collapsing morphism
$\widetilde {\mathcal {H}}_g^o\to \overline {\mathcal {H}}_g^o$
to a morphism to
$\widehat {\mathcal {H}}_g^o$
. Second, we will show that the lifted morphism is actually an isomorphism.
For the first point, we need to show that if
$(C\to S,\iota )$
is a family of admissible coverings, and
$(\overline {C}\to S,\overline {\iota })$
is its stabilization, then the divisor
$D\subset S$
parametrizing those fibers in
$\overline {C}$
with a nonseparating node is of the form
$2E$
for some other divisor
$E\subset S$
. Here, we are implicitly assuming that D is a divisor and does not coincide with the whole S nor is empty.
Let
$Y\subset \overline {C}$
be the singular locus of
$\overline {C}\to S$
supported over D. Up to passing to a Zariski cover of S, we can assume that
$D=\{f=0\}$
is principal, and that étale-locally around Y the family
$\overline {C}\to S$
has equation
$x'y'+f = 0$
. After a change of coordinates, this can be rewritten as
$$ \begin{align*} y^2-x^2+f=0. \end{align*} $$
A key point here is that for every geometric point
$s\in D$
, the two nodes of
$C_s$
are switched by the admissible involution. This implies that the two branches of
$\overline {C}_s$
at the node are switched by the hyperelliptic involution, hence at the level of coordinates
$x'\mapsto y'$
and
$y'\mapsto x'$
or equivalently
$x\mapsto x$
and
$y\mapsto -y$
.
This means that, locally around Y, the equation
$y^2=x^2-f$
describes
$\overline {C}\to S$
as a double cover of
$\overline {P}\to S$
with branching divisor
$W\subset \overline {P}$
of equation
$x^2-f=0$
. Let
$W'\subset P$
be the branching divisor of the admissible double covering
$C\to P$
: then around Y, the induced map
$W'\to W$
is an isomorphism away from the locus supported over
$D\subset S$
, because by construction
$W^{\prime }_{|D}\to W_{|D}$
is of degree
$2$
. In other terms, locally around Y, the map
$W'\to W$
is a normalization.
This implies that W must be singular along
$D=\{f=0\}$
: a straightforward computation shows that then the vanishing locus of f must be everywhere singular, hence
$f=g^k$
for some
$k>1$
. Moreover, the geometry of the normalization morphism
$W'\to W$
forces
$k=2n$
for some
$n\geq 1$
. This shows that
$D=2E$
, where E is the divisor of equation
$g^n=0$
.
Observe that the Cartier divisor E descends along the Zariski cover that we considered at the beginning of the proof, so we actually defined a morphism
$\widetilde {\mathcal {H}}_g^o(S) \to \widehat {\mathcal {H}}_g^o(S)$
for every S such that
$D\subset S$
is not empty nor equal to S. If
$D=\emptyset $
, there is nothing to prove. If
$D=S$
, we can consider a deformation
$S'$
of S: as we just proved that
$S=2E$
, we deduce that
$\mathcal {O}(E)_{|S}$
is a square root of the normal bundle
$\mathcal {O}(S)_{|S}$
, implying that as above we have a lifting
$\widetilde {\mathcal {H}}_g^o(S)\to \widehat {\mathcal {H}}_g^o(S)$
.
We claim that
$\widetilde {\mathcal {H}}_g^o\to \widehat {\mathcal {H}}_g^o$
is representable, finite, and birational. Concerning representability, it is enough to show that the induced homomorphism between the automorphism groups of the objects is always injective. Let
$C_s\to P_s\to \operatorname {\mathrm {Spec}}(k)$
be an admissible double covering, and let
$\overline {C}_s\to \operatorname {\mathrm {Spec}}(k)$
be its stabilization: the kernel of the induced homomorphism
$$ \begin{align*} \underline{{\mathrm{Aut}}}_k((C_s,\iota_s)) \longrightarrow \underline{{\mathrm{Aut}}}_k((\overline{C}_s,\overline{\iota}_s)) \end{align*} $$
is trivial unless
$\overline {C}_s$
has a nonseparating node, in which case the kernel is generated by the involution
$\varphi _s$
that on the genus zero component of
$C_s$
fixes the two nodes and on the other component is the identity.
Consider a generic one-parameter deformation
$\overline {C}'\to S'$
of
$\overline {C}_s$
, where
$S'$
is a DVR with parameter t. Let
$S\to S'$
be the degree 2 ramified cover
$S\to S'$
given by
$t\mapsto t^2$
, and let
$\overline {C}\to S$
be the pullback of
$\overline {C}'\to S'$
. In this case, the divisor D is supported on the closed point, and it is defined by the equation
$\{t^2=0\}$
. In particular,
$D=2E$
, where E is the divisor of equation
$\{t=0\}$
.
Call
$p\in \overline {C}_s$
the nonseparating node: for what we have seen, locally around p, the double cover
$\overline {C}\to \overline {P}\to S$
is described by the equation
$y^2=(x^2-t^2)$
. The blowup of
$\overline {C}$
at p is a family of admissible double coverings
$C\to S$
, whose central fiber is isomorphic to
$C_s$
.
The involution
$S\to S$
defined by
$t\mapsto -t$
induces an involution
$\varphi $
of
$C\to S$
which on the central fiber coincides with
$\varphi _s$
. By construction, the morphism
$\widetilde {\mathcal {H}}_g^o(S) \to \widehat {\mathcal {H}}_g^o(S)$
sends
$$ \begin{align*}(C\to S,\iota)\longmapsto(\overline{C}\to S,\overline{\iota},\mathcal{O}\cdot t^{-1}, t),\end{align*} $$
hence the automorphism
$\varphi _s$
is sent to the automorphism of
$((\overline {C}_s,\overline {\iota }),k\cdot t^{-1},t)$
that acts as the identity on
$\overline {C}_s$
and as
$-1$
on
$k\cdot t^{-1}$
. We deduce that the homomorphism between the automorphism groups is injective and that the map of stacks is representable.
The same argument can be used to show that
$\widetilde {\mathcal {H}}_g^o\to \widehat {\mathcal {H}}_g^o$
is proper and quasi-finite, hence finite. As this map is representable, birational, and finite, and the target is normal, we deduce that it is actually an isomorphism, thus concluding the proof.▪
Remark 4.6 Proposition 4.5 can also be deduced from [Reference Scavia23, Proposition 4.4.5].
5 Computation of cohomological invariants
We start by recalling some facts on the cohomological invariants of the stack of smooth hyperelliptic curves. A more in depth discussion of this subject can be found in the introduction to [Reference Di Lorenzo and Pirisi9].
Let
$\acute{\rm e}{{\rm t}}_n$
be the stack of étale algebras of degree n. It is isomorphic to the classifying stack
${\mathrm{BS}}_n$
. Given a family of smooth hyperelliptic curves
$C\to S$
with involution
$\iota $
, the ramification divisor of
$C\to C/\langle \iota \rangle $
is the spectrum of an étale
$\mathcal {O}_S$
-algebra of degree
$2g+2$
. This defines a map
$$ \begin{align*}\mathcal{H}_g \longrightarrow {\acute{\rm e}{\rm t}}_{2g+2} \simeq {\mathrm{BS}}_{2g+2},\end{align*} $$
which is smooth-Nisnevich, so that the pullback
$\operatorname {\mathrm {Inv^{\bullet }}}({\mathrm{BS}}_{2g+2},{\mathrm{M}}) \to \operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {H}_g,{\mathrm{M}})$
is injective. The cohomological invariants of
${\mathrm{BS}}_{2g+2}$
are isomorphic to
$$ \begin{align*} {\mathrm{M}}^{\bullet}(k) \oplus \alpha_1\cdot {\mathrm{M}}^{\bullet}(k)_2 \oplus \cdots \oplus \alpha_{g+1}\cdot {\mathrm{M}}^{\bullet}(k)_2\,, \end{align*} $$
where the classes
$\alpha _i$
are the arithmetic Stiefel–Whitney classes coming from the invariants of
${\mathrm{BO}}_{2g+2}$
(see [Reference Garibaldi, Merkurjev and Serre12, p.53 23.3] for the case
${\mathrm{M}}={\mathrm{H}}_D$
, and [Reference Di Lorenzo and Pirisi9, Proposition 5.10] for the general case).
Theorem 5.1 [Reference Di Lorenzo and Pirisi9, Theorem 6.9 and Remark 6.10]
Let
${\mathrm{M}}$
be an
$\ell $
-torsion cycle module, and let
$I_g$
be the submodule of
$\operatorname {\mathrm {Inv^{\bullet }}}({\mathrm{BS}}_{2g+2},{\mathrm{M}})$
given by
$$ \begin{align*} I_g={\mathrm{M}}^{\bullet}(k) \oplus \bigoplus_{i=2}^{g+1} \alpha_i \cdot {\mathrm{M}}^{\bullet}(k)_2. \end{align*} $$
Let
$n_g$
be equal to
$4g+2$
if g is even and
$8g + 4$
if g is odd. If g is even, we have
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{H}_g,{\mathrm{M}}) = \alpha^{\prime}_1\cdot {\mathrm{M}}^{\bullet}(k)_{n_g} \oplus I_g \oplus \beta_{2g+2}\cdot {\mathrm{M}}^{\bullet}(k)_2\,, \end{align*} $$
where
$\alpha ^{\prime }_1$
and
$\beta _{2g+2}$
are elements of degrees, respectively,
$1$
and
$2g+2$
.
If g is odd, there is an exact sequence
$$ \begin{align*} 0 \to {\alpha^{\prime}_1}\cdot {\mathrm{M}}^{\bullet}(k)_{n_g} \oplus I_g \oplus w_2\cdot {\mathrm{M}}^{\bullet}(k)_2 \to \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{H}_g,{\mathrm{M}}) \to {\mathrm{M}}^{\bullet}(k)_2\,, \end{align*} $$
where
$\alpha ^{\prime }_1$
and
$w_2$
are elements of degrees, respectively,
$1$
and
$2,$
and the last map lowers degree by
$g+2$
. When
$k=\overline {k}$
, the last map is surjective and the sequence splits.
As the pullback of an open immersion is injective, the invariants of
$\widetilde {\mathcal {H}}_g$
inject into those of
$\mathcal {H}_g$
. We want to understand which cohomological invariants extend to the stack
$\widetilde {\mathcal {H}}_g^o$
introduced in Section 4, which in turn has the same invariants as
$\widetilde {\mathcal {H}}_g$
as they differ by a subset of codimension
$2$
.
Recall that for
$g\geq 3$
odd, there is a morphism
$\mathcal {H}_g\to {\mathrm{B}}\operatorname {\mathrm {PGL}}_2$
that sends an object
$(C\to S,\iota )$
to the Severi–Brauer variety
$P=C/\iota \to S$
(for g even, this map still exists, but it is not very interesting, as in this case
$C/\iota $
is always the projectivization of a rank 2 vector bundle on S).
Proposition 5.2 We have:
-
(1) For every
$g\geq 2$
and for every
$\ell $
-torsion cycle module
${\mathrm{M}}$
, the cohomological invariants of
$\mathcal {H}_g$
with coefficients in
${\mathrm{M}}$
that is pulled back from
${\mathrm{B}}S_{2g+2}$
extend to cohomological invariants of
$\overline {\mathcal {H}}^o_g\backslash \Delta _0^o$
. -
(2) For
$g\geq 3$
odd, the invariant
$w_2$
pulled back along
$\mathcal {H}_g\to {\mathrm{B}}\operatorname {\mathrm {PGL}}_2$
does not extend to an invariant of
$\overline {\mathcal {H}}^o_g\backslash \Delta _0^o$
.
Proof To prove (1), we show that the map
$\mathcal {H}_g\to {\mathrm{B}}S_{2g+2}$
extends to the whole of
$\overline {\mathcal {H}}^o_g\backslash \Delta _0^o$
. Because of the isomorphism
$\overline {\mathcal {H}_g}^o \simeq \mathcal {F}_g^o$
of Proposition 3.1, we can equivalently prove that there is a morphism
$\mathcal {F}_g^o\backslash \mathcal {D}_0^o\to {\mathrm{B}}S_{2g+2}$
extending the morphism on
$\mathcal {H}_g$
.
This last task is readily accomplished: to every object
$(P\to S,L,s)$
of
$\mathcal {F}_g^o\backslash \mathcal {D}_0^o$
, we associate the vanishing locus of s, which by construction is étale over S of degree
$2g+2$
, determining in this way the desired map
$\overline {\mathcal {H}}^o_g\backslash \Delta _0^o\to {\mathrm{B}}S_{2g+2}$
.
Point (2) has already been shown in the proof of [Reference Di Lorenzo and Pirisi8, Theorem A.1].▪
Proposition 5.3 Let
${\mathrm{M}}$
be a
$p^n$
-primary torsion cycle module, with
$2, {\mathrm{char}}(k)\nmid p$
. Then,
$$ \begin{align*}\operatorname{\mathrm{Inv^{\bullet}}}(\overline{\mathcal{H}}_g^o\backslash\Delta_0^o,{\mathrm{M}})\simeq{\mathrm{M}}^{\bullet}(k). \end{align*} $$
Proof We know from [Reference Di Lorenzo and Pirisi9, Proposition 7.5] that
$$ \begin{align} \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{H}_g,{\mathrm{M}})\simeq {\mathrm{M}}^{\bullet}(k)\oplus{\mathrm{M}}^{\bullet}(k)_{p^m}[1] \end{align} $$
as
${\mathrm{M}}^{\bullet }(k)$
-modules, where
$p^m={\mathrm{gcd}}(p^n,2g+1)$
.
Let
$[\mathcal {L}]$
be the generator of the
$p^n$
-torsion subgroup of
$$ \begin{align*}\operatorname{\mathrm{Pic}}(\mathcal{H}_g)\simeq \mathbb{Z}/2^{1+r_g}(2g+1)\mathbb{Z},\end{align*} $$
where
$r_g\in \{0,1\}$
is such that
$r_g\equiv g$
mod
$2$
. Recall [Reference Di Lorenzo and Pirisi9, Lemma 1.13] that
${\mathrm{Inv}}^1(\mathcal {H}_g,{\mathrm{K}}_\ell ) = {\mathrm{H}}^1_{\mathrm{\acute{e}t}}(\mathcal {H}_g, \mu _\ell )$
. Then, in (5.1) above, we have
$$ \begin{align*} {\mathrm{M}}^{\bullet}(k)_{p^m}[1]\simeq {\mathrm{M}}^{\bullet}(k)_{p^{m}}\cdot [\mathcal{L}], \end{align*} $$
with the product induced by the
${\mathrm{K}}_{\ell }$
-module structure of
${\mathrm{M}}$
.
First, suppose that
${\mathrm{M}}={\mathrm{K}}_{p^n}$
. In order to prove that
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
has only trivial invariants, it is enough to check that its Picard group has trivial
$p^n$
-torsion: this easily follows from the computation of
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g)$
(see [Reference Cornalba6, Theorem 2] in characteristic
$0$
and [Reference Di Lorenzo and Pirisi8, Theorem B.1] in positive characteristic). Moreover, this also shows that the ramification of
$[\mathcal {L}]$
along the boundary divisor
$\cup \Delta _i^o \cup \Xi _j^o$
, regarded as an element of
$\mathbb {Z}/p^m\mathbb {Z}$
, is invertible.
For a general
$p^n$
-primary torsion cycle module
${\mathrm{M}}$
and
$\gamma \in {\mathrm{M}}^{\bullet }_{p^m}(k)$
, let
$[\mathcal {L}]\cdot \gamma $
be an invariant of
$\mathcal {H}_g$
that extends to an invariant of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
: its ramification is equal to
$\gamma $
times the ramification of
$[\mathcal {L}]$
, which we know to be invertible as an element of
$\mathbb {Z}/p^m\mathbb {Z}$
. This implies
$\gamma =0$
, and we are done.▪
Corollary 5.4 Let
${\mathrm{M}}$
be a
$p^n$
-primary torsion Galois module, with
${\mathrm{char}}(k), 2\nmid p$
. Then,
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\widetilde{\mathcal{H}}_g,{\mathrm{M}})\simeq{\mathrm{M}}^{\bullet}(k). \end{align*} $$
Proof The stack
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
is an open substack of
$\widetilde {\mathcal {H}}_g$
, hence the invariants of the latter inject into the ring of invariants of the former, which is trivial by Proposition 5.3.▪
5.1 The g even case
Proposition 5.5 Let
$g\geq 2$
be even, and let
${\mathrm{M}}$
be a
$2^n$
-primary torsion Galois module. Then, the exceptional invariant
$\beta _{g+2}$
of
$\mathcal {H}_g$
extends to
$\overline {\mathcal {H}}^o_g\backslash \Delta _0^o$
.
Proof We give the proof for
${\mathrm{M}}={\mathrm{K}}_2$
, the general case follows easily from this one.
Let us recall how
$\beta _{g+2}$
is constructed. Let
$\mathcal {C}\to \mathcal {H}_g$
be the universal smooth conic, and let
$\mathcal {L}$
and
$\mathbf {s}$
be, respectively, the universal line bundle on
$\mathcal {C}$
and the universal global section
$\mathbf {s}$
of
$\mathcal {L}^{\otimes (-2)}$
. The existence of such objects over
$\mathcal {H}_g$
is a consequence of [Reference Arsie and Vistoli3, Proposition 2.2].
Let
$\mathcal {U}$
be the complement in
$\mathcal {C}$
of the vanishing locus
$\mathcal {V}$
of
$\mathbf {s}$
: in particular, we have that
$\mathcal {L}^{\otimes 2}$
is trivial on
$\mathcal {U}$
or equivalently
$\mathcal {L}_{\mathcal {U}}$
belongs to
${\mathrm{H}}^1(\mathcal {U},\mu _2)$
.
Then, we can define a degree
$1$
cohomological invariant t of
$\mathcal {U}$
as follows: given a morphism
$p:\operatorname {\mathrm {Spec}}(K)\to \mathcal {U}$
we set
$t(p)=[\mathcal {L}_K]$
, which belongs to
${\mathrm{H}}^1(\operatorname {\mathrm {Spec}}(K),\mu _2)$
.
Another way to see the invariant t is the following. Given a morphism
$p:\operatorname {\mathrm {Spec}}(K)\to \mathcal {U}$
, consider any lifting
$p':\operatorname {\mathrm {Spec}}(K) \to \mathcal {L}^{\otimes (-2)}$
. The element
$p^{\prime *}(\mathbf {s}) \in K^*$
is well defined up to squares, and it is easy to see that this glues to the cohomological invariant t. In particular, this shows that the boundary of t in the complement of
$\mathcal {U}$
is equal to
$1$
.
This invariant does not extend to a global invariant of
$\mathcal {C}$
, but
$\alpha _{g+1}\cdot t$
does, where
$\alpha _{g+1}$
is the degree
$g+1$
generator of
$\operatorname {\mathrm {Inv^{\bullet }}}({\mathrm{B}}S_{2g+2},{\mathrm{M}})$
. The reason is that
$$ \begin{align*} \partial_{\mathcal{V}}(t \cdot \alpha_{g+1})=\alpha_{g+1}, \end{align*} $$
and the restriction of
$\alpha _g$
to
$\mathcal {V}$
is zero. This is explained in detail in [Reference Di Lorenzo and Pirisi8, Section 2].
Using the well-known property that
$\mathcal {C}$
is actually the projectivization of a rank
$2$
vector bundle, we deduce that
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {C})\simeq \operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {H}_g)$
, hence the construction above gives the exceptional invariant
$\beta _{g+2}$
of
$\mathcal {H}_g$
.
The construction of the exceptional invariant for
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
proceeds along the same lines. Let
$\mathcal {P}_{sm}\to \overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
be the smooth locus of the universal conic, and let
$\mathcal {L}$
and
$\mathbf {s}$
be, respectively, the universal line bundle restricted to
$\mathcal {P}_{sm}$
and the universal global section of
$\mathcal {L}^{\otimes (-2)}$
. Again, the existence of such objects is assured by the isomorphism
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o\simeq \mathcal {F}_g^o$
of Proposition 3.1.
Observe that
$\mathcal {P}_{sm}\to \overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
is a smooth-Nisnevich covering of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
, hence the cohomological invariants of the latter inject into
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {P}^{sm}, {\mathrm{M}})$
via pullback.
Moreover, if an invariant of
$\mathcal {C}$
, the universal smooth conic, can be extended to
$\mathcal {P}_{sm}$
, then it necessarily comes from
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
. Indeed, let
$\xi $
be such an invariant: due to
$\mathcal {P}_{sm}\to \overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
being a smooth-Nisnevich cover, we have that
$\xi $
comes from
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
if and only if it glues, i.e.,
$\operatorname {\mathrm {pr}}_1^*\xi -\operatorname {\mathrm {pr}}_2^*\xi =0$
in
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {P}_{sm}\times _{\overline {\mathcal {H}}_g^o\backslash \Delta _0^o}\mathcal {P}_{sm}, {\mathrm{M}})$
.
On the other hand, the restriction of
$\operatorname {\mathrm {pr}}_1^*\xi -\operatorname {\mathrm {pr}}_2^*\xi $
to
$\mathcal {C}\times _{\mathcal {H}_g}\mathcal {C}$
is zero, as we already know that the invariant glues over
$\mathcal {H}_g$
. As
$\mathcal {C}\times _{\mathcal {H}_g}\mathcal {C}$
is an open substack of
$\mathcal {P}_{sm}\times _{\overline {\mathcal {H}}_g^o\backslash \Delta _0^o}\mathcal {P}_{sm}$
, the invariants of the latter injects into the invariants of the former. Therefore,
$\operatorname {\mathrm {pr}}_1^*\xi -\operatorname {\mathrm {pr}}_2^*\xi $
is zero as well, hence
$\xi $
glues to an invariant of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
, as claimed.
The same argument used to produce the invariant
$\beta _{g+2}$
of
$\mathcal {C}$
applies in this situation, giving us an invariant on
$\mathcal {P}_{sm}$
which obviously extends
$\beta _{g+2}$
. More precisely, one considers the complement of the vanishing locus of
$\mathbf {s}$
in
$\mathcal {P}_{sm}$
: here, we have the invariant t of cohomological degree
$1$
given by
$$ \begin{align*} t= [\mathcal{L}]\in {\mathrm{H}}^1(\mathcal{P}_{sm}\backslash\{\mathbf{s}=0\},\mu_2). \end{align*} $$
The product
$\alpha _{g+1}\cdot t$
, due to the very same reason for which
$\alpha _{g+1}\cdot t$
extended in the smooth case, extends to a global invariant of
$\mathcal {P}_{sm}$
. With a little abuse of notation, this invariant is also denoted
$\beta _{g+2}$
. Then, for what we observed a few lines above, the invariant
$\beta _{g+2}$
must come from
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
. This concludes the proof.▪
Putting together Propositions 5.2 and 5.5, we derive the following result.
Corollary 5.6 Let
$g\geq 2$
be an even integer and
${\mathrm{M}}$
a
$2$
-primary torsion cycle module. Then, we have
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{H}_g,{\mathrm{M}})\simeq \operatorname{\mathrm{Inv^{\bullet}}}(\overline{\mathcal{H}}_g^o\backslash\Delta_0^o,{\mathrm{M}}). \end{align*} $$
We can now apply the formula for the invariants of a root stack to compute the invariants of
$\widetilde {\mathcal {H}}_g$
.
Theorem 5.7 Let
$g\geq 2$
be even and
${\mathrm{M}}$
a cycle module of
$p^n$
-primary torsion, with
$p\neq c$
prime. Then, if
$p=2$
, we have
$$ \begin{align*} \operatorname{\mathrm{Inv^{\bullet}}}(\widetilde{\mathcal{H}}_g,{\mathrm{M}})\simeq \operatorname{\mathrm{Inv^{\bullet}}}(\mathcal{H}_g,{\mathrm{M}}), \end{align*} $$
otherwise the invariants of
$\widetilde {\mathcal {H}}_g$
are trivial.
Proof The complement in
$\widetilde {\mathcal {H}}_g$
of the open substack
$\widetilde {\mathcal {H}}_g^o$
has codimension
$2$
, hence these two stacks have the same cohomological invariants.
We know from Proposition 4.5 that
$\widetilde {\mathcal {H}}_g^o$
is a root stack of order
$2$
of
$\overline {\mathcal {H}}_g^o$
along the divisor
$\Delta _0^o$
. Therefore, by Theorem 2.4, the cohomological invariants of
$\widetilde {\mathcal {H}}_g^o$
are equal to the cohomological invariants of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
whose ramification along
$\Delta _0^o$
is of
$2$
-torsion.
It follows from Corollary 5.6 that the ramification of all the cohomological invariants of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
along
$\Delta _0^o$
is of
$2$
-torsion. This concludes the proof.▪
Remark 5.8 Theorem 5.7 should be compared with [Reference Di Lorenzo and Pirisi8, Theorem A.1], where it is shown that the cohomological invariants of
$\overline {\mathcal {H}}_g$
are all trivial. This does not come as a surprise: the compactification of
$\mathcal {H}_g$
by admissible double coverings preserves most of the structure needed to construct the cohomological invariants of smooth hyperelliptic curves. On the other hand, this is not the case for the compactification by stable hyperelliptic curves.
5.2 The g odd case
When g is odd and
${\mathrm{M}}$
is a
$2^n$
-primary torsion cycle module with
$n\geq 2$
, the stack
$\mathcal {H}_g$
has basically two more invariants than in the even genus case (see Theorem 5.1). One of the two is the invariant
$w_2$
of cohomological degree
$2$
pulled back from
${\mathrm{B}}\operatorname {\mathrm {PGL}}_2$
, and the other comes from the
$4$
-primary torsion of
$\operatorname {\mathrm {Pic}}(\mathcal {H}_g)$
, which for g odd is equal to
$\mathbb {Z}/4(2g+1)\mathbb {Z}$
. This second invariant, denoted
$\alpha _1'$
, is a square root of
$\alpha _1$
, the degree 1 invariant coming from
${\mathrm{B}}S_{2g+2}$
.
We have already seen in Proposition 5.2 that
$w_2$
does not extend to an invariant of
$\overline {\mathcal {H}}_g^o\backslash \Delta _0^o$
, so in order to have a complete picture in degree
${\leq} g+1$
of
$\operatorname {\mathrm {Inv^{\bullet }}}(\overline {\mathcal {H}}_g^o\backslash \Delta _0^o,{\mathrm{M}})$
, we only need to understand if
$\alpha _1'$
extends or not. A bit surprisingly, the value of the genus plays a role here.
Proposition 5.9 Let
$g\geq 3$
be an odd integer, and let
${\mathrm{M}}$
be a
$2$
-primary torsion cycle module. There are submodules
${\mathrm{N}}^{\bullet }_g$
of
${\mathrm{M}}^{\bullet }(k)_2$
such that:
-
(1) If
$g\geq 5$
, we have an exact sequence such that the inverse image of a nonzero element in
$$ \begin{align*} 0 \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}({\mathrm{B}} S_{2g+2},{\mathrm{M}}) \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}(\overline{\mathcal{H}}_g^o\backslash\Delta_0^o,{\mathrm{M}}) \rightarrow {\mathrm{N}}^{\bullet}_g \rightarrow 0\end{align*} $$
${\mathrm{N}}^{\bullet }_g$
has degree at least
$g+2$
.
-
(2) If
$g=3$
, define Then, there is an exact sequence
$$ \begin{align*}I = \alpha_2 \cdot {\mathrm{M}}^{\bullet}(k)_2 \oplus \alpha_3 \cdot {\mathrm{M}}^{\bullet}(k)_2 \oplus \alpha_4 \cdot {\mathrm{M}}^{\bullet}(k)_2 \subset \operatorname{\mathrm{Inv^{\bullet}}}({\mathrm{BS}}_4,{\mathrm{M}}).\end{align*} $$
such that the inverse image of a nonzero element in
$$ \begin{align*}0 \rightarrow {\mathrm{M}}^{\bullet}(k) \oplus {\mathrm{M}}^{\bullet}(k)_4\!\left[1\right] \oplus I \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}(\overline{\mathcal{H}}_3^o\backslash\Delta_0^o,{\mathrm{M}}) \rightarrow {\mathrm{N}}^{\bullet}_3 \rightarrow 0\end{align*} $$
$K_3$
has degree at least
$5$
. The
${\mathrm{M}}^{\bullet }(k)_4\!\left [1\right ]$
is equal to
$\alpha ^{\prime }_1 \cdot {\mathrm{M}}^{\bullet }(k)_4$
, where
$\alpha ^{\prime }_1$
is a square root of
$\alpha _1$
.
Proof First suppose
${\mathrm{M}}={\mathrm{K}}_{2^n}$
,
$n\geq 2$
.
When g is odd, the Picard group of
$\mathcal {H}_g$
is generated by the Hodge line bundle
$\lambda $
[Reference Gorchinskiy and Viviani14]. Consequently,
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g)$
is generated by
$\lambda $
, which extends to the compactification, and the boundary divisor classes
$\left [ \Xi _j \right ], \left [\Delta _i\right ]$
.
In [Reference Cornalba and Harris7], Cornalba and Harris showed that, over a field of characteristic zero, the group
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g) \otimes \mathbb {Q}$
is defined by a single relation. The proof of
$\mathbb {Q}$
-linear independence of the boundary divisors in
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g) \otimes \mathbb {Q}$
contained in [Reference Arbarello, Cornalba and Griffiths2, Chapter 13, Section 8], where is formulated over fields of characteristic zero, can be easily verified to hold true also over fields of characteristic
$c>2$
.
Concerning the Cornalba–Harris relation, this was proved in positive characteristic
$c>2$
by Kausz [Reference Kausz18, Theorem 3.1] and by Yamaki [Reference Yamaki26, Theorem 1.7] in characteristic 2. These results imply that
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g) \otimes \mathbb {Q}$
has rank g also over fields of positive characteristic
$c>2$
.
From [Reference Cornalba6, Theorem 2] in characteristic zero and [Reference Di Lorenzo and Pirisi8, Theorem B.1] in positive characteristic
$c> 2$
, we moreover know that the Picard group of
$\overline {\mathcal {H}}_g$
is torsion-free, and thus free of rank g.
Consider the restriction map
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g^o)_4 \rightarrow \operatorname {\mathrm {Pic}}(\mathcal {H}_g)_4=\mathbb {Z}/4\mathbb {Z}$
. Using the equality
${\mathrm{H}}^1(\mathcal {X},\mu _\ell )={\mathrm{Inv}}^1(\mathcal {X},{\mathrm{K}}_\ell )$
[Reference Di Lorenzo and Pirisi9, Lemma 1.13] and the Kummer exact sequence, we get a commutative diagram:
The rows are exact sequences, the first vertical arrow is an isomorphism (both are equal to
$k^*/(k^*)^4$
), and the second vertical arrow is injective. We conclude that a primitive
$4$
-torsion element in
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g^o\backslash \Delta _0^o)$
, if any, would have to map to a generator of the
$4$
-torsion of
$\operatorname {\mathrm {Pic}}(\mathcal {H}_g)$
, which is generated by
$(2g+1)\lambda $
. In particular, if there is a primitive
$4$
-torsion element in
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g^o\backslash \Delta _0^o)$
, we may get one in the form
$(2g+1)\lambda + \beta $
, where
$\beta $
is a linear combination of the boundary divisors.
If
$g=3$
, it is immediate to verify that
$(2g+1)\lambda + 2\left [ \Xi _1 \right ] + 2\left [\Delta _1\right ]$
is a primitive
$4$
-torsion element. Then, the description of
$\operatorname {\mathrm {Inv^{\bullet }}}(\mathcal {H}_3,D)$
of Theorem 5.1, combined with Proposition 5.2 and the observations above, yields the desired result.
If
$g> 3$
, let
$(2g+1)\lambda + \beta $
be a
$4$
-torsion element. Then,
$$ \begin{align*} (8g+4)\lambda - (8g+4)\lambda - 4\beta = 2\sum_{j=1}^{\lfloor \frac{g-1}{2} \rfloor} (j+1)(g-j)[\Xi_j] + 4\sum_{i=1}^{\lfloor \frac{g}{2} \rfloor} i(g-i)[\Delta_i]. \end{align*} $$
Note that the coefficient of
$\left [\Xi _2\right ]$
on the RHS is equal to
$6(g-2)$
, which is not divisible by
$4$
. This gives a nontrivial relation between the
$\left [\Xi _i\right ]$
and
$\left [\Delta _j\right ]$
classes, which is impossible as we know that
$\operatorname {\mathrm {Pic}}(\overline {\mathcal {H}}_g^o)\otimes \mathbb {Q}$
has rank
$g-1$
.
What we have just shown implies that for
$g>3$
, the ramification of
$\alpha _1'$
along the boundary divisor is of
$2$
-torsion, because we know that
$\alpha _1=2\alpha _1'$
can be extended: this implies that an invariant
$\gamma \cdot \alpha _1'$
of
${\mathrm{M}}^{\bullet }(k)_4\cdot \alpha _1'$
is unramified if and only if
$\gamma $
is a multiple of
$2$
, hence the subgroup of
${\mathrm{M}}^{\bullet }(k)_4\cdot \alpha _1'$
of unramified elements coincides with
${\mathrm{M}}^{\bullet }(k)_2\cdot \alpha _1$
.
As in the genus 3 case, combining this with Proposition 5.2 and Theorem 5.1, we get the claimed description.▪
This allows us to give an almost complete description of the cohomological invariants of
$\widetilde {\mathcal {H}}_g$
for g odd.
Theorem 5.10 Let
$g\geq 3$
be an odd integer, and let
${\mathrm{M}}$
be a
$p^n$
-primary torsion cycle module, with
$p\neq c$
prime. If
$p=2$
, there is a (possibly trivial) submodule
${\mathrm{N}}^{\bullet }_g$
of
${\mathrm{M}}^{\bullet }(k)_2$
and an exact sequence
$$ \begin{align*} 0 \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}({\mathrm{B}} S_{2g+2},{\mathrm{M}}) \rightarrow \operatorname{\mathrm{Inv^{\bullet}}}(\widetilde{\mathcal{H}}_g,{\mathrm{M}}) \rightarrow {\mathrm{N}}^{\bullet}_g \rightarrow 0\end{align*} $$
such that the inverse image of a nonzero element in
${\mathrm{N}}^{\bullet }_g$
has degree at least
$g+2$
. If
$p>2$
, then the cohomological invariants of
$\widetilde {\mathcal {H}}_g$
are trivial.
Proof For
$g\geq 5$
, we can apply exactly the same argument that we used to prove Theorem 5.7.
In the case
$g=3$
, the ramification of the invariant
$\alpha _1'$
along
$\Delta _0$
cannot be of
$2$
-torsion, otherwise we would be able to extend the invariant
$\alpha _1=2\alpha _1'$
to
$\overline {\mathcal {H}}_g$
. Therefore, this case can be worked out as in the proof of Theorem 5.7 as well.▪
5.3 Brauer groups
Theorems 5.7 and 5.10 give us all the information needed to determine the prime-to-the-characteristic part of the Brauer group of
$\widetilde {\mathcal {H}}_g$
.
Theorem 5.11 Let
$g\geq 2$
and set
$c={\mathrm{char}}(k)$
. Let
$^c{\mathrm{Br}}(-)$
denote the prime-to-c part of the Brauer group. Then:
$$ \begin{align*} ^c{\mathrm{Br}}(\widetilde{\mathcal{H}}_g) \simeq {}^{c}{\mathrm{Br}}(k) \oplus H^1(k,\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}/2\mathbb{Z}. \end{align*} $$
In particular, when
$c=0$
, the formula above describes the whole Brauer group of
$\widetilde {\mathcal {H}}_g$
.
Proof The proof works exactly as in [Reference Di Lorenzo and Pirisi9, Theorem 8.1]. The starting point is the isomorphism
$$ \begin{align} {\mathrm{Inv}}^2(\widetilde{\mathcal{H}}_g,\mathbb{Z}/\ell\mathbb{Z}(-1))\simeq {\mathrm{Br}}'(\widetilde{\mathcal{H}}_g)_\ell, \end{align} $$
where, on the right, we have the
$\ell $
-torsion part of the cohomological Brauer group.
Theorems 5.7 and 5.10 can then be used to compute the left-hand side of (5.2): one has only to observe that all the elements in the cohomological Brauer group that do not come from the ground field are of
$2$
-primary torsion.
To conclude, by results of Edidin–Hassett-Kresch–Vistoli and Kresch–Vistoli [Reference Di Lorenzo and Pirisi9, Theorem 1.1], we have that the prime-to-
${\mathrm{char}}(k)$
part of the cohomological Brauer group of a smooth, quasi-projective, tame Deligne Mumford stack belongs to the Brauer group, so in particular
$^c{\mathrm{Br}}(\widetilde {\mathcal {H}}_g)= {^c}{\mathrm{Br}}'(\widetilde {\mathcal {H}}_g)$
.▪
