Torsion in Griffiths Groups]Torsion in Griffiths Groups
Theodosis Alexandrou
Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover , Germany.

[2010]primary 14C25 ; secondary 14J28

We show that for any integer $n\geq2$ there is a smooth complex projective variety $X$ of dimension $5$ whose third Griffiths group $\Griff^{3}(X)$ contains infinitely many torsion elements of order $n$. This generalises a recent theorem of Schreieder who proved the result for $n=2$.

§ INTRODUCTION

The Griffiths group $\Griff^i(X)$ of a smooth complex projective variety $X$ is the group of nullhomologous codimension $i$ cycles modulo algebraic equivalence. For divisors and zero-cycles these two equivalence relations of algebraic cycles coincide and so $\Griff^i(X)$ is interesting only if $2\leq i \leq \dim X-1$.The terminology of this group is due to the famous example of Griffiths [85], who showed that $\Griff^2(X)$ is not a torsion group for a very general quintic $X\subset\mathbb{P}^4_{\mathbb{C}}$. Clemens [43] proved later that in this case $\Griff^2(X)$ has in fact infinite rank. Further examples of infinite dimensionality of $\Griff^i(X)\otimes \mathbb{Q}$ were constructed by Nori [141] and Voisin [195].Totaro [189] showed that the vector space $\Griff^{i}(X)/\ell$ can be infinite dimensional for all prime numbers $\ell$ and all $2\leq i\leq \dim X-1$. This builds on earlier works of Schoen [176] and Rosenschon-Srinivas [160], where there are constraints on the prime $\ell$. The result of Totaro also relies on Griffiths' method [85], a theorem of Bloch-Esnault [32] and Nori's approach [141] of the proof of Clemens' theorem.Schoen [173] proved that the torsion subgroup of the Griffiths groups is in general not zero. Merkurjev-Suslin [132] used Bloch's map [27] to show that the $n$-torsion in $\Griff^{2}(X)$ is always finite. It was then a natural problem in the field whether $\Griff^{i}(X)$ for $i\geq3$ can have infinite $n$-torsion, cf. [173]. This was recently solved by Schreieder [171], who produced examples with infinite 2-torsion in their third Griffiths group. The question of infinite $\ell$-torsion for odd primes $\ell$ was left open. Building on Schreieder’s approach, we solve in this paper the problem in the following strong sense:

Let $n\geq2$ be an integer. There is a smooth complex projective variety $X$ of dimension 5 whose third Griffiths group $\Griff^{3}(X)$ contains infinitely many torsion elements of order $n$. Moreover, these elements are linearly independent modulo $n$, i.e. in $\Griff^{3}(X)/n$.
The above result implies that for any finite set P of prime numbers, there is a smooth complex projective $5$-fold $X$, such that the vector space $\Griff^{3}(X)[\ell^{\infty}]/\ell$ is infinite dimensional for all $\ell\in\textbf{P}$.

Taking products with projective spaces, we obtain the following generalisation in higher dimensions.
For any integers $m\geq 5$ and $n\geq 2$, there is a smooth complex projective $m$-fold $X$ whose $j$-th Griffiths group $\Griff^{j}(X)$ contains infinitely many torsion elements of order $n$ for all $3\leq j\leq m-2$. Moreover, these elements are linearly independent in $\Griff^{j}(X)/n$.
The examples in Theorem <ref> are given by a product $X:=S\times JC$, where $S$ is a $\Z/n$-quotient of a carefully chosen smooth complete intersection of multidegree $(n,n,n)$ in $\mathbb{P}^{5}_{\C}$ (see $\S$<ref>) and $C$ is a very general genus $3$ curve. The surface $S$ is somewhat special as such admits an exceptional degeneration, see $\S \ref{sb:degenerations}$ below. For instance, if $n=2$, then $S$ is an Enriques surface (as used in [171]) that degenerates to a flower pot. For $n>2$, the surface $S$ is of general type; its geometric genus is positive and grows polynomially in $n$, see Proposition <ref>.
Let $Y$ be a smooth complex projective variety and let $\NS(Y):=\Pic(Y)/\Pic^{0}(Y)$ denote its Néron Severi group. The following injectivity result is crucial for us.
Let $n\geq2$ be an integer. There is a smooth complex projective surface $S$ with $\NS(S)_{\tors}\cong\Z/n$, such that for every smooth complex projective variety $Z$, the exterior product map
\begin{align}\label{ext-prod-griffiths}\Griff^{j}(Z)/n\longrightarrow\frac{\Griff^{j+1}(S\times Z)[n]}{n\Griff^{j+1}(S\times Z)[n^{2}]},\ [z]\mapsto [\mathcal{L}\times z]\end{align}
is injective, where $0\neq\mathcal{L}\in\NS(S)_{\tors}$ is any generator.
Theorem <ref> follows from Theorem <ref> because the Jacobian of a very general genus $3$ curve $C$ has the property that $\Griff^{2}(JC)/n$ has infinitely many elements of order $n$ by a recent result of Totaro [189] and the Merkurjev-Suslin theorem [132] (see Theorem <ref>).

§.§ Special degenerations of surfaces
Apart from a general injectivity result that works for all primes, the second main ingredient in [171] was the construction of degenerations of Enriques surfaces such that the Brauer class of the geometric generic fibre extends across the whole family, while the special fibre splits up into several components that are rational and hence the Brauer class restricts to zero on the components of the special fibre. These degenerations of Enriques surfaces are known in the literature as flower-pot, see <cit.>. They are exceptional as such do not admit (up to birational equivalence) an étale 2:1 cover by a Kulikov degeneration (see [120]) of $K3$ surfaces.
The main task within this paper is thus to generalize the above construction to other surfaces. Namely, to prove Theorem <ref> we need to exhibit for every $n\geq2$, a surface $S$ with $\NS(S)_{\tors}\cong\Z/n$ that admits a similar degeneration, i.e. there is a Brauer class $\alpha\in\Br(S)[n]$, whose image via the Bockstein map generates the torsion in $\bigoplus_{\ell}H^{3}(S,\Z_{\ell}(1))$ and extends across the whole family, while its restriction to the components of the special fibre is zero. This is achieved by the following result.
Let $\kappa$ be an algebraically closed field of characteristic zero and let $n\geq2$ be an integer. Fix an algebraic closure $k$ of the fraction field $\kappa((t))$. There is a regular flat projective scheme $\mathcal{S}\to\Spec\kappa[[t]]$ with geometrically connected fibres of relative dimension two, such that:

*  the geometric generic fibre $S_{\bar{\eta}}$ is a smooth projective surface with $\NS(S_{\bar{\eta}})_{\tors}\cong\Z/n$.

*  the special fibre $S_{0}=\sum_{i=1}^{r}S_{0i}$ is a reduced simple normal crossing divisor whose dual graph $\Gamma$ is a chain, i.e. the double intersection $S_{0i}\cap S_{0i+1}$ is a smooth and irreducible curve while all intersections $S_{0i}\cap S_{0j}$ with $j\notin\{i-1,i,i+1\}$ are empty.

* there is a class $\alpha\in\Br(\mathcal{S})[n]$, such that

* $\delta(\alpha|_{S_{\bar{\eta}}})\in \bigoplus_{\ell}H^{3}(S_{\bar{\eta}},\Z_{\ell}(1))_{\tors}\cong\Z/n$ is a generator, where $\delta$ is the Bockstein map defined on the Brauer group $\Br(S_{\bar{\eta}})$ and such that
*  for any component $S_{0i}$ of the special fibre, $\alpha|_{S_{0i}}=0\in\Br(S_{0i})$.
The existence of the above degenerations seems to be slightly counter intuitive as one typically expects that Brauer classes ramify when the varieties break up into components. Their construction is inspired by the one of flower pot of Enriques that was given by Persson, see <cit.>.It would be interesting to know if Theorem <ref> holds for some degenerations of some well-known surfaces e.g. Godeaux surface. For instance, the property (<ref>) completely fails (see <cit.>) for the degenerations of Godeaux surfaces constructed in <cit.>.

§.§ Outline of the main arguments
We briefly explain some of the main steps of the proof of Theorem <ref>. Let $Y\subset\mathbb{P}^{5}_{k}$ be a smooth complete intersection of multidegree $(n,n,n)$ that is invariant and fixed point free by the automorphism $\varphi_{n}$ from (<ref>). Then the crucial step is to degenerate $Y$ to some special singular complete intersection $Y_{0}\subset\mathbb{P}^{5}_{\kappa}$ (see Lemma <ref>) that is invariant by $\varphi_{n}$. The singular locus of $Y_{0}$ consists of a single point $p_{0}$ of multiplicity $n$ and the automorphism $\varphi_{n}$ fixes exactly this point in $Y_{0}$. The total space $\mathcal{Y}$ of this degeneration can be chosen to be regular and so blowing up $p_{0}$ in our family we obtain $\mathbb{P}^{2}_{\kappa}$ with multiplicity $n$. The automorphism $\varphi_{n}$ lifts naturally on the blow-up and we will see that in fact acts trivially on the exceptional divisor. Quotiening out yields the desired degeneration.
For $n>2$, the property (<ref>) is more subtle. Namely, the surface $S:=S_{\bar{\eta}}$ has positive geometric genus and thus a class $\alpha\in\Br(S)[n]$ with the property that $\delta(\alpha)$ generates the torsion in $\bigoplus_{\ell}H^{3}(S,\Z_{\ell}(1))$ is not canonically given any longer. Instead, the functoriality of the Bockstein map (see (<ref>)) with respect to pull-backs reduces our task to showing
$$H^{3}(\mathcal{S},\Z_{\ell}(1))_{\tors}\to H^{3}(S,\Z_{\ell}(1))_{\tors}$$
is surjective for all primes $\ell$ dividing $n$. One of the advantages of this reduction step is that we can apply the proper base change theorem to compute $H^{3}(\mathcal{S},\Z_{\ell}(1))$ (see Claim <ref>) and in particular its torsion subgroup. The existence of a lift $\beta\in H^{3}(\mathcal{S},\Z_{\ell}(1))$ of a generator of the cyclic group $H^{3}(S,\Z_{\ell}(1))_{\tors}$ follows from the Gysin sequence. The intricate part of the proof is then to show that the class $\beta$ can be chosen to be torsion, so that it comes from a Brauer class on $\mathcal{S}$.The property (<ref>) of Theorem <ref> is more involved than the case of flower pot of Enriques, i.e. $n=2$. This is because the special fibre $S_{0}$ contains a single component (say $W$), which is not ruled if $n>2$ (see Proposition <ref>). We will be able to take care of this matter by adding some integral cohomology class to a lift $\tilde{\alpha}\in H^{2}(\mathcal{S},\mu_{n})$ of the Brauer class on $\mathcal{S}$ from (<ref>), i.e. $\tilde{\alpha}|_{W}=0$ up to adding a cohomology class from $H^{2}(\mathcal{S},\Z_{n}(1)):=\prod_{\ell|n}H^{2}(\mathcal{S},\Z_{\ell}(1))$. Note that this modification on the class $\alpha$ has no effect on its value via the Bockstein map (cf. (<ref>)).

§ PRELIMINARIES

§.§ Notations
The $n$-torsion subgroup of an abelian group $G$ is denoted by $G[n]$. The subgroup $G[n^{\infty}]$ consists of the elements that are annihilated by some power of $n$. We also denote by $G_{div}$ the maximal divisible subgroup of $G$. If $\psi\colon H\to G$ is a homomorphism of abelian groups, then we denote by slight abuse of notation $G/H:=\coker(\psi)$.Let $k$ be a field. An algebraic scheme $X$ over $k$ is a seperated scheme of finite multidegree over $k$ and it is equi-dimensional if its irreducible components have all the same dimension. A variety over $k$ is an integral algebraic $k$-scheme. For an equi-dimensional algebraic $k$-scheme $X$ and any $0\leq i\leq \dim X$, we let $X^{(i)}:=\{x\in X|\codim_{X}(x):=\dim X-\dim\overline{\{x\}}=i\}$ be the set of all codimension $i$ points in $X$.

§.§ Borel-Moore cohomology
Fix a field $k$ and a prime $\ell$ invertible in $k$. For an algebraic $k$-scheme $X$ of dimension $d_{X}$ and $A\in\{\Z/\ell^{r},\Z_{\ell},\Q_{\ell},\Q_{\ell}/\Z_{\ell}\}$, we let
be the twisted Borel-Moore pro-étale cohomology, as defined in <cit.> (see <cit.>). The main properties of this functor are listed in <cit.>.If $X$ is smooth and equi-dimensional, then by <cit.>, there are canonical isomorphisms
\begin{align}\label{eq:borel-moore} H^{i}(X,\Z/\ell^{r}(n))\cong H^{i}(X_{\text{\'et}},\mu_{\ell^{r}}^{\otimes n})\ \text{and}\ H^{i}(X,\Z_{\ell}(n))\cong H^{i}_{cont}(X_{\text{\'et}},\Z_{\ell}(n)), \end{align}
where $H^{i}_{cont}$ denotes Jannsen's continuous étale cohomology (see [104]).More generally, if $X$ admits a closed embedding $\iota\colon X\hookrightarrow Y$ into a smooth and equi-dimensional algebraic $k$-scheme $Y$ of dimension $d_{Y}$, then there is a canonical identification with continuous étale cohomology with support
\begin{align}\label{eq:borel-moore-support} H^{i}(X,A(n))\cong H^{i+2c}_{X,cont}(Y,A(n+c)),\end{align}
where $c=d_{Y}-d_{X}$ (see <cit.> and <cit.>).

§.§ Refined unramified cohomology
For an algebraic $k$-scheme $X$, we have an increasing filtration:
$$F_{0}X\subset F_{1}X\subset F_{2}X\subset\dots\subset F_{\dim X}X=X,$$
where $F_{j}X$ is given by the inverse limit of all open subsets $U\subset X$, with $\dim(X\setminus U)<\dim X-j$. Recall that for an open subset $U\subset X$, with $\dim U=\dim X$, there are restriction maps $H^{*}(X,A(n))\to H^{*}(U,A(n))$ and one defines
$$H^{*}(F_{j}X,A(n)):=\lim_{\substack{\longrightarrow\\ F_{j}X\subset U\subset X}}H^{*}(U,A(n)).$$
Here the direct limit is taken over the open subsets $U\subset X$ that make up $F_{j}X$ (see <cit.>).If $m\geq j$, then we have restriction maps $H^{*}(F_{m}X,A(n))\to H^{*}(F_{j}X,A(n))$ and we set
$$F^{m}H^{*}(F_{j}X,A(n)):=\im(H^{*}(F_{m}X,A(n))\to H^{*}(F_{j}X,A(n))).$$
We define the $j$-th refined unramified cohomology of $X$ with values in $A(n)$ by
For a scheme point $x\in X$, we write $H^i(x,A(n)):=H^i(F_0\overline{\{ x\}},A(n))$, where $\overline{\{ x\}}\subset X$ denotes the closure of $x$. Note that  $H^0(x,A(0))=A\cdot [x]$, where $[x]\in H^0(x,A(0))$ denotes the fundamental class of $x$, cf. <cit.>.

The Gysin sequence induces the following long exact sequence (see <cit.>)

\begin{align} \label{eq:les}
\longrightarrow
H^i(F_jX,A(n))\longrightarrow   H^i(F_{j-1}X,A(n))  \stackrel{\del}\longrightarrow \bigoplus_{x\in X^{(j)}}H^{i+1-2j}(x,A(n-j)) \stackrel{\iota_\ast} \longrightarrow    H^{i+1}(F_jX,A(n)).
\end{align}

We have $H^i(X,A(n))=H^i(F_jX,A(n))$ for all $j\geq \lceil i/2\rceil$ by <cit.>. Moreover, if $X$ is smooth and equi-dimensional $H^{*}_{0,nr}(X,A(n))$ coincides with the classical [169] unramified cohomology $H^{*}_{nr}(X,A(n))$.

§.§ Algebraic cycle groups
Let $k$ be a field and let $\ell$ be a prime number invertible in $k$. For an algebraic $k$-scheme $X$ of dimension $d_{X}$, we denote by $\CH^{i}(X):=\CH_{d_{X}-i}(X)$ the Chow group of cycles of dimension $d_{X}-i$ on $X$. We set $\CH^i(X)_{\Z_\ell}:=\CH^i(X)\otimes_{\Z} \Z_\ell$.
There is a cycle class map (see <cit.>)

\cl^i_X:\CH^i(X)_{\Z_\ell}\longrightarrow H^{2i}(X,\Z_\ell(i)),

that is induced by the following pushforward map that appears in the above long exact sequence (<ref>):

\bigoplus_{x\in X^{(i)}}H^{0}(x,\Z_\ell(0))= \bigoplus_{x\in X^{(i)}} \Z_\ell[x] \stackrel{\iota_\ast} \longrightarrow    H^{2i}(F_iX,\Z_\ell(i))=H^{2i}(X,\Z_\ell(i)).

If $X$ is smooth and equi-dimensional, then the above cycle class map agrees with Jannsen's cycle class map in continuous étale cohomology from [104], see <cit.>.

There is a natural coniveau filtration $N^\ast$ on $\CH^i(X)_{\Z_\ell}$, given by the condition that a cycle $[z]$ lies in $N^j$ if and only if
there is a closed subset $Z\subset X$ of codimension $j$ such that $z$ is rationally equivalent to a homologously trivial cycle on $Z$, i.e.

[z]\in \im(\ker(\cl_Z^{i-j})\longrightarrow \CH^i(X)_{\Z_\ell} ) ,

see <cit.>.
In view of <cit.>, we define

A^i(X)_{\Z_\ell}:=\CH^i(X)_{\Z_\ell}/N^{i-1}\CH^i(X)_{\Z_\ell} \ \ \text{and}\ \ A^i(X)[\ell^\infty]:=A^i(X)_{\Z_\ell}[\ell^\infty].

If the field $k$ is algebraically closed, then $N^{i-1}\CH^i(X)_{\Z_\ell}=\CH^{i}(X)_{\alg}\otimes_{\Z}\Z_{\ell}$ by <cit.> and therefore in this case $A^i(X)_{\Z_\ell}=(\CH^i(X)/\sim_{\alg})\otimes_{\Z}{\Z_\ell}$.

§.§ Brauer groups
For a scheme $X$, we let $\Br(X):=H^{2}(X_{\text{\'et}},\mathbb{G}_{m})$ be the Brauer group of $X$. If $n$ is a positive integer that is invertible on $X$, then the Kummer sequence yields a short exact sequence
\begin{align}\label{eq:kummer} 0\longrightarrow\Pic(X)/n\overset{\cl^{1}}{\longrightarrow} H^{2}(X_{\text{\'et}},\mu_{n})\longrightarrow\Br(X)[n]\longrightarrow 0. \end{align}
We also recall that $\Br(X)[n]=H^{2}_{nr}(X_{\text{\'et}},\mu_{n})$ for a regular variety $X$ (cf. <cit.>). Let $X$ be an algebraic scheme over a field $k$ and fix a prime number $\ell$ invertible in $k$. There is a long exact Bockstein sequence
\begin{align}\label{eq:bockstein-sequence}
\dots\longrightarrow H^{i}(X,\Z_{\ell}(n))\overset{\times \ell^{r}}{\longrightarrow}H^{i}(X,\Z_{\ell}(n))\longrightarrow H^{i}(X,\Z/\ell^{r}(n))\overset{\delta}{\longrightarrow}H^{i+1}(X,\Z_{\ell}(n))\overset{\times \ell^{r}}{\longrightarrow}\dots\end{align}
where $H^{i}(X,\Z_{\ell}(n))\longrightarrow H^{i}(X,\Z/\ell^{r}(n))$ is given by functoriality in the coefficients and where $\delta$ is called the Bockstein map (see <cit.>).
Let $X$ be an algebraic scheme over a field $k$ and let $\ell$ be a prime invertible in $k$. Then the Bockstein map $\delta\colon H^{2}(X,\Q_{\ell}/\Z_{\ell}(1))\longrightarrow H^{3}(X,\Z_{\ell}(1))_{\tors}$ descends to a well-defined map $\delta\colon H^{2}_{0,nr}(X,\Q_{\ell}/\Z_{\ell}(1))\longrightarrow H^{3}(X,\Z_{\ell}(1))_{\tors}$ that fits into a short exact sequence
\begin{align}\label{eq:short-exact-sequence}0\longrightarrow \frac{H^{2}_{0,nr}(X,\Q_{\ell}(1))}{H^{2}_{0,nr}(X,\Z_{\ell}(1))}\longrightarrow H^{2}_{0,nr}(X,\Q_{\ell}/\Z_{\ell}(1))\overset{\delta}{\longrightarrow} H^{3}(X,\Z_{\ell}(1))_{\tors}\longrightarrow 0.\end{align}
Pick a class $\alpha\in H^{2}(X,\Z/\ell^{r}(1))$ whose image in $H^{2}_{0,nr}(X,\Z/\ell^{r}(1))$ is zero. It follows from the Gysin sequence (<ref>) that $\alpha=\iota_{*}\xi$ for some $\xi\in\bigoplus_{x\in X^{(1)}}\Z/\ell^{r}\cdot[x]$. By functoriality of the Bockstein sequence (<ref>) with respect to proper pushforwards (see <cit.>) we obtain $\delta(\iota_{*}\xi)=\iota_{*}\delta(\xi)$. Note that the element $\delta(\xi)\in \bigoplus_{x\in X^{(1)}}H^{1}(x,\Z_{\ell}(0))$ is torsion. The group $H^{1}(x,\Z_{\ell}(0))$ being torsion free (see <cit.>) thus yields $\delta(\xi)=0$. In particular, the Bockstein map induces a well-defined map $\delta\colon H^{2}_{0,nr}(X,\Z/\ell^{r}(1))\longrightarrow H^{3}(X,\Z_{\ell}(1))[\ell^{r}]$ and taking the direct limit over $r$ one gets $\delta\colon H^{2}_{0,nr}(X,\Q_{\ell}/\Z_{\ell}(1))\longrightarrow H^{3}(X,\Z_{\ell}(1))_{\tors}$.Consider the following commutative diagram with exact rows

0 [r]     H^2(X,_ℓ(1))/ℓ^r [r] [d, "π_1", two heads]     H^2(X,/ℓ^r(1)) [r, "δ"] [d, "π_2", two heads]     H^3(X,_ℓ(1))[ℓ^r] [r] [d, "π_3", two heads]       0

H^2_0,nr(X,_ℓ(1))/ℓ^r [r]                              H^2_0,nr(X,/ℓ^r(1)) [r]                                          H^2_0,nr(X,/ℓ^r(1))/H^2_0,nr(X,_ℓ(1)) [r]     0,

where the upper row corresponds to the short exact sequence induced by (<ref>) and where the $\pi_{i}$ are surjective simply because $H^{2}(X,A(n))=H^{2}(F_{1}X,A(n))\twoheadrightarrow H^{2}_{0,nr}(X,A(n))$ is onto.We wish to show that the lower row can be completed to a short exact sequence. It suffices to prove injectivity for $H^{2}_{0,nr}(X,\Z_{\ell}(1))/\ell^{r}\longrightarrow H^{2}_{0,nr}(X,\Z/\ell^{r}(1))$. To this end, pick a class $\alpha\in H^{2}(X,\Z_{\ell}(1))$ and assume that its reduction $\bar{\alpha}$ modulo $\ell^{r}$ restricts to zero in $H^{2}_{0,nr}(X,\Z/\ell^{r}(1))$. It follows from the Gysin sequence (<ref>) that $\bar{\alpha}=\iota_{*}\bar{\xi}$ for some $\xi\in\bigoplus_{x\in X^{(1)}}\Z_{\ell}\cdot[x]$, where we denote by $\bar{\xi}$ the reduction of this cycle modulo $\ell^{r}$. Therefore $\alpha=\iota_{*}\xi\in H^{2}(X,\Z_{\ell}(1))/\ell^{r}$ and by exactness of (<ref>), the image of the class $\alpha$ in $H^{2}_{0,nr}(X,\Z_{\ell}(1))/\ell^{r}$ is zero.Next we prove that $\pi_{3}$ is an isomorphism. Recall that this map is onto and so we only need to prove is injective. In the argument of the previous paragraph, we have seen that the kernel $\ker(\pi_{1})$ surjects onto $\ker(\pi_{2})$. Since $\pi_{1}$ is surjective, the result is an immediate consequence of the serpents lemma.Finally, (<ref>) follows from the short exact sequence
\begin{align}\label{eq:short-exact-sequence-2}
H^{2}_{0,nr}(X,\Z_{\ell}(1))/\ell^{r}\longrightarrow H^{2}_{0,nr}(X,\Z/\ell^{r}(1))\overset{\delta}{\longrightarrow} H^{3}(X,\Z_{\ell}(1))[\ell^{r}]\longrightarrow 0,\end{align}
by taking the direct limit over $r$. This completes the proof.
If $X$ is smooth and equi-dimensional, then (<ref>) yields
\begin{align}\label{eq:extension-brauer}
0\longrightarrow \Br(X)[\ell^{\infty}]_{div}\longrightarrow\Br(X)[\ell^{\infty}]\overset{\delta}{\longrightarrow} H^{3}_{cont}(X,\Z_{\ell}(1))_{\tors}\longrightarrow 0,\end{align}
where $\Br(X)[\ell^{\infty}]_{div}:=\frac{H^{2}_{0,nr}(X,\Q_{\ell}(1))}{H^{2}_{0,nr}(X,\Z_{\ell}(1))}$ is a divisible group (cf. <cit.>).

§.§ Strictly semi-stable models
Let $R$ be a discrete valuation ring with fraction field $k$ and residue field $\kappa$. A proper flat $R$-scheme $\mathcal{X}\to\Spec R$ is called strictly semi-stable if the generic fibre $X_{\eta}:=\mathcal{X}\times_{R} k$ is smooth and the special fibre $X_{0}:=\mathcal{X}\times_{R}\kappa$ is a geometrically reduced simple normal crossing divisor on $\mathcal{X}$, i.e. the irreducible components $X_{0i}$ are all smooth Cartier divisors and for every $n$, the intersection of $n$ distinct components is either empty or smooth and equi-dimensional of codimension $n$ in $\mathcal{X}$. Note that the total space $\mathcal{X}$ is in fact regular (see <cit.>). We say that the dual graph $\Gamma$ of the special fibre $X_{0}$ is a chain if the scheme-theoretic intersection of two consecutive components $X_{0i}\cap X_{0i+1}$ is non-empty while all intersections $X_{0i}\cap X_{0j}$ with $j\notin\{i-1,i,i+1\}$ are empty.

§.§ An injectivity result of Schreieder
Let $Y$ be a smooth projective variety over an algebraically closed field $k$ and let $n$ be a positive integer invertible in $k$. We shall use the notation
\begin{align} E^{j}_{n}(Y):=\ker(A^{j}(Y)/n\overset{\cl^{j}_{Y}}{\longrightarrow} H^{2j}(Y,\mu_{n}^{\otimes j})),\end{align}
where we recall $A^{j}(Y):=\CH^{j}(Y)/\sim_{\alg}$.
The following result is contained in [171].
$($[171]$)$ Let $\kappa$ be an algebraically closed field of characteristic zero and fix an algebraic closure $k$ of the fraction field $\kappa((t))$. Let $\mathcal{X}\to\Spec\kappa[[t]]$ be a proper strictly semi-stable scheme whose geometric generic fibre $X_{\bar{\eta}}$ is a surface with $\NS(X_{\bar{\eta}})_{\tors}\cong\Z/n$ for some $n\geq2$. Assume that the following properties hold

(P1) there is a class $\alpha\in\Br(\mathcal{X})[n]$, such that $\delta(\alpha|_{X_{\bar{\eta}}})\in \bigoplus_{\ell|n}H^{3}(X_{\bar{\eta}},\Z_{\ell}(1))_{\tors}\cong\Z/n$ is a generator, where $\delta$ is the Bockstein map defined on $\Br(X_{\bar{\eta}})$ and such that

(P2) for each component $X_{0i}$ of the special fibre, $\alpha|_{X_{0i}}=0\in\Br(X_{0i})$.
Then for every smooth projective variety $Z$ over $\kappa$ the exterior product map
\begin{align}\label{eq:exterior-product-a}
E^{j}_{n}(Z)/n\longrightarrow A^{j+1}(X_{\bar{\eta}}\times_{k}Z_{k})/n,\ [z]\mapsto [\mathcal{L}\times z]\end{align}
is injective, where $\mathcal{L}\in\NS(X_{\bar{\eta}})_{\tors}$ is any generator.
By <cit.> and (<ref>), the conditions $(\text{C}1)$ and $(\text{C}2)$ in <cit.> can be replaced by $(\text{P}1)$ and $(\text{P}2)$, respectively. Write $n=\ell^{r_{1}}_{1}\ell^{r_{2}}_{2}\dots\ell^{r_{s}}_{s}$ for the prime factorisation. Let $\tilde{\delta}\colon H^{1}(X_{\bar{\eta}},\mu_{\ell_{i}^{r_{i}}})\to H^{2}(X_{\bar{\eta}},\mu_{\ell_{i}^{r_{i}}})$ be the composition of the Bockstein map with reduction modulo $\ell_{i}^{r_{i}}$ and recall by <cit.> that
$$\tilde{\delta}(H^{1}(X_{\bar{\eta}},\mu_{\ell_{i}^{r_{i}}}))\cong A^{1}(X_{\bar{\eta}})[\ell_{i}^{r_{i}}]/\ell_{i}^{r_{i}}A^{1}(X_{\bar{\eta}})[\ell_{i}^{2r_{i}}].$$
Since $A^{1}(X_{\bar{\eta}})_{\tors}=\NS(X_{\bar{\eta}})_{\tors}\cong\Z/n$, we find $\tilde{\delta}(H^{1}(X_{\bar{\eta}},\mu_{\ell_{i}^{r_{i}}}))\cong A^{1}(X_{\bar{\eta}})[\ell_{i}^{r_{i}}]\cong\Z/\ell_{i}^{r_{i}}$. The injectivity of the composite
\begin{align*} A^{1}(X_{\bar{\eta}})[\ell_{i}^{r_{i}}]\otimes E^{j}_{\ell_{i}^{r_{i}}}(Z)\longrightarrow A^{1}(X_{\bar{\eta}})[\ell_{i}^{r_{i}}]\otimes E^{j}_{\ell_{i}^{r_{i}}}(Z_{k})\longrightarrow A^{j+1}(X_{\bar{\eta}}\times_{k}Z_{k})/\ell_{i}^{r_{i}}\end{align*}
thus follows from <cit.> applied to the free $\Z/\ell_{i}^{r_{i}}$-module $M=\tilde{\delta}(H^{1}(X_{\bar{\eta}},\mu_{\ell_{i}^{r_{i}}}))$ and from the commutativity of the diagram <cit.> (see <cit.>). The exterior product map (<ref>) is injective by the Chinese remainder theorem. The proof is complete.

§.§ Griffiths groups modulo $n$
We need the following variation of Totaro's result in [189].$($[189]$)$ There is a smooth quartic curve $C\subset\mathbb{P}^{2}_{\C}$, such that for every $n\geq2$ the group $\Griff^{2}(JC)/n$ contains infinitely many elements of order $n$.
Clearly by the Chinese remainder theorem it suffices to prove the theorem for all integers of the form $n=\ell^{r}$, where $\ell$ is a prime and $r\geq1$ an integer. By <cit.>, there is a smooth quartic curve $C\subset\mathbb{P}^{2}_{\C}$, such that $\CH^{2}(JC)/\ell$ is infinite for all prime numbers $\ell$. Recall that the group of algebraically trivial cycles modulo rational equivalence is divisible and so $A^{2}(JC)/\ell\cong\CH^{2}(JC)/\ell$. In particular, we find that $A^{2}(JC)/\ell$ is infinite for all primes $\ell$. The infiniteness of $A^{2}(JC)/\ell$ in this case arises from pulling back the Ceresa cycle by infinitely many distinct isogenies. These cycles are known to be homologically trivial (see [40]) and thus $\Griff^{2}(JC)/\ell$ is infinite dimensional for all primes $\ell$.It remains to treat the case $r\geq2$. Assume that there is a prime $\ell_{0}$ and an integer $r_{0}\geq2$, such that $\Griff^{2}(JC)/\ell_{0}^{r_{0}}$ contains only finitely many elements of order $\ell_{0}^{r_{0}}$. Fix an infinite set $\mathfrak{I}\subset\Griff^{2}(JC)$ whose elements are linearly independent modulo $\ell_{0}$ and denote by $\overline{\mathfrak{I}}\subset\Griff^{2}(JC)/\ell_{0}$ its image. We claim that all but finitely many cycles in $\overline{\mathfrak{I}}$ lie in the image of the map $\Griff^{2}(JC)[\ell_{0}^{r_{0}}]\to\Griff^{2}(JC)/\ell_{0}$. Indeed, if the image of $[z]\in\mathfrak{I}$ in $\Griff^{2}(JC)/\ell_{0}^{r_{0}}$ is killed by $\ell_{0}^{r}$ with $1\leq r< r_{0}$, then $\ell_{0}^{r}([z]-\ell_{0}^{r_{0}-r}[z'])=0\in\Griff^{2}(JC)$ for some $[z']\in\Griff^{2}(JC)$ and in particular $[z_{0}]:=[z]-\ell_{0}^{r_{0}-r}[z']\in\Griff^{2}(JC)[\ell_{0}^{r_{0}}]$. Since $r_{0}-r>0$, we find that the reduction of $[z_{0}]$ modulo $\ell_{0}$ coincides with the one of $[z]$. The claim follows as only finitely many images of elements of $\mathfrak{I}$ in $\Griff^{2}(JC)/\ell_{0}^{r_{0}}$ can have order $\ell_{0}^{r_{0}}$.To conclude the argument recall by <cit.> that the group $\Griff^{2}(JC)[\ell_{0}^{r_{0}}]$ is finite. This in turn contradicts the infiniteness of $\overline{\mathfrak{I}}$. The proof is complete.

§ A CYCLIC QUOTIENT OF A COMPLETE INTERSECTION

In this section, we introduce the surfaces that will use as a substitute of the Enriques surface used in [171].Let $k$ be an algebraically closed field of characteristic zero. Let $n\geq 2$ be an integer and fix a primitive $n$-th root of unity $1\neq\omega\in k$. Consider the automorphism $\varphi_{n}$ on $\mathbb{P}^{5}_{k}$ defined by

\begin{align}\label{eq:varphi}(x_{0} : x_{1} : x_{2} : x_{3} : x_{4} : x_{5})\longmapsto (x_{0} : \omega x_{1} : x_{2} : \omega x_{3} : x_{4} :\omega x_{5}).\end{align}
Its fixed locus consists of two disjoint planes given by
\begin{align}\label{eq:fixed-locus}
P_{1}:=\{x_{0}=x_{2}=x_{4}=0\}\ \text{and} \ P_{2}:=\{x_{1}=x_{3}=x_{5}=0\}.
\end{align}

The complete linear system $|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|$ that parametrizes hypersurfaces $H\subset\mathbb{P}^{5}_{k}$ of degree $n$, splits under $\varphi_{n}$ into $n$ disjoint eigenspaces corresponding to the eigenvalues $\omega^{i}$, $i=0,1,\ldots,n-1$. An easy check shows that the ones that are correlated with the eigenvalue $1$ form projectively a $d$-dimensional family, where $d:=(n+1)(n+2)-1$. We denote this linear system by ${|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|}^{\varphi_{n}}$ and call its elements invariant hypersurfaces.Note that the above is a base-point free linear system, since the sections $x_{0}^{n},x_{1}^{n},\ldots,x_{5}^{n}$ are all invariant. It follows by Bertini's theorem that for general choices of hypersurfaces $H_{1},H_{2},H_{3}\in {|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|}^{\varphi_{n}}$, the complete intersection $Y:=H_{1}\cap H_{2}\cap H_{3}$ is smooth and $\varphi_{n}$ acts freely on $Y$.A concrete example that works for any $n$ is given by the $3$ equations
\begin{align*}\label{concrete-example}\tag{3.3}
x_{0}^{n}+x_{1}^{n}+x_{3}^{n}+x_{4}^{n}+x_{1}x_{5}^{n-1}+x_{3}x_{5}^{n-1}=0,\\ 2x_{0}^{n}+x_{1}^{n}+3x_{2}^{n}-x_{3}^{n}+5x_{4}^{n}+x_{1}x_{5}^{n-1}-x_{3}x_{5}^{n-1}=0,\\ x_{0}^{n}-3x_{1}^{n}+x_{2}^{n}+x_{3}^{n}-2x_{4}^{n}+7x_{5}^{n}=0.
\end{align*}

We let $S:=Y/\varphi_{n}$ be the $\Z/n$-quotient of $Y$ by the automorphism $\varphi_{n}$. This is a smooth projective surface because the quotient map $\pi\colon Y\to S$ is finite and étale. For instance, if $n=2$, then $S$ is an Enriques surface and $Y$ is its canonical double cover, which is a K3 surface. For $n\geq 3$, the surface $S$ is always of general type with $q(S):=h^{0,1}(S)=0$ and topological fundamental group $\pi_{1}(S)\cong\Z/n$. For the sake of completeness, in the following proposition we compute some of its invariants:
Let $k$ be an algebraically closed field of characteristic zero and let $n\geq2$ be an integer. Let $Y\subset\mathbb{P}^{5}_{k}$ be a smooth complete intersection of three invariant hypersurfaces $H_{i}\in{|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|}^{\varphi_{n}}$, such that $\varphi_{n}$ from (<ref>) acts freely on $Y$ and set $S:=Y/\varphi_{n}$. Then the following assertions are true:

* $S$ is a smooth projective surface of general type if $n\geq 3$.

* The topological fundamental group $\pi_{1}(S)$ is cyclic of order $n$.

* The Hodge numbers of $S$ are given by $h^{2,0}=\frac{n^{2}}{4}(5n^{2}-18n+17)-1$, $h^{1,1}=\frac{n^{2}}{2}(7n^{2}-18n+13),$ and $h^{1,0}=0$.

The computations are straightforward. Recall that any smooth complete intersection of dimension $2$ is simply connected. Hence $\pi_{1}(S)\cong\Z/n$ and $q(S)=\frac{1}{2}b_{1}(S)=0$. The geometric genus of $S$ is computed from the Euler characteristic formula $\chi(\mathcal{O}_{Y})=n\chi(\mathcal{O}_{S})$, where we use $\chi(\mathcal{O}_{Y})=\frac{n^{3}}{4}(5n^{2}-18n+17)$ (cf. <cit.>).The Hirzebruch-Riemann-Roch theorem yields $h^{1,1}(S)=10\chi(\mathcal{O}_{S})- K_{S}^{2}$. We note that $K_{Y}^{2}=n K_{S}^{2}$, whereas $K_{Y}^{2}=9n^{3}(n-2)^{2}$ (see <cit.>). An obvious substitution then gives the desired relation for $h^{1,1}(S)$.It remains to check that $S$ is indeed a surface of general type as long as $n\geq 3$. Recall that $K_{Y}=\mathcal{O}_{Y}(3n-6)$ is ample and  $K_{Y}=\pi^{*}(K_{S}\otimes \mathcal{L}^{\otimes n-1})$ (see <cit.>) for some generator $\mathcal{L}\in\Pic(S)[n]\cong\Z/n$. We thus find by the projection formula that $\pi^{*}C.K_{Y}=\pi^{*}C.\pi^{*}(K_{S}\otimes \mathcal{L}^{\otimes n-1})=nC.K_{S}>0$ for all irreducible curves $C\subset S$. In particular $K_{S}$ is ample. This concludes the proof.
We note that any étale $\Z/n$-quotient of a smooth complete intersection of multidegree $(n,n,n)$ in $\mathbb{P}^{5}_{\C}$ has the same invariants with the surface of Proposition <ref>.

§ DEGENERATION
Let $k$ be an algebraically closed field of characteristic zero and fix an integer $n\geq 2$. Let $Y\subset \mathbb{P}^{5}_{k}$ be a smooth complete intersection of three very general invariant hypersurfaces $H_{i}\in|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|^{\varphi_{n}}$ and set $S:=Y/\varphi_{n}$.We aim to exhibit an example of (semi-stable) degenerations for the surface $S$ that will satisfy the properties of Theorem <ref>. The main result of this section is as follows.
Let $\kappa$ be an algebraically closed field of characteristic zero and fix an integer $n\geq 2$. Let $k$ denote an algebraic closure of the fraction field $\kappa((t))$. There is a regular flat projective scheme $\mathcal{S}\to\Spec\kappa[[t]]$, such that:

* the geometric generic fibre $S_{\bar{\eta}}$ is a quotient of the form $Y/\varphi_{n}$, where $Y\subset\mathbb{P}^{5}_{k}$ is a smooth complete intersection of three very general invariant hypersurfaces $H_{i}\in|\mathcal{O}_{\mathbb{P}^{5}_{k}}(n)|^{\varphi_{n}}$ and $\varphi_{n}$ is the automorphism from (<ref>).

* the special fibre $S_{0}=V\cup W$ is a reduced simple normal crossing divisor on $\mathcal{S}$ and consists of two irreducible components, where $V=\mathbb{P}^{2}_{\kappa}$ and $W$ is a surface with $q(W)=0$ and $H^{3}(W,\Z_{\ell}(1))=0$ for all prime factors $\ell$ of $n$. If we set $C:=V\cap W$, then $C\subset V$ is a smooth plane curve of degree $n$.

We shall need the following lemma. Let $k$ be an algebraically closed field of characteristic zero and fix an integer $n\geq2$. Consider the following three homogeneous polynomials $F_{i}\in\kappa[x_{0},x_{1},x_{2},x_{3},x_{4},x_{5}]$ of degree $n$
\begin{align*}
where $\alpha_{i},\beta_{i},\gamma_{i}\in k$. Set $T:=\{F_{1}=F_{2}=0\}$ and $Y:=\{F_{1}=F_{2}=F_{3}=0\}$ for the corresponding intersections inside $\mathbb{P}^{5}_{k}$. Then there are $\alpha_{i},\beta_{i},\gamma_{i}\in k$, such that the following assertions hold true:

*  $T$ is a smooth complete intersection of dimension $3$.

*  $Y$ is a reduced and irreducible surface whose singular locus consists only of the point $p_{0}:=(0:0:0:0:0:1)$. In addition, the multiplicity of $Y$ at $p_{0}$ is $n$.

*  The blow-up $\tilde{Y}:=Bl_{p_{0}}Y$ is smooth and the exceptional divisor $E$ is a smooth plane curve of degree $n$.

* The resolution $\tilde{Y}$ can be realized as an (n-1)-fold cyclic branched cover of a smooth complete intersection $Z\subset\mathbb{P}^{4}_{\kappa}$ of multidegree $(n,n+1)$, branched along $P'+E'$, where $P'$ and $E'$ correspond to $Y\cap\{x_{5}=0\}$ and $E$, respectively.

An easy check shows that if we set $\alpha_{i}=1$ for all $i$, then the hypersurface $S:=\{F_{1}=0\}$ is smooth. Let $L\subset|\mathcal{O}_{\mathbb{P}_{k}^{5}}(n)|$ be the linear system spanned by the sections $x_{0}^{n},x_{1}^{n},x_{3}^{n},x_{4}^{n}$ and $x_{5}^{n-1}(x_{1}+x_{3})$. When restricted to $S$, we find that this linear system has only $p_{0}$ as its base point, whereas the image of the corresponding morphism $\varphi_{L}\colon S\setminus\{p_{0}\}\rightarrow\mathbb{P}^{4}_{k}$ has dimension $>1$. Hence Bertini's theorem implies that for general $\beta_{i}\in k$, the intersection $T$ is reduced, irreducible and can only happen to have singularities at $p_{0}$. Note however that $T$ is always smooth at $p_{0}$. Indeed, the tangent space of $T$ at $p_{0}$ is given by $\{x_{1}=x_{3}=0\}\subset k^{5}$, which is $3$-dimensional. This proves (<ref>).We prove (<ref>) and (<ref>) together. A Bertini argument as above yields $\gamma_{i}\in k$, such that $Y$ is an irreducible and reduced complete intersection of dimension $2$ that acquires a unique singularity at the point $p_{0}$.Next we compute the blow-up $\pi\colon Bl_{p_{0}}Y\to Y$. We work over the chart $\{x_{5}\neq0\}$ and by abuse of notation we may identify $x_{i}\leftrightarrow \frac{x_{i}}{x_{5}}$. Consider the open sets $U_{i}:=\{(\xi_{0}:\xi_{1}:\xi_{2}:\xi_{3}:\xi_{4})\in\mathbb{P}^{4}_{k}\ |\ \xi_{i}\neq0\}$ and denote by $V_{i}\subset U_{i}\times \mathbb{A}^{1}_{k}$ the closed subset defined by the equations
\begin{align}
\{\zeta^{n-1}(\xi_{0}^{n}+\xi_{1}^{n}+\xi_{2}^{n}+\xi_{3}^{n}+\xi_{4}^{n})+\xi_{i}^{n-1}(\xi_{1}-\xi_{3})=0\}\\ \{\zeta^{n-1}(\beta_{0}\xi_{0}^{n}+\beta_{1}\xi_{1}^{n}+\beta_{3}\xi_{3}^{n}+\beta_{4}\xi_{4}^{n})+\xi_{i}^{n-1}(\xi_{1}+\xi_{3})=0\}\\ \{\gamma_{0}\xi_{0}^{n}+\gamma_{1}\xi_{1}^{n}+\gamma_{2}\xi_{2}^{n}+\gamma_{3}\xi_{3}^{n}+\xi_{4}^{n}=0\}.\end{align}
Then the $\{V_{i}\}$ form an open covering of $Bl_{p_{0}}Y$ and the restriction of $\pi$ to $V_{i}$ is given by
\begin{align}((\xi_{0}:\xi_{1}:\xi_{2}:\xi_{3}:\xi_{4}),\zeta)\mapsto (\zeta \frac{\xi_{0}}{\xi_{i}},\zeta \frac{\xi_{1}}{\xi_{i}},\zeta \frac{\xi_{2}}{\xi_{i}},\zeta \frac{\xi_{3}}{\xi_{i}},\zeta \frac{\xi_{4}}{\xi_{i}}).
\end{align}

Note here that two elements $(p,\zeta_{i})\in V_{i}$ and $(q,\zeta_{j})\in V_{j}$ are identified if and only if $p=q$ and $\zeta_{j}=\frac{\xi_{j}}{\xi_{i}}\zeta_{i}$.A straightforward calculation then shows $E:=\{(\xi_{0}:0:\xi_{2}:0:\xi_{4})\in\mathbb{P}^{4}_{k}\ |\ \gamma_{0}\xi_{0}^{n}+\gamma_{2}\xi_{2}^{n}+\xi_{4}^{n}=0 \}$ is the exceptional divisor of this blow-up. Since $E$ is a smooth (may assume $\gamma_{0}\gamma_{2}\neq0$) Cartier divisor, we find that $Bl_{p_{0}}Y$ is also smooth. The multiplicity of $Y$ at $p_{0}$ coincides with the degree of the obvious embedding $E\subset\mathbb{P}^{2}_{k}$ and thus equals $n$. In particular, $E^{2}=-n$ (cf. [156]).It remains to prove (<ref>). It is readily checked that
$$V_{i}=\{\xi_{0}^{n}+\xi_{1}^{n}+\xi_{2}^{n}+\xi_{3}^{n}+\xi_{4}^{n}\neq0\}\cup \{\beta_{0}\xi_{0}^{n}+\beta_{1}\xi_{1}^{n}+\beta_{3}\xi_{3}^{n}+\beta_{4}\xi_{4}^{n}\neq 0\},$$
as long as $\beta_{4}\gamma_{0}+\gamma_{2}(\beta_{0}-\beta_{4})-\beta_{0}\neq 0$. Over $\{\xi_{0}^{n}+\xi_{1}^{n}+\xi_{2}^{n}+\xi_{3}^{n}+\xi_{4}^{n}\neq0\}$ (similarly over the other open subset), we can rewrite the equations of $V_{i}$ as follows
\begin{align}
\zeta^{n-1}+\frac{\xi_{i}^{n-1}(\xi_{1}-\xi_{3})}{\xi_{0}^{n}+\xi_{1}^{n}+\xi_{2}^{n}+\xi_{3}^{n}+\xi_{4}^{n}}=0\\
(\xi_{3}-\xi_{1})(\beta_{0}\xi_{0}^{n}+\beta_{1}\xi_{1}^{n}+\beta_{3}\xi_{3}^{n}+\beta_{4}\xi_{4}^{n})+(\xi_{1}+\xi_{3})(\xi_{0}^{n}+\xi_{1}^{n}+\xi_{2}^{n}+\xi_{3}^{n}+\xi_{4}^{n})=0\label{eq:Z_{1,2}_1} \\
\gamma_{0}\xi_{0}^{n}+\gamma_{1}\xi_{1}^{n}+\gamma_{2}\xi_{2}^{n}+\gamma_{3}\xi_{3}^{n}+\xi_{4}^{n}=0\label{eq:Z_{1,2}-2}.
\end{align}

We let $Z\subset\mathbb{P}^{4}_{k}$ be the complete intersection of multidegree $(n,n+1)$ defined by (<ref>) and (<ref>). Then the natural projection $\pr_{1}\colon Bl_{p_{0}}Y\to\mathbb{P}^{4}_{k}$ factors throught $Z$ and the above equations show that over $\pi^{-1}(\{x_{5}\neq0\})$, the map $\pr_{1}$ locally has the form of a cyclic covering of degree $n-1$ whose ramification locus is $E$. Note that $\pr_{1}$ is compatible with $Y\setminus\{p_{0}\}\to Z$ that is induced from the obvious projection $\mathbb{P}^{5}_{k}\setminus\{p_{0}\}\to\mathbb{P}^{4}_{k}$. Similarly, one then shows $Y\setminus\{p_{0}\}\to Z$ is also of the desired form and ramified along $P:=Y\cap\{x_{5}=0\}$. To conclude observe that $Z$ is smooth because its cyclic (branched) cover $Bl_{p_{0}}Y$ is smooth and the ramification locus is a smooth divisor (cf. <cit.>). This finishes the proof.
In the above setting, if we set $\alpha_{0}=2,\alpha_{1}=1,\alpha_{2}=3,\alpha_{3}=-1,\alpha_{4}=5$ and $\beta_{i}=1$ for all $i$, then $T$ is smooth as long as $n$ is even. If $n$ is odd, then $T$ has isolated singularities. The singular locus of $T$ in this case is given by the equations:
A Bertini argument however provides us with $\gamma_{i}\in k$, such that $Y\subset\mathbb{P}^{5}_{k}$ satisfies (<ref>),(<ref>) and (<ref>) from Lemma <ref>. For instance, one can take $\gamma_{0}=1,\gamma_{1}=-3,\gamma_{2}=1,\gamma_{3}=1,\gamma_{4}=-2$ if $n\in\{2,3,4,5,6\}$.
We may work over $\C$. Fix $\alpha_{i},\beta_{i},\gamma_{i}\in\C$, so that the consequence of Lemma <ref> is true. Then $p_{0}:=(0:0:0:0:0:1)$ is the only fixed point of $\varphi_{n}$ in $Y$. Pick a hypersurface $H:=\{F=0\}\in|\mathcal{O}_{\mathbb{P}^{5}}(n)|^{\varphi_{n}}$ such that the intersection $T\cap H$ is smooth and such that $\varphi_{n}$ acts freely on $T\cap H$.Consider the one-parameter family
$$\mathcal{Y}:=\{F_{3}+t(F-F_{3})=0\}\subset T\times \mathbb{A}^{1}.$$
We note that $\bar{p}_{0}=(p_{0},0)\in\mathcal{Y}$ is a smooth point, as the tangent space is $\{x_{1}=x_{3}=F(p_{0})t=0\}\subset k^{6}$, which is $3$-dimensional ($F(p_{0})\neq0$). It follows that if we perform a base change with respect to the completion of the local ring at $0\in\A^{1}$, then the resulting projective flat family $\mathcal{Y}\to\Spec\C[[t]]$ is regular and the geometric generic fibre $Y_{\bar{\eta}}$ is a smooth complete intersection of multidegree $(n,n,n)$, such that $\varphi_{n}$ acts freely on it.Let $\pi\colon Bl_{\bar{p}_{0}}\mathcal{Y}\to \mathcal{Y}$ be the blow-up at $\bar{p}_{0}\in\mathcal{Y}$ and denote by $\tilde{\varphi}_{n}$ the natural lift of $\varphi_{n}$. Consider the open subset
$$U_{i}:=\{(\xi_{j})\in \mathbb{P}^{5}\ |\ \xi_{i}\neq0\}\times \A^{1}\subset Bl_{\bar{p}_{0}}(\mathbb{P}^{5}\times\A^{1}).$$
The exceptional divisor $V$ is a projective plane and as one can easily check it is given by the equations $\{\xi_{1}=\xi_{3}=\xi_{5}=\zeta=0\}\subset U_{i}$ for $i\in\{0,2,4\}$. If $i\in\{0,2,4\}$, then the restriction $\tilde{\varphi}_{n}|_{U_{i}}$ has the form

from which immediately one obtains that $V$ constitutes the fix point locus of $\tilde{\varphi}_{n}$ inside $Bl_{\bar{p}_{0}}\mathcal{Y}$.It is clear how to proceed now. Set $\tilde{\mathcal{Y}}:=Bl_{\bar{p}_{0}}\mathcal{Y}$ and note by (<ref>) that the component $V$ appears with multiplicity $n$ in the special fibre of $\tilde{\mathcal{Y}}\to\Spec\C[[t]]$. Since $\Fix(\tilde{\varphi}_{n})=V$, quotiening out by $\tilde{\varphi}_{n}$ yields a regular flat projective family $\mathcal{S}:=\tilde{\mathcal{Y}}/\tilde{\varphi}_{n}\to\Spec\C[[t]]$ with geometric generic fibre $S_{\bar{\eta}}=Y_{\bar{\eta}}/\varphi_{n}$ degenerating into two components that are both clearly reduced. By abuse of notation we let $V\subset S_{0}$ denote the component corresponding to the branch locus of $\tilde{\mathcal{Y}}\to\mathcal{S}$. The other component $W$ of the special fibre $S_{0}$ is the quotient surface $\tilde{Y}/\tilde{\varphi}_{n}$, which is smooth by (<ref>). Note that the assertion $C:=V\cap W$ is a smooth plane curve of degree $n$ in $V$ follows from (<ref>), as well.Next we show $q(W)=0$. To this end, consider the cyclic branched covering $f\colon\tilde{Y}\to Z$ from (<ref>) that is ramified along $P+E$, where $P:=Y\cap\{x_{5}=0\}$ and $E$ is the exceptional divisor of the blow-up $\tilde{Y}\to Y$. Let $P'$ and $E'$ denote the images in $Z$ of $P$ and $E$, respectively. The intersection number $P'^{2}=(n-1)n^{3}$ is computed by:
In particular, the prime divisor $P'$ is nef and big. Let $\mathcal{L}$ be the line bundle on $Z$ that determines the above $(n-1)$-fold cyclic branched covering, i.e. $\mathcal{L}^{\otimes(n-1)}=\mathcal{O}_{Z}(B)$, where we set $B:=P'+E'$. Fix a Weil divisor $D$ on $Z$, such that $\mathcal{L}\cong\mathcal{O}_{Z}(D)$. Then $D\sim_{\Q}\frac{1}{n-1}B$ and Kawamata-Viehweg vanishing theorem (see <cit.>) yields
$$H^{1}(Z,\mathcal{L}^{\otimes -i})=H^{1}(Z,\mathcal{O}_{Z}(-iD))=0$$
for all $1\leq i\leq n-2$. It follows $H^{1}(\mathcal{O}_{\tilde{Y}})=H^{1}(\mathcal{O}_{Z})=0$ (see <cit.>). Since the Albanese map $\Alb(\tilde{Y})\to\Alb(W)$ is surjective, we find $q(W)=0$, as claimed.It remains to check $H^{3}(W,\Z_{\ell}(1))=0$ for all prime factors $\ell$ of $n$. From <cit.>, we get $\coker(H_{1}(C,\Z)\to H_{1}(W,\Z))=0$. By Poincare duality, the latter is of course equivalent with $\iota_{*}\colon H^{1}_{sing}(C,\Z(0))\to H^{3}_{sing}(W,\Z(1))$ being surjective, where $\iota\colon C\hookrightarrow W$ is the inclusion. Especially, the comparison theorem (see <cit.>) implies $\iota_{*}\colon H^{1}(C,\Z_{\ell}(0))\to H^{3}(W,\Z_{\ell}(1))$ is surjective for all primes $\ell$. Thus our task is reduced to show that the push-forward $\iota_{*}$ is the zero map for all factors $\ell$ of $n$. In what follows, we shall identify the branch locus $C\subset W$ of the cyclic cover $\tilde{Y}\to W$ with its ramification locus $E$. By <cit.>, we have the commutative diagram

H^1(C,_ℓ(0)) [d, "×(n-1)"] [r, "j'_*"]     H^3(Z,_ℓ(1)) [d, "f^*"]

H^1(C,_ℓ(0)) [r, "j_*"]                                H^3(Ỹ,_ℓ(1)),

where we note that $H^{3}(Z,\Z_{\ell}(1))=0$ because the surface $Z$ is a smooth complete intersection in $\mathbb{P}^{4}$ as well as $\times(n-1)\colon H^{1}(C,\Z_{\ell}(0))\to H^{1}(C,\Z_{\ell}(0))$ is isomorphism, since the integer $n-1$ is invertible in $\Z_{\ell}$. Consequently $j_{*}\colon H^{1}(C,\Z_{\ell}(0))\to H^{3}(\tilde{Y},\Z_{\ell}(1))$ is the zero map. To conclude observe that $\iota_{*}\colon H^{1}(C,\Z_{\ell}(0))\to H^{3}(W,\Z_{\ell}(1))$ factors through $j_{*}$. The proof is complete.

For the sake of completeness, let us compute the geometric genus of $W$. In the notation of Theorem <ref>, the geometric genus of $W$ is given by the formula

By Proposition <ref>, it suffices to show $g(S_{\bar{\eta}})=g(C)+g(W)$, where $S_{\bar{\eta}}$ is the geometric generic fibre of the degeneration $\mathcal{S}\to\Spec\C[[t]]$ from Theorem <ref>. To this end, consider the morphism $\pi\colon W\to W'$ obtained by contracting the $-n^{2}$ curve $C$. In the notation of Lemma <ref>, the surface $W'$ is the quotient $Y/\varphi_{n}$. We claim that the singular point $p_{0}\in W'$ is Du Bois, i.e. the canonical map
\begin{align}\label{du-bois}R^{i}\pi_{*}\mathcal{O}_{W}\to H^{i}(C,\mathcal{O}_{C})\end{align}
is isomorphism for $i>0$. Since $\pi$ is isomorphism outside $p_{0}$, we find that the coherent sheaf $\mathcal{F}^{i}:=R^{i}\pi_{*}\mathcal{O}_{W}$ is supported at $p_{0}$ if $i>0$ and so $\mathcal{F}^{i}=\hat{\mathcal{F}}^{i}$, where $\hat{\mathcal{F}}^{i}$ is the completion of the stalk at $p_{0}$ and $i>0$. Let $C_{m}$ denote the closed subscheme of $W$ defined by the ideal sheaf $\mathfrak{J}^{m}:=\mathcal{O}_{W}(-mC)$. The theorem on Formal functions <cit.> then yields
There are canonical short exact sequences
for each $m$. We have $\mathfrak{J}^{m}/\mathfrak{J}^{m+1}\cong\mathcal{N}_{C/W}^{\otimes -m}$, where $\mathcal{N}_{C/W}$ is the normal bundle. It follows by Serre duality
$$H^{1}(C,\mathfrak{J}^{m}/\mathfrak{J}^{m+1})\cong H^{0}(C,\mathcal{N}_{C/W}^{\otimes m}\otimes\omega_{C})=0,$$
where the last term vanishes because $\deg_{C}(\mathcal{N}_{C/W}^{\otimes m}\otimes\omega_{C})=-mn^{2}+(n-1)(n-2)-2<0$. The long exact sequence of cohomology thus gives $H^{i}(C_{m},\mathcal{O}_{C_{m}})\cong H^{i}(C,\mathcal{O}_{C})$ for all $m\geq 1$ and $i\geq 1$. In particular, $\hat{\mathcal{F}}^{1}\cong H^{1}(C,\mathcal{O}_{C})$ and $\hat{\mathcal{F}}^{i}\cong 0$ for $i>2$. This proves (<ref>).It is easy to see that the Leray spectral sequence
$$E^{i,j}_{2}:=H^{i}(W',R^{j}\pi_{*}\mathcal{O}_{W})\implies H^{i+j}(W,\mathcal{O}_{W})$$
yields the exact sequence
\begin{align}\label{spectral-sequence}0\to H^{1}(W',\mathcal{O}_{W'})\to H^{1}(W,\mathcal{O}_{W})\to H^{1}(C,\mathcal{O}_{C})\to H^{2}(W',\mathcal{O}_{W'})\to H^{2}(W,\mathcal{O}_{W})\to0.\end{align}
Recall by Theorem <ref> that $H^{1}(W,\mathcal{O}_{W})=0$. In addition, by <cit.>, we obtain $H^{i}(W',\mathcal{O}_{W'})\cong H^{i}(S_{\bar{\eta}},\mathcal{O}_{S_{\bar{\eta}}})$. Altogether, from (<ref>) we find $g(S_{\bar{\eta}})=g(W)+g(C)$. The proof is complete.

§ EXTENDING BRAUER CLASSES

In this section, we aim to prove Theorem <ref>.
Let $\kappa$ be an algebraically closed field of characteristic zero and let $n\geq 2$ be an integer. Write $n=\ell_{1}^{r_{1}}\ell_{2}^{r_{2}}\ldots\ell_{s}^{r_{s}}$ for its prime factorisation. Let $\mathcal{S}\to\Spec\kappa[[t]]$ denote the degeneration from Theorem <ref>. Then up to base change, followed by a resolution of $\mathcal{S}$ the following holds:For all $1\leq i\leq s$, there is a class $\alpha_{i}\in\Br(\mathcal{S})[\ell_{i}^{r_{i}}]$, such that

*  $\delta(\alpha_{i}|_{S_{\bar{\eta}}})\in H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))\cong\Z/\ell_{i}^{r_{i}}$ is a generator, where $\delta$ is the Bockstein map defined on the Brauer group $\Br(S_{\bar{\eta}})$ and such that
*  for every component $S_{0j}$ of the special fibre, $\alpha_{i}|_{S_{0j}}=0\in\Br(S_{0j})$.
Let $\mathcal{S}\to\Spec R$ be a proper strictly semi-stable scheme over a discrete valuation ring $R$ with geometrically connected fibres of relative dimension $2$. When performing a finite ramified base change $\tilde{R}/R$, the model $\mathcal{S}_{\tilde{R}}:=\mathcal{S}\times_{R}\tilde{R}$ becomes singular. Following Hartl (proof of <cit.>), one finds that the family $\mathcal{S}_{\tilde{R}}\to\Spec\tilde{R}$ can be made again into a strictly semi-stable $\mathcal{\tilde{S}}\to\Spec\tilde{R}$ by repeatedly blowing up all non-Cartier divisors of the special fibre. In particular, if $S_{0}=\bigcup_{i=1}^{n}S_{0i}$ is a chain, then so is the special fibre
\begin{align}\label{special-fibre}
\tilde{S}_{0}=S_{01}\cup (\bigcup_{j=1}^{s}R_{1,j})\cup S_{02}\cup (\bigcup_{j=1}^{s}R_{2,j})\cup \dots\cup S_{0n}\end{align}
of the resulting model. The new components $R_{i,j}$ are all minimal ruled surfaces over $C_{i,i+1}:=S_{0i}\cap S_{0i+1}$ and we shall denote by $\pi_{i,j}\colon R_{i,j}\to C_{i,i+1}$ the corresponding projection. Moreover, the double intersections $S_{0i}\cap R_{i,1}$, $R_{i,s}\cap S_{0i+1}$ and $R_{i,j}\cap R_{i,j+1}$ are isomorphic to $C_{i,i+1}$ for all $1\leq i< n$ and all $1\leq j< s$, i.e. constitute sections.

Write $S:=S_{\eta}$ for the generic fibre. Recall by <cit.>
$$\Br(S_{\bar{\eta}})=\lim_{\substack{\longrightarrow\\ \kappa((t))\subset K}}\Br(S_{K}),$$
where the direct limit runs over all finite field extensions $K/\kappa((t))$. Thus we may find a finite extension $K$ of $\kappa((t))$, such that $\Br(S_{K})[n]\twoheadrightarrow \Br(S_{\bar{\eta}})[n]$ is surjective. Let $R\subset K$ be the normalization of $\kappa[[t]]$ in $K$. Then $R$ is a henselian discrete valuation ring that is finite over $\kappa[[t]]$. Consider the base change $\mathcal{S}_{R}\to\Spec R$. By <cit.>, the family $\mathcal{S}_{R}\to\Spec R$ can be turned into strictly semi-stable by repeatedly blowing up all non Cartier components of the special fibre. Let $\tilde{\mathcal{S}}\to\Spec R$ denote the induced semi-stable model. The completion of $R$ is isomorphic to a formal power series ring over $\kappa$. Fix an isomorphism $\hat{R}\cong\kappa[[t']]$ and observe that the model $\tilde{\mathcal{S}}_{\kappa[[t']]}\to\Spec\kappa[[t']]$ remains strictly semi-stable (see <cit.>). The invariance of étale cohomology under extensions of seperably closed ground fields then yields $\Br(S_{k'})[n]\twoheadrightarrow \Br(S_{\bar{k'}})[n]$ is also surjective, where $k':=\kappa((t'))$.It follows from the above discussion that we may assume $\Br(S_{\eta})[n]\twoheadrightarrow\Br(S_{\bar{\eta}})[n]$ is surjective. Note however that the special fibre $S_{0}$ takes the form $S_{0}=W\cup (\bigcup_{j=1}^{N} R_{j})\cup V$, where $N\geq 1$ and where the notation is consistent with (<ref>).Pick a class $\alpha_{i}\in\Br(S_{\bar{\eta}})[\ell^{r_{i}}_{i}]$, such that $\delta(\alpha_{i})\in H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))\cong\Z/\ell_{i}^{r_{i}}$ is a generator and denote by the same symbol a lift of this class on $\Br(S_{\eta})[\ell_{i}^{r_{i}}]$. We wish to show that $\alpha_{i}\in \Br(S_{\eta})[\ell_{i}^{r_{i}}]$ extends to an honest class on $\Br(\mathcal{S})[\ell_{i}^{r_{i}}]$. As a first step, we prove the following: We may assume that $\alpha_{i}\in\Br(S_{\eta})[\ell^{r_{i}}_{i}]$ extends to a class $\tilde{\alpha}_{i}\in\Br(\mathcal{S}\setminus \bigcup_{i=1}^{N}R_{j})[\ell_{i}^{r_{i}}]$, where we can take $N=dn^{r}-1$ for some integers $r\geq 2$ and $d>0$. We perform the $n^{2}:1$ base change $t\mapsto t^{n^{2}}$, followed by resolving the singularities as above and let $f\colon\tilde{\mathcal{S}}\to\mathcal{S}_{\kappa[[t]]}\to\mathcal{S}$ be the induced semi-stable model. Then the special fibre $\tilde{S}_{0}$ has the form (see (<ref>))
\begin{align}
\tilde{S}_{0}=W\cup(\bigcup_{j=1}^{n^{2}-1}R_{1,j})\cup R_{1}\cup (\bigcup_{j=1}^{n^{2}-1}R_{2,j})\cup R_{2} \cup \dots \cup R_{N}\cup (\bigcup_{j=1}^{n^{2}-1}R_{N +1,j})\cup V,\end{align}
and the number of ruled components has increased to $N':=N + (N+1)(n^{2}-1)=n^{2}(N+1)-1$. We set $U:=\mathcal{S}\setminus \bigcup_{i=1}^{N}R_{j}$ and let $\tilde{U}:=f^{-1}(U)$ be the open subset obtained by removing the ruled components of $\tilde{S}_{0}$ from $\tilde{\mathcal{S}}$. By <cit.>, there is a commutative diagram

0 [r]     (U)[ℓ_i^r_i] [r] [d, "f^*"]     (S_η)[ℓ_i^r_i] [r, "∂"] [d, "f^*"]     H^1(κ(V),/ℓ_i^r_i)×H^1(κ(W),/ℓ_i^r_i)  [d, "n^2·f^*=0"]

0 [r]     (Ũ)[ℓ_i^r_i] [r]                (S̃_η)[ℓ_i^r_i] [r, "∂"]                H^1(κ(V),/ℓ_i^r_i)×H^1(κ(W),/ℓ_i^r_i) ,
where the rows are exact (see <cit.>). It follows that the class $\tilde{\alpha}_{i}:=f^{*}\alpha_{i}\in\Br(\tilde{S}_{\eta})[\ell_{i}^{r_{i}}]$ extends on $\tilde{U}$. This proves the claim. Let $R:=\bigcup_{j=1}^{N}R_{j}\subset\mathcal{S}$ be the union of ruled components in the special fibre $S_{0}$ and consider the class $\bar{\alpha}_{i}:=\delta(\tilde{\alpha}_{i})\in H^{3}_{cont}(\mathcal{S}\setminus R,\Z_{\ell_{i}}(1))[\ell_{i}^{r_{i}}]$. We aim to show that $\bar{\alpha}_{i}$ lifts to a torsion class on $H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$. To this end, consider the Gysin sequence
\begin{align}\label{eq:Gysin-sequence-H^{3}}
H^{1}_{BM}(R,\Z_{\ell_{i}}(0))\overset{\iota_{*}}{\to}H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))\to H^{3}_{cont}(\mathcal{S}\setminus R,\Z_{\ell_{i}}(1))\overset{\partial}{\to}H^{2}_{BM}(R,\Z_{\ell_{i}}(0))
\end{align}
The group $H^{2}_{BM}(R,\Z_{\ell_{i}}(0))$ is torsion free. Moreover, the push-forward maps $(\iota_{j})_{*}\colon H^{1}(R_{j},A(0))\to H^{1}_{BM}(R,A(0))$, where $A\in\{\Z/\ell_{i}^{m},\Z_{\ell_{i}},\Q_{\ell_{i}},\Q_{\ell_{i}}/\Z_{\ell_{i}}\}$ yield an isomorphism
\begin{align}\label{eq:isomorphism-R}
\bigoplus_{j=1}^{N} H^{1}(R_{j},A(0))\overset{\cong}{\to}H^{1}_{BM}(R,A(0)).\end{align}
We first prove (<ref>). We proceed by induction on $N$. Clearly the isomorphism (<ref>) holds if $N=1$. Suppose $N\geq 2$ and set $R^{N-1}:=\bigcup_{j=1}^{N-1}R_{j}$. We claim that the push-forward map $\iota_{*}\colon H^{1}(R^{N-1},A(0))\to H^{1}(R,A(0))$ is injective. Indeed, by <cit.>, there is a commutative diagram

H^0(R,A(0)) [r, two heads]                                                H^0(R_N∖C,A(0)) [r]                              H^1(R^N-1,A(0)) [r, "ι_*"]     H^1(R,A(0))

A=H^0(R_N,A(0)) [u, "(ι_N)_*"] [r, "𝕀"]     H^0(R_N∖C,A(0))=A, [u, "𝕀"', shift left]

where the upper row is the Gysin sequence of the pair $(R,R^{N-1})$ and where $C:=C_{N-1,N}$. Exactness of this sequence then yields $\iota_{*}\colon H^{1}(R^{N-1},A(0))\to H^{1}(R,A(0))$ is injective.Consider the commutative diagram

0 [r]     H^1(R^N-1,A(0)) [r]     H^1(R,A(0)) [r]                                                                         H^1(R_N∖C,A(0))

0 [u] [r]                     H^1(R_N,A(0)) [u, "(ι_N)_*"', shift left] [r] [r]     H^1(R_N∖C,A(0)), [u, "𝕀"]

where the two rows are induced by the Gysin sequences of the pairs $(R,R^{N-1})$ and $(R_{N},C)$, respectively. To deduce (<ref>) thus remains to prove $H^{1}(R_{N},A(0))\to H^{1}(R_{N}\setminus C,A(0))$ is surjective. By exactness of the Gysin sequence of $(R_{N},C)$, it suffices to show $H^{0}(C,A(-1))\to H^{2}(R_{N},A(0))$ is injective. To see this just note that the composite
$$H^{j}(C,A(m))\overset{s_{*}}\to H^{j+2}(R_{N},A(m+1))\overset{(\pi_{N})_{*}}{\to}H^{j}(C,A(m))$$
is the identity.Next we prove that $H^{2}_{BM}(R,\Z_{\ell_{i}}(0))$ is torsion free. In the above argument we have already seen that $H^{1}(R,\Z_{\ell_{i}}(0))\to H^{1}(R_{N}\setminus C,\Z_{\ell_{i}}(0))$ is surjective. Hence exactness of the Gysin sequence of $(R,R^{N-1})$ gives that $\iota_{*}\colon H^{2}(R^{N-1},\Z_{\ell_{i}}(0))\to H^{2}(R,\Z_{\ell_{i}}(0))$ is injective. As before we have a commutative diagram

0 [r]     H^2(R^N-1,_ℓ_i(0)) [r]     H^2(R,_ℓ_i(0)) [r]                                                                                H^2(R_N∖C,_ℓ_i(0))

H^2(R_N,_ℓ_i(0)) [u, "(ι_N)_*"', shift left] [r] [r, two heads]     H^2(R_N∖C,_ℓ_i(0)), [u, "𝕀"]

where the surjectivity of $H^{2}(R_{N},\Z_{\ell_{i}}(0))\to H^{2}(R_{N}\setminus C,\Z_{\ell_{i}}(0))$ follows again from the exactness of the Gysin sequence of $(R_{N},C)$ (and where we use that $H^{1}(C,\Z_{\ell_{i}}(-1))\to H^{3}(R_{N},\Z_{\ell_{i}}(0))$ is injective). Thus we obtain a short exact sequence
$$0\to H^{2}(R^{N-1},\Z_{\ell_{i}}(0))\to H^{2}(R,\Z_{\ell_{i}}(0))\to H^{2}(R_{N}\setminus C,\Z_{\ell_{i}}(0))\to 0,$$
where $H^{2}(R^{N-1},\Z_{\ell_{i}}(0))$ is torsion free by the induction hypothesis and $H^{2}(R_{N}\setminus C,\Z_{\ell_{i}}(0))\cong \Z_{\ell_{i}}\cdot f$ is free of rank $1$, generated by the class of a fibre. This concludes the proof. Since $H^{2}_{BM}(R,\Z_{\ell_{i}}(0))$ is torsion free by Claim <ref>, we find that the residue $\partial\bar{\alpha}_{i}=0$ vanishes and so the class $\bar{\alpha}_{i}$ extends to $H^{2}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ (see (<ref>)). The technical part of the proof is then to show that we can choose the lift $\bar{\alpha}_{i}\in H^{2}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ to be torsion. For this reason, let us compute the torsion in $H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$. There is a canonical extension
\begin{align}\label{canonical-extension}
0\to\Z/\ell_{i}^{r_{i}}\to H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))\overset{\{\iota_{j}^{*}\}_{j}}{\to}\bigoplus_{j=1}^{N}H^{3}_{cont}(R_{j},\Z_{\ell_{i}}(1))\to 0.\end{align}
In particular, $H_{cont}^{3}(\mathcal{S},\Z_{\ell_{i}}(1))_{\tors}\cong\Z/\ell_{i}^{r_{i}}$. The proper base change theorem (see <cit.>) combined with <cit.> yield the canonical isomorphism $H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))\cong H^{3}_{cont}(S_{0},\Z_{\ell_{i}}(1))$. The Mayer-Vietoris sequence gives rise to the following short exact sequence
\begin{align}\label{MV} 0\to A\to H^{3}_{cont}(S_{0},\Z_{\ell_{i}}(1))\overset{\{\iota_{j}^{*}\}_{j}}{\to} \bigoplus_{j=1}^{N}H^{3}(R_{j},\Z_{\ell_{i}}(1))\to 0,\end{align}
where $A$ is defined as the cokernel of the map
\begin{align}\label{A} H^{2}(W,\Z_{\ell_{i}}(1))\oplus(\bigoplus_{j=1}^{N}H^{2}(R_{j},\Z_{\ell_{i}}(1)))\oplus H^{2}(V,\Z_{\ell_{i}}(1))\to \bigoplus_{j=0}^{N} H^{2}(C_{j,j+1},\Z_{\ell_{i}}(1))\cong \Z_{\ell_{i}}^{N+1}\end{align}
and where $C_{j,j+1}$ stands for the double intersection of two consecutive components of the special fibre $S_{0}$, i.e. $C_{0,1}:=W\cap R_{1},\ C_{j,j+1}:=R_{j}\cap R_{j+1}$ for $1\leq j<N$ and $C_{N,N+1}:= R_{N}\cap V$.To show that $\{\iota_{j}^{*}\}_{j}$ in (<ref>) is onto, we use that $H^{3}(C_{j,j+1},\Z_{\ell_{i}}(1))=0$ (see <cit.>). The terms $H^{3}(W,\Z_{\ell_{i}}(1))$ and $H^{3}(V,\Z_{\ell_{i}}(1))$ do not appear in (<ref>), as both vanish (see Theorem <ref>).It remains to compute $A$. We claim that the summation map $\Z_{\ell_{i}}^{N+1}\to\Z_{\ell_{i}}$ yields an isomorphism $A\cong \Z/\ell_{i}^{r_{i}}$. To this end, recall $H^{2}(R_{j},\Z_{\ell_{i}}(1))=\Z_{\ell_{i}}\cdot\cl^{1}([C_{j,j+1}])\times \Z_{\ell_{i}}\cdot f_{j}$, where $f_{j}$ denotes the cycle class of a fibre of the projection $\pi_{j}\colon R_{j}\to C_{1,2}$. Via (<ref>) the tuple $(0,\ldots,0,f_{j},0,\ldots,0)$ maps to
$$(0,\ldots,0,-f_{j}\cdot C_{j-1,j},f_{j}\cdot C_{j,j+1},0,\ldots,0)=(0,\ldots,0,-1,1,0,\ldots,0).$$
In addition, the self-intersection $C_{j,j+1}^{2}=\pm n^{2}$, i.e. is a multiple of $\ell_{i}^{r_{i}}$, and
$$\gcd (\gamma_{1}\cdot C_{0,1},\gamma_{2}\cdot C_{N,N+1})=\ell_{i}^{r_{i}},$$
where the cycles $\gamma_{1}$ and $\gamma_{2}$ run through $H^{2}(W,\Z_{\ell_{i}}(1))$ and $H^{2}(V,\Z_{\ell_{i}}(1))$, respectively. For the second claim recall that $W$ admits an $n$-fold cyclic covering branched along $C_{0,1}$ and so $C_{0,1}\sim n\cdot D$ for some divisor $D$, whereas $C_{N,N+1}$ sits as a degree $n$ plane curve in $V$. It follows that the summation map induces a well-defined map $A\twoheadrightarrow\Z/\ell_{i}^{r_{i}}$. The injectivity condition can be easily checked and thus details are omitted. Fix the projection $\pi_{j}\colon R_{j}\to C$ of the ruled surface $R_{j}$ for all $1\leq j\leq N$ and consider the composite
\begin{align}\label{prod}
\Phi\colon H^{1}(C,A(0))^{\oplus N}\overset{\{\pi_{j}^{*}\}}{\to}\bigoplus_{j=1}^{N}H^{1}(R_{j},A(0))\overset{\{\iota_{j}^{*}\}\circ\{\iota_{j*}\}}{\to}\bigoplus_{j=1}^{N}H^{3}(R_{j},A(1))\overset{\{\pi_{j*}\}}{\to} H^{1}(C,A(0))^{\oplus N}.\end{align}
We shall prove a series of properties of $\Phi$ that will be needed later. The map $\Phi$ is explicitly given by
$$(\alpha_{1},\alpha_{2},\ldots,\alpha_{N-1},\alpha_{N})\mapsto (\alpha_{2}-2\alpha_{1},\alpha_{1}+\alpha_{3}-2\alpha_{2},\ldots,\alpha_{j-1}+\alpha_{j+1}-2\alpha_{j},\ldots,\alpha_{N-1}-2\alpha_{N}).$$
We shall frequently cite properties of cohomology with support from <cit.>, which applies to our setting by (<ref>). Let $\tilde{\Phi}:=\{\iota_{j}^{*}\}\circ\{\iota_{j*}\}\circ\{\pi_{j}^{*}\}$ and note that
\begin{align}\label{tilde-prod}
\tilde{\Phi}(\alpha_{1},\alpha_{2},\ldots,\alpha_{N-1},\alpha_{N})=(\iota_{1}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j}),\iota_{2}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j}),\ldots,\iota_{N}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})).\end{align}

We compute $\iota_{t}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})$ for all $1\leq t\leq N$. We have
$$\iota_{t}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})=\sum_{j=1}^{N}(\iota_{j*}\pi_{j}^{*}\alpha_{j}\cup [R_{t}])=\sum_{j=1}^{N}\iota_{j*}(\pi_{j}^{*}\alpha_{j}\cup \iota_{j}^{*}[R_{t}]),$$
where $[R_{t}]\in H^{0}(R_{t},\Z_{\ell_{i}}(0))=H^{2}_{R_{t},cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ denotes the fundamental class and where in the last equality we use the projection formula (cf. <cit.>). Since $R_{t}\cap R_{j}=\emptyset$ for $j\notin\{t-1,t,t+1\}$, we find
\begin{align*}
\iota_{1}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})=\iota_{1*}(\pi_{1}^{*}\alpha_{1}\cup \iota_{1}^{*}[R_{1}])+\iota_{2*}(\pi_{2}^{*}\alpha_{2}\cup \iota_{2}^{*}[R_{1}])\\
\iota_{t}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})=\sum_{j=t-1}^{t+1}\iota_{j*}(\pi_{j}^{*}\alpha_{j}\cup \iota_{j}^{*}[R_{t}])\ \text{for}\ 1<t<N \\\iota_{N}^{*}(\sum_{j=1}^{N}\iota_{j*}\pi_{j}^{*}\alpha_{j})=\iota_{N-1*}(\pi_{N-1}^{*}\alpha_{N-1}\cup \iota_{N-1}^{*}[R_{N}])+\iota_{N*}(\pi_{N}^{*}\alpha_{N}\cup \iota_{N}^{*}[R_{N}]).
\end{align*}
Observe that $\iota_{t-1*}(\pi_{t-1}^{*}\alpha_{t-1}\cup \iota_{t-1}^{*}[R_{t}])=\iota_{t-1*}(\pi_{t-1}^{*}\alpha_{t-1}\cup [C_{t-1,t}])=s^{t}_{t-1*}(\alpha_{t-1})$, where $s^{t}_{t-1}$ denotes the section $C\to C_{t-1,t}\hookrightarrow R_{t}$. Similarly, one obtains
where $s^{t}_{t+1}$ is defined accordingly. It remains to compute $\iota_{t*}(\pi_{t}^{*}\alpha_{t}\cup\iota_{t}^{*}[R_{t}])$. By functoriality of the Gysin sequence for continuous étale cohomology with support, we get $\iota_{t}^{*}[R_{t}]=\iota_{t}^{*}\cl_{\mathcal{S}}^{1}([R_{t}])\in H^{2}(R_{t},\Z_{\ell_{i}}(1))$. Since $[V]+\sum_{j=1}^{N}[R_{j}]+[W]\sim 0$ on $\mathcal{S}$, we find
$$\cl^{1}_{\mathcal{S}}([R_{t}])=-\cl^{1}_{\mathcal{S}}([V])-\sum_{j\neq t}\cl^{1}_{\mathcal{S}}([R_{j}])-\cl^{1}_{\mathcal{S}}([W]).$$
We thus deduce the equality $\iota_{t}^{*}([R_{t}])=-\cl^{1}_{R_{t}}([C_{t-1,t}])-\cl^{1}_{R_{t}}([C_{t,t+1}])$ for $1\leq t\leq N$. By making the above substitution, we get
where the last equality follows again by the projection formula. Finally, $\tilde{\Phi}(\alpha_{1},\alpha_{2},\ldots,\alpha_{N})$ is given by
\begin{align*}(s^{1}_{2*}(\alpha_{2})-s^{1}_{0*}(\alpha_{1})-s^{1}_{2*}(\alpha_{1}),\ldots,s^{t}_{t-1*}(\alpha_{t-1})+s^{t}_{t+1*}(\alpha_{t+1})-s^{t}_{t-1*}(\alpha_{t})-s^{t}_{t+1*}(\alpha_{t}),\ldots)\end{align*}
and pushing forward with respect to $\{\pi_{j}\}$ yields the desired element. The proof is complete.
Write $N=\tilde{d}\ell^{r}_{i}-1$ for some positive integers $r$ and $\tilde{d}$ with $\gcd (\tilde{d},\ell_{i})=1$. For $A=\Z_{\ell_{i}}$ the map $\Phi$ is injective. If $A=\Z/\ell_{i}^{r}$, then $\ker(\Phi)\cong J(C)[\ell_{i}^{r}]$.

By Lemma <ref>, we find that $(\alpha_{1},\alpha_{2},\ldots,\alpha_{N})\in\ker(\Phi)$ if and only if $\alpha_{j}=j\cdot\alpha_{1}$ for all $1\leq j\leq N$ and $\alpha_{N-1}-2\alpha_{N}=(N-1)\alpha_{1}-2N\alpha_{1}=-(N+1)\alpha_{1}=-\tilde{d}\ell^{r}_{i}\alpha_{1}=0$. Since $H^{1}(C,\Z_{\ell_{i}}(0))$ is torsion free, we conclude that $\Phi$ is injective in this case. For $A=\Z/\ell_{i}^{r}$, the map
$$\alpha\mapsto (\alpha,2\alpha,\ldots,j\alpha,\ldots,N\alpha)$$
yields an isomorphism $J(C)[\ell_{i}^{r}]\cong\ker(\Phi)$, where we use that $J(C)[\ell_{i}^{r}]\cong H^{1}(C,\Z/\ell_{i}^{r})$.
In the setting of Lemma <ref>, we have an isomorphism $\coker(\Phi)\cong J(C)[\ell_{i}^{r}]$ for $A\in\{\Z_{\ell_{i}},\Z/\ell^{r}_{i}\}$. We begin by showing that $\coker(\Phi)$ is annihilated by $\ell_{i}^{r}$. Pick an element $\alpha\in H^{1}(C,\Z_{\ell_{i}}(0))$ and consider the tuple $\alpha_{j}\in H^{1}(C,\Z_{\ell_{i}}(0))^{\oplus N}$ whose $j$-entry is the element $\alpha$ and all the others are zero. A simple induction argument then yields
where the non-trivial entries of this tuple are $m$ and $m+1$ and where $1\leq m\leq N-1$. From this follows $\Phi(\sum_{j=1}^{N}j\cdot\alpha_{j})=(0,0,\ldots,0,-N\alpha,(N-1)\alpha)+\Phi(N\cdot\alpha_{N})=(0,\ldots,0,-(N+1)\alpha)=(0,\ldots,0,-\tilde{d}\ell_{i}^{r}\alpha)$. Since the integer $\tilde{d}$ is invertible in $\Z_{\ell_{i}}$, we find
Let $\beta:=-{(\tilde{d})}^{-1}\sum_{j=1}^{N}j\cdot\alpha_{j}$ and consider the system of equations
\begin{align*}
\Phi(\ell_{i}^{r}\alpha_{N})=\ell_{i}^{r}\cdot\alpha_{N-1}-2\Phi(\beta),\\ \Phi(\ell_{i}^{r}\alpha_{j})=\ell_{i}^{r}\cdot\alpha_{j-1}-2\ell_{i}^{r}\cdot\alpha_{j}+\ell_{i}^{r}\cdot\alpha_{j+1}\ \text{for}\ 2\leq j\leq N-1 \\
\Phi(\ell_{i}^{r}\cdot\alpha_{1})=-2\ell_{i}^{r}\cdot\alpha_{1}+\ell_{i}^{r}\cdot\alpha_{2}.\end{align*}
A recursive induction on the above system yields $\ell^{r}_{i}\cdot\alpha_{j}\in\im(\Phi)$ for all $1\leq j\leq N$. In particular, $\coker(\Phi)$ is $\ell_{i}^{r}$-torsion.Next we consider the commutative diagram with exact rows

0 [r]     H^1(C,_ℓ_i(0))^⊕N [r, "×ℓ_i^r"] [d, "Φ"]     H^1(C,_ℓ_i(0))^⊕N [r] [d, "Φ"]     H^1(C,/ℓ_i^r(0))^⊕N [r] [d, "Φ"]     0

0 [r]     H^1(C,_ℓ_i(0))^⊕N [r, "×ℓ_i^r"]                      H^1(C,_ℓ_i(0))^⊕N [r]                      H^1(C,/ℓ_i^r(0))^⊕N [r]                                 0,

where $\overline{\Phi}$ stands for $\Phi$ with $A=\Z/\ell_{i}^{r}$. Since $\ker(\Phi)=0$, serpents Lemma gives the exact sequence
from which immediately one deduces the isomorphisms
$$\ker(\overline{\Phi})\cong\coker(\Phi)\ \text{and}\ \coker(\Phi)\cong\coker(\overline{\Phi}).$$
To conclude, recall by Lemma <ref> that $\ker(\overline{\Phi})\cong J(C)[\ell_{i}^{r}]$. This finishes the proof.

Finally, we are in the position to conclude the proof of Theorem <ref>. Recall that we have a class $\bar{\alpha}_{i}\in H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ whose image in $H^{3}_{cont}(\mathcal{S}\setminus R,\Z_{\ell_{i}}(1))$ is killed by $\ell_{i}^{r_{i}}$ and further restricts to a generator $0\neq\delta(\alpha_{i})\in H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))\cong\Z/\ell_{i}^{r_{i}}$. By Claim <ref>, there is a canonical extension
\begin{align}
\label{extension}0\to\Z/\ell_{i}^{r_{i}}\to H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))\to\bigoplus_{j=1}^{N}H^{3}(R_{j},\Z_{\ell_{i}}(1))\to 0.\end{align}
The push-forward map $\iota_{*}\colon H^{1}_{BM}(R,\Z_{\ell_{i}}(0))\to H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ can be identified with
(see (<ref>)), where the last map is injective by Lemma <ref>. Hence (<ref>) yields a short exact sequence
\begin{align} \label{final-extension} 0\longrightarrow\Z/\ell_{i}^{r_{i}}\longrightarrow \frac{H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))}{\im(\iota_{*})}\overset{g:=\{\iota^{*}_{j}\}}{\longrightarrow} J(C)[\ell_{i}^{r}] \longrightarrow 0,
\end{align}
where we use $\coker(\{\iota^{*}_{j}\}\circ\{\iota_{j*}\})\cong\coker(\Phi)\cong J(C)[\ell_{i}^{r}]$ (see Lemma <ref>) and where $r\geq 2r_{i}$ by Claim <ref>.The element $[\bar{\alpha}_{i}]\in\coker(\iota_{*})\subset H^{3}(\mathcal{S}\setminus R,\Z_{\ell_{i}}(1))$ is $\ell_{i}^{r_{i}}$-torsion. Thus we may write $g([\bar{\alpha}_{i}])=\ell_{i}^{r_{i}}c$ for some $c\in J(C)[\ell_{i}^{2r_{i}}]\subset J(C)[\ell_{i}^{r}]$. Pick a class $[\gamma]\in\coker(\iota_{*})$ with $g([\gamma])=c$. Then $g([\bar{\alpha}_{i}-\ell_{i}^{r_{i}}\gamma])=0$ and therefore $[\bar{\alpha}_{i}-\ell_{i}^{r_{i}}\gamma]=m[\tau]$ for some generator $\tau$ of $\Z/\ell_{i}^{r_{i}}\hookrightarrow H^{3}_{cont}(\mathcal{S},\Z_{\ell_{i}}(1))$ and integer $0\leq m<\ell_{i}^{r_{i}}$. It follows in $H^{3}_{cont}(\mathcal{S}\setminus R,\Z_{\ell_{i}}(1))$, we have the equality $\bar{\alpha}_{i}=\ell_{i}^{r_{i}}\gamma+m\tau$. Since $H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))\cong\Z/\ell_{i}^{r_{i}}$, restricting further to $S_{\bar{\eta}}$ yields $\delta(\alpha_{i})=m\tau$. In particular, this implies that the restriction map
$$\Z/\ell_{i}^{r_{i}}\cong H^{3}(\mathcal{S},\Z_{\ell_{i}}(1))_{\tors}\to H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))$$
is isomorphism. Observe that any $\alpha\in H^{2}(\mathcal{S},\mu_{\ell^{r_{i}}_{i}})$, such that $\delta(\alpha)\in H^{3}(\mathcal{S},\Z_{\ell_{i}}(1))_{\tors}$ is a generator yields a Brauer class on $\mathcal{S}$ with the property, $\delta(\alpha|_{S_{\bar{\eta}}})$ generates $H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))$. This proves (<ref>).It remains to show (<ref>). We need only to choose our lift $\alpha_{i}\in\Br(\mathcal{S})[\ell^{r_{i}}_{i}]$ from (<ref>) in such a way that the restriction to $W$ vanishes, i.e. $\alpha_{i}|_{W}=0$, since the other components of $S_{0}$ are ruled. To this end, we consider the following commutative diagram with exact rows

0 [r]     H^2(𝒮,_ℓ_i(1))⊗/ℓ_i^r_i [r] [d, "ρ"]     H^2(𝒮,μ_ℓ_i^r_i) [r, "δ"] [d, "ρ̅"]     H^3(𝒮,_ℓ_i(1))_ [r] [d, "=0"]     0

0 [r]     H^2(W,_ℓ_i(1))⊗/ℓ_i^r_i [r]                                 H^2(W,μ_ℓ_i^r_i) [r, "δ"]                                       H^3(W,_ℓ_i(1))_=0 [r]                             0.

We claim that both restriction maps $\rho$ and $\bar{\rho}$ are surjective. Clearly it suffices to prove surjectivity only for $\rho$. The proper base change theorem provides with $H^{2}(\mathcal{S},\Z_{\ell_{i}}(1))\cong H^{2}(S_{0},\Z_{\ell_{i}}(1))$. Mayer-Vietoris exact sequence then yields
\begin{align}\label{MV-2}
H^{2}(S_{0},\Z_{\ell_{i}}(1))\to H^{2}(W,\Z_{\ell_{i}}(1))\oplus(\bigoplus_{j=1}^{N}H^{2}(R_{j},\Z_{\ell_{i}}(1))\oplus H^{2}(V,\Z_{\ell_{i}}(1))\to \bigoplus_{j=0}^{N}H^{2}(C_{j,j+1},\Z_{\ell_{i}}(1)). \end{align}
Recall that $W$ admits an $n$-fold cyclic branched covering along $C:=C_{0,1}\subset W$ and so $C\sim n\cdot D$ for some divisor $D$ on $W$. Pick a class $\beta\in H^{2}(W,\Z_{\ell_{i}}(1))$ and consider the intersection number $d:=(D\cdot\beta)\in\Z$. Let $f_{j}\in H^{2}(R_{j},\Z_{\ell_{i}}(1))$ be the class of a fibre of the projection $\pi_{j}\colon R_{j}\to C$ and let $H\in H^{2}(V,\Z_{\ell_{i}}(1))$ be the class of a hyperplane. It is easy to check that the tuple $(\beta,(nd)f_{1},\ldots,(nd)f_{N},dH)$ maps to zero via the last map of (<ref>). Exactness of (<ref>) thus yields an element $\tilde{\beta}\in H^{2}(S_{0},\Z_{\ell_{i}}(1))$, such that $\tilde{\beta}|_{W}=\beta$. In particular, we find that $\rho$ is surjective.Assume that a lift $\alpha_{i}\in H^{2}(\mathcal{S},\mu_{\ell_{i}^{r_{i}}})$ such that $\delta(\alpha_{i})\in H^{3}(\mathcal{S},\Z_{\ell_{i}}(1))_{\tors}$ is a generator, satisfies $0\neq\alpha_{i}|_{W}\in H^{2}(W,\mu_{\ell_{i}^{r_{i}}})$. Since $\rho$ is surjective and $H^{2}(W,\Z_{\ell_{i}}(1))\otimes\Z/\ell_{i}^{r_{i}}\cong H^{2}(W,\mu_{\ell_{i}^{r_{i}}})$, we can choose $\tilde{\alpha}_{i}\in H^{2}(\mathcal{S},\Z_{\ell_{i}}(1))\otimes\Z/\ell_{i}^{r_{i}}$, such that $\tilde{\alpha}_{i}|_{W}=\alpha_{i}|_{W}$. Moreover, we have $\delta(\alpha_{i}-\tilde{\alpha}_{i})=\delta(\alpha_{i})$ and thus replacing the Brauer class of $\alpha_{i}$ by the one of $\alpha_{i}-\tilde{\alpha}_{i}$, we obtain the desired lift. The proof of Theorem <ref> is finally complete.
Let $n=\ell_{1}^{r_{1}}\ell^{r_{2}}_{2}\dots\ell^{r_{s}}_{s}$ be the prime factorisation. By Theorem <ref>, there is a semi-stable degeneration $\mathcal{S}\to\Spec\kappa[[t]]$ having the properties (<ref>), (<ref>) and in addition for all $1\leq i\leq s$, there is a class $\alpha_{i}\in\Br(\mathcal{S})[\ell^{r_{i}}_{i}]$, such that $\delta(\alpha_{i}|_{S_{\bar{\eta}}})\in H^{3}(S_{\bar{\eta}},\Z_{\ell_{i}}(1))_{\tors}\cong\Z/\ell^{r_{i}}_{i}$ is a generator and such that for every component $S_{0i}$ of the special fibre, $\alpha_{i}|_{S_{0i}}=0\in\Br(S_{0i})$. It follows that the class $\alpha:=\sum_{i=1}^{s}\alpha_{i}\in\Br(\mathcal{S})[n]$ satisfies (<ref>). The proof is complete.
Let $\kappa$ be an algebraically closed field of characteristic zero and let $n\geq 2$ be an integer. Let $\mathcal{S}\to\Spec\kappa[[t]]$ denote the degeneration from Theorem <ref> and fix an algebraic closure $k$ of the fraction field $\kappa((t))$. Then the following holds: For any smooth projective variety $Z$ over $\kappa$ the exterior product map
\begin{align}\label{eq:exterior-product}
E^{j}_{n}(Z)/n\longrightarrow A^{j+1}(S_{\bar{\eta}}\times_{k}Z_{k})/n,\ [z]\mapsto [\mathcal{L}\times z]\end{align}
is injective, where $0\neq\mathcal{L}\in\NS(S_{\bar{\eta}})_{\tors}=\Pic(S_{\bar{\eta}})_{\tors}\cong\Z/n$ is any generator.
The properties $(\text{P}1)$ and $(\text{P}2)$ in Theorem <ref> correspond to (<ref>) and (<ref>) in Theorem <ref>, respectively. Thus the result follows immediately from Theorem <ref>.

§ PROOF OF THE MAIN RESULTS

Corollary <ref> will be deduced from the following. Let $n\geq 2$ be an integer. There is a smooth complete intersection $Y\subset\mathbb{P}^{5}_{\mathbb{C}}$ of three invariant hypersurfaces $H_{i}\in|\mathcal{O}_{\mathbb{P}^{5}}(n)|^{\varphi_{n}}$, such that $\varphi_{n}$ from (<ref>) acts freely on $Y$ and if we set $S:=Y/\varphi_{n}$ then, the exterior product map
\begin{align}\label{ext-prod-A^{i}}
A^{j}(Z)/n\longrightarrow A^{j+1}(S\times Z)/n,\ [z]\mapsto [\mathcal{L}\times z]\end{align}
is injective for all smooth complex projective varieties $Z$ and where $0\neq\mathcal{L}\in \Pic(S)_{\tors}\cong\Z/n$ is any generator.
We argue as in the proof of <cit.>. Let $Y\subset\mathbb{P}^{5}_{\C}$ be a smooth complete intersection of three hypersurfaces $H_{i}\in|\mathcal{O}_{\mathbb{P}^{5}}(n)|^{\varphi_{n}}$, such that $\varphi_{n}$ from (<ref>) acts freely on $Y$. As a first step, we show that the kernel of the exterior product map (<ref>) is contained in $E^{j}_{n}(Z)$. To see this, consider the natural projections $p\colon S\times Z\to S$ and $q\colon S\times Z\to Z$ and note
$$\cl^{j+1}_{S\times Z}([\mathcal{L}\times z])=p^{*}\cl_{S}^{1}([\mathcal{L}])\cup q^{*}\cl_{Z}^{j}([z])\in H^{2j+2}(S\times Z,\Z/n(j+1)).$$
Set $\alpha:=\cl_{S}^{1}([\mathcal{L}])\in H^{2}(S,\Z/n(1))$ and $\beta:=\cl_{Z}^{j}([z])\in H^{2j}(Z,\Z/n(j))$. Since $\alpha\neq 0$, we find $p^{*}\alpha\cup q^{*}\beta=0$ if and only if $\beta=0$. Indeed, Poincare duality yields a class $\bar{\alpha}\in H^{2}(S,\Z/n(1))$ with $\alpha\cup\bar{\alpha}=\cl^{2}_{S}(pt)$ and we can recover the class $\beta$ by
$$q_{*}(p^{*}\bar{\alpha}\cup p^{*}\alpha \cup q^{*}\beta)=q_{*}(p^{*}\cl_{S}^{2}(pt)\cup q^{*}\beta)=\beta\in H^{2j}(Z,\Z/n(j)).$$
Hence, we see $[\mathcal{L}\times z]=0$ implies $[z]\in E^{j}_{n}(Z)$.Our task is thus reduced to show
$$\times\colon E_{n}^{j}(Z)\longrightarrow A^{j+1}(S\times Z)/n$$
is injective for some $S$. Recall that Chow groups modulo algebraic equivalence are countable. Therefore, we may find a countable algebraically closed field $\kappa\subset\C$, such that $Z=Z_{\kappa}\times_{\kappa}\C$ for some smooth projective variety $Z_{\kappa}$ over $\kappa$ and such that the natural map
\begin{align}\label{eq:base-change}
A^{j}(Z_{\kappa})\longrightarrow A^{j}(Z)\end{align}
is surjective. Since we may express $\C$ as the direct limit of (smooth) finitely generated $\kappa$-algebras with $\kappa$ algebraically closed, a well-known specialization argument yields that (<ref>) is always injective and thus in our case an isomorphism. In addition, the invariance of étale cohomology under seperably closed field extensions allows one to identify the reductions modulo $n$ of the cycle class maps $\cl^{j}_{Z_{\kappa}}$ and $\cl^{j}_{Z}$. Consequently (<ref>) yields the isomorphism
\begin{align}\label{eq:base-change-2}
E_{n}^{j}(Z_{\kappa})\longrightarrow E_{n}^{j}(Z).\end{align}
Fix an algebraic closure $k$ of the fraction field $\kappa((t))$ and let $\mathcal{S}\to\Spec\kappa[[t]]$ be the degeneration from Theorem <ref>. By Corollary <ref>, the exterior product map
$$E_{n}^{j}(Z_{\kappa})\longrightarrow A^{j+1}(S_{\bar{\eta}}\times_{k} Z_{k})/n$$
is injective. As the field $\kappa$ is countable, the embedding $\kappa\subset \C$ factors through $k\subset\C$. A specialization argument as before then shows
$$A^{j+1}(S_{\bar{\eta}}\times_{k} Z_{k})/n\ \hooklongrightarrow A^{j+1}(S\times Z)/n$$
is injective, where $S:=S_{\bar{\eta}}\times_{k}\C$. Combining the above, we finally obtain
$$E_{n}^{j}(Z)\cong E_{n}^{j}(Z_{\kappa})\longrightarrow A^{j+1}(S\times Z)/n$$
is injective. The proof is complete.
Let $X$ be a smooth complex projective variety. Recall that we have a transcendental Abel-Jacobi map (see <cit.>) on torsion cycles
\begin{align}\label{Abel-Jacobi-map} \lambda^{i}_{tr}\colon \Griff^{i}(X)_{\tors}\longrightarrow H^{2i-1}(X,\Q/\Z(i))/N^{i-1}H^{2i-1}(X,\Q(i)),\end{align}
where $N^{*}$ denotes Grothendieck's coniveau filtration. By <cit.>, it coincides with Griffiths transcendental Abel-Jacobi map (see [85]) on torsion cycles. We let

In the setting of Theorem <ref>, there is a canonical injection
$$E_{n}^{j}(Z)\ \hooklongrightarrow\mathcal{T}^{j+1}(S\times Z)[n].$$
By Theorem <ref>, we have an injection
\begin{align}\label{c-i}
E_{n}^{j}(Z)\ \hooklongrightarrow A^{j+1}(S\times Z)[n],\ [z]\mapsto[\mathcal{L}\times z].\end{align}
We claim that this map factors through $\mathcal{T}^{j+1}(S\times Z)[n]$. Let $p\colon S\times Z\to S$ and $q\colon S\times Z \to Z$ be the two projections. Note that for any $[z]\in E_{n}^{j}(Z)$, the class
$$\cl^{j+1}_{S\times Z}([\mathcal{L}\times z])=p^{*}\cl^{1}_{S}([\mathcal{L}])\cup q^{*}\cl^{j}_{Z}([z])$$
is zero, since $\mathcal{L}$ is annihilated by $n$, whereas $\cl^{j}_{Z}([z])$ is divisible by $n$. It follows that the image of (<ref>) is contained in $\Griff^{j+1}(S\times Z)[n]$. The formula (cf. <cit.>)
$$\lambda^{j+1}_{tr}([\mathcal{L}\times z])=p^{*}\lambda^{1}_{tr}([\mathcal{L}])\cup q^{*}\overline{\cl^{j}_{Z}([z])},$$
where $\overline{\cl^{j}_{Z}([z])}\in H^{2j}(Z,\Q/\Z(j))$ denotes the image of the reduction modulo $n$ of $\cl^{j}_{Z}([z])$ via the natural map $H^{2j}(Z,\Z/n(j))\to H^{2j}(Z,\Q/\Z(j))$ yields $[\mathcal{L}\times z]\in\mathcal{T}^{j+1}(S\times Z)[n]$ for all $[z]\in E_{n}^{j}(Z)$. This concludes the proof. By Theorem <ref>, and the proof of Corollary <ref>, we have a canonical injection
$$E_{n}^{j}(Z)\ \hooklongrightarrow\frac{\Griff^{j+1}(S\times Z)[n]}{n\Griff^{j+1}(S\times Z)[n^{2}]},\ [z]\mapsto[\mathcal{L}\times z].$$
Moreover, note that $\Griff^{j}(Z)/n\subset E_{n}^{j}(Z)$ (see <cit.>) and so the injectivity of (<ref>) follows.
Let $C\subset\mathbb{P}^{2}_{\C}$ be a smooth quartic curve and write $JC$ for its Jacobian variety. By Corollary <ref>, we have a canonical injection
$$\Griff^{2}(JC)/n\ \hooklongrightarrow \frac{\Griff^{3}(S\times JC)[n]}{n\Griff^{3}(S\times JC)[n^{2}]},\ [z]\mapsto[\mathcal{L}\times z],$$
where $0\neq\mathcal{L}\in\Pic(S)_{\tors}\cong\Z/n$ is any generator. The result (where we take $X:=S\times JC$) thus follows from Theorem <ref>, since we can choose $C$, such that $\Griff^{2}(JC)/n$ contains infinitely many elements of order $n$.
Note that in Theorem <ref>, the infinitely many torsion elements in $\Griff^{3}(X)$ have all trivial Abel-Jacobi invariant, by Corollary <ref>.
Fix an integer $m\geq 5$. Then the result is an immediate consequence of the projective bundle formula (see <cit.>) applied to the smooth complex projective variety $X:=S\times JC\times\mathbb{P}^{m-5}_{\C}$, where $C$ and $S$ are as in the proof of Theorem <ref>.

§ ACKNOWLEDGEMENTS

I am grateful to my advisor Stefan Schreieder for giving me this problem. I thank him for answering my questions and for helpful comments during the preparation of the paper. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 948066 (ERC-StG RationAlgic).

[4]
The Stacks project authors, The Stacks project,
<https://stacks.math.columbia.edu>, 2021.

[10]
Th. Alexandrou, On the Brauer group of a generic Godeaux surface, to appear in J. London Math. Soc. (2023), <https://doi.org/10.1112/jlms.12727>.
[11]
Th. Alexandrou and S. Schreieder, On Bloch’s map for torsion cycles over non-closed fields, Preprint 2022, arXiv:2210.03201.

[18]
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265.

[19]
W. P. Barth, K. Hulek, C. A. M. Peters, A. Ven, Compact Complex Surfaces, A Series of Modern Surveys in Math., Springer, Berlin, Heidelberg, 2004.

[23]
B. Bhatt and P. Scholze, The pro-étale topology of schemes,  Astérisque 369 (2015),  99–201.

[27]
S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Compositio Mathematica 39 (1979), 107–127.

[28]
S. Bloch, Lectures on algebraic cycles,  Duke University Mathematics Series, IV. Duke University, Mathematics Department, Durham, N.C., 1980. 182 pp.
2nd ed., Cambridge 2010.

[31]
S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér., 7 (1974), 181–201.

[32]
S. Bloch and H. Esnault, The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann. 304 (1996), 303–314.

[40]
G. Ceresa, $C$ is not algebraically equivalent to $C^{-}$ in its Jacobian, Ann. of Math. 117(1983), 285–291.

[43]
H. Clemens,
Homological equivalence modulo algebraic equivalenceis not finitely generated, Publications mathématiques de l’I.H.É.S. 58 (1983),  19–38.

[46]
J.-L. Colliot-Thélène and A. N. Skorobogatov The Brauer-Grothendieck Group, Vol. 71. Ergeb. Math. Grenzgeb., 3. Folge. Cham: Springer, 2021.

[58]
A.J.  de  Jong, The period-index problem for the Brauer group of an algebraic surface,  Duke Math. J. 123 (2004), 71–94.

[74]
W. Fulton, Intersection theory, Springer–Verlag, 1998.

[85]
P. Griffiths, On the periods of certain rational integrals I, II, Annals of Math. 90 (1969), 460–541.

[86]
A. Grothendieck, Le groupe de Brauer I, II, III, in: Dix exposés sur la cohomologie des schémas, Advanced Studies in Pure Mathematics vol. 3, Masson et North-Holland, 1968.

[87]
U.T. Hartl., Semi-stability and base change, Arch. Math. 77 (2001), 215-221.

[88]
R. Hartshorne, Algebraic Geometry, vol. 52 Grad. Texts Math., Springer New York, NY, 1977.

[104]
U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), 207–245.

[120]
V. S. Kulikov, Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izv. 11 (1977),  957–989.

[123]
R. Lazarsfeld,Positivity in algebraic geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48,49, Springer-Verlag, Berlin, 2004.

[132]
A. Merkurjev and A. Suslin, K-cohomology of Severi–Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136. Eng. trans., Math. USSR Izv. 21 (1983), 307–340.

[136]
J.S. Milne, Étale cohomology, Princeton University Press, Princeton, NJ, 1980.

[141]
M. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), 349–373.

[146]
K.H. Paranjape, Curves on threefolds with trivial canonical bundle,  Proc. Indian Acad. Sci. (Math. Sci.)  101  (1991), 199–213.

[151]
U. Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189, xv+144.

[156]
C.P. Ramanujam, On a geometric interpretation of multiplicity, Invent. Math. 22 (1973), 63–67.

[160]
A. Rosenschon and V. Srinivas, The Griffiths group of the generic abelian 3-fold, Cycles, motives and Shimura varieties, 449–467. Tata Inst. Fund. Res., Mumbai, 2010.

[169]
S. Schreieder, Unramified cohomology, algebraic cycles and rationality,
in: G. Farkas et al. (eds), Rationality of Varieties, Progress in Mathematics 342, Birkhäuser (2021), 345–388.

[170]
S. Schreieder, Refined unramified cohomology of schemes, to appear in Compositio Mathematica. Preprint 2021, arXiv:2010.05814v4.

[171]
S. Schreieder, Infinite torsion in Griffiths groups, to appear in Journal of the EMS. Preprint 2021, arXiv:2011.15047v3.

[172]
S. Schreieder, A moving lemma for cohomology with support, Preprint 2022, arXiv:2207.08297.

[173]
C. Schoen, Some examples of torsion in the Griffiths group, Math. Ann. 293 (1992), 651–679.

[176]
C. Schoen, Complex varieties for which the Chow group mod n is not finite, J. Alg. Geom. 11 (2002), 41–100.

[181]
J. H. M. Steenbrink, Cohomologically insignificant degenerations, Compositio Mathematica 42 (1980), 315–320.

[189]
B. Totaro, Complex varieties with infinite Chow groups modulo 2, Annals of Mathematics 183 (2016), 363–375.

[195]
C. Voisin, The Griffiths group of a general Calabi-Yau threefold is not finitely generated, Duke Math. J. Vol. 102, No 1 (2000).