2.1 Relative moduli spaces of sheaves on surfaces

Let $S\xrightarrow {r} T$ be a smooth family of projective surfaces, i.e. $r$ is smooth and projective. We are mostly interested in the case where the relative anti-canonical bundle $-K_{S/T}$ is $r$-ample or trivial; we then say that $S$ is del Pezzo, respectively K3, over $T$.

For us $T$ will be either $\mathsf {Spec} (\mathcal {O})$, or $\mathsf {Spec}\,F,$ or $\mathsf {Spec}\,k$ where $F$ is a p-adic field, $\mathcal {O}$ its ring of integers, and $k$ a field, possibly of positive characteristic.

2.1.1 Construction of the moduli spaces

Let us fix $\beta$ on $S$ the class of an $r$-effective divisor, ample and base point free, i.e. $\beta _{t}$ is ample and base point free for each $t \in T$. Given $\mathcal {F}$ a sheaf on $S$, flat over $T$, we say that $\mathcal {F}$ is one-dimensional over $T$ of class $\beta$ if

\[ \operatorname{Supp}(\mathcal{F}):=\operatorname{Fit}_0(\mathcal{F})= C \]

with $C\in \mathbb {P}_T(g_*\mathcal {O}_S(\beta ))$. Since the Fitting support is compatible with pull-back [Sta, Tag 07Z6] this means that $\mathcal {F}_{t}$ is supported on a curve of class $\beta _{t}$ on $S_{t}$ for each $t \in T$.

Let $\mathcal {O}_S(1)$ be a $r$-polarization. Given $\mathcal {F}$ a sheaf on $S$ flat over $T,$ the Hilbert polynomial

\[ P_r(n):=\chi(S_{t}, \mathcal{F}_{t}\otimes\mathcal{O}_{S_t}(n))=(\beta_{t}\cdot H_{t})n +\chi(\mathcal{F}_{t}) \]

for $H_{t}\in |\mathcal {O}_{S_t}(1)|$ is locally constant on $T$ (see, for example, [Reference HartshorneHar77, Theorem 9.9]); in particular, $\chi (\mathcal {F}_{t})$ is locally constant.

We denote by $p_r(n)$ the reduced relative Hilbert polynomial obtained from $P_r(n)$ dividing by the coefficient of the leading term.

Definition 2.1.1 (see [Reference Bayer, Lahoz, Macrì, Nuer, Perry and StellariBLM⁺21, Definition 21.11])

Let $\beta,\chi$ and $\mathcal {O}_S(1)$ be as above. We denote by

\begin{align*} \mathbb{M}_{\beta,\chi}(S)\colon(\operatorname{Sch}/T)^{{\rm op}}\rightarrow\operatorname{Gpds} \end{align*}

the functor whose value on $W\to T$ consist of sheaves $\mathcal {F}$ on $S\times _T W,$ flat over $W$ such that for all $w\in W$, $\mathcal {F}_w$ is a Gieseker semi-stable one-dimensional sheaf of class $\beta _w$ and Euler characteristic $\chi \in \mathbb {Z}$. We recall that $\mathcal {F}_w$ is said Gieseker semi-stable if for all proper subsheaves $\mathcal {G}\in \mathcal {F}_w$ we have

(2.1.2)\begin{equation} n +\frac{\chi(\mathcal{G})}{[\operatorname{Supp}(\mathcal{F}_w)]\cdot H_{w}} \leq p_r(n), \end{equation}

and Gieseker stable if the inequality is strict. Finally, we say that $\mathcal {F}_w$ is geometrically Gieseker stable if its pull-back $\mathcal {F}_{\overline {w}}$ to the algebraic closure is Gieseker stable.

An object $\mathcal {F}\in \mathbb {M}_{\beta,\chi }(S)(W)$ is called a family of Gieseker semi-stable sheaves of class $(\beta,\chi )$ over $W$.

Here $\mathbb {M}^{\rm st}_{\beta,\chi }(S)$ is the sub-functor of families of Gieseker geometrically stable objects.

Theorem 2.1.3 [Reference LangerLan04b, Reference LangerLan04a, Reference Bayer, Lahoz, Macrì, Nuer, Perry and StellariBLM⁺21]

In the notation above:

1. (i) $\mathbb {M}_{\beta,\chi }(S)$ and $\mathbb {M}^{\rm st}_{\beta,\chi }(S)$ are algebraic stacks of finite type over $T$; the same statement hold for the $\mathbb {G}_m$-rigidification $\mathcal {M}_{\beta,\chi }(S)$ of $\mathbb {M}_{\beta,\chi }(S)$;

2. (ii) there exists a projective $T$-scheme $\operatorname {M}_{\beta,\chi }(S)$ which uniformly represent the set valued functor over $T$-schemes associating to $W$ the set of $\text {S}$-equivalence classes of families of Gieseker semi-stable sheaves on the fibers of $S\times _T W\to W$ with Hilbert polynomial $p_r(n)$;

3. (iii) $\mathbb {M}^{\rm st}_{\beta,\chi }(S)$ is a $\mathbb {G}_m$ gerbe over its coarse moduli space $\operatorname {M}^{\rm st}_{\beta,\chi }(S)$, which is a quasi-projective scheme over $T$ representing isomorphism classes of Gieseker geometrically stable sheaves on the fibers of $S\times _T W\to W$ with Hilbert polynomial $p_r(n)$;

4. (iv) $\mathbb {M}_{\beta,\chi }(S)$ is universally closed over $T$ and, thus, universally closed over $\operatorname {M}_{\beta,\chi }(S)$.

Proof. (i) A far more general version of this statement is proved in [Reference Bayer, Lahoz, Macrì, Nuer, Perry and StellariBLM⁺21, Theorem 21.24]. For the benefit of the reader, we recall the key steps of the proof in the special case of Langer [Reference LangerLan04a], which is the only one we need.

Let $E$ be a coherent sheaf on $S$ flat on $T$, and let $\operatorname {Quot}_{T,p_r(n)}(E)\colon (\operatorname {Sch}/T)^{{\rm op}}\rightarrow \operatorname {Sets}$ be the relative Quot functor, whose $W\to T$ valued points are quotients $E_W\to Q\to 0$ with $Q$ a coherent sheaf on $S\times _T W$, flat over $W$, with Hilbert polynomial $p_r(n)$ along the fibers of $S\times _T W\xrightarrow {r_W} W$. As proved for example in [Sta, Tag 09TQ], $\operatorname {Quot}_{T,p_r(n)}(E)$ is represented by an algebraic space over $T$ locally of finite presentation.

By Langer's results [Reference LangerLan04a, § 3], families of Gieseker semi-stable sheaves on the fibers of $S\xrightarrow {r} T$ with fixed Hilbert polynomial $p_r(n)$ are bounded, i.e. any $\mathcal {F}\in \mathbb {M}_{\beta,\chi }(S)$ is a quotient of $E=\mathcal {O}_S(-N)^{\oplus F(N)}$ for some fixed $N.$

Furthermore, as shown in [Reference LangerLan04a, § 4], the open subsets $\mathcal {R}^{\rm st}\subseteq \mathcal {R}^{{\rm ss}}\subset \operatorname {Quot}_{T,p_r(n)}(E)$ of Gieseker stable/semi-stable sheaves on the fibers of $S\xrightarrow {r} T$, correspond to the opens of stable/semi-stable points with respect to the natural action of the $T$-smooth group $\operatorname {GL}(F(N))$.

It thus follows that $\mathbb {M}_{\beta,\chi }(S)\cong [ \mathcal {R}^{{\rm ss}}/ \operatorname {GL}(F(N))]$ and $\mathbb {M}^{\rm st}_{\beta,\chi }(S)\cong [ \mathcal {R}^{\rm st}/ \operatorname {GL}(F(N))]$ as stacks over $\operatorname {Sch}/T$. The statement for the rigidification $\mathcal {M}_{\beta,\chi }(S)$ now follows immediately as the latter is simply given by $[ \mathcal {R}^{{\rm ss}}/ \operatorname {PGL}(F(N))]$.

(ii) The projective $T$-scheme $\operatorname {M}_{\beta,\chi }(S)$ is the GIT quotient $\mathcal {R}^{{\rm ss}}/ / \operatorname {GL}(F(N))$ this exists and has the claimed property by [Reference SeshadriSes77, Theorem 4], as also argued by Langer in the proof of [Reference LangerLan04a, Theorem 4.1].

(iii) The statement follows by the construction explained in statement (i).

(iv) The universal closedness over $T$ follows from a relative version of Langton's theorem. The reader can see [Reference LangtonLan75] or [Reference Huybrechts and LehnHL97, Theorem 2.B.1] for the proof over an algebraic closed field, and [Reference Bayer, Lahoz, Macrì, Nuer, Perry and StellariBLM⁺21, Lemma 21.22] for the general case.

Given the GIT construction of the moduli space recalled above, the universal closedness of $\mathbb {M}_{\beta,\chi }(S)\to \operatorname {M}_{\beta,\chi }(S)$ also follows from [Reference Alper, Halpern-Leistner and HeinlothAHH23, Theorem A.8].

2.1.2 Deformation theory

Let $S\xrightarrow {r} T$ be as in the previous section, and let $\mathcal {F}$ be a coherent sheaf on $S$ flat over $T$. We denote by $\operatorname {Ext}^i_S(\mathcal {F},\mathcal {F})$ the $T$ sheaf obtained by taking the $i$th cohomology of the complex $Rr_*\operatorname {R}\mathcal {H}om(\mathcal {F},\mathcal {F}),$ i.e.

\[ \operatorname{Ext}^i_S(\mathcal{F},\mathcal{F}):= \text{H}^i(Rr_*\operatorname{R}\mathcal{H}om(\mathcal{F},\mathcal{F})). \]

By [Reference LieblichLie06, § 3] (see also [Reference Huybrechts and ThomasHT10, § 4]) the sheaves $\operatorname {Ext}^i_S(\mathcal {F},\mathcal {F})$ control the deformation theory of $\mathcal {F}$ over $T$, namely:

1. (i) there is a natural isomorphism $\operatorname {Ext}^0_S(\mathcal {F},\mathcal {F})\cong \operatorname {End}(\mathcal {F},\mathcal {F})$;

2. (ii) the first-order deformations of $\mathcal {F}$ are given by $\operatorname {Ext}^1_S(\mathcal {F},\mathcal {F})$;

3. (iii) the obstruction to deformations are contained in $\operatorname {Ext}^2_S(\mathcal {F},\mathcal {F})$.

Lemma 2.1.4 Let $S\xrightarrow {r} T$ be a smooth family of del Pezzo surfaces. Then ${\mathbb {M}}_{\beta,\chi }(S)$ is smooth over $T$.

Proof. It suffices to show that the obstruction $\operatorname {Ext}^2_S(\mathcal {F},\mathcal {F})$ is zero. We argue that $\operatorname {Ext}^2_S(\mathcal {F},\mathcal {F})\otimes k(\bar {t})=0$ for any geometric point $\bar {t}\in T$. By [Reference Altman and KleimanAK80, Theorem 1.9], the cohomology and base change property for Ext sheaves implies

\[ \operatorname{Ext}^2_S(\mathcal{F},\mathcal{F})\otimes k(\bar{t})= \operatorname{Ext}^2_{S_{\bar{t}}}(\mathcal{F}_{\bar{t}},\mathcal{F}_{\bar{t}}), \]

where $\mathcal {F}_{\bar {t}}$ is, by definition, a Gieseker semi-stable sheaf of class $\beta _{\bar {t}}$ and Euler characteristic $\chi$ on $S_{\bar {t}}.$ The vanishing of the right-hand side is then classical; see, for example, [Reference Maulik and ShenMS23, Lemma 2.5].

2.1.4 Hilbert–Chow map

We keep the notation of the previous section. Let us denote by $B_T=\mathbb {P}_T(g_*\mathcal {O}_S(\beta )),$ and by $\mathcal {C}\to B_T$ the universal curve. As for the case $T=\mathsf {Spec}(\mathbb {C})$ there is a morphism

\[ \mathbf{h}_{\chi}\colon \mathbb{M}_{\beta,\chi}(S)\to B_T, \]

which associate to $\mathcal {F}$ its fitting support and factors through the moduli space

\[ h_{\chi}\colon \operatorname{M}_{\beta,\chi}(S)\to B_T. \]

We call $h_{\chi }$ the Hilbert–Chow morphism and $\mathbf {h}_{\chi }$ the stacky Hilbert–Chow morphism. Since both $\operatorname {M}_{\beta,\chi }(S)$ and $B_T$ are projective over $T,$ $h_{\chi }$ is proper over $T$.

For $b\colon T\to B_T$ a section such that $\mathcal {C}_b/T$ is smooth, the fiber $\mathbf {h}_{\chi }^{-1}(b)$ is the Picard stack $\mathfrak {P}ic^{\chi +g-1}_{\mathcal {C}_b/T}$ where $g$ is the genus of the curve, and $h^{-1}_{\chi }(b)$ the Picard space $\operatorname {Pic}^{\chi +g-1}_{\mathcal {C}_b/T},$ i.e. the $\mathbb {G}_m$-rigidification.

For $b\colon T\to B_T$, such that $\mathcal {C}_b/T$ is integral, the fiber of the stacky Hilbert–Chow morphism is some universally closed stack over $T$ containing $\mathfrak {P}ic^{\chi +g-1}_{\mathcal {C}_b/T}$ as an open sub-stack, and $h^{-1}_{\chi }(b)$ is the corresponding GIT quotient, which is the compactification of the Picard space $\operatorname {Pic}^{\chi +g-1}_{\mathcal {C}_b/T}$ constructed by Altman and Kleiman [Reference Altman and KleimanAK80].

In general, for $\mathcal {C}_b/T$ non-integral not all line bundles of total degree $\operatorname {totdeg}=\chi +g-1$ are Gieseker semi-stable and one cannot say much about the fiber, except that it is a moduli stack of semi-stable sheaves of pure dimension 1. In the reduced case, moduli stacks and moduli spaces of rank-one semi-stable torsion free sheaves on $\mathcal {C}_b/T$ have been studied, and in the following we recall some known results.

2.2 The open sub-stack of invertible sheaves

We denote by

\[ \mathfrak{U}_{\beta,\chi}\subseteq\mathfrak Pic^{\chi+g-1}_{\mathcal{C}/B_T} \]

the open sub-stack corresponding to line bundles of total degree $\chi +g-1$, which are Gieseker semi-stable on the fibers of $\mathcal {C}/B_T$.

The rigidification $\mathcal {U}_{\beta,\chi }\subseteq \mathcal {M}_{\beta,\chi }(S)$ is stabilizer free, and $\mathfrak {U}_{\beta,\chi }\to \mathcal {U}_{\beta,\chi }$ is simply a $\mathbb {G}_m$ gerbe. It follows from Theorem 2.1.3, that if we further restrict to geometrically stable line bundles $\mathfrak {U}^{\rm st}_{\beta,\chi }$, the rigidification $\mathcal {U}^{\rm st}_{\beta,\chi }$ is a good moduli space for the moduli stack and it is isomorphic to an open in $\operatorname {M}_{\beta,\chi }(S)$.

The opens $\mathcal {U}^{\rm st}_{\beta,\chi }\subseteq \mathcal {U}_{\beta,\chi }$ will play a crucial role in the construction of the canonical measure. To this end, we need to prove the following codimension estimate.

The proof of this statement is somewhat involved and it will occupy the rest of the section. The reader who wish to get to the core of the paper quickly may skip this in a first reading.

Working on $T=\mathsf {Spec}(\bar {k})$ allows us to appeal to the results of [Reference Maulik and ShenMS23, Reference YuanYua23]. It is proved in [Reference YuanYua23], with techniques inspired by [Reference SchiffmannSch16], that for smooth projective surfaces $S$ over an algebraically closed field $\bar {k}$ with canonical $K_S\leq 0$ the stacky Hilbert–Chow morphism $\mathbf {h}_{\chi }$ (and, thus, also $h_{\chi }$) is equidimensional.

Theorem 2.2.3 [Reference YuanYua23, Theorem 1.2, Corollary 1.3]

Let $S$ be a smooth projective surface and $\beta =\sum _{i=1}^n m_i\beta _i$ an ample effective curve class with $\beta _i$ pairwise distinct. If $K_S\cdot \beta _i\leq 0$ for all $i$, then

\[ \dim(\mathbf{h}^{-1}_{\chi}(b))\leq g-1 \]

for any $b\in |\beta |$, where $g=\frac {1}{2} \beta \cdot (\beta +K_S)+1$.

Using the dimension estimate, Maulik and Shen proved the following.

We use the equidimensionality of the (stacky) Hilbert–Chow as follows. First, by Bertini's theorem, since $\beta$ is ample and base point free, there is a dense open $B^{sm}\subset B$ such that for $b\in B^{sm}$ the curve ${\mathcal {C}}_b$ is a non-singular, integral curve of genus $g=\frac {1}{2} \beta \cdot (\beta +K_S)+1$. In this case, $\mathbf {h}_{\chi }^{-1}(b)\cong \mathfrak Pic^{\chi +g-1}_{{\mathcal {C}}_b},$ which is simply a $\mathbb {G}_m$ gerbe over the degree $\chi +g-1$ component of the Picard group.

In particular, $\mathbf {h}_{\chi }^{-1}(B^{sm})$ is contained in the open sub-stack of (stable) line bundles. Note, furthermore, that since the number of connected components is lower semi-continuous in a proper flat family (see, for example, [Reference Deligne and MumfordDM69, Theorem 4.17]) all the curves in the linear system are connected.

To prove the codimension estimate, we now look at the fibers of the Hilbert–Chow morphism over the locus of singular curves.

The complement of $B^{sm}$ in $B$ has codimension 1, and by equidimensionality of $\mathbf {h}_{\chi }$ so does $\mathbf {h}^{-1}_{\chi }(B\setminus B^{sm})$.

Let us denote by $V\subseteq B\setminus B^{sm}$ the locus of non-reduced curves. We prove in § 2.2.1 that this has always codimension at least 2 in $B$ for $S$ del Pezzo, provided $\beta$ is a base point free ample class. For $S$ a K3, the codimension of $V$ is at least two by assumption; see Remark 2.1.8 for examples where the assumption is verified. Then $\mathbf {h}^{-1}_{\chi }(V)$ has codimension at least two, and it can be discarded for the purposes of estimating the codimension of the complement of $\mathfrak {U}_{\beta,\chi }$.

Finally, we study the fibers over $(B\setminus B^{sm})\setminus V$ in § 2.2.2 and prove that the locus of line bundles in $\mathbf {h}_{\chi }^{-1}(b)$ is dense for all $b\in (B\setminus B^{sm})\setminus V$.

In conclusion, the complement of $\mathfrak {U}_{\beta,\chi }$ is contained in the union of $\mathbf {h}^{-1}_{\chi }(V)$ and a divisor $\mathfrak {D}\subset \mathbf {h}^{-1}_{\chi }(B\setminus B^{sm})$. These have both codimension at least two, which allows us to conclude.

2.2.1 Excluding the non-reduced locus

Consider the stratification of $B$ given by

\[ B=\bigsqcup_{\underline{\beta}} B_{\underline{\beta}}, \]

where $\underline {\beta }=\{m_1\beta _1,\ldots,m_s\beta _s\}$ with $s\geq 1$, $m_i\geq 1$, and $\beta _i$ are distinct curve classes such that there exists an integral curve in $|\beta _i|$ for each $i$; $B_{\underline {\beta }}$ is the locally closed strata in $B$ of curves of type $\underline {\beta }$, i.e.

\begin{align*} B_{\underline{\beta}}=\bigg\{C=\sum_i m_i C_i\in |\beta|,\ C_i\in |\beta_i|\bigg\}, \end{align*}

with $C_i$ integral.

Let $Z$ be an irreducible component in $\overline {B}_{\underline {\beta }}$; we denote by $\delta _Z$ the dimension of the affine part of the Jacobian of $\mathcal {C}_b$ for $b\in Z^{\circ }\subseteq \overline {B}_{\underline {\beta }}$ generic, i.e.

\[ \delta_Z=\dim(\operatorname{Pic}^0(\mathcal{C}_b))^{{\rm Aff}}. \]

Lemma 2.2.7 Let $\beta$ be a base point free ample class on a smooth del Pezzo surface. The locus $V\subset B$ of non-reduced curves has codimension at least two.

Proof. The locus $V$ of non-reduced curves is the union of closed strata $\overline {B}_{\underline {\beta }}$ with $\underline {\beta }$ having at least one $m_i\geq 2$. We want to show that the codimension of any irreducible component $Z$ of such a stratum $\overline {B}_{\underline {\beta }}$ is at least two in $B$. The proof follows from the following inequality which is proved in [Reference Maulik and ShenMS23, Proposition 4.3]:

(2.2.8)\begin{equation} \operatorname{codim}(Z)\geq -K_S\cdot \bigg(\sum_i(m_i-1)\beta_i\bigg)+\delta_Z\geq 1+\delta_Z, \end{equation}

where the second inequality holds since $-K_S$ is ample, $m_i-1\geq 0$ for each $i$ and $m_i-1\geq 1$ for at least one $i$ and, thus,

\[ -K_S\cdot \bigg(\sum_i(m_i-1)\beta_i\bigg)\geq 1; \]

since $-K_S$ is ample, the equality can hold only if all but one $m_i$ are equal to $1$.

If the latter inequality is strict, we are done; otherwise we can assume that $m_1=2$ and $m_i=1$ for any other $i$. Again, if $\delta _Z\geq 1$ we are done.

To complete the proof we only need to look the cases where $m_1=2$, $m_i=1$ for any other $i$, and $\delta _Z=0$.

Let us denote by $C:=\mathcal {C}_b$ for $b\in B_{\{2\beta _1,\beta _2,\ldots,\beta _s\}}$ a generic point in this stratum, by $C'\subseteq C$ the reduced subcurve, and by $\beta '=\beta -\beta _1$ its class.

We have two short exact sequences:

\begin{align*} & 0\to \mathcal{O}_{C_1}(-\beta'\cdot\beta_1)\to \mathcal{O}_{C}\to\mathcal{O}_{C'}\to 0\\ & 0\to O_{C'}\to \oplus\bigoplus_{i=1}^s\mathcal{O}_{\widetilde{C}_i}\to \bigoplus_{p\in C^{\prime{\rm sing}}} \mathcal{O}_p^{\delta(p)}\to 0. \end{align*}

The latter is simply the normalization sequence for $\widetilde {C'}=\sqcup _{i=1}^s\widetilde {C}_i\to C'$ (see [Reference LiuLiu06, § 7.5]). We denoted by

\begin{align*} \delta(p)=\operatorname{length}_k(\mathcal{O}_{\widetilde{C'},p}/\mathcal{O}_{C'}) \end{align*}

the so-called delta invariant of the singularity.

By [Reference LiuLiu06, Lemma 5.11], there is a surjective morphism of abelian groups

\[ \operatorname{Pic}^0(C)\to \operatorname{Pic}^0(C')\to 0 \]

with kernel an affine unipotent group of dimension $\operatorname {dim}\operatorname {H}^1(C_1,\mathcal {O}_{C_1}(-\beta '\cdot \beta _1))$. Moreover, by [Reference LiuLiu06, Theorem 5.19], $\operatorname {Pic}^0(C)$ is an extension of $\times \operatorname {Pic}^0(\widetilde {C}_i)$ by an affine group of dimension

(2.2.9)\begin{equation} \bigg(\sum_{p\in C^{\prime{\rm sing}}} \delta (p)\bigg) -s+1, \end{equation}

and, thus,

\[ \delta_Z= \bigg(\sum_{p\in C^{\prime{\rm sing}}}{\delta (p)}\bigg) -s+1+ \operatorname{dim}\operatorname{H}^1(C_1,\mathcal{O}_{C_1}(-\beta'\cdot\beta_1)). \]

One can prove by induction on the number $s$ of irreducible components that $\sum _{p\in C^{\prime{\rm sing}}} \delta (p) -s+1\geq 0$ with equality holding only if the only singularities are nodes and $C'$ is of compact type, namely its dual graph is a tree. For the induction step look at the partial normalization of $C'$ given by $R\sqcup \overline {C'\setminus R}$ for $R$ an irreducible component.

Let us also analyze the degree of $\mathcal {O}_{C_1}(-\beta '\cdot \beta _1)$. We can write $\beta '=\beta ''+\beta _1$ with $\beta ''$ and effective curve class, and we can assume that $\beta ''\cdot \beta _1>0$ otherwise $C'$ is not connected and we can estimate $\delta _Z$ separately on each connected component. From $0\leq g(\beta _1)=\frac {1}{2} K_S\cdot \beta _1+ \frac {1}{2} \beta ^2_1+1$ and the fact that $K_S$ is anti-ample, it follows that $\beta ^2_1\geq -1$. In particular, $-\beta '\cdot \beta _1=- \beta ''\cdot \beta _1- \beta ^2_1\leq 0$ with equality holding only in the case $\beta _1^2=-1$, and $\beta ''\cdot \beta _1=1$. However, if the equality is verified, then $\beta \cdot \beta _1 <0$, contradicting the ampleness of $\beta$.

We can thus assume that $\mathcal {O}_{C_1}(-\beta '\cdot \beta _1)$ has strictly negative degree $d.$ By Riemann–Roch,

\[ \operatorname{dim}\operatorname{H}^1(C_1,\mathcal{O}_{C_1}(-\beta'\cdot\beta_1))=g(C_1)-d-1, \]

which is always positive unless $g(C_1)=0$ and $d=-1$. In particular, in this case we must have: $\beta ''\cdot \beta _1=1$, $\beta _1^2=0$ and $K_S\cdot \beta _1=-2$. We can thus give a better estimate of the codimension of the component $Z$ in $B$ corresponding to such curves. By [Reference Maulik and ShenMS23, Lemma 2.1], given $\gamma$ an effective curve class, the dimension of the linear system $|\gamma |$ is $\frac {1}{2}\gamma \cdot (\gamma -K_S)$. Thus, $\dim B=\frac {1}{2}\beta \cdot (\beta -K_S)$, whereas the dimension of $Z$ is $\frac {1}{2}\beta '\cdot (\beta '-K_S)$. We compute

\begin{align*} \operatorname{codim}(Z)&=\tfrac{1}{2}\beta\cdot(\beta-K_S)-\tfrac{1}{2}\beta'\cdot(\beta-K_S)\\ &=\tfrac{1}{2}[(\beta'+\beta_1)^2 -K_S\cdot\beta-\beta'^2+K_S\cdot\beta')\\ &=\tfrac{1}{2}[2\beta'\cdot\beta_1-K_S\cdot\beta_1]=2. \end{align*}

The assumption in case (ii) and Lemma 2.2.7 in case (i) of Proposition 2.2.1 allows us to disregard completely the locus $V$ of non-reduced curve.

2.2.2 Reduced curves

If ${\mathcal {C}}_b$ is integral, then any rank-one torsion free sheaf $\mathcal {F}$ is strictly stable, independently from the choice of polarization. This is easily seen noting that for any proper sub-sheaf $\mathcal {G}\hookrightarrow \mathcal {F}$, the cokernel is supported on a dimension $0$ sub-scheme, which imply the strict inequality in (2.1.2). In particular, $\mathbf {h}_{\chi }^{-1}(b)$ is a $\mathbb {G}_m$ gerbe over the compactification $\overline {\mathrm {Pic}}^{\chi +g-1}_{{\mathcal {C}}_b}$of $\mathrm {Pic}^{\chi +g-1}_{{\mathcal {C}}_b}$ of Altman and Kleiman [Reference Altman and KleimanAK80]. It was proved in [Reference RegoReg80, Theorem A] that for an integral curve with only planar singularities $\overline {\mathrm {Pic}}^{\chi +g-1}_{{\mathcal {C}}_b}$ is irreducible and, thus, the open of line bundles is dense.

We are left to study the fiber $\mathbf {h}^{-1}_\chi (b)$ for $b \in |\beta |$ a reduced but possibly reducible curve.

Proposition 2.2.10 Let $C$ be a reduced curve with planar singularities and polarization $H_C$. Let ${\mathbb {M}}_{\chi }(C)$ be the moduli stack of Gieseker semi-stable rank-one torsion free sheaves on $C$. Then the open substack $\mathfrak {U}_{\chi }$ of semi-stable invertible sheaves is dense in ${\mathbb {M}}_{\chi }(C)$.

Proof. To show that $\mathfrak {U}_{\chi }\subseteq {\mathbb {M}}_{\chi }(C)$ is dense, we need to show that any point $F$ in ${\mathbb {M}}_{\chi }(C)$ is the limit of a family of line bundles on $C$ in a one-parameter family. This now follows from the deformation theory of rank-one sheaves on planar curves. Let $q_1,\ldots q_k$ be the points (necessarily) in the singular locus $C^{{\rm sing}}$ where $F$ fails to be a line bundle. We look at the forgetful morphism

\[ \operatorname{Def}_{F}\xrightarrow{l}\prod_{i=1,\dots k} \operatorname{Def}_{F_{q_i}}, \]

where $F_{q_i}$ is the stalk of $F$ at $q_i$. Since we are looking at curves with planar singularities, the forgetful morphism is smooth [Reference Fantechi, Goettsche and Van SratenFGV99]. Thus, it is sufficient to show that the generic element in each local deformation space $R_{F_{q_i}}$ corresponds to a invertible module. However, this follows from the analogous statement for compactified Jacobians of integral planar curves [Reference RegoReg80]. More explicitly: in order to describe $R_{F_{q_i}}$, consider an integral curve $C_{q_i}$ with $q_i$ as unique singular point. By [Reference RegoReg80] the Picard is dense in the compactified Jacobian, which in particular implies that the generic point in $R_{F_{q_i}}$ parametrizes a line bundle.

This concludes the proof of Proposition 2.2.1.

Remark 2.2.11 When there are no strictly semi-stables, and thus ${\mathbb {M}}_{\chi }(C)$ is a $\mathbb {G}_m$ gerbe on $h_{\chi }^{-1}(C)=\operatorname {M}_{\chi }(C)$, it is stated in [Reference Melo, Rapagnetta and VivianiMRV17, Corollary 2.20] or also in [Reference López-MartínLop05, Theorem 4.5] that stable line bundles are dense in the moduli space.

When there are strictly semi-stables (which is the most interesting case for us) the density statement at the level of moduli spaces $\operatorname {M}_{\chi }(C)$ is false; the description of the moduli space in [Reference López-MartínLop05, Theorem 4.5] explains this fact.

2.3 Moduli of Higgs bundles

Let $C\to T$ be a smooth projective curve. As for the case of sheaves on surfaces, there is an adaptation of theory of moduli stack and moduli spaces of Higgs bundles relative to a base $T$. The case where $T$ is a field of positive characteristic is already considered in [Reference NgôNgô06].

We refer to [Reference Groechenig, Wyss and ZieglerGWZ20b, § 7.1] and references therein for the adaptation of the classical theory to the case of smooth curves over $T=\mathsf {Spec} (O_F),\mathsf {Spec}(F)$.

We denote by ${\mathbb {M}}_{r,\chi }(C)$ and $\operatorname {M}_{r,\chi }(C)$ the moduli stack and moduli space of semi-stable Higgs bundles $(\mathcal {E}, \Theta \colon \mathcal {E}\to \mathcal {E}\otimes \mathcal {O}_C(D))$ of rank $r$ and Euler characteristic $\chi$ on the fibers of $C\to T$. Here $D$ is a relative effective divisor of degree $d=\text {deg}(D) \geq 2g-2$ on the fibers of $C\to T$. If equality holds, we assume that $D$ is the canonical divisor and $g \geq 2$.

The stability is the (fiber-wise) Gieseker stability for sub-bundles preserved by the Higgs field, i.e. $(\mathcal {E},\Theta )$ is semi-stable if for any $\mathcal {F}\subseteq \mathcal {E}$ such that $\Theta \rvert _{\mathcal {F}}\colon \mathcal {F}\to \mathcal {F}\otimes \mathcal {O}_C(D)$

\[ \frac{\chi}{r(\mathcal{F})}\leq\frac{\chi}{r}. \]

The Hitchin fibration is the morphism:

\[ H\colon \operatorname{M}_{r,\chi}(C)\to B_T:=\bigoplus_{i=1}^r H^0(C,\mathcal{O}_C(id)) \]

associating to $(\mathcal {E},\Theta )$ the coefficients of its characteristic polynomial.

Via the spectral correspondence [Reference HitchinHit87], ${\mathbb {M}}_{r,\chi }(C)$ can be interpreted as a moduli stack of one-dimensional sheaves $\mathcal {F}$ on the non-compact surface

\[ S_C:=\operatorname{Tot}(\mathcal{O}_C(D))\xrightarrow{\pi} C \]

with $[\operatorname {Supp}(\mathcal {F})]=r[C]$. The hypothesis on the degree of $D$ implies that $S_C$ is del Pezzo, which, in turn, guarantees the smoothness of the moduli stack ${\mathbb {M}}_{r,\chi }(C)$ and its ${\mathbb {G}}_m$-rigidification ${\mathcal {M}}_{r,\chi }(C)$ for each $(r,\chi )$ similar to Lemma 2.1.4.

Moreover, via the correspondence we can identify the fibers of the Hitchin fibration $H$ with the Simpson compactified Jacobians of the spectral curve $C_{P(t)}\subseteq S_C$ cut out by the polynomial $\text {det}(\pi ^*\Theta -t\text {Id})$.

In the following theorem we collect the analogous statements of Theorem 2.1.3 and Proposition 2.2.1 for the moduli stack of meromorphic Higgs bundles. Since the proof is essentially the same as in the previous section we do not repeat it here.

Remark 2.3.2 In the case of Higgs bundle, no extra assumption is necessary to show normality of the moduli stack. We can understand that from at least two points of view: in the language of [Reference Davison, Hennecart and MejiaDHM22], the moduli stack of Higgs bundles satisfies the totally negative CY2 property [Reference Davison, Hennecart and MejiaDHM22, § 7]; under this additional property the normality follows from the results of [Reference VernetVer22].

Otherwise, we note that the locus $V$ corresponding to non-reduced spectral curves in the Hitchin base $V$ has always codimension at least two and, thus, as in Proposition 2.2.1, the locus of Higgs bundle whose corresponding spectral sheaf is a line bundle has codimension at least two in the stack, which allows us to argue as in Theorem 2.1.7.

The estimate on the codimension on $V$ can be given by an explicit calculation:

By Riemann–Roch the base $B$ of the Hitchin fibration has dimension

\[ \dim B=d\frac{r(r+1)}{2}+r(1-g), \]

where $r$ is the fixed rank and $d=\text {deg}\mathcal {O}_C(D)\geq 2g-2$. The non-reduced locus is the (closure of) the strata $V_{kr_1,r_2} \subseteq B$ whose points correspond to polynomials in $t$ admitting a factorization of the form

(2.3.3)\begin{align} P(t)=P_1(t)^kP_2(t),\quad \text{deg}(P_1(t))=r_1\geq 1,\enspace k\geq 2,\quad \text{deg}(P_2(t))=r_2\geq 0,\enspace kr_1+r_2=r. \end{align}

Thus, the codimension of the non-reduced locus is the minimum of the codimension of the strata $V_{kr_1,r_2}$. Now, the dimension of the strata is

\[ \dim V_{kr_1,r_2}=d\frac{r_1(r_1+1)}{2}+r_1(1-g)+d\frac{r_2(r_2+1)}{2}+r_2(1-g), \]

and it is easy to see by direct computation that

\begin{align*} \dim B-\dim \dim V_{kr_1,r_2}\geq \frac{4g+1}{2}, \end{align*}

for $d\geq 2g-1$ and

\begin{align*} \dim B-\dim \dim V_{kr_1,r_2}= 3(g-1), \end{align*}

for $d=2g-2$.

3.1 The measure on the moduli spaces

Let ${\mathcal {M}}$ be a normal algebraic stack with a finite-type morphism to $\mathsf {Spec}({\mathcal {O}})$ of dimension $n$. We assume that there exists a (quasi-)projective variety $\operatorname {M}$ and a morphism $\pi :{\mathcal {M}} \rightarrow M$ satisfying the following conditions: there exists a dense open substack $U' \subset {\mathcal {M}}$, smooth over ${\mathcal {O}}$, that is stabilizer free, i.e. an algebraic space, and that the complement of $U'$ has codimension at least $2$ in ${\mathcal {M}}$; there exists a second dense open $U \subset U'$ such that $\pi$ restricted to $U$ induces an isomorphism onto its image in $M$ which we also denote by $U$.

The goal of this section is to construct a canonical measure on the analytic manifold

\[ M^\natural = M({\mathcal{O}}) \cap U(F). \]

For this we need the following extra assumption.

Note that Assumption 3.1.1 is, in particular, satisfied if ${\mathcal {M}} \to M$ is universally closed [Sta, 01KA].

We fix a Zariski-open cover $U'= \bigcup _{i\in I} U'_i$ such that on each $U'_i$ the relative canonical bundle $\Omega _{U'/{\mathcal {O}}}^{n}$ is trivial and pick a trivializing (i.e. non-vanishing) $n$-form $\omega _i$ on $U'_i$. Let $U_i = U \cap U'_i$. We have a decomposition

\begin{align*} M^\natural = \bigcup_i M_i^\natural, \end{align*}

with $M_i^\natural = M({\mathcal {O}}) \cap U_i(F)$ and we claim that the measures $\mu _{\omega _i}$ on $M_i^\natural$ glue to a measure on $M^\natural$ that is independent of the choice of $(\omega _i)_i$.

Proposition 3.1.2 (The canonical measure on $M^\natural$)

For all $i,j \in I$ and any measurable subset $A \subset M_i^\natural \cap M_j^\natural$ we have

\[ \mu_{\omega_i}(A) = \mu_{\omega_j}(A). \]

In particular, the family $(\mu _{\omega _i})_{i\in I}$ glues to a Borel measure $\mu _{{\rm can},{\mathcal {M}}}$ on $M^\natural$. Furthermore, $\mu _{{\rm can},{\mathcal {M}}}$ is independent of the choice of $(\omega _i)_{i\in I}$.

Proof. Since $\omega _i$ and $\omega _j$ are both non-vanishing sections of the same invertible sheaf over $U'$ we have $\omega _{j|U'_i \cap U'_j} = f \cdot \omega _{i|U'_i \cap U'_j}$ with $f \in H^0(U'_i\cap U'_j, {\mathcal {O}}^*_{U'_i \cap U'_j})$. We claim that for every $x \in M_i^\natural \cap M_j^\natural \subset (U_i \cap U_j)(F)$ we have that $f(x) \in {\mathcal {O}}_F^*$. This implies the first part of the proposition, since $\mu _{\omega _i}$ and $\mu _{\omega _j}$ are given by locally integrating the absolute value of $\omega _i$ and $\omega _j$.

To prove the claim, let $L/F$ be a finite extension such that $x_L$ lifts to an ${\mathcal {O}}_L$-point $x'_L$ of ${\mathcal {M}}_L$ as in Assumption 3.1.1. Let $p: X \to {\mathcal {M}}$ be a smooth atlas. By passing to an even bigger finite extension of $F$ if necessary, we may further assume that the closed point $\mathsf {Spec}(k_L) \to {\mathcal {M}}_{k_L}$ lifts to the atlas $X_{k_L}$. Since $X_L \to {\mathcal {M}}_L$ is smooth we can then lift the whole $x'_L$ to a morphism $\tilde {x}_L: \mathsf {Spec}({\mathcal {O}}_L) \to X_L$ by Hensel's lemma. Since $F^* \cap {\mathcal {O}}_L = {\mathcal {O}}^*$ it is enough to show that $f(x_L) = f \circ \pi \circ p (\tilde {x}_L) \in {\mathcal {O}}_L^*$.

Write $V_i = p^{-1}(U'_i)$. The point is now that $f \circ p_{|V_i \cap V_j}$ is a regular non-vanishing function on $V_i \cap V_j$ and the complement of $V_i\cap V_j$ in $X$ has codimension at least $2$ by assumption. Since $X$ is normal, Hartogs’ theorem applies and $f$ thus extends to a regular function on $X$ which is still non-vanishing, as the $0$-locus would have codimension $1$. Therefore,

\[ f(x_L) = f \circ p (\tilde{x}_L) \in {\mathcal{O}}_L^*, \]

as claimed.

The proof of the independence of $\mu _{{\rm can},{\mathcal {M}}}$ from $(\omega _i)_{i\in I}$ is essentially the same.

If the stack ${\mathcal {M}}$ is clear from the context we often write $\mu _{\rm can}$ instead of $\mu _{{\rm can},{\mathcal {M}}}$.

Example 3.1.3 Consider the quotient stack ${\mathcal {M}} = [{\mathbb {A}}^2/{\mathbb {G}}_m]$ where ${\mathbb {G}}_m$ acts linearly on ${\mathbb {A}}^2$ with weights $(1,-1)$. In the above notation we have

\[ U' = [({\mathbb{A}}^2 \setminus \{0\}) /{\mathbb{G}}_m],\quad U = [({\mathbb{A}}^2 \setminus \{xy=0\}) / {\mathbb{G}}_m] \cong {\mathbb{G}}_m,\quad M \cong {\mathbb{A}}^1. \]

Then $U'$ is isomorphic to the affine line with doubled origin, which admits an open cover by two copies of ${\mathbb {A}}^1$. The standard $1$-forms on these ${\mathbb {A}}^1$ glue to a global non-vanishing $1$-form on $U'$ and $\mu _{\rm can}$ is given by integrating ${{\rm d}x}$ (i.e. the standard Haar measure) on $M^\natural = {\mathcal {O}} \cap F^* = {\mathcal {O}} \setminus \{0\}$.

Remark 3.1.4 If ${\mathcal {M}} = M$ is stabilizer-free we have $M^\natural = M({\mathcal {O}})$ and the construction of $\mu _{\rm can}$ goes back to Weil [Reference WeilWei12]. The volume is related to the number of $k$-rational points by the formula [Reference WeilWei12, Theorem 2.2.5]

\[ \int_{M({\mathcal{O}})} \mu_{\rm can} = \frac{|M(k)|}{q^n}. \]

3.2 The canonical measure on moduli spaces of sheaves

Let $S$ be a smooth projective relative surface over $\mathsf {Spec}({\mathcal {O}})$ and ${\mathcal {M}}_{\beta,\chi }$ the ${\mathbb {G}}_m$-rigidified moduli stack of semi-stable one-dimensional sheaves on $S$ with moduli space $\operatorname {M}_{\beta,\chi }$ as in § 2.

We take the open substacks $\mathcal {U}_{\beta,\chi }^{st}\subset \mathcal {U}_{\beta,\chi }\subset {\mathcal {M}}_{\beta,\chi }$ of geometrically stable line bundles and of line bundles, respectively, as substacks $U\subset U'$ in the notation of § 3.1. From now on we always assume that $S \to \mathsf {Spec}({\mathcal {O}})$ is either a smooth projective del Pezzo surface or a K3 surface satisfying Assumption 3.2.1.

By Theorem 2.1.3, Lemma 2.1.4 and Proposition 2.2.1 all the assumptions for Proposition 3.1.2 to hold are satisfied and we obtain a canonical measure $\mu _{\rm can}$ on $\operatorname {M}_{\beta,\chi }^\natural$.

Also, the smooth base $B$ of the Hilbert–Chow morphism $h_\chi :\operatorname {M}_{\beta,\chi } \to B$ admits a canonical measure $\mu _{{\rm can},B}$ and it will be useful later to have an explicit description of the relative measure $\mu _{\rm can}/\mu _{{\rm can},B}$ on smooth fibers of $h_\chi$, as in [Reference Groechenig, Wyss and ZieglerGWZ20b, § 6.3].

An analogous construction can be done for moduli spaces of Higgs bundles using Theorem 2.3.1.

4.1 ${\mathbb {G}}_m$ gerbes

Let $X\to V$ be a stack for the fppf-topology. Then $X$ is said to be a gerbe if:

1. (i) for every scheme $T$ over $V$ and objects $x,y \in X(T)$, there exists some fppf-cover $f\colon U \to T$ and an isomorphism $x_{|U}\cong y_{|U}$;

2. (ii) there exists an fppf-cover $U\to V$ such that the groupoid $X(U)$ is non-empty.

For every $T\to V$ and isomorphism $\phi \colon x \to y$ in $X(T)$, conjugation by $\phi$ induces an isomorphism $\mathsf {Aut}_T(x)\to \mathsf {Aut}_T(y)$. Suppose that every object of $X$ has abelian automorphism group in its fiber category; in this case we say that the gerbe $X\to V$ is abelian. Then for every $T\to V$ and objects $x,y \in X(T)$, there is a canonical isomorphism $\mathsf {Aut}_T(x)\cong \mathsf {Aut}_T(y)$. These isomorphisms determine a sheaf of groups $Band(X/V)$ over $V$.

We say that an abelian gerbe $X \rightarrow V$ is a ${\mathbb {G}}_m$ gerbe if there is an isomorphism ${\mathbb {G}}_{m,V}\to Band(X/V)$.

By [Reference GiraudGir20] equivalence classes of ${\mathbb {G}}_m$ gerbes on $V$ are in bijection with the group $H^2(V_{fppf},{\mathbb {G}}_m)$, the Brauer group of $V$. A gerbe $X/V$ is trivial if it is equivalent to $[X/{\mathbb {G}}_{m,V}]$, where ${\mathbb {G}}_{m,V}$ acts trivially on $X$.

We recall that for $F$ a local field, the Brauer group $H^2(F_{fppf},{\mathbb {G}}_m)$ is isomorphic to $\mathbb {Q}/\mathbb {Z}$ by mean of the Hasse invariant; see, for example, [Reference SerreSer68, Proposition XIII.6].

Given $X/V$ a ${\mathbb {G}}_m$ gerbe, for any point $F\xrightarrow {s} V$ we get a ${\mathbb {G}}_m$ gerbe $s^*X/F$ by pullback. This defines a complex valued function:

(4.1.1)\begin{equation} \varphi_V\colon V(F)\to H^2(F_{fppf},{\mathbb{G}}_m)\cong\mathbb{Q}/\mathbb{Z}\xrightarrow{e^{2\pi i} }\mathbb{C} \end{equation}

4.2 Picard stacks/schemes of curves

Let $V$ be a scheme and $c\colon X\to V$ a smooth proper curveFootnote ³ with fibers of genus $g\geq 0$.

We denote by $\mathfrak {Pic}_{X/V}$ the Picard stack of $X/V$; in other words, for an $V$-scheme $T$, $\mathfrak {Pic}_{X/V}(T)$ is the groupoid of invertible sheaves on $X\times _VT$. The group of isomorphism classes of objects of $\mathfrak {Pic}_{X/V}(T)$ is therefore the classical Picard group $\mathrm {Pic}(X\times _VT)$. For each integer $d$, we denote by $\mathfrak {Pic}^d_{X/V}$ the substack of line bundles of degree $d$.

We denote by $\mathrm {Pic}_{X/V}$ the sheaf $R^1_{fppf}c_*{\mathbb {G}}_m$ on $V$; it can be equivalently defined as the sheafification of the presheaf $T \mapsto \mathrm {Pic}(X\times _VT)$. The natural morphism

(4.2.1)\begin{equation} \alpha \colon \mathfrak{Pic}_{X/V} \to \mathrm{Pic}_{X/V} \end{equation}

is a gerbe [Sta, 0DME] banded by ${\mathbb {G}}_m$.

The degree-map factors through $\alpha$ and we denote by $\mathrm {Pic}^d_{X/V}$ the corresponding component of degree $d$ line bundles. The degree-zero part $\mathrm {Pic}^0_{X/V}$ is represented by an abelian scheme of relative dimension $g$, the relative Jacobian, and each $\mathrm {Pic}^d_{X/V}$ is an $V$-torsor under it.

In general, a $T$-point of $\mathrm {Pic}_{X/V}$ need not correspond to a line bundle on $X\times _VT$; this property gets lost when the sheafification of $T\mapsto \mathrm {Pic}(X\times _VT)$ is taken.

Lemma 4.2.2 [Reference Bosch, Lütkebohmert and RaynaudBLR12, Proposition 8.4]

For every $V$-scheme $T$, there is a canonical exact sequence of abelian groups, functorial in $T$,

\[ 0\to \mathrm{Pic}(T) \to \mathrm{Pic}(X\times_VT) \to \mathrm{Pic}_{X/V}(T) \to H^2(T,{\mathbb{G}}_m) \to H^2(X\times_VT,{\mathbb{G}}_m). \]

Since in our case $\mathrm {Pic}_{X/V}$ is a $V$-scheme, we may look at the particular case $T=\mathrm {Pic}_{X/V}$. Then the identity in $\mathrm {Pic}_{X/V}(\mathrm {Pic}_{X/V})$ gets mapped to the element of $H^2(\mathrm {Pic}_{X/V},{\mathbb {G}}_m)$ corresponding to the ${\mathbb {G}}_m$ gerbe $\alpha$ of (4.2.1). By Lemma 4.2.2, $\alpha$ is trivial if and only if there is a universal Poincaré bundle $\mathcal {L}$ on $X\times _V\mathrm {Pic}_{X/V}$. For example, this is the case when $X\to V$ has a section, since then the map

\[ H^2(\mathrm{Pic}_{X/V},{\mathbb{G}}_m) \to H^2(X\times_V\mathrm{Pic}_{X/V},{\mathbb{G}}_m) \]

is injective.

More generally, for an $V$-scheme $T$, the map $c_T\colon \mathrm {Pic}_{X/V}(T) \to H^2(T,{\mathbb {G}}_m)$ sends $x\colon T\to \mathrm {Pic}_{X/V}$ to the equivalence class of the ${\mathbb {G}}_m$ gerbe on $T$ obtained by pulling back $\alpha$ along $x$. Such a gerbe is trivial if and only if $x$ comes from a line bundle on $X\times _VT$ (that is, is in the essential image of $\alpha$).

4.3 Tate duality

We consider now the special case where the base $V$ is the spectrum of a non-archimedean local field $F$. As recalled above, the Brauer group $H^2(F,{\mathbb {G}}_m)$ is identified with ${\mathbb {Q}}/{\mathbb {Z}}$ via the Hasse invariant. We denote by $f\colon \mathrm {Pic}_{X/F}(F)\to {\mathbb {Q}}/{\mathbb {Z}}$ the group homomorphism of Lemma 4.2.2, and for each integer $d\in {\mathbb {Z}}$, we call $f_d\colon \mathrm {Pic}^d_{X/F}(F)\to {\mathbb {Q}}/{\mathbb {Z}}$ the restriction of $f$.

Note that any $x \in \mathrm {Pic}^d_{X/F}(F)$ gives rise to an isomorphism $s_x: \mathrm {Pic}^0_{X/F} \to \mathrm {Pic}^d_{X/F}$ and since $f$ is a group homomorphism we have

(4.3.1)\begin{equation} f_d \circ s_x = f_d(x) + f_0. \end{equation}

The following proposition gives some control over $f_0$ in terms of torsor instead of gerbes.

Similar statements have been used crucially in [Reference Groechenig, Wyss and ZieglerGWZ20b, Reference Groechenig, Wyss and ZieglerGWZ20a] and the following discussion is essentially taken from there.

The proof of Proposition 4.3.2 relies on Tate duality over local fields. Let $A$ be an abelian variety over $F$ and $A^t$ its dual.

Theorem 4.3.3 [Reference MilneMil86, Theorem 3.7.8]

There is a canonical perfect pairing

\[ A(F)\otimes H^{1}(F,A^t) \to H^2(F,{\mathbb{G}}_m)={\mathbb{Q}}/{\mathbb{Z}}. \]

The pairing admits the following geometric description, see [Reference Groechenig, Wyss and ZieglerGWZ20b, Remark 3.11]. First from [Reference MilneMil86, Lemma 3.1] we have an isomorphism

(4.3.4)\begin{equation} H^{1}(F,A^t) \cong \mathsf{Ext}^2(A,{\mathbb{G}}_m). \end{equation}

Now, working in the category of commutative group stacks considered in [Reference BrochardBro21], elements of $\mathsf {Ext}^2(A,{\mathbb {G}}_m)$ can be represented by ${\mathbb {G}}_m$ gerbes on $A$ with a group structure.

Given $x \in \mathsf {A}(F)$, $\tau \in H^{1}(F,A^t)$ and $\alpha _\tau \in \mathsf {Ext}^2(A,{\mathbb {G}}_m)$ the image of $\tau$ under (4.3.4), the paring of $x$ with $\tau$ from Theorem 4.3.3 equals the class of the ${\mathbb {G}}_m$ gerbe $x^*\alpha _\tau$ on $\mathsf {Spec}(F)$ under the Hasse invariant isomorphism $H^2(F,{\mathbb {G}}_m) \cong {\mathbb {Q}}/{\mathbb {Z}}$.

Thus, for $A= \mathrm {Pic}^0_{X/F} = A^t$ the homomorphism $f_0$ is simply the Tate-duality pairing with the ${\mathbb {G}}_m$ gerbe $\alpha _0: \mathfrak {Pic}^0_{X/F} \rightarrow \mathrm {Pic}^0_{X/F}$.

Proof of Proposition 4.3.2 To simplify the notation we drop the subscript $X/F$. By the previous discussion and Theorem 4.3.3 we see that $d_0\cdot f_0 \equiv 0$ if and only if $\alpha _0^{d_0}$ is equivalent to the trivial ${\mathbb {G}}_m$ gerbe on $\mathrm {Pic}^0$. We thus need to show that the latter holds if and only if $\mathrm {Pic}^{d_0}(F)$ is non-empty.

We consider the category of dualizable commutative group stacks $\mathrm {DCGS}_F$ over $\mathsf {Spec}(F)$; see, for example, [Reference BrochardBro21] for a detailed account. The internal hom-functor $D(\cdot ) = \mathcal {H}om(\cdot,B\mathbb {G}_m)$ induces an anti-equivalence on $\mathrm {DCGS}_F$ satisfying $D\circ D = Id$ and extending the usual duality functor on abelian varieties. Furthermore, there is an auto-equivalence $D(\mathfrak {Pic}) \cong \mathfrak {Pic}$ (see [Reference TravkinTra16, § 3.2]). Thus, applying $D$ to the short exact sequence

\[ 0 \to B\mathbb{G}_m \to \mathfrak{Pic} \to \mathrm{Pic} \to 0, \]

we get the sequence

\[ 0 \to D(\mathrm{Pic}) \to \mathfrak{Pic} \xrightarrow{\beta} {\mathbb{Z}} \to 0. \]

Here exactness on the right follows from [Reference BrochardBro21, Proposition 3.18] and the fact that $\mathcal {E}xt^2(\mathrm {Pic})=0$ (see [Reference BrochardBro21, Corollary 11.5]). Since $\beta$ is an epimorphism with kernel $\mathfrak {Pic}^0$ we get an equivalence $D(\mathrm {Pic}) \cong \mathfrak {Pic}^0$. From this we deduce that the sequences

(4.3.5)\begin{align} 0 \to \mathrm{Pic}^0 \to \mathrm{Pic} \to {\mathbb{Z}} \to 0 \end{align}

(4.3.6)\begin{align} 0 \to B{\mathbb{G}}_m \to \mathfrak{Pic}^0 \to \mathrm{Pic}^0 \to 0 \end{align}

are exchanged by duality.

Now (4.3.5) is the extension of ${\mathbb {Z}}$ by $\mathrm {Pic}^0$ corresponding to the $\mathrm {Pic}^0$-torsor $\mathrm {Pic}^1$ under the equivalence [Reference BrochardBro21, Proposition 5.8]. In particular, $\mathrm {Pic}^1$ has a rational point if and only (4.3.5) splits. Dually (4.3.6) is the $B{\mathbb {G}}_m$-torsor on $\mathrm {Pic}^0$ corresponding to $\alpha _0$ under the equivalence [Reference BrochardBro21, Proposition 5.11] and splits if and only if $\alpha _0$ is trivial. Since $D(\cdot )$ is an anti-equivalence, we have thus shown that $\mathrm {Pic}^1$ has a rational point if and only if $\alpha _0$ is trivial.

To extend this to $\mathrm {Pic}^{d_0}$ and $\alpha _0^{d_0}$, we note that their associated short exact sequences are given by the $d_0$-fold Baer sum [Sta, 010I] of (4.3.5) and (4.3.6), respectively. By definition, $D(\cdot )$ commutes with direct sums and exchanges the addition map with the diagonal, i.e. for every $G \in \mathrm {DCGS}_F$

\[ D(G \oplus G \xrightarrow{+} G) \cong D(G) \xrightarrow{\Delta} D(G) \oplus D(G), \]

and, hence, also commutes with Baer sums. We thus deduce that $\mathrm {Pic}^{d_0}$ has a rational point if and only if $\alpha _0^{d_0}$ is trivial, which proves the proposition.