# The maximal unipotent finite quotient, unusual torsion in Fano threefolds,
and exceptional Enriques surfaces

Andrea Fanelli  Institut de mathématiques de Bordeaux (IMB), CNRS, Université
de Bordeaux, 33405 Talence cedex, France<EMAIL_ADDRESS>and
Stefan Schröer  Mathematisches Institut, Heinrich-Heine-Universität, 40204
Düsseldorf, Germany<EMAIL_ADDRESS>
  *     * scAbstract. We introduce and study the maximal unipotent finite quotient for algebraic group schemes in positive characteristics. Applied to Picard schemes, this quotient encodes unusual torsion. We construct integral Fano threefolds where such unusual torsion actually appears. The existence of such threefolds is surprising, because the unusual torsion vanishes for del Pezzo surfaces. Our construction relies on the theory of exceptional Enriques surfaces, as developed by Ekedahl and Shepherd-Barron.

scKeywords. Fano varieties, Enriques surfaces, group schemes

sc2020 Mathematics Subject Classification. 14J45, 14J28, 14L15, 14C22

sc[Français]

scLe quotient fini unipotent maximal, torsion inhabituelle dans les solides de
Fano et surfaces d’Enriques exceptionnelles.

scRésumé. Nous introduisons et étudions le quotient fini unipotent maximal
pour les schémas en groupes algébriques en caractéristique positive. Appliqué
aux schémas de Picard, ce quotient encode la torsion inhabituelle. Nous
construisons des solides de Fano intègres qui font effectivement apparaître
cette torsion inhabituelle. L’existence de ces solides est surprenante, compte
tenu du fait que la torsion inhabituelle s’annule pour les surfaces de Del
Pezzo. Notre construction repose sur la théorie des surfaces d’Enriques
développée par Ekedahl et Shepherd-Barron.

  *
cMay 25, 2020Received by the Editors on February 24, 2020.
Accepted on July 3, 2020.

Institut de mathématiques de Bordeaux (IMB), CNRS, Université de Bordeaux,
33405 Talence cedex, France

sce-mail<EMAIL_ADDRESS>
Mathematisches Institut, Heinrich-Heine-Universität, 40204 Düsseldorf, Germany

sce-mail<EMAIL_ADDRESS>
This research was conducted in the framework of the research training group
_GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology_ ; we
wish to thank the Deutsche Forschungsgemeinschaft for financial support. The
first-named author is currently funded by the _Fondation Mathématique Jacques
Hadamard._

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 The maximal finite unipotent quotient
  2. 2 Picard scheme and Bockstein operators
  3. 3 The case of surfaces
  4. 4 Enriques surfaces and del Pezzo surfaces
  5. 5 Exceptional Enriques surfaces
  6. 6 Cones and Fano varieties
  7. 7 Fano threefolds with unusual torsion

## Introduction

In algebraic geometry over ground fields $k$ of characteristic $p>0$, unusual
behavior of certain algebraic schemes is often reflected by the structure of
unipotent torsion originating from the Picard group. For example, an elliptic
curve $E$ is _supersingular_ if and only if the kernel $E[p]$ for
multiplication-by-$p$ is unipotent. An even more instructive case are Enriques
surfaces $Y$, which have $c_{1}=0$ and $b_{2}=10$. Then the Picard scheme
$P=\operatorname{Pic}^{\tau}_{Y/k}$ of numerically trivial invertible sheaves
has order two. In characteristic $p=2$, this gives the three possibilities. In
case $P=\mu_{2}$, the Enriques surface is called ordinary, and behaves like in
characteristic zero. Otherwise, we have $P=\mathbb{Z}/2\mathbb{Z}$ or
$P=\alpha_{2}$, which is a unipotent group scheme, and $Y$ is a _simply-
connected_ Enriques surfaces. Their geometry and deformation theory is more
difficult to understand. The crucial difference to the case of elliptic curves
is that the unipotent torsion can be regarded as a _quotient object_ , and not
only as a subobject, which makes it more “unusual”.

The first goal of this paper is to introduce a general measure for unipotent
torsion, the _maximal finite unipotent quotient_ ${}_{Y/k}={}_{P}$ of the
algebraic group scheme $P=\operatorname{Pic}^{\tau}_{Y/k}$. Over algebraically
closed fields, this comprises the $p$-primary torsion part of the Néron–Severi
group $\operatorname{NS}(Y)$ and the part of the local group scheme
$P^{0}/P^{0}_{\operatorname{red}}$ whose Cartier dual is also local. Such
finite group schemes are often called _local-local_. This actually extends to
algebraic group scheme that are not necessarily commutative. Our approach
builds on the work of Brion [Bri17]. It turns out that Y/k is useful in
various situations. For example, it easily explains that the reduced part of
an algebraic group scheme is not necessarily a subgroup scheme. Of course,
this may only happen over imperfect fields.

The second goal is to construct Fano varieties whose Picard scheme actually
contains such unipotent torsion. Roughly speaking, a _Fano variety_ is a
Gorenstein scheme $Y$ that is proper and equi-dimensional, and whose dualizing
sheaf $\omega_{Y}$ is anti-ample. This notion indeed goes back to Fano
[Fan31]. Usually, one also assumes that $Y$ is integral. We write $n=\dim(Y)$
for the dimension. The structure and classification of Fano varieties is an
interesting subject of its own. Fano varieties play an important role in
representation theory, because proper homogeneous spaces $Y=G/H$ for linear
groups schemes in characteristic $p=0$ are Fano varieties. Moreover, they are
crucial for the minimal model program, because they arise as generic fibers
$Y=X_{\eta}$ in Mori fibrations $X\rightarrow B$.

Suppose that $Y$ is a smooth Fano variety in characteristic zero. Then

$H^{i}(Y,\mathscr{O}_{Y})=H^{i}(Y,{}^{n}_{Y/k}\otimes\omega_{Y}^{\otimes-1})=0$

for $i>0$, by Kodaira–Akizuki–Nakano Vanishing (see [DI87] for an algebraic
proof). In particular, the Lie algebra $H^{1}(Y,\mathscr{O}_{Y})$ of the
Picard scheme $\operatorname{Pic}_{Y/k}$ vanishes, so that the connected
component $\operatorname{Pic}^{0}_{Y/k}$ is trivial. Moreover, the Euler
characteristic is $\chi(\mathscr{O}_{Y})=1$, whence all connected étale
coverings are trivial. Consequently, the algebraic fundamental group vanishes,
and the Néron–Severi group $\operatorname{NS}(Y)$ is torsion-free. Summing up,
the group scheme $\operatorname{Pic}^{\tau}_{Y/k}$ of numerically trivial
invertible sheaves vanishes. Its fundamental role was emphasized by
Grothendieck [FGA4].

It is a natural question to what extent the vanishing for
$\operatorname{Pic}^{\tau}_{Y/k}$ holds true in positive characteristics, or
for singular Fano varieties. Vanishing holds in dimension $n=1$ for integral
Gorenstein curves with $\omega_{Y}^{\otimes-1}$ ample, basically by
Riemann–Roch. In dimension $n=2$, the classification of regular del Pezzo
surfaces is independent of the characteristic, and we again have vanishing.
For normal del Pezzo surfaces, the second author observed in [Sch01] that
$h^{1}(\mathscr{O}_{Y})=0$. The situation becomes much more challenging for
non-normal del Pezzo surfaces. Here Reid [Rei94] constructed for each prime
$p>0$ examples with $h^{1}(\mathscr{O}_{Y})\neq 0$. In [Sch07], some normal
locally factorial del Pezzo surfaces over imperfect fields in characteristic
$p=2$ with $h^{1}(\mathscr{O}_{Y})\neq 0$ were constructed, and even regular
examples exist, as shown by Maddock [Mad16]. For smooth Fano threefolds, the
vanishing of $\operatorname{Pic}^{\tau}_{Y/k}$ was shown by Shepherd-Barron
[SB97]. However, Cascini and Tanaka [CT19] constructed a klt Fano threefold in
characteristic $p=2$ with $h^{2}(\mathscr{O}_{Y})\neq 0$; other examples with
$p\geq 3$ were found by Bernasconi [Ber19].

Moreover, Tanaka [Tan16] constructed Mori fibrations $X\rightarrow B$ on
threefolds in characteristic $p=2,3$ where the generic fiber $Y=X_{\eta}$ is a
normal del Pezzo surface whose Picard group contains elements of order
$p=2,3$. Very recently, Bernasconi and Tanaka [BT20] provided effective bounds
for the torsion on del Pezzo–type surfaces over imperfect fields.

However, in all the above examples the maximal unipotent finite quotient Y/k
of $\operatorname{Pic}^{\tau}_{Y/k}$ still vanishes. We therefore regard Y/k
as a measure for _unusual torsion in Fano varieties_. The main results of this
paper are as follows. First, we give general criteria for the vanishing of
Y/k, which relies on the theory of Bockstein operators:

######  Theorem.

(see Thm. 2.2) We have ${}^{0}_{Y/k}=0$ provided the Frobenius map on
$H^{2}(Y,\mathscr{O}_{Y})$ has maximal Hasse–Witt rank.

From this we deduce in Theorem 3.3 that ${}_{Y/k}=0$ for any reduced surface
whose dualizing sheaf $\omega_{Y}$ is negative, in a suitable sense. Note that
this requires no further restrictions on the singularities. It applies in
particular to del Pezzo surfaces. In light of this, it is surprising that
unusual torsion does appear in dimension $n\geq 3$:

######  Theorem.

(see Thm. 7.3) There are integral Fano threefolds $Y$ in characteristic two
such that ${}_{Y/k}=\operatorname{Pic}^{\tau}_{Y/k}$ is isomorphic to the
group scheme $\mathbb{Z}/2\mathbb{Z}$ or $\alpha_{2}$.

The construction relies on the theory of _exceptional Enriques surfaces_ , as
developed by Ekedahl and Shepherd-Barron [ESB04]. These are simply-connected
Enriques surfaces containing very strange configurations of $(-2)$-curves. The
geometry of such surfaces was already studied in the monograph of Cossec and
Dolgachev ([CD89], Chapter III, §3–4), although their existence was
established only later by Salomonsson [Sal03].

The idea for our construction is rather simple: start with an exceptional
Enriques surface $S$ that contains exactly ten $(-2)$-curves with a certain
dual graph. We then contract all but one of these curves, creating a normal
Enriques surface $Z$. Its normalized K3-like covering $Z^{\prime}\rightarrow
Z$ turns out to be a normal del Pezzo surface with a unique singularity, which
is a rational double point of type $D_{5}$. This relies on the classification
of normal del Pezzo surfaces, as explained by Dolgachev [Dol12]. We take the
$\mathbb{P}^{1}$-bundle $X=\mathbb{P}(\mathscr{E})$ for the locally free sheaf
$\mathscr{E}=\mathscr{O}_{Z^{\prime}}\oplus\omega_{Z^{\prime}}^{\otimes-1}$,
contract the negative section $E=\mathbb{P}(\omega_{Z^{\prime}}^{\otimes-1})$,
and denormalize along the positive section
$Z^{\prime}=\mathbb{P}(\mathscr{O}_{Z})$ with respect to the purely
inseparable double covering $\nu:Z^{\prime}\rightarrow Z$. The resulting
scheme $Y=X\amalg_{Z}Z^{\prime}$ is an integral Fano threefold with Euler
characteristic, degree and index

$\chi(\mathscr{O}_{Y})=1,\quad\deg(Y)=4\quad\text{and}\quad\operatorname{ind}(Y)=1.$

A detailed analysis of the normalized K3-like covering
$\nu:Z^{\prime}\rightarrow Z$ reveals that the morphism is flat, except over a
rational double point $a\in Z$ of type $E_{8}$. From this we infer that our
integral Fano threefold $Y$ has invertible dualizing sheaf and satisfies Serre
Condition $(S_{2})$, but _fails to be Cohen–Macaulay_. Such schemes might be
called _quasi-Fano varieties_ , and deserve further study.

Note that recently Totaro [Tot19] constructed three-dimensional terminal
singularities in positive characteristics that are not Cohen–Macaulay, and
further examples were constructed by Yasuda [Yas19]. In turn, the above
examples of non-normal Fano threefolds $Y$ may admit twisted forms
$Y^{\prime}$ whose local rings are normal $\mathbb{Q}$-factorial klt
singularities, and thus could occur as generic fibers in Mori fiber spaces.
See [FS20] for our analysis of non-normal del Pezzo surfaces having twisted
forms whose local rings are regular.

The paper is organized as follows: In Section 1 we discuss various smallest
normal subgroup schemes $N\subset G$ whose quotients have certain properties,
introduce the maximal finite unipotent quotient ${}_{G}=G/N$, and establish
its basic properties. We apply this in Section 2 to Picard schemes
$G=\operatorname{Pic}^{\tau}_{Y/k}$, and give general criteria for the
vanishing of ${}_{Y/k}={}_{G}$. In Section 3 we show that Y/k vanishes for
reduced surfaces with negative dualizing sheaf. In Section 4 we recall various
notions of Enriques surfaces, and describe how del Pezzo surfaces arise as
normalization of K3-like coverings. The following Section 5 contains a
detailed analysis for the case of exceptional Enriques surfaces of type
$T_{2,3,7}$. In Section 6 we discuss the cone construction and its Gorenstein
properties, and show how it leads to new Fano varieties. The final Section 7
contains the construction of integral Fano threefolds whose maximal finite
unipotent quotient Y/k is non-trivial.

Acknowledgements: We like to thank Fabio Bernasconi, Michel Brion, Igor
Dolgachev, Hiromu Tanaka and the referee for many valuable suggestions.

## 1\. The maximal finite unipotent quotient

In order to study unusual torsion in Picard groups, we shall introduce the
maximal finite unipotent quotient for general algebraic group schemes, which
are not necessarily commutative. These results seem to be of independent
interest, and mainly rely on the theory of algebraic groups.

Let $k$ be a ground field of characteristic $p\geq 0$. An _algebraic group
scheme_ is a group scheme $G$ where the structure morphism
$G\rightarrow\operatorname{Spec}(k)$ is of finite type. Note that the
underlying scheme is automatically separated. We say that $G$ is _finite_ if
the structure morphism is finite. Then the _order_ is defined as
$\operatorname{ord}(G)=h^{0}(\mathscr{O}_{G})=\dim_{k}H^{0}(G,\mathscr{O}_{G})$.
One says that $G$ is _of multiplicative type_ if the base-change $G\otimes
k^{\operatorname{alg}}$ to the algebraic closure is isomorphic to
$\operatorname{Spec}k^{\operatorname{alg}}[M]$, where $M$ is a finitely
generated abelian group. In other words, $G$ is commutative and the base-
change has a filtration whose subquotients are isomorphic to the
multiplicative group $\mathbb{G}_{m}$, or the constant groups
$\mu_{l}=\mathbb{Z}/l\mathbb{Z}$ for some prime $l\neq p$, or the local group
scheme $\mu_{p}$. One also says that $G$ is _multiplicative_. The algebraic
group $G$ is called _unipotent_ if $G\otimes k^{\operatorname{alg}}$ admits a
filtration whose subquotients are isomorphic to subgroup schemes of the
additive group $\mathbb{G}_{a}$. In characteristic $p>0$, this means that
after refinement the subquotients are $\mathbb{G}_{a}$, the constant group
$\mathbb{Z}/p\mathbb{Z}$ or the local group scheme $\alpha_{p}$. Note that
unipotent group schemes are not necessarily commutative. For more on these
notions, see [SGA3-II, exposé IX and exposé XVII].

######  Theorem 1.1.

For each of the following properties (i)-(vi) and each algebraic group scheme
$G$, there is a smallest normal subgroup scheme $N\subset G$ such that the
quotient $G/N$ has the property in question:

  1. (i)

étale;

  2. (ii)

affine;

  3. (iii)

proper;

  4. (iv)

finite;

  5. (v)

finite and multiplicative;

  6. (vi)

finite and unipotent.

Moreover, the subgroup schemes $N\subset G$ and the quotients $G/N$ commute
with separable field extensions $k\subset k^{\prime}$.

###### Proof.

By [SGA3-I, Exposé $\rm{IV_{A}}$, Section 2], the connected component
$G^{0}\subset G$ of the origin is a normal subgroup scheme such that $G/G^{0}$
is étale, and it is indeed the smallest. According to Brion’s analysis
([Bri17, Theorem 1 and Theorem 2]), there are the smallest normal subgroup
schemes $N_{1},N_{2}\subset G$ such that the resulting quotients are affine
and proper, respectively. Let $N\subset G$ be the normal subgroup scheme
generated by $N_{1}$ and $N_{2}$. Then the resulting quotient $G/N$ is both
proper and affine, hence finite. Moreover, every homomorphism $G\rightarrow K$
into some finite group scheme contains $N_{1}$ and $N_{2}$ in its kernel. It
follows that $N$ is the desired smallest normal subgroup scheme with finite
quotient. This settles the cases (i)–(iv).

Consider the ordered family of normal subgroup schemes $H_{\lambda}\subset G$,
$\lambda\in L$ whose quotients $G/H_{\lambda}$ are finite and unipotent. The
group scheme $G$ itself belongs to this family, and each member contains $N$.
We first check that for any two members $H_{\lambda}$ and $H_{\mu}$, the
intersection $K=H_{\lambda}\cap H_{\mu}$ also belongs to the family. We have
an exact sequence

$0\longrightarrow H_{\lambda}/K\longrightarrow G/K\longrightarrow
G/H_{\lambda}\longrightarrow 0.$

By the Isomorphism Theorem, the term on the left is isomorphic to
$(H_{\lambda}\cdot H_{\mu})/H_{\mu}$, which is contained in $G/H_{\mu}$. The
latter is finite and unipotent, so the same holds for the subgroup scheme
$(H_{\lambda}\cdot H_{\mu})/H_{\mu}$ and the extension $G/K$.

Seeking a contradiction, we assume that there is no smallest member. Since the
family is filtered, this means that it contains an infinite descending
sequence $H_{0}\supsetneqq H_{1}\supsetneqq\ldots$ such that the quotients
$U_{n}=G/H_{n}$ have unbounded orders. On the other hand, all of them are
quotients of the finite group scheme $G/N$, hence the orders are bounded,
contradiction. Hence, there is a smallest normal subgroup scheme whose
quotient is finite and unipotent. This settles (vi). The argument for (v) is
similar and left to the reader.

We now prove the second part of the assertion: let $k\subset k^{\prime}$ be a
separable extension, and $N^{\prime}\subset G^{\prime}\otimes k^{\prime}$ be
the smallest subgroup scheme over $k^{\prime}$ such that the quotient has the
property $\mathcal{P}$ in question. This gives an inclusion $N^{\prime}\subset
N\otimes k^{\prime}$, and we have to verify that it is an equality. Suppose
first that $k\subset k^{\prime}$ is algebraic. By fpqc descent, it suffices to
check $N^{\prime}=N\otimes k^{\prime}$ after enlarging the field extension. It
thus suffices to treat the case that $k\subset k^{\prime}$ is Galois, with
Galois group $\Gamma=\operatorname{Gal}(k^{\prime}/k)$. Then for each element
$\sigma\in\Gamma$ we have $\sigma(N^{\prime})=N^{\prime}$, by the uniqueness
of $N^{\prime}\subset G\otimes k^{\prime}$. Galois descent gives a closed
subscheme $N_{0}\subset N$ with $N_{0}\otimes k^{\prime}=N^{\prime}$. This
subscheme is a subgroup scheme and normal in $G$, and the base-change
$G/N_{0}\otimes k^{\prime}=(G\otimes k^{\prime})/N^{\prime}$ has property
$\mathcal{P}$. For each of the cases (i)–(vi), this implies that $G/N_{0}$ has
property $\mathcal{P}$. This gives $N_{0}=N$, and in turn the desired equality
$N^{\prime}=N\otimes k^{\prime}$.

Now write $k^{\prime}=\bigcupop\displaylimits k_{\lambda}$ as the filtered
union of finitely generated subextensions. The structure morphism
$G^{\prime}\rightarrow\operatorname{Spec}(k^{\prime})$ is of finite
presentation. Hence, [EGA4-III, Theorem 8.8.2] ensures that there is some
index $\lambda$ so that the closed subscheme $N^{\prime}$ is the base-change
of some closed subscheme $N_{\lambda}\subset G\otimes k_{\lambda}$. Moreover,
this subscheme is a normal subgroup scheme, and we have $N_{\lambda}\subset
N\otimes k_{\lambda}$. This reduces our problem to the case that $k^{\prime}$
is finitely generated. Choose an integral affine scheme $S$ of finite type
with function field $\kappa(\eta)=k^{\prime}$. Since $k\subset k^{\prime}$ is
separable, the scheme $S$ is geometrically reduced. Passing to some dense open
set, we may assume that $S$ is smooth.

Consider the relative group scheme $G_{S}$, with generic fiber
$G_{\eta}=G\otimes k^{\prime}$. Seeking a contradiction, we assume that
$N^{\prime}\subset N\otimes k^{\prime}$ is not an equality. Then there is a
homomorphism $f_{\eta}\colon G_{\eta}\rightarrow H_{\eta}$ to some algebraic
group scheme $H_{\eta}$ having property $\mathcal{P}$, such that $N_{\eta}$ is
not contained in the kernel. Again by [EGA4-III, théorème 8.8.2], there is a
dense open set $U\subset S$, a relative group scheme $H_{U}$ where the
structure morphism $H_{U}\rightarrow U$ is of finite type, and a homomorphism
$f_{U}\colon G_{U}\rightarrow H_{U}$ inducing $f_{\eta}$. If the scheme
$H_{\eta}$ is affine, proper or finite, we may assume that the respective
property holds for the morphism $H_{U}\rightarrow U$, by [EGA4-III, théorème
8.10.5]. Moreover, if $H_{\eta}$ is finite and unipotent, we may assume that
there is a finite étale covering $U^{\prime}\rightarrow U$ so that
$H\times_{U}U^{\prime}$ is a successive extension of $\mathbb{Z}/p\mathbb{Z}$
and $\alpha_{p}$, such that all fibers $H_{U}\rightarrow U$ are finite and
unipotent. The situation for properties (i) and (v) is similar. Summing up, we
may assume that all fibers of $H_{U}\rightarrow U$ have the property
$\mathcal{P}$ in question. Passing to a dense open set again, we reduce to
$U=S$. Consider the $S$-scheme

$K=\operatorname{Ker}(f|N)=N\times_{G}\\{e_{S}\\}.$

This is a relative group scheme, and the structure morphism $g\colon
K\rightarrow S$ is of finite type. The generic fiber is equi-dimensional, say
of dimension $n=\dim(K_{\eta})$. By [SGA3-I, exposé ${\rm VI_{B}}$,
proposition 4.1], the set of points $a\in S$ with $\dim(K_{a})=n$ is
constructible. Replacing $S$ by some dense open set, we may assume that all
fibers $K_{a}$ are $n$-dimensional. By Bertini’s Theorem ([Jou83, théorème
6.3]), there are closed points $a\in S$ such that the finite extension
$k\subset\kappa(a)$ is separable. Since the fiber $H_{a}$ has property
$\mathcal{P}$ in question, the kernel $K_{a}$ is trivial, and thus $n=0$. In
turn, the morphism $g\colon K\rightarrow S$ is quasi-finite. By Zariski’s Main
Theorem, there is a closed embedding $K\subset X$ into some finite $S$-scheme
$X$, with $K_{\eta}=X_{\eta}$. After replacing $S$ by a dense open set, we may
assume that $g\colon K\rightarrow S$ is finite, and furthermore flat, say of
degree $d=\deg(K/S)$. Looking again at fibers $K_{a}$, we see that $d=1$. In
turn, $K_{\eta}$ is trivial, contradiction. ∎

Let us say that $G/N$ is _the maximal quotient_ with the property
$\mathcal{P}$ in question. By construction, any homomorphism $G\rightarrow H$
into some algebraic group scheme with property $\mathcal{P}$ uniquely factors
over $G/N$. Thus $G/N$ is functorial in $G$, and the functor $G\mapsto G/N$ is
the left adjoint for the inclusion $H\mapsto H$ of the category of algebraic
group schemes with property $\mathcal{P}$ into the category of all algebraic
group schemes.

Given a field extension $k\subset k^{\prime}$, we set $G^{\prime}=G\otimes
k^{\prime}$ and let $N^{\prime}\subset G^{\prime}$ be the normal subgroup
scheme giving the maximal quotient over $k^{\prime}$. This gives an inclusion
$N^{\prime}\subset N\otimes k^{\prime}$, and a resulting _base-change map_
$G^{\prime}/N^{\prime}\rightarrow(G/N)\otimes k^{\prime}$. It may or may not
be an isomorphism, as we shall see below. However, we have the following
immediate fact:

######  Proposition 1.2.

In the above situation, the base-change map is an epimorphism. In particular,
the algebraic group scheme $G/N$ vanishes if $G^{\prime}/N^{\prime}$ vanishes.

The maximal affine quotient is indeed the _affine hull_ in the sense of scheme
theory, and written as $G^{\text{\rm
aff}}=\operatorname{Spec}\Gamma(G,\mathscr{O}_{G})$. The kernel $N$ for the
homomorphism $G\rightarrow G^{\text{\rm aff}}$ is _anti-affine_ , which means
that the inclusion $k\subset H^{0}(N,\mathscr{O}_{N})$ is an equality ([DG70,
Chapter III, §3, Theorem 8.2]). This notion was introduced by Brion [Bri09],
and implies that $N$ is _semi-abelian_ , and in particular smooth and
connected ([Bri17, Proposition 5.5.1]). The maximal étale quotient is usually
denoted by ${}_{G}=G/G^{0}$, and called the _group scheme of components_. Both
$G^{\text{\rm aff}}$ and G actually commute with arbitrary field extensions
$k\subset k^{\prime}$.

Throughout this paper we are mainly interested in the _maximal finite
unipotent quotient_

${}_{G}=G/N.$

Any subgroup scheme of $G$ that is multiplicative, or smooth and connected, or
merely integral vanishes in G. In fact, the following holds:

######  Lemma 1.3.

Suppose the algebraic group scheme $G$ has a filtration whose subquotients are
generated by integral group schemes and group schemes of multiplicative type.
Then all homomorphisms $f\colon G\rightarrow U$ into finite unipotent group
schemes $U$ are trivial.

###### Proof.

By induction on the length of the filtration, it suffices to treat the two
cases that $G$ is integral, or of multiplicative type. In the latter case, the
statement follows from [SGA3-II, exposé XVII, proposition 2.4]. Now suppose
that $G$ is integral. Replacing $U$ by the image
$\operatorname{Im}(f)=G/\operatorname{Ker}(f)$, we may assume that $f\colon
G\rightarrow U$ is surjective and schematically dominant. Setting
$A=H^{0}(G,\mathscr{O}_{G})$ and $R=H^{0}(U,\mathscr{O}_{U})$, we see that the
canonical map $R\rightarrow A$ is injective. These rings are integral, because
the scheme $G$ is integral. Moreover, $R$ is an Artin ring, because the group
scheme $U$ is finite. The neutral element $e\in U$ shows that the residue
field is $R/\mathfrak{m}_{R}=k$. In turn, we have $R=k$, thus
$U=\operatorname{Spec}(k)$ is trivial. ∎

Each algebraic group scheme $G$ yields a _Galois representation_
$G(k^{\operatorname{sep}})$ of the Galois group
$\Gamma=\operatorname{Gal}(k^{\operatorname{sep}}/k)$ on the abstract group
$G(k^{\operatorname{sep}})$, which might be infinitely generated. However,
this construction yields an equivalence between the category of algebraic
group schemes that are étale and continuous Galois representations on finite
groups ([SGA4-II, exposé VIII, proposition 2.1]). Note also that for finite
group schemes $H$, the group scheme $H^{0}$ is local, and the finite universal
homeomorphism $H\rightarrow{}_{H}$ yields an equality
$H(k^{\prime})={}_{H}(k^{\prime})$ for all field extensions $k\subset
k^{\prime}$. This applies in particular to the maximal finite unipotent
quotient $H={}_{G}$. The following observation thus computes its group scheme
of components:

######  Proposition 1.4.

The Galois module ${}_{G}(k^{\operatorname{sep}})$ is the quotient of the
finite group ${}_{G}(k^{\operatorname{sep}})$ by the subgroup generated by all
$l$-Sylow groups for the primes $l\neq p$.

###### Proof.

Base-changing to $k^{\operatorname{sep}}$, we are reduced to the case that the
ground field $k$ is separably closed. Write ${}^{\prime}_{G}$ for the quotient
of the constant group scheme G by the subgroup generated by the
Sylow-$l$-subgroups, and set $U={}_{G}$. Then ${}_{U}(k)=U(k)$ is a finite
$p$-group. It follows that the canonical surjection ${}_{G}\rightarrow{}_{U}$
factors over ${}^{\prime}_{G}$. The resulting map
$f\colon{}^{\prime}_{G}\rightarrow{}_{U}$ admits a section, by the universal
properties of G and U. This section $s\colon{}_{U}\rightarrow{}^{\prime}_{G}$
is surjective, because $G\rightarrow{}^{\prime}_{G}$ is surjective. In turn,
$f$ and $s$ are inverse to each other, and give the desired identification
${}^{\prime}_{G}(k)={}_{U}(k)={}_{G}(k)$. ∎

It is much more difficult to understand the component of the origin
${}^{0}_{G}$, because the base-change map ${}_{G\otimes
k^{\prime}}\rightarrow{}_{G}\otimes k^{\prime}$ might fail to be an
isomorphism, for inseparable extensions $k^{\prime}$. Here is a typical
example, over imperfect fields $k$:

######  Proposition 1.5.

Suppose $G$ is an integral group scheme such that some base-change
$G^{\prime}=G\otimes k^{\prime}$ becomes isomorphic to
$\mathbb{G}_{a}\oplus\alpha_{p}$. Then we have ${}_{G}=0$, whereas
${}_{G^{\prime}}=\alpha_{p}$.

###### Proof.

According to Proposition 1.3, the quotient G is trivial, whereas
$G^{\prime}\rightarrow{}_{G^{\prime}}$ is the projection onto the second
factor. ∎

For example, $G$ could be the kernel for the homomorphism
$h\colon\mathbb{G}_{a}\times\mathbb{G}_{a}\rightarrow\mathbb{G}_{a}$ given by
the additive map $(x,y)\longmapsto x^{p}+ty^{p}$, for some scalar $t\in k$
that is not a $p$-th power. Note that certain fiber products
$G=\mathbb{G}_{a}\times_{\mathbb{G}_{a}}\mathbb{G}_{a}$ were systematically
studied by Russel [Rus70] to describe twisted forms of the additive group. In
our concrete example the compactification in $\mathbb{P}^{2}$ is the Fermat
curve with homogeneous equation $X^{p}+tY^{p}+Z^{p}=0$, as analyzed in [Sch10,
§4]. The scheme $G$ is integral, with singular locus $G(k)=\\{0\\}$, and its
normalization is the affine line over the height-one extension $E=k(t^{1/p})$.
It has another peculiar feature:

######  Proposition 1.6.

Notation as above. For the connected scheme $H=G\times G$, the closed
subscheme $H_{\operatorname{red}}\subset H$ is not a subgroup scheme.

###### Proof.

This already appeared in [SGA3-I, exposé VI, exemple 1.3.2]. Let us give an
independent short proof based on our maximal finite unipotent quotient:
Suppose $H_{\operatorname{red}}$ were a subgroup scheme. Then
$H/H_{\operatorname{red}}$ is a local unipotent group scheme, so the
projection factors over ${}_{H}={}_{G}\times{}_{G}=0$, consequently
$H_{\operatorname{red}}=H$. On the other hand, the normalization
$\mathbb{A}^{1}_{E}\rightarrow G$ shows that $H$ is birational to the affine
plane over the non-reduced ring $E\otimes E=E[t]/(t^{p})$, hence
$H_{\operatorname{red}}\subsetneqq H$, contradiction. ∎

If the reduced part $G_{\operatorname{red}}\subset G$ of an algebraic group
scheme is _geometrically reduced_ , then the product
$G_{\operatorname{red}}\times G_{\operatorname{red}}$ remains reduced, and the
group multiplication factors over $G_{\operatorname{red}}$. In turn, the
closed subscheme $G_{\operatorname{red}}\subset G$ is a subgroup scheme. This
frequently fails, as we saw above. Other examples for this behavior are the
non-split extensions $0\rightarrow\alpha_{p}\rightarrow
G\rightarrow\mathbb{Z}/p\mathbb{Z}\rightarrow 0$, where the fiber
$\operatorname{pr}^{-1}(a)$ for the projection $\operatorname{pr}\colon
G\rightarrow\mathbb{Z}/p\mathbb{Z}$ is reduced if and only if $a\neq 0$. Such
extensions exist over imperfect fields: the abelian group of all _central
extensions_ contains the flat cohomology group
$H^{1}(\operatorname{Spec}(k),\alpha_{p})=k/k^{p}$ as a subgroup, by [DG70,
Chapter III, §6, Proposition in 3.5]. Also note that if
$G_{\operatorname{red}}$ is a subgroup scheme, it need not be normal, for
example in semidirect products like $G=\alpha_{p}\rtimes\mathbb{G}_{m}$.

The situation simplifies somewhat for commutative group schemes. Recall that
if $G$ is commutative and affine, then there is a _maximal multiplicative
subgroup scheme_ $G^{{\text{\rm mult}}}\subset G$, and the quotient is
unipotent ([DG70, Chapter IV, §3, Theorem 1.1]). If the resulting quotient is
finite, Lemma 1.3 gives ${}_{G}=G/G^{{\text{\rm mult}}}$. In general, we get:

######  Proposition 1.7.

Suppose that $G$ is commutative, that the scheme $G^{0}_{\operatorname{red}}$
is geometrically reduced, and that the projection $G\rightarrow{}_{G}$ admits
a section. For the local group scheme $L=G^{0}/G^{0}_{\operatorname{red}}$, we
get an identification ${}^{0}_{G}=L/L^{{\text{\rm mult}}}$.

###### Proof.

The section gives a decomposition of commutative algebraic groups
$G=G^{\prime}\oplus G^{\prime\prime}$, where the first factor is connected and
the second factor is étale. In turn, we have
${}_{G}={}_{G^{\prime}}\times{}_{G^{\prime\prime}}$. From the universal
property we infer that ${}_{G^{\prime\prime}}$ is étale, so we may assume from
the start that $G$ is connected, and have to show that the projection
$G\rightarrow G/G_{\operatorname{red}}=L$ induces an isomorphism
${}_{G}\rightarrow{}_{L}=L/L^{{\text{\rm mult}}}$. Since
$G_{\operatorname{red}}$ is geometrically reduced, the inclusion
$G_{\operatorname{red}}\subset G$ is a subgroup scheme, which is smooth and
connected. It lies in the kernel of any homomorphism $G\rightarrow U$ to some
finite unipotent scheme $U$, by Lemma 1.3. From the universal properties we
infer that ${}_{G}\rightarrow{}_{L}$ is an isomorphism. ∎

This leads to the following structure result:

######  Theorem 1.8.

If $G$ is commutative and $k$ is perfect, we get an identification

${}_{G}=L/L^{{\text{\rm mult}}}\times{}_{G}[p^{\infty}],$

with the local group scheme $L=G^{0}/G^{0}_{\operatorname{red}}$ and the
$p$-primary part ${}_{G}[p^{\infty}]\subset{}_{G}$. Moreover, the kernel $N$
for the projection $G\rightarrow{}_{G}$ has a three-step filtration with
$N/N_{2}$ multiplicative, $N_{2}/N_{1}$ smooth unipotent, and $N_{1}$ anti-
affine.

###### Proof.

Since G is commutative, the quotient of ${}_{G}(k^{\operatorname{sep}})$ by
the subgroup generated by the Sylow-$l$-groups gets identified with the
$p$-primary torsion in ${}_{G}(k^{\operatorname{sep}})$. From Proposition 1.4
we infer that ${}_{G}[p^{\infty}]$ is the group scheme of components for G.
Since $k$ is perfect, the scheme $G^{0}_{\operatorname{red}}$ is geometrically
reduced, and Proposition 1.7 gives ${}^{0}_{G}=L/L^{\text{\rm mult}}$.
Moreover, the reduced part of G yields a section for the group scheme of
components, and we get the decomposition ${}_{G}=L/L^{{\text{\rm
mult}}}\times{}_{G}[p^{\infty}]$. Applying the Five Lemma to the commutative
diagram

$\begin{CD}0@>{}>{}>G^{0}@>{}>{}>G@>{}>{}>{}_{G}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ 0@>{}>{}>L/L^{\text{\rm
mult}}@>{}>{}>{}_{G}@>{}>{}>{}_{G}[p^{\infty}]@>{}>{}>0,\end{CD}$

we see that the kernel $N$ for $G\rightarrow{}_{G}$ has a filtration
$F_{i}\subset N$ with

$F_{5}=N\quad\text{and}\quad F_{4}=\operatorname{Ker}(G^{0}\rightarrow
L/L^{\text{\rm mult}})\quad\text{and}\quad
F_{3}=\operatorname{Ker}(G^{0}\rightarrow L)=G^{0}_{\operatorname{red}}.$

Then $F_{5}/F_{4}$ is the sum of the $l$-primary parts in G for the primes
$l\neq p$, and $F_{4}/F_{3}=L^{\text{\rm mult}}$. The kernel $F_{1}$ for the
affinization $F_{3}\rightarrow F_{3}^{\text{\rm aff}}$ is anti-affine.
Moreover, the multiplicative part of $F_{3}^{\text{\rm aff}}$ has a unique
complement, because $k$ is perfect, and this complement is smooth unipotent.
Let $F_{2}\subset F_{3}$ be its preimage.

Summing up, we have constructed a five-step filtration
$0=F_{0}\subset\ldots\subset F_{5}=N$. Setting $N_{2}=F_{2}$ and $N_{1}=F_{1}$
we see that $N/N_{1}$ is multiplicative, $N_{2}/N_{1}$ is smooth unipotent,
and $N_{1}$ is anti-affine. ∎

Note that the epimorphism $G\rightarrow{}_{G}$ does not admit a section in
general: suppose that $N$ is either the additive group, or a supersingular
elliptic curve. With respect to the scalar multiplication of
$\operatorname{End}(\alpha_{p})=k$, the abelian group
$\operatorname{Ext}^{1}(\alpha_{p},N)$ becomes a one-dimensional vector space,
provided that $k$ is algebraically closed ([Oor66, table on page II.14-2]). In
the ensuing non-split extensions $0\rightarrow N\rightarrow
G\rightarrow\alpha_{p}\rightarrow 0$, the projection coincides with
$G\rightarrow{}_{G}$, according to Lemma 1.3.

## 2\. Picard scheme and Bockstein operators

Let $k$ be a ground field of characteristic $p>0$, and $Y$ be a proper scheme.
Then the _Picard scheme_ $P=\operatorname{Pic}_{Y/k}$ exists, and this group
scheme is locally of finite type ([SGA6, exposé XII, corollaire 1.5 and exposé
XIII, proposition 3.2]). The Galois module for the étale group scheme
${}_{P}=P/P^{0}$ is the _Néron–Severi group_
${}_{P}(k^{\operatorname{sep}})=\operatorname{NS}(Y\otimes_{k}k^{\operatorname{alg}})$,
which is finitely generated. In particular, the torsion part in P is a finite
group scheme, hence its inverse image $G=\operatorname{Pic}^{\tau}_{Y/k}$ in
the Picard scheme is an algebraic group scheme. We now consider the maximal
finite unipotent quotient

${}_{Y/k}={}_{G}={}_{\operatorname{Pic}^{\tau}_{Y/k}},$

and regard this in many situations as a measure for _unusual_ behavior of
torsion in the Picard scheme. From Proposition 1.4 we get:

######  Proposition 2.1.

The Galois module ${}_{Y/k}(k^{\operatorname{sep}})$ is the $p$-torsion part
of the Néron–Severi group $\operatorname{NS}(Y\otimes
k^{\operatorname{alg}})$.

We now seek to understand the component of the origin ${}_{Y/k}^{0}$. For this
the Frobenius map $f\mapsto f^{p}$ on the structure sheaf $\mathscr{O}_{Y}$ is
crucial. This map is additive, and becomes $k$-linear when one re-defines
scalar multiplication on the range as $\lambda\cdot f=\lambda^{p}f$. Such
additive maps are called _$p$ -linear_. We now consider the induced $p$-linear
maps on the cohomology groups $H^{i}(Y,\mathscr{O}_{Y})$.

To understand this better, suppose we have an arbitrary finite-dimensional
$k$-vector space $V$, together with a $p$-linear map $f\colon V\rightarrow V$.
Choose a basis $a_{1},\ldots,a_{n}\in V$. Then $f$ is determined by the images
$f(a_{j})$, and the expansion $f(a_{j})=\sumop\displaylimits\lambda_{ij}a_{i}$
gives a matrix $A=(\lambda_{ij})\in\operatorname{Mat}_{n}(k)$. A different
basis $b_{1},\ldots,b_{n}\in V$ yields another matrix $B=SAT$, where the base-
change matrix $S=(\sigma_{ij})$ is defined by
$a_{j}=\sumop\displaylimits\sigma_{ij}b_{i}$, and $T$ is obtained from the
inverse of $S$ by applying Frobenius to the entries. In turn, the rank of $A$
depends only on $f$. Let us call this integer the _Hasse–Witt rank_
$\operatorname{rank}_{\text{\rm HW}}(f)\geq 0$ of the $p$-linear map $f\colon
V\rightarrow V$. Note also that one may define the _Hasse–Witt determinant_
$\det_{\text{\rm HW}}(f)=\det(A)$ as a class in the monoid
$k/k^{\times(p-1)}$. Furthermore, we may regard the datum $(V,f)$ as a _left
module_ over the associative ring $k[F]$, where the relations
$F\lambda=\lambda^{p}F$ hold, by setting $F\cdot a=f(a)$. The Hasse–Witt rank
and determinant then become invariants of this module. All these
considerations go back to Hasse and Witt [HW36], who studied
$V=H^{1}(C,\mathscr{O}_{C})$ for a smooth algebraic curve $C$ over an
algebraically closed field $k$. Compare also the recent discussion of Achter
and Howe [AH19] for a discussion of historical developments, and widespread
inaccuracies in the literature.

Given a field extension $k\subset k^{\prime}$, we see that there is a unique
$p$-linear extension $f^{\prime}$ of $f$ to $V^{\prime}=V\otimes k$, given by
$f(a_{j}\otimes\lambda)=\lambda^{p}f(a_{j})$. Obviously, the $p$-linear maps
$f$ and $f^{\prime}$ have the same Hasse–Witt rank. If $k$ is perfect, the
subgroup $U=f(V)$ inside $V$ is actually a vector subspace with respect to the
original scalar multiplication, and we have $\operatorname{rank}_{\text{\rm
HW}}(f)=\dim_{k}(U)$. If $\operatorname{rank}_{\text{\rm HW}}(f)=\dim(V)$, we
say that $f$ has _maximal Hasse–Witt rank_. This means that $f(V)$ generates
the vector space $V$ with respect to the original scalar multiplication.
Equivalently, for some and hence all perfect field extensions $k\subset
k^{\prime}$ the $p$-linear extension $f^{\prime}\colon V^{\prime}\rightarrow
V^{\prime}$ is bijective. We now come to the main result of this section:

######  Theorem 2.2.

We have ${}_{Y/k}^{0}=0$ provided the Frobenius map on the second cohomology
group $H^{2}(Y,\mathscr{O}_{Y})$ has maximal Hasse–Witt rank.

The proof is deferred to the end of this section. It relies on Bockstein
operators, a theory introduced by Serre [Ser58], which we like to discuss
first. Write $W_{m}(k)$ be the ring of _Witt vectors_
$(\lambda_{0},\ldots,\lambda_{m-1})$ of length $m$. This is a ring endowed
with two commuting additive maps _Frobenius_ $F$ and _Verschiebung_ (“shift”)
$V$, given by the formula

$F(\lambda_{0},\ldots,\lambda_{m-1})=(\lambda_{0}^{p},\ldots,\lambda_{m-1}^{p})\quad\text{and}\quad
V(\lambda_{0},\ldots,\lambda_{m-1})=(0,\lambda_{0},\ldots,\lambda_{m-2}).$

We refer to [Bou83, chapitre 9, §1] for the general theory of Witt vectors.
The canonical projection $W_{m}(k)\rightarrow W_{m-n}(k)$ is a homomorphism of
rings, whose kernel we denote by

$V_{m}^{n}(k)=V^{m-n}W_{m}(k).$

Note that this kernel has length $m-n$, and is stable under Frobenius, by the
relation $FV=VF$. Likewise, we have a short exact sequence of abelian sheaves

$0\longrightarrow V^{n}_{m}(\mathscr{O}_{Y})\longrightarrow
W_{m}(\mathscr{O}_{Y})\longrightarrow W_{m-n}(\mathscr{O}_{Y})\longrightarrow
0,$

where the maps are $W_{m}(k)$-linear and compatible with Frobenius. To
simplify notation, write $W_{m}=W_{m}(\mathscr{O}_{Y})$ and
$V_{m}^{n}=V_{m}^{n}(\mathscr{O}_{Y})$, such that $W_{1}=\mathscr{O}_{Y}$.
Combining the long exact sequences for the short exact sequences $0\rightarrow
V^{r}_{r+1}\rightarrow W_{r+1}\rightarrow W_{r}\rightarrow 0$ and
$0\rightarrow V^{1}_{r}\rightarrow W_{r}\rightarrow\mathscr{O}_{Y}\rightarrow
0$, we get for each $i\geq 0$ a commutative diagram

${H^{i}(W_{r+1})}$${H^{i}(V^{1}_{r})}$${H^{i}(W_{r})}$${H^{i}(\mathscr{O}_{Y})}$${H^{i+1}(V^{r}_{r+1})}$

with exact row and column. This gives a canonical $W_{r}(k)$-linear map

$\operatorname{Im}(H^{i}(W_{r})\rightarrow
H^{i}(\mathscr{O}_{Y}))\;\stackrel{{\scriptstyle\beta_{r}}}{{\longrightarrow}}\;\operatorname{Coker}(H^{i}(V^{1}_{r})\rightarrow
H^{i+1}(V^{r}_{r+1}))$

called _Bockstein operator_ , by sending the image of $x\in H^{i}(W_{r})$ in
$H^{i}(\mathscr{O}_{Y})$ to the image of $x$ in $H^{i+1}(V^{r}_{r+1})$ modulo
the image of $H^{i}(V^{1}_{r})$. Obviously, the kernel of $\beta_{r}$
comprises those cohomology classes in $H^{i}(\mathscr{O}_{Y})$ that extend to
$H^{i}(W_{r+1})$. These form a decreasing sequence. We write
$H^{i}(Y,\mathscr{O}_{Y})[\beta]$ for their common intersection, and call it
the _Bockstein kernel_. By construction, this is a vector subspace of
$H^{i}(Y,\mathscr{O}_{Y})$ invariant under Frobenius.

Now consider the case $i=1$. Then $\mathfrak{g}=H^{1}(Y,\mathscr{O}_{Y})$ is
the _Lie algebra_ for the Picard scheme, hence also for the algebraic group
scheme $G=\operatorname{Pic}^{0}_{Y/k}$. As such, it has an additional
structure, namely the _$p$ -power map_ $x\mapsto x^{[p]}$ obtained from the
$p$-fold composition of derivations in the associative algebra of differential
operators. This turns $\mathfrak{g}$ into a _restricted Lie algebra_. The
interplay between $p$-power map, Lie bracket and scalar multiplication is
regulated by three axioms ([DG70, Chapter II, §7]). In our situation, the
$p$-power map on the Lie algebra coincides with the Frobenius on cohomology,
so the Bockstein kernel
$\mathfrak{g}_{\operatorname{red}}=H^{1}(Y,\mathscr{O}_{Y})[\beta]$ is a
restricted Lie subalgebra. Here we are interested in the _Bockstein cokernel_
$\mathfrak{h}=\mathfrak{g}/\mathfrak{g}_{\operatorname{red}}$, with its
inherited structure of restricted Lie algebra.

Recall that for every algebraic group scheme $H$, the relative Frobenius is a
homomorphism $F\colon H\rightarrow H^{(p)}$ of group schemes, and its kernel
is a local group scheme $H[F]$. Local group schemes $H$ with $H=H[F]$ are
called _of height $\leq 1$_. According to [DG70, Chapter II, §7, Theorem 4.2],
the functor $H\mapsto\operatorname{Lie}(H)$ is an equivalence between the
categories of local groups schemes of height $\leq 1$ and finite-dimensional
restricted Lie algebras.

######  Proposition 2.3.

Suppose that the Frobenius map on $H^{2}(Y,\mathscr{O}_{Y})$ has maximal
Hasse–Witt rank. Then the group scheme $H$ of height $\leq 1$ corresponding to
the Bockstein cokernel
$\mathfrak{h}=H^{1}(Y,\mathscr{O}_{Y})/H^{1}(Y,\mathscr{O}_{Y})[\beta]$ is
multiplicative. If the reduced part of $G=\operatorname{Pic}^{0}_{Y/k}$ is
geometrically reduced, then the resulting local group scheme
$L=G/G_{\operatorname{red}}$ is also multiplicative.

###### Proof.

It suffices to treat the case that $k$ is perfect. According to Mumford’s
analysis in [Mum66, Lecture 27], the Lie algebra of the smooth connected group
scheme $G_{\operatorname{red}}$ coincides with the Bockstein kernel. In turn,
the Bockstein cokernel is the Lie algebra for the local group scheme
$L=G/G_{\operatorname{red}}$, which gives an identification $H=L[F]$. But $L$
is multiplicative if and only if its Frobenius kernel $H$ is multiplicative,
because the higher Frobenius kernels $L[F^{i}]$ give a filtration on $L$ whose
subquotients are isomorphic to $H$.

Seeking a contradiction, we assume that the inclusion $H^{\text{\rm
mult}}\subset H$ is not an equality. Since $k$ is perfect, the projection
$H\rightarrow H/H^{\text{\rm mult}}$ admits a section, and we get
$H=H^{\text{\rm mult}}\oplus U$ for some non-trivial unipotent local group
scheme $U$. According to [DG70, Chapter IV, §2, Corollary 2.13], the Lie
algebra $\operatorname{Lie}(U)$ contains a vector $b\neq 0$ with $b^{[p]}=0$.
Choose a vector $a\neq 0$ from $\mathfrak{g}=H^{1}(Y,\mathscr{O}_{Y})$ mapping
to $b$. Then $a^{[p]}\in\mathfrak{g}_{{\operatorname{red}}}$.

Now recall that the $p$-power map equals the Frobenius map. By construction
$a$ does not lie in the Bockstein kernel. Hence there is a largest integer
$r\geq 0$ such that $a$ belongs to the image of $H^{1}(Y,W_{r})\rightarrow
H^{1}(Y,\mathscr{O}_{Y})$. For the Bockstein operator, this means
$\beta_{r}(a)\neq 0$. Since $F(a)=a^{[p]}$ lies in the Bockstein kernel, we
have $0=\beta_{r}(F(a))=F(\beta_{r}(a))$.

By definition, the range of the Bockstein $\beta_{r}$ is a quotient of
$H^{2}(V^{r}_{r+1})$, and $V^{r}_{r+1}$ consists of tuples
$(0,\ldots,0,\lambda)$. In turn, there is an identification
$V^{r}_{r+1}=\mathscr{O}_{Y}$ of abelian sheaves, compatible with Frobenius.
By assumption, the Frobenius is bijective on $H^{2}(Y,\mathscr{O}_{Y})$. With
Lemma 2.4 below we infer that it is also bijective on the the range of the
Bockstein operator. This gives $\beta_{r}(a)=0$, contradiction. ∎

In the above arguments, we have used the following simple observation:

######  Lemma 2.4.

Let $V^{\prime}\rightarrow V\rightarrow V^{\prime\prime}$ be an exact sequence
of $k[F]$-modules whose underlying $k$-vector spaces are finite-dimensional.
If $V^{\prime}$ and $V^{\prime\prime}$ have maximal Hasse–Witt rank, the same
holds for $V$.

###### Proof.

It suffices to treat the case that $k$ is perfect. We then have to check that
the $p$-linear map $F:V\rightarrow V$ is surjective. Clearly, its image
contains the image of $V^{\prime}\rightarrow V$. This reduces us to the case
$V^{\prime}=0$, such that $V\subset V^{\prime\prime}$. Choose a vector space
basis $a_{1},\ldots,a_{r}\in V$ and extend it to a basis
$a_{1},\ldots,a_{n}\in V^{\prime\prime}$. Then
$F(a_{j})=\sumop\displaylimits\lambda_{ij}a_{i}$ defines a matrix
$A^{\prime\prime}=(\lambda_{ij})$. In turn, the Hasse–Witt determinant
$\det(A^{\prime\prime})$ is non-zero. Since $V\subset V^{\prime\prime}$ is an
$F$-invariant subspace, we see that $\lambda_{ij}=0$ for $1\leq j\leq r<i$,
and conclude that also $F:V\rightarrow V$ has non-zero Hasse–Witt determinant.
∎

_Proof of Theorem 2.2._ Suppose that the Frobenius has maximal Hasse–Witt rank
on $H^{2}(Y,\mathscr{O}_{Y})$. We have to check that ${}_{G}^{0}=0$ for the
algebraic group scheme $G=\operatorname{Pic}^{\tau}_{Y/k}$. In light of
Proposition 1.2, it suffices to treat the case that $k$ is perfect. Then
$G^{0}_{\operatorname{red}}$ is a subgroup scheme, and by Theorem 1.8 we have
to verify that the local group scheme $L=G^{0}/G^{0}_{\operatorname{red}}$ is
multiplicative. This holds by Proposition 2.3. ∎

## 3\. The case of surfaces

We keep the assumptions of the previous section, so that Y/k is the maximal
unipotent quotient of $G=\operatorname{Pic}^{\tau}_{Y/k}$, where $Y$ is a
proper scheme over our ground field $k$ of characteristic $p>0$. The goal now
is to apply the general results of the previous section to certain classes of
$Y$, and establish vanishing results. For simplicity, we assume that $Y$ is
equi-dimensional, of dimension $n\geq 0$. Our first observation is:

######  Proposition 3.1.

We have ${}_{Y/k}=0$ provided that $Y$ is a curve.

###### Proof.

The Néron–Severi group $\operatorname{NS}(X\otimes k^{\operatorname{alg}})$ is
a free group, according to [BLR90, Section 9.4, Corollary 14]. Hence by
Proposition 2.1, the group scheme of components for Y/k is trivial.
Furthermore, the group $H^{2}(Y,\mathscr{O}_{Y})$ vanishes by dimension
reason, so the Frobenius has a priori maximal Hasse–Witt rank. The the
assertion thus follows from Theorem 2.2. ∎

Let $\omega_{Y}$ be the _dualizing sheaf_ , with its trace map
$\operatorname{tr}\colon H^{n}(Y,\omega_{Y})\rightarrow k$. In turn, for every
coherent sheaf $\mathscr{F}$ we get a pairing

$H^{i}(Y,\mathscr{F})\times\operatorname{Ext}^{n-i}(\mathscr{F},\omega_{Y})\stackrel{{\scriptstyle{\rm\text{can}}}}{{\longrightarrow}}H^{n}(Y,\omega_{Y})\stackrel{{\scriptstyle\operatorname{tr}}}{{\longrightarrow}}k,$

which is non-degenerate for $i=n$, regardless of the singularities. The
pairings remain non-degenerate in all degrees $i\leq n$ provided that $Y$ is
Cohen–Macaulay. We record:

######  Proposition 3.2.

We have ${}_{Y/k}^{0}=0$ provided that the scheme $Y$ has dimension $n\geq 2$,
is Cohen–Macaulay, and has the property $H^{n-2}(Y,\omega_{Y})=0$.

###### Proof.

The above Serre Duality gives $h^{2}(\mathscr{O}_{Y})=h^{n-2}(\omega_{Y})=0$,
and the assertion follows from Theorem 2.2. ∎

Now suppose that $Y$ is a surface. For every invertible sheaf $\mathscr{L}$,
the Euler characteristic $\chi(\mathscr{L}^{\otimes t})$ is a numerical
polynomial of degree two in the variable $t$, which can be written as

$\chi(\mathscr{L}^{\otimes
t})=\frac{(\mathscr{L}\cdot\mathscr{L})}{2}t^{2}-\frac{(\mathscr{L}\cdot\omega_{Y})}{2}t+\chi(\mathscr{O}_{Y}).$

The quadratic term is determined by the self-intersection number
$(\mathscr{L}\cdot\mathscr{L})$, whereas _the linear term defines an integer_
$(\mathscr{L}\cdot\omega_{Y})$, which coincides with the usual intersection
number of invertible sheaves provided that $Y$ is Gorenstein.

######  Theorem 3.3.

Let $Y$ be a reduced surface. Suppose there is an invertible sheaf
$\mathscr{L}$ such that for each irreducible component $Z\subset Y$, the
restriction $\mathscr{L}_{Z}$ is nef and has
$(\mathscr{L}_{Z}\cdot\omega_{Z})<0$. Then the cohomology group
$H^{2}(Y,\mathscr{O}_{Y})$ vanishes, the Néron–Severi group
$\operatorname{NS}(Y\otimes k^{\operatorname{alg}})$ is free, and the maximal
unipotent finite quotient Y/k is trivial.

###### Proof.

The statement on the maximal unipotent quotient follows from the assertions on
cohomology and Néron–Severi group, using Proposition 2.1 and Theorem 2.2. To
proceed, we first reduce to the case of integral surfaces that are
Cohen–Macaulay. Let $X_{1},\ldots,X_{r}$ be the $S_{2}$-ization of the
irreducible components $Y_{1},\ldots,Y_{r}\subset Y$, and write $X$ for their
disjoint union (see for example [SV04] for details on the $S_{2}$-ization).
The resulting finite morphism $f\colon X\rightarrow Y$ is surjective, hence
induces an injection on Néron–Severi groups. Moreover, the cokernel in the
short exact sequence $0\rightarrow\mathscr{O}_{Y}\rightarrow
f_{*}(\mathscr{O}_{X})\rightarrow\mathscr{F}\rightarrow 0$ is at most zero-
dimensional (see [EGA4-II, corollaire 5.10.15]). In the resulting long exact
sequence

$H^{1}(Y,\mathscr{F})\longrightarrow H^{2}(Y,\mathscr{O}_{Y})\longrightarrow
H^{2}(X,\mathscr{O}_{X})\longrightarrow H^{2}(Y,\mathscr{F}),$

the outer terms vanish, and we get an identification
$H^{2}(Y,\mathscr{O}_{Y})=H^{2}(X,\mathscr{O}_{X})$. Now fix some irreducible
component $Z=Y_{i}$, and consider the resulting connected component
$Z^{\prime}=X_{i}$. Its dualizing sheaf is given by
$f_{*}(\omega_{Z^{\prime}})=\underline{\operatorname{Hom}}(f_{*}(\mathscr{O}_{Z^{\prime}}),\omega_{Z})$,
and we get a short exact sequence $0\rightarrow
f_{*}(\omega_{Z^{\prime}})\rightarrow\omega_{Z}\rightarrow\mathscr{G}\rightarrow
0$ where the cokernel is at most zero-dimensional. Tensoring with
$\mathscr{L}^{\otimes t}$ and comparing linear terms in the numerical
polynomials, we see that
$(\mathscr{L}_{Z^{\prime}}\cdot\omega_{Z^{\prime}})=(\mathscr{L}_{Z}\cdot\omega_{Z})$.
Clearly, the pullback of $\mathscr{L}$ under the finite surjection
$Z^{\prime}\rightarrow Z$ remains nef. Summing up, it suffices to treat the
case that $Y$ is integral and Cohen–Macaulay.

Next, we verify that $h^{2}(\mathscr{O}_{Y})=0$. Seeking a contradiction, we
assume that there is a short exact sequence
$0\rightarrow\mathscr{O}_{Y}\rightarrow\omega_{Y}\rightarrow\mathscr{F}\rightarrow
0$, where $\mathscr{F}$ is a torsion sheaf. Tensoring with
$\mathscr{L}^{\otimes t}$ and using Serre Duality, we get

(1) $\chi(\mathscr{L}^{\otimes t})+\chi(\mathscr{L}^{\otimes
t}\otimes\mathscr{F})=\chi(\mathscr{L}^{\otimes
t}\otimes\omega_{Y})=\chi(\mathscr{L}^{\otimes-t}).$

Let $\zeta_{i}\in Y$ be the generic points of
$\operatorname{Supp}(\mathscr{F})$ whose closures $C_{i}\subset Y$ are one-
dimensional, write $m_{i}\geq 1$ for the length of the finite
$\mathscr{O}_{Y,\zeta_{i}}$-module $\mathscr{F}_{a_{i}}$, and let $C$ be the
union $\bigcupop\displaylimits C_{i}$. By the results in [Kle66, Section
II.2], we have $\chi(\mathscr{L}^{\otimes
t}\otimes\mathscr{F})=\sumop\displaylimits m_{i}(\mathscr{L}\cdot
C_{i})t+\chi(\mathscr{O}_{C})$, with $\sumop\displaylimits
m_{i}(\mathscr{L}\cdot C_{i})\geq 0$. In equation (1), the linear term on the
left has coefficient
$-\frac{1}{2}(\mathscr{L}\cdot\omega_{Y})+\sumop\displaylimits
m_{i}(\mathscr{L}\cdot C_{i})>0$, whereas the coefficient on the right is
$\frac{1}{2}(\mathscr{L}\cdot\omega_{Y})<0$, contradiction. This shows
$h^{2}(\mathscr{O}_{Y})=0$.

It remains to verify the assertion on the Néron–Severi group. Let $f\colon
X\rightarrow Y$ be some resolution of singularities. This proper surjective
morphism induces an inclusion
$\operatorname{NS}(Y)\subset\operatorname{NS}(X)$. Furthermore, we obtain a
short exact sequence $0\rightarrow
f_{*}(\omega_{X})\rightarrow\omega_{Y}\rightarrow\mathscr{F}\rightarrow 0$ for
some torsion sheaf $\mathscr{F}$. Tensoring with $\mathscr{L}^{\otimes t}$,
and using Serre duality alongside the Leray–Serre spectral sequence, we get

$\chi(\mathscr{L}^{\otimes-t})=\chi(\mathscr{L}_{X}^{\otimes-t})-\chi(\mathscr{L}^{\otimes
t}\otimes R^{1}f_{*}\mathscr{O}_{X})+\chi(\mathscr{L}^{\otimes
t}\otimes\mathscr{F}).$

Looking at the linear terms and using the notation of the preceding paragraph,
we get

$\frac{1}{2}(\mathscr{L}\cdot\omega_{Y})=\frac{1}{2}(\mathscr{L}_{X}\cdot\omega_{X})+\sumop\displaylimits
m_{i}(\mathscr{L}\cdot C_{i}),$

and conclude that $(\mathscr{L}_{X}\cdot\omega_{X})<0$. This reduces us to the
case of regular irreducible surfaces $Y$. Likewise, one easily reduces to the
case that $k$ is separably closed.

The base-change to the algebraic closure $k^{\operatorname{alg}}$ is not
necessarily regular or normal, not even reduced. Let $X\rightarrow Y\otimes
k^{\operatorname{alg}}$ be the normalization of the reduction, and consider
the composite morphism $f\colon X\rightarrow Y$. According to a result of
Tanaka ([Tan18, Theorem 4.2], see also [PW17, Theorem 1.1]), we have the
following equality $\omega_{X}=f^{*}(\omega_{Y})\otimes\mathscr{O}_{X}(-R)$
for some curve $R\subset X$. In turn, $(\mathscr{L}_{X}\cdot\omega_{X})<0$.
Now let $r\colon S\rightarrow X$ be the minimal resolution of singularities,
with exceptional divisor $E=E_{1}+\ldots+E_{r}$, and
$K_{S/X}=\sumop\displaylimits\lambda_{i}E_{i}$ be the unique
$\mathbb{Q}$-divisor with $(K_{S/X}\cdot E_{i})=(K_{S}\cdot E_{i})$. Since the
resolution is minimal, we must have $\lambda_{i}\leq 0$. Now choose some Weil
divisor $K_{X}$ representing $\omega_{X}$, and consider the rational pullback
$r^{*}(K_{X})$ in the sense of Mumford [Mum61], compare also [Sch19]. We then
have

$(\mathscr{L}_{S}\cdot\omega_{S})=f^{*}(\mathscr{L})\cdot(K_{S/X}+f^{*}K_{X})=(\mathscr{L}_{X}\cdot\omega_{X})<0.$

Using that $\mathscr{L}_{S}$ is nef, we infer that the plurigenera
$h^{0}(\omega_{S}^{\otimes n})$, $n\geq 1$ of the smooth surface $S$ vanish,
so its Kodaira dimension must be $\operatorname{kod}(S)=-\infty$. By the
Enriques classification, the surface is either $S=\mathbb{P}^{2}$ or admits a
ruling. In both cases, $\operatorname{NS}(S)$ is free. In turn, the same holds
for the subgroup $\operatorname{NS}(Y\otimes k^{\operatorname{alg}})$. ∎

This applies in particular for reduced _del Pezzo surfaces_ $Y$, which by
definition are Gorenstein, with $\omega_{Y}$ anti-ample, and have
$h^{0}(\mathscr{O}_{Y})=1$.

######  Corollary 3.4.

Let $Y$ be a reduced del Pezzo surface. Then Y/k is trivial.

Note that Miles Reid [Rei94] has classified reduced non-normal del Pezzo
surfaces over algebraically closed fields. Then the algebraic group
$G=\operatorname{Pic}^{\tau}_{Y/k}$ is smooth, and there are cases with
$H^{1}(Y,\mathscr{O}_{Y})\neq 0$.

Tanaka constructed Mori fiber spaces in characteristic $p\leq 3$, where the
generic fiber is a normal projective surfaces $Y$ with
$h^{0}(\mathscr{O}_{Y})=1$ having only $\mathbb{Q}$-factorial klt-terminal
singularities, the $\mathbb{Q}$-divisor $K_{Y}$ is anti-ample, yet the Picard
group contains elements of order $p$ ([Tan16, Theorem 1.2], with further
investigation in [BT20]). From Theorem 3.3, we see that such elements must
come from the component $\operatorname{Pic}^{0}_{Y/k}$ of the origin.

Note also that the reducedness assumption in our results is indispensable:
Suppose that $S$ is an irreducible smooth surface, and let $\mathscr{L}$ be an
invertible sheaf such that its dual is ample. After passing to suitable
multiples, we achieve that
$h^{1}(\mathscr{L}^{\vee})=h^{2}(\mathscr{L}^{\vee})=0$, and
$\mathscr{L}\otimes\omega_{S}$ becomes anti-ample. Consider the quasicoherent
$\mathscr{O}_{S}$-algebra
$\mathscr{A}=\mathscr{O}_{Y}\oplus\mathscr{L}^{\vee}$, with multiplication
$(f,s)\cdot(f^{\prime},s^{\prime})=(ff^{\prime},fs^{\prime}+f^{\prime}s)$, and
let $Y=\operatorname{Spec}(\mathscr{A})$ be its relative spectrum. The
structure morphism $f\colon Y\rightarrow S$ has a canonical section, which is
given by the projection $\mathscr{A}\rightarrow\mathscr{O}_{Y}$ and identifies
$S$ with $Y_{\operatorname{red}}$. One also says that $Y$ is a _ribbon_ on
$S$. The resulting short exact sequence of abelian sheaves
$0\rightarrow\mathscr{L}^{\vee}\rightarrow\mathscr{O}_{Y}^{\times}\rightarrow\mathscr{O}_{S}^{\times}\rightarrow
1$ shows that the inclusion $S\subset Y$ induces an identification of Picard
schemes. Moreover, the relative dualizing sheaf for $f\colon Y\rightarrow S$
is given by
$f_{*}(\omega_{Y/S})=\underline{\operatorname{Hom}}(\mathscr{A},\mathscr{O}_{Y})=\mathscr{A}\otimes\mathscr{L}$.
In turn, we have $\omega_{Y}=f^{*}(\mathscr{L}\otimes\omega_{S})$, which is
anti-ample. Summing up, $Y$ is an irreducible non-reduced del Pezzo surface
with ${}_{Y}={}_{S}$. The latter easily becomes non-trivial, e.g. if $S$ is a
simply-connected Enriques surface in characteristic $p=2$.

## 4\. Enriques surfaces and del Pezzo surfaces

Let $k$ be an algebraically closed ground field of characteristic $p=2$. In
this section, we construct certain normal Enriques surfaces $Z$ where the
numerically trivial part of the Picard scheme is unipotent of order two,
together with a finite universal homeomorphism $\nu\colon
Z^{\prime}\rightarrow Z$ of degree two from a normal del Pezzo surface of
degree four with Picard number one. Theses surfaces will arise from very
special simply-connected Enriques surfaces.

Throughout, $S$ denotes an _Enriques surface_. This means that $S$ is a
regular connected surface with $c_{1}=0$ and $b_{2}=10$. The group
$\operatorname{Num}(S)=\operatorname{Pic}(S)/\operatorname{Pic}^{\tau}(S)$ of
numerical classes is a free abelian group of rank $\rho=10$ called the
_Enriques lattice_. The intersection form is isomorphic to $E_{8}\oplus H$,
comprising the root lattice of type $E_{8}$ and the hyperbolic lattice
$H=(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix})$. The group scheme
$P=\operatorname{Pic}^{\tau}_{S/k}$ of numerically trivial invertible sheaves
is finite of order two. In characteristic $p=2$, there are three possibilities
for $P$, namely

$\mu_{2}\quad\text{and}\quad\mathbb{Z}/2\mathbb{Z}\quad\text{and}\quad\alpha_{2}.$

The respective Enriques surfaces $S$ are called _ordinary_ , _classical_ and
_supersingular_. The inclusion $P\subset\operatorname{Pic}_{S/k}$ yields a
$G$-torsor $\epsilon\colon{\tilde{S}}\rightarrow S$, where
$G=\underline{\operatorname{Hom}}(P,\mathbb{G}_{m})$ is the Cartier dual. For
ordinary $S$ the Cartier dual $G=\mathbb{Z}/2\mathbb{Z}$ is étale, and the
total space ${\tilde{S}}$ is a _K3 surface_. In the other two cases $P$ is
unipotent, $G$ is local, and the integral Gorenstein surface $\tilde{S}$
necessarily acquires singularities. One says that $S$ is a _simply-connected
Enriques surface_ , and that ${\tilde{S}}$ is its _K3-like covering_. Note
that the relation between inclusions $P\subset\operatorname{Pic}_{S/k}$ and
$G$-torsors ${\tilde{S}}\rightarrow S$ goes back to Raynaud [Ray70]. The
trichotomy of Enriques surfaces for $p=2$ was developed by Bombieri and
Mumford [BM76]. For more information on K3-like coverings, see for example
[Sch17].

We write $\gamma\colon S^{\prime}\rightarrow\tilde{S}$ for the normalization.
Since $S$ is regular and $S^{\prime}$ and ${\tilde{S}}$ are Cohen–Macaulay,
both projections $\nu\colon S^{\prime}\rightarrow S$ and
$\epsilon\colon{\tilde{S}}\rightarrow S$ are finite and flat of degree two
([EGA4-II, proposition 6.1.5]). Hence all fibers take the form
$\operatorname{Spec}\kappa(s)[\epsilon]$, which ensures that both $S^{\prime}$
and ${\tilde{S}}$ are Gorenstein. The relative dualizing sheaf for
$\gamma\colon S^{\prime}\rightarrow\tilde{S}$ is given by

$\gamma_{*}(\omega_{S^{\prime}/{\tilde{S}}})=\underline{\operatorname{Hom}}(\gamma_{*}(\mathscr{O}_{S^{\prime}}),\mathscr{O}_{\tilde{S}}).$

It is also the ideal sheaf for the _ramification locus_ $R\subset S^{\prime}$
of the normalization map. This Weil divisor must be Cartier, because
$S^{\prime}$ and ${\tilde{S}}$ are Gorenstein. As explained in [ESB04], there
is a unique effective divisor $C\subset S$ with $R=\nu^{-1}(C)$. Ekedahl and
Shepherd-Barron call $C\subset S$ the _conductrix_ of the Enriques surface
$S$. Note that $S$ is simply-connected if $C$ is non-empty.

Now suppose that $E=E_{1}+\ldots+E_{r}$ is a configuration of $(-2)$-curves
whose intersection matrix $(E_{i}\cdot E_{j})$ is negative-definite. We also
say that $E$ is an _ADE-curve_. Let $f\colon S\rightarrow Z$ be its
contraction. The resulting surfaces $Z$ are called _normal Enriques surfaces_.
They are projective and their singularities are rational double points, such
that $\omega_{Z}=f_{*}(\omega_{S})$ and $\omega_{S}=f^{*}(\omega_{Z})$. By the
Hodge Index Theorem, we have $r\leq 9$.

######  Proposition 4.1.

The numerical group $\operatorname{Num}(Z)$ if free of rank $\rho=10-r$, and
the homomorphism of group schemes
$f^{*}\colon\operatorname{Pic}^{\tau}_{Z/k}\rightarrow\operatorname{Pic}^{\tau}_{S/k}$
is an isomorphism.

###### Proof.

The Picard group of $Z$ can be viewed as the orthogonal complement of the
curves $E_{1},\ldots,E_{r}\in\operatorname{Pic}(S)$, via the preimage map
$f^{*}\colon\operatorname{Pic}(Z)\rightarrow\operatorname{Pic}(S)$. It follows
that $\operatorname{Pic}(Z)$ is finitely generated of rank $\rho=10-r$, and
the assertion on the numerical group $\operatorname{Num}(Z)$ follows.

The subgroup $\operatorname{Pic}^{\tau}(Z)\subset\operatorname{Pic}(Z)$ is the
torsion part. This is cyclic of order two provided that $S$ is classical, and
then the assertion on $\operatorname{Pic}^{\tau}_{Z/k}$ is already contained
in the preceding paragraph. It remains to treat the case that $S$ is ordinary
or supersingular. In other words, the group scheme
$P=\operatorname{Pic}^{\tau}_{Z/k}$ is local of height one, with one-
dimensional tangent space $\operatorname{Lie}(P)=H^{1}(S,\mathscr{O}_{S})$.
The Leray–Serre spectral sequence for the contraction $f\colon S\rightarrow Z$
gives an exact sequence

$0\longrightarrow H^{1}(Z,\mathscr{O}_{Z})\longrightarrow
H^{1}(S,\mathscr{O}_{S})\longrightarrow H^{0}(Z,R^{1}f_{*}(\mathscr{O}_{S})).$

The term on the right vanishes, because $Z$ has only rational singularities,
and it follows that
$\operatorname{Pic}_{Z/k}\rightarrow\operatorname{Pic}_{S/k}$ induces a
bijection on tangent spaces. This homomorphism of groups schemes must be a
monomorphism, because $\mathscr{O}_{S}=f_{*}(\mathscr{O}_{X})$. In turn, the
inclusion
$\operatorname{Pic}^{\tau}_{Z/k}\subset\operatorname{Pic}^{\tau}_{S/k}$ is an
isomorphism. ∎

Set $P=\operatorname{Pic}^{\tau}_{S/k}=\operatorname{Pic}^{\tau}_{Z/k}$ and
let $G=\underline{\operatorname{Hom}}(P,\mathbb{G}_{m})$ be the Cartier dual.
The inclusion of $P\subset\operatorname{Pic}_{Z/k}$ corresponds to a
$G$-torsor ${\tilde{Z}}\rightarrow Z$, and the total space $Z^{\prime}$ is an
integral Gorenstein surface. Let $Z^{\prime}\rightarrow{\tilde{Z}}$ be its
normalization. Of course, we have analogous constructions for the Enriques
surface $S$, and by naturality we get a commutative diagram

(2) $\begin{CD}S^{\prime}@>{}>{}>{\tilde{S}}@>{}>{}>S\\\
@V{}V{}V@V{}V{}V@V{}V{f}V\\\ Z^{\prime}@>{}>{}>{\tilde{Z}}@>{}>{}>Z\end{CD}$

of projective integral surfaces. We write $\nu\colon Z^{\prime}\rightarrow Z$
for the composition of the lower arrows. The image $D=f(C)$ is a Weil divisor
on the normal surface $Z$. We call it the _conductrix of the normal Enriques
surface_.

######  Theorem 4.2.

Suppose the Enriques surface $S$ and the chosen ADE-curve $E\subset S$ satisfy
the following three conditions:

  1. (i)

The conductrix $C\subset S$ is not supported by the exceptional divisor $E$.

  2. (ii)

There are integers $m_{1},\ldots,m_{r}$ such that $(C\cdot
E_{j})=\sumop\displaylimits m_{i}(E_{i}\cdot E_{j})$ for all $1\leq j\leq r$.

  3. (iii)

The ADE curve $E=E_{1}+\ldots+E_{r}$ has $r=9$ irreducible components.

Then the conductrix $D\subset Z$ is Cartier, and $Z^{\prime}$ is a normal del
Pezzo surface of degree $K_{Z^{\prime}}^{2}=2D^{2}$, with canonical class
$K_{Z^{\prime}}=-\nu^{-1}(D)$, irregularity
$h^{1}(\mathscr{O}_{Z^{\prime}})=0$ and Picard group
$\operatorname{Pic}(Z^{\prime})=\mathbb{Z}$. Moreover, all singularities on
$Z^{\prime}$ are rational double points.

###### Proof.

Since the horizontal maps in the diagram (2) are universal homeomorphisms and
the scheme $Z$ is $\mathbb{Q}$-factorial, the schemes $Z^{\prime},{\tilde{Z}}$
and $Z$ have the same Picard number, so Condition (iii) yields
$\rho(Z^{\prime})=\rho(Z)=1$. Condition (i) ensures that the Weil divisor
$D\subset Z$ is non-empty, and (ii) means that it is Cartier.

By definition, the preimage of the conductrix $C\subset S$ on $S^{\prime}$ is
the ramification locus for the normalization
$S^{\prime}\rightarrow{\tilde{S}}$. Let $U\subset{\tilde{Z}}$ be the preimage
of the regular locus $\operatorname{Reg}(Z)$. Since
${\tilde{S}}\rightarrow{\tilde{Z}}$ becomes an isomorphism over $U$, the
preimage of $D$ in ${\tilde{Z}}$ and the branch locus in ${\tilde{Z}}$ for the
normalization $Z^{\prime}\rightarrow{\tilde{Z}}$ coincide, at least over $U$.
But the preimage $\nu^{-1}(D)$ and the ramification curve $R\subset
Z^{\prime}$ have no embedded components, and $Z^{\prime}$ is normal, so
$\nu^{-1}(D)=R$. As $D\subset Z$ is Cartier, the same holds for $R\subset
Z^{\prime}$. Since $\omega_{\tilde{Z}}$ and
$\omega_{Z^{\prime}/{\tilde{Z}}}=\mathscr{O}_{Z^{\prime}}(-R)$ are invertible,
the normal surface $Z^{\prime}$ is Gorenstein. We have $-(K_{Z^{\prime}}\cdot
R)=\nu^{*}(D)^{2}=\deg(\nu)\cdot D^{2}=2D^{2}>0$. By the Nakai criterion,
$\omega_{Z^{\prime}}$ is anti-ample, thus $Z^{\prime}$ is a normal del Pezzo
surface.

Seeking a contradiction, we now assume that
$H^{1}(Z^{\prime},\mathscr{O}_{Z^{\prime}})\neq 0$. This is the tangent space
to the Picard scheme, so $A=\operatorname{Pic}^{0}_{Z^{\prime}/k}$ is non-
zero. The latter is smooth, because the obstruction group
$H^{2}(Z^{\prime},\mathscr{O}_{Z^{\prime}})\simeq
H^{0}(Z^{\prime},\omega_{Z^{\prime}})$ vanishes, compare [Mum66, Lecture 27].
In turn, the group scheme $A\neq 0$ is an abelian variety, and we conclude
that for each prime $l\neq 2$, there is a $\mu_{l}$-torsor
$Z^{\prime\prime}\rightarrow Z^{\prime}$ with connected total space. The
projection $\nu\colon Z^{\prime}\rightarrow Z$ is a universal homeomorphism,
so by [SGA1, exposé IX, théorème 4.10], the finite étale Galois covering
$Z^{\prime\prime}\rightarrow Z^{\prime}$ is the base-change of some finite
étale Galois covering of $Z$. This implies that $\operatorname{Pic}(Z)$
contains an element of order $l$, contradiction. Thus
$h^{1}(\mathscr{O}_{Z^{\prime}})=0$.

It then follows $\operatorname{Pic}(Z^{\prime})=\mathbb{Z}$ by [Sch01, Lemma
2.1]. It remains to check that the singularities on $Z^{\prime}$ are rational
double points. Since $Z^{\prime}$ is Gorenstein, the task is to show that they
are rational. Seeking a contradiction, we assume that there is at least one
non-rational singularity. Consider the minimal resolution of non-rational
singularities $r\colon Y\rightarrow Z^{\prime}$. According to _loc. cit._ ,
Theorem 2.2, there exists a fibration $\varphi\colon Y\rightarrow B$ over some
curve of genus $g>0$, and $r\colon Y\rightarrow Z^{\prime}$ is the contraction
of some section $E\subset Y$. Set $V=\operatorname{Reg}(Z)$, and fix some
prime $l\neq 2$ such that the finitely generated abelian group
$\operatorname{Pic}(V)=\operatorname{Pic}(S)/\sumop\displaylimits\mathbb{Z}E_{i}$
contains no element of order $l$. As in the preceding paragraph, we find some
$\mu_{l}$-torsor $Y^{\prime}\rightarrow Y$, which yields a finite étale Galois
covering of $V$. This implies that $\operatorname{Pic}(V)$ contains an element
of order $l$, contradiction. ∎

######  Corollary 4.3.

Keep the assumptions of Theorem 4.2. Then the normal del Pezzo surface
$Z^{\prime}$ is not regular, we have
$\operatorname{Pic}(Z^{\prime})=\mathbb{Z}K_{Z^{\prime}}$, and the Cartier
divisor $D\subset Z$ has selfintersection $1\leq D^{2}\leq 4$.

###### Proof.

Suppose $Z^{\prime}$ were regular. It contains no $(-1)$-curves, because
$\rho=1$. According to [Dol12, Theorem 8.1.5], the regular del Pezzo surface
$Z^{\prime}$ is isomorphic to $\mathbb{P}^{2}$. But $K^{2}_{\mathbb{P}^{2}}=9$
is odd, whereas $K_{Z^{\prime}}^{2}=2D^{2}$ is even, contradiction.

Let $r\colon X\rightarrow Z^{\prime}$ be the minimal resolution of
singularities. Then $X$ is a _weak del Pezzo surface_ , which for integral
surfaces means that it is Gorenstein and the inverse of the dualizing sheaf is
nef and big. Let $Y$ be a minimal model, obtained from a successive
contraction $X=X_{0}\rightarrow\ldots\rightarrow X_{n}=Y$ of $(-1)$-curves.
Then $Y$ is either the projective plane or a Hirzebruch surface, and in both
cases we have $K_{Y}^{2}\leq 9$. This gives
$2D^{2}=K_{Z^{\prime}}^{2}=K_{X}^{2}=K_{Y}^{2}-n\leq 9$, and in turn $1\leq
D^{2}\leq 4$. In particular, the degree $K_{Z^{\prime}}^{2}$ of the normal del
Pezzo surface belongs to the set $\\{2,4,6,8\\}$.

Suppose the canonical class does not generate the Picard group. Then we are in
the situation $K_{Z^{\prime}}=2A$ for some Cartier divisor $A$, and either
$A^{2}=1$ or $A^{2}=2$. In case $K_{Z^{\prime}}^{2}=8$, the surface $X$ is the
Hirzebruch surface with invariant $e=2$, and $Z^{\prime}$ is obtained by
contraction of the $(-2)$-curve. It then follows that $K_{Z^{\prime}}$
generates the Picard group, contradiction. Now suppose that
$K_{Z^{\prime}}^{2}=4$, with intersection numbers $A^{2}=1$ and $A\cdot
K_{Z^{\prime}}=-2$. In turn, the Euler characteristic
$\chi(\mathscr{L})=A\cdot(A-K_{Z^{\prime}})/2+\chi(\mathscr{O}_{Z^{\prime}})$
of the invertible sheaf $\mathscr{L}=\mathscr{O}_{Z^{\prime}}(A)$ is not an
integer, contradiction. ∎

For later use, we also record the following vanishing result:

######  Corollary 4.4.

Keep the assumptions of Theorem 4.2. Then
$h^{1}(\omega_{Z^{\prime}}^{\otimes-i})=0$ for all integers $i$.

###### Proof.

By Serre Duality, it suffices to verify this for $i\geq 0$. Let $r\colon
X\rightarrow Z^{\prime}$ be the minimal resolution of singularities. Since
$Z^{\prime}$ has only rational double points, we have
$\omega_{X}^{\otimes-i}=r^{*}(\omega_{Z^{\prime}}^{\otimes-i})$ and
$h^{1}(\omega_{X}^{\otimes-i})=h^{1}(\omega_{Z^{\prime}}^{\otimes-i})$. The
regular surface $X$ is obtained from its minimal model $Y$, which is the
projective plane or a Hirzebruch surface, by a sequence of blowing-ups of
closed points. It follows that $X$ lifts to characteristic zero. In
particular, we may apply Raynaud’s vanishing result ([DI87, Corollary 2.8])

$H^{1}(X,\omega_{X}^{\otimes-i})=H^{1}(X,{}_{X/k}^{2}\otimes\mathscr{L})=0$

for the nef and big invertible sheaf $\mathscr{L}=\omega_{X}^{\otimes-(i+1)}$.
∎

## 5\. Exceptional Enriques surfaces

Now suppose that $S$ is a simply-connected Enriques surface whose conductrix
takes the form

$C=2C_{1}+3C_{3}+5C_{4}+2C_{2}+4C_{5}+4C_{6}+3C_{7}+3C_{8}+2C_{0}+C_{9},$

where the ten irreducible components $C_{0},\ldots,C_{9}$ are $(-2)$-curves
with simple normal crossings having the following dual graph :

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One also says that $S$ is a _exceptional Enriques surface of type
$T_{2,3,7}$_. Here the indices $2,3,7$ denote the length of the terminal
chains in the star-shaped tree , including the central vertex
$C_{4}\in\Gamma$. Such Enriques surfaces were already considered in the
monograph of Cossec and Dolgachev [CD89, Chapter III, §4].

The general notion of _exceptional Enriques surfaces_ was introduced by
Ekedahl and Shepherd-Barron [ESB04], who studied them in detail. They can be
characterized in terms of the conductrix $C\subset S$, and also by properties
of the Hodge ring $\bigoplusop\displaylimits_{ij}H^{i}(S,{}_{S}^{j})$.
Explicit equations for birational models were found by Salomonsson [Sal03].
For examples, the equation

$z^{2}+(y^{4}+x^{4})x^{3}y^{3}s^{4}+\lambda
x^{5}y^{3}s^{3}t+xyt=0,\quad\lambda\neq 0,$

as well as

$z^{2}+x^{3}y^{7}s^{4}+\mu x^{8}s^{3}t+xyt^{4}=0,\quad\mu\neq 0$

define birational models for exceptional Enriques surfaces of type $T_{2,3,7}$
as inseparable double covering of a Hirzebruch surface with coordinates
$x,y,s,t$. The first equation gives classical, the second equation
supersingular Enriques surfaces.

The reduced curve $F=C_{0}+\ldots+C_{8}$ on the Enriques surface $S$ supports
a curve of canonical type with Kodaira symbol $\text{\rm II}^{*}$. Let
$\varphi\colon S\rightarrow\mathbb{P}^{1}$ be the resulting genus-one
fibrations. This fibration is quasielliptic, and there is no other genus-one
fibration, according to [ESB04, Theorem C]. The fiber corresponding to $F$ is
multiple, because otherwise $2\leq(C_{9}\cdot F)=(C_{9}\cdot C_{0})=1$,
contradiction. Since $b_{2}=10$, all other fibers are irreducible, thus have
Kodaira symbol II. If $S$ is classical, there must be another multiple fiber.
In the supersingular case, all other fibers are simple ([CD89, Theorem
5.7.2]).

Let $f\colon S\rightarrow Z$ be the contraction of the ADE-curves
$C_{1}+\ldots+C_{8}$ and $C_{9}$. Then $Z$ is a normal Enriques surface with
$\operatorname{Sing}(Z)=\\{a,b\\}$, where the first local ring
$\mathscr{O}_{Z,a}$ is a rational double point of type $E_{8}$, and the second
local ring $\mathscr{O}_{Z,b}$ is a rational double point of type $A_{1}$.
Write $D_{0}=f(C_{0})$ for the image of the remaining $(-2)$-curve, which is a
Weil divisor. The conductrix of the normal Enriques surface is
$D=f(C)=2D_{0}$. One easily sees that Theorem 4.2 applies, so $D\subset Z$ is
Cartier, and we get a a normal del Pezzo surface $Z^{\prime}$ as an
inseparable double covering $\nu\colon Z^{\prime}\rightarrow Z$. The goal of
this section is to study the geometry of these surfaces in detail.

Following Hartshorne [Har94], we write $\operatorname{APic}(X)$ for the group
of isomorphism classes of reflexive rank-one sheaves, on a given normal
noetherian scheme $X$. It is called the _almost Picard group_ , and could also
be seen as the group of 1-cycles modulo linear equivalence. If $X$ is a proper
surface, the group $\operatorname{APic}(X)$ is endowed with Mumford’s
_rational selfintersection numbers_ [Mum61], which extend the usual
intersection numbers for invertible sheaves. If $X=\operatorname{Spec}(R)$ is
local, we use the more traditional
$\operatorname{Cl}(R)=\operatorname{APic}(X)$. In the global case, we prefer
$\operatorname{APic}(X)$, because it emphasizes the relation to the Picard
group.

######  Proposition 5.1.

The group $\operatorname{APic}(Z)$ is generated by $D_{0}$ and the canonical
class $K_{Z}$, with selfintersection number $D_{0}^{2}=1/2$. Moreover, the
subgroup $\operatorname{Pic}(Z)$ has index two, and is generated by the
conductrix $D$, which has $D^{2}=2$, together with $K_{Z}$.

###### Proof.

Via the pullback map
$f^{*}\colon\operatorname{Pic}(Z)\rightarrow\operatorname{Pic}(S)$, we may
regard $\operatorname{Pic}(Z)$ as the orthogonal complement of the nine curves
$C_{1},\ldots,C_{9}\in\operatorname{Pic}(S)$. This orthogonal complement has
rank one, and is generated by the numerically trivial $K_{Z}$ and the linear
combination

$2C_{0}+2(2C_{8}+3C_{7}+4C_{6}+5C_{5}+6C_{4}+3C_{2}+2C_{1}+4C_{3})+C_{9}.$

This coincides with $f^{*}(D)$, and gives the selfintersection
$D^{2}=f^{*}(D)^{2}=2$. The almost Picard group can be seen as the cokernel
for the inclusion
$\sumop\displaylimits_{i=1}^{9}\mathbb{Z}C_{i}\subset\operatorname{Pic}(S)$.
Since $C_{0},\ldots,C_{9}\in\operatorname{Pic}(S)$ form a basis modulo the
torsion part, the assertion on $\operatorname{APic}(Z)$ follows. ∎

Let ${\tilde{Z}}\rightarrow Z$ be the canonical covering and
$Z^{\prime}\rightarrow{\tilde{Z}}$ be its normalization, as considered in the
previous section. Consider the composite morphism $\nu\colon
Z^{\prime}\rightarrow Z$, which is finite of degree two. According to Theorem
4.2, the total space $Z^{\prime}$ is a normal del Pezzo surface of degree
$K_{Z^{\prime}}^{2}=4$ with $\operatorname{Pic}(Z^{\prime})=\mathbb{Z}$. In
the next sections, we will embed $Z^{\prime}$ into some normal threefold, and
use $\nu$ as a gluing map for a denormalization. Our goal here is to
understand the geometry of the double covering $\nu\colon
Z^{\prime}\rightarrow Z$. The main task is to understand what happens over the
conductrix $D=f(C)$.

The following terminology will be useful: The _rational cuspidal curve_ is the
projective scheme
$\operatorname{Spec}k[t^{2},t^{3}]\cup\operatorname{Spec}k[t^{-1}]$, which is
the integral singular curve of genus one whose local rings are unibranch. A
_ribbon_ on a scheme $X$ is a closed embedding $X\subset Y$ whose ideal sheaf
$\mathscr{N}\subset\mathscr{O}_{Y}$ satisfies $\mathscr{N}^{2}=0$, such that
the $\mathscr{O}_{Y}$-module $\mathscr{N}$ is actually an
$\mathscr{O}_{X}$-module, and that $\mathscr{N}$ is invertible as
$\mathscr{O}_{X}$-module. This terminology is due to Bayer and Eisenbud
[BE95].

######  Proposition 5.2.

The scheme $D_{\operatorname{red}}$ is the rational cuspidal curve, and the
conductrix $D=f(C)$ is a ribbon on $D_{\operatorname{red}}$ with ideal sheaf
$\mathscr{N}$ of degree $\deg(\mathscr{N})=-1$. The cuspidal point of
$D_{\operatorname{red}}$ is located at the $E_{8}$-singularity $a\in Z$.

###### Proof.

The morphism $f\colon S\rightarrow Z$ factors over the contraction $g\colon
S\rightarrow\bar{S}$ of the ADE-curve $C_{1}+\ldots+C_{8}$. This creates a
rational double point $\bar{a}\in\bar{S}$ of type $E_{8}$, and we have
$\mathscr{O}_{\bar{S},\bar{a}}=\mathscr{O}_{Z,a}$. Since the normal surface
$\bar{S}$ is locally factorial, the integral curve $\bar{D}_{0}=g(C_{0})$
remains Cartier. It contains $\bar{a}$, and the local ring
$\mathscr{O}_{\bar{S},\bar{a}}$ is singular, whence
$\mathscr{O}_{\bar{D}_{0},\bar{a}}$ is singular as well. This singularity on
$\bar{D}_{0}$ must be unibranch, in light of the dual graph (3).

Since the fibration $\varphi\colon S\rightarrow\mathbb{P}^{1}$ factors over
$\bar{S}$, we must have $(\bar{D}_{0}\cdot\bar{D}_{0})=0$. The Adjunction
Formula gives
$\deg(\omega_{\bar{D}_{0}})=(K_{\bar{S}}+\bar{D}_{0})\cdot\bar{D}_{0}=0$. We
have $h^{0}(\mathscr{O}_{\bar{D}_{0}})=1$, because the curve is integral,
hence $h^{1}(\mathscr{O}_{\bar{D}_{0}})=1$. The classification of integral
curves of genus one shows that $\bar{D}_{0}$ is the rational cuspidal curve.

The induced morphism $h\colon\bar{S}\rightarrow Z$ is the contraction of the
$(-2)$-curve $\bar{C}_{9}$ corresponding to $C_{9}\subset S$, resulting in the
rational double point $b\in Z$ of type $A_{1}$. We have
$\operatorname{Spec}(k)=\bar{D}_{0}\cap\bar{C}_{9}=\bar{D}_{0}\cap h^{-1}(b)$,
the latter by [Art66, Theorem 4]. According to the Nakayama Lemma, the induced
morphism $h\colon\bar{D}_{0}\rightarrow D_{0}$ is an isomorphism. In turn,
$D_{\operatorname{red}}=D_{0}$ is the rational cuspidal curve.

The Adjunction Formula for $D\subset Z$ yields
$\deg(\omega_{D})=(K_{Z}+D)\cdot D=2$, thus $\chi(\mathscr{O}_{D})=-1$. The
normal surface $Z$ satisfies Serre’s Condition $(S_{2})$, so the Cartier
divisor $D$ satisfies $(S_{1})$. Consequently, the ideal sheaf
$\mathscr{N}\subset\mathscr{O}_{D}$ for the closed subscheme $D_{0}\subset D$
is torsion-free. It is invertible at $a\in D_{0}$, where $D_{0}$ is Cartier,
and has rank one, hence $\mathscr{N}$ is invertible as sheaf on $D_{0}$. Thus
$D$ is a ribbon on the rational cuspidal curve $D_{0}$. The short exact
sequence
$0\rightarrow\mathscr{N}\rightarrow\mathscr{O}_{D}\rightarrow\mathscr{O}_{D_{0}}\rightarrow
0$ yields
$\chi(\mathscr{N})=\chi(\mathscr{O}_{D})-\chi(\mathscr{O}_{D_{0}})=-1$. In
turn, $\deg(\mathscr{N})=-1$. ∎

Now consider the preimage $D^{\prime}=\nu^{-1}(D)$ of the conductrix $D\subset
Z$ on the normal del Pezzo surface with respect to the double covering
$\nu\colon Z^{\prime}\rightarrow Z$.

######  Proposition 5.3.

The scheme $D^{\prime}$ is a ribbon on
$D^{\prime}_{\operatorname{red}}=\mathbb{P}^{1}$, for the invertible sheaf
$\mathscr{M}=\mathscr{O}_{\mathbb{P}^{1}}(-2)$. The morphism $\nu\colon
D^{\prime}_{\operatorname{red}}\rightarrow D_{\operatorname{red}}$ factors as
the normalization map $\mathbb{P}^{1}\rightarrow D_{\operatorname{red}}$
followed by the relative Frobenius
$F\colon\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. Moreover, the induced map

(4) $\mathscr{O}_{\mathbb{P}^{1}}(-1)^{\otimes
2}=F^{*}(\mathscr{O}_{\mathbb{P}^{1}}(-1))=\nu^{*}(\mathscr{N})\longrightarrow\mathscr{M}=\mathscr{O}_{\mathbb{P}^{1}}(-2)$

is bijective.

###### Proof.

Let $\zeta\in D$ be the generic point. Then
$\mathscr{O}_{D,\zeta}=F[\epsilon]$, where $F=\kappa(\zeta)$ is the function
field of $D_{\operatorname{red}}$ and $\epsilon$ is an indeterminate subject
to $\epsilon^{2}=0$. The induced extension
$F\subset\mathscr{O}_{D^{\prime},\zeta}/(\epsilon)$ has degree two. Since
$Z^{\prime}$ is normal, the local ring $\mathscr{O}_{Z^{\prime},\zeta}$ is a
discrete valuation ring, and the fiber $\nu^{-1}(\zeta)$ has embedding
dimension at most one. It follows that the local Artin ring
$\mathscr{O}_{D^{\prime},\zeta}/(\epsilon)$ is a field, which must be purely
inseparable over $F$. This shows that
$D^{\prime}=2D^{\prime}_{\operatorname{red}}$.

The short exact sequence
$0\rightarrow\mathscr{O}_{Z^{\prime}}(-D^{\prime}_{\operatorname{red}})\rightarrow\mathscr{O}_{Z^{\prime}}\rightarrow\mathscr{O}_{D^{\prime}_{\operatorname{red}}}\rightarrow
0$ yields an exact sequence

$H^{1}(Z^{\prime},\mathscr{O}_{Z^{\prime}})\longrightarrow
H^{1}(D^{\prime}_{\operatorname{red}},\mathscr{O}_{D^{\prime}_{\operatorname{red}}})\longrightarrow
H^{2}(Z^{\prime},\mathscr{O}_{Z^{\prime}}(-D^{\prime}_{\operatorname{red}})).$

According to Theorem 4.2, the term on the left vanishes, whereas the term on
the right is Serre dual to
$H^{0}(Z^{\prime},\mathscr{O}_{Z^{\prime}}(-D^{\prime}_{\operatorname{red}}))$,
which vanishes as well. Thus
$h^{1}(\mathscr{O}_{D^{\prime}_{\operatorname{red}}})=0$, and it follows that
$D^{\prime}_{\operatorname{red}}=\mathbb{P}^{1}$.

The Adjunction Formula gives
$\deg(\omega_{D^{\prime}})=(D^{\prime}-D^{\prime})\cdot D^{\prime}=0$, and
this implies $\chi(\mathscr{O}_{D^{\prime}})=0$. The ideal sheaf $\mathscr{M}$
for the inclusion $D^{\prime}_{\operatorname{red}}\subset D$ has
$\chi(\mathscr{M})=\chi(\mathscr{O}_{D^{\prime}})-\chi(\mathscr{O}_{D^{\prime}_{\operatorname{red}}})=-1$.
Furthermore, it is torsion-free and of rank one as sheaf on
$D^{\prime}_{\operatorname{red}}=\mathbb{P}^{1}$, thus
$\mathscr{M}=\mathscr{O}_{\mathbb{P}^{1}}(-2)$. In turn,
$D^{\prime}=\nu^{-1}(D)$ is a ribbon on $\mathbb{P}^{1}$ with respect to the
dualizing sheaf $\omega_{\mathbb{P}^{1}}=\mathscr{O}_{\mathbb{P}^{1}}(-2)$.
The morphism $\nu\colon Z^{\prime}\rightarrow Z$ induces the map (4), which
must be injective. It is bijective, because both sides have the same Euler
characteristic. ∎

Note that a similar situation already occurred in the study of Beauville’s
Kummer varieties in characteristic two ([Sch09, Proposition 7.3]). We now
clarify the flatness properties of the normal del Pezzo surface over the
normal Enriques surface:

######  Corollary 5.4.

The double covering $\nu\colon Z^{\prime}\rightarrow Z$ is flat precisely over
the complement of the $E_{8}$-singularity $a\in Z$.

###### Proof.

According to [Har77, Chapter III, Theorem 9.9], our finite morphism is flat at
all points $z\in Z$ where the fiber $\nu^{-1}(z)$ has length two. Since
$Z^{\prime}$ is Cohen–Macaulay, this automatically holds when the local ring
$\mathscr{O}_{Z,z}$ is regular ([EGA4-II, proposition 6.1.5]), that is, for
$z\neq a,b$. In light of our description of the induced map $\nu\colon
D^{\prime}_{\operatorname{red}}\rightarrow D_{\operatorname{red}}$ as a
composition of relative Frobenius with the normalization, this fiber over the
$E_{8}$-singularity $a\in Z$ has length four, whereas the fiber of the
$A_{1}$-singularity $b\in Z$ has length two. ∎

It remains to determine the singularities on $Z^{\prime}$, and then to compute
the group $\operatorname{APic}(Z^{\prime})$. Recall that the normal Enriques
surface $Z$ contains a rational double point $b\in Z$ of type $A_{1}$. Let
$b^{\prime}\in Z^{\prime}$ be the corresponding point on the normal del Pezzo
surface. Around this point, the double covering $\nu\colon
Z^{\prime}\rightarrow Z$ is flat, so the local ring
$\mathscr{O}_{Z^{\prime},b^{\prime}}$ must be singular. Moreover, all
singularities must be rational double points, according to Theorem 4.2. We now
make a preliminary observation:

######  Lemma 5.5.

The point $b^{\prime}\in Z^{\prime}$ is the only singularity on the normal del
Pezzo surface $Z^{\prime}$ that lies on the preimage $\nu^{-1}(D)$ of the
conductrix. If the Enriques surface $S$ is supersingular, we actually have
$\operatorname{Sing}(Z^{\prime})=\\{b^{\prime}\\}$.

###### Proof.

The reduction $D_{0}\subset Z$ of the conductrix is Cartier away from the
singularity $b\in Z$, because the other singularity is factorial. In turn, its
preimage $\nu^{-1}(D_{0})\subset Z^{\prime}$ is Cartier away from
$b^{\prime}\in Z^{\prime}$. According to Proposition 5.3, this preimage is
regular, hence the local rings $\mathscr{O}_{Z^{\prime},x}$ are regular when
$x\in Z^{\prime}$ maps to $D\smallsetminus\\{b\\}$.

For the second statement, consider the unique genus-one fibration
$\varphi\colon S\rightarrow\mathbb{P}^{1}$, and the normalization
$S^{\prime}\rightarrow S$ of the K3-like covering. The arguments from [Sch17,
Proposition 8.1] show that the Stein factorization of the composite map
$S^{\prime}\rightarrow\mathbb{P}^{1}$ is given by the relative Frobenius map
$F\colon\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. In particular, $S^{\prime}$
is regular over the relative smooth locus
$\operatorname{Reg}(S/\mathbb{P}^{1})$. Now suppose that $S$ is supersingular.
Then there is only one multiple fiber, whose reduction is
$C_{0}+\ldots+C_{8}$, with Kodaira symbol $\text{\rm II}^{*}$. All other
geometric fibers are rational cuspidal curves, and $C_{9}\subset S$ is the
curve of cusps. Since $C_{1}+\ldots+C_{8}$ and $C_{9}$ are contracted by
$f\colon S\rightarrow Z$, we see that the singular locus of $Z^{\prime}$ lies
over $D_{0}=f(C_{0})$. ∎

We now can unravel the picture completely:

######  Proposition 5.6.

The rational double point $\mathscr{O}_{Z^{\prime},b^{\prime}}$ has type
$D_{5}$, and this is the only singularity on the normal del Pezzo surface
$Z^{\prime}$.

###### Proof.

Let $r\colon X\rightarrow Z^{\prime}$ be the minimal resolution of
singularities. Then $X$ is a weak del Pezzo surface of degree
$K_{X}^{2}=K_{Z^{\prime}}^{2}=4$. It is obtained from $\mathbb{P}^{2}$ by
blowing-up $5=9-4$ points. The Picard number is $\rho(X)=6=1+5$, so the
morphism $r\colon X\rightarrow Z^{\prime}$ contracts five $(-2)$-curves
$E_{1},\ldots,E_{5}\subset X$. The possible configurations of rational double
points on normal del Pezzo surfaces of degree four were classified by
Dolgachev [Dol12, Section 8.3.6]. There are only two possibilities with five
exceptional divisors, namely $D_{5}$ or the configuration $A_{3}+A_{1}+A_{1}$.

Seeking a contradiction, we assume that $\operatorname{Sing}(Z^{\prime})$ is
given by $A_{3}+A_{1}+A_{1}$. In light of Lemma 5.5, the Enriques surface $S$
must be classical. The unique genus-one fibration $\varphi\colon
S\rightarrow\mathbb{P}^{1}$ thus has two multiple fibers. Without restriction,
these occur over the points $0,\infty\in\mathbb{P}^{1}$, with respective
Kodaira symbols $\text{\rm II}^{*}$ and II. Write $C_{\infty}$ for the reduced
part of the multiple fiber $\varphi^{-1}(\infty)$, and
$D_{\infty}=f(C_{\infty})$ for its image on the normal Enrique surface $Z$.
This is the rational cuspidal curve with self-intersection
$D_{\infty}^{2}=1/2$. Its preimage on the canonical covering
$\tilde{Z}\rightarrow Z$, which coincides with the canonical covering for the
inclusion $\alpha_{2}\subset\operatorname{Pic}_{D_{\infty}/k}$, must be a
ribbon on the projective line, according to [Sch17, Lemma 4.2]. In turn, the
preimage on the normal del Pezzo surface is of the form
$\nu^{-1}(D_{\infty})=2\Theta$, where $\Theta=\mathbb{P}^{1}$ and rational
selfintersection ${}^{2}=1/4$.

Moreover, the Weil divisor $\nu^{-1}(D_{0})=\mathbb{P}^{1}$ is linearly
equivalent to $\nu^{-1}(D_{\infty})=2\Theta$. Since both pass through the
singular point $b^{\prime}\in Z^{\prime}$, the local class group
$\operatorname{Cl}(\mathscr{O}_{Z^{\prime},b^{\prime}})$ is not annihilated by
two, and we conclude that $b^{\prime}\in Z^{\prime}$ has type $A_{3}$ rather
than $A_{1}$. Furthermore, the germ ${}_{b^{\prime}}$ generates the local
class group. The five exceptional curves $E_{i}$ and the strict transform
${}^{*}\subset X$ of $\Theta\subset Z^{\prime}$ have normal crossings, because
the scheme is regular. After reordering, their dual graph takes this form:

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It follows that the rational pull-back in the sense of Mumford [Mum61] is
given by

$r^{*}(\Theta)={}^{*}+\frac{1}{4}(E_{1}+2E_{2}+3E_{3}+2E_{4}+2E_{5}),$

which yields $1/4={}^{2}=r^{*}(\Theta)^{2}=({}^{*}\cdot{}^{*})+3/4+2/4+2/4$.
Consequently $({}^{*}\cdot{}^{*})=-6/4$, contradicting that this
selfintersection number on the regular surface $X$ is an integer. ∎

######  Proposition 5.7.

The normal del Pezzo surface has $\operatorname{APic}(Z^{\prime})=\mathbb{Z}$,
and the subgroup $\operatorname{Pic}(Z)=\mathbb{Z}K_{Z^{\prime}}$ has index
four. In particular, the generator $\Theta\in\operatorname{APic}(Z^{\prime})$
has selfintersection ${}^{2}=1/4$.

###### Proof.

Let $r\colon X\rightarrow Z^{\prime}$ be the minimal resolution of
singularities, and $E_{1},\ldots,E_{5}\subset X$ the five exceptional curves
over the rational double point $b^{\prime}\in Z^{\prime}$ of type $D_{5}$, and
set $\operatorname{Div}_{E}(X)=\bigoplusop\displaylimits\mathbb{Z}E_{i}$.
Consider the commutative diagram

$\begin{CD}0@>{}>{}>\operatorname{Div}_{E}(X)@>{}>{}>\operatorname{Pic}(Z^{\prime})\oplus\operatorname{Div}_{E}(X)@>{\operatorname{pr}}>{}>\operatorname{Pic}(Z^{\prime})@>{}>{}>0\\\
@V{}V{{\operatorname{id}}}V@V{}V{(r^{*},{\rm\text{can}})}V@V{}V{{\rm\text{can}}}V\\\
0@>{}>{}>\operatorname{Div}_{E}(X)@>{}>{}>\operatorname{Pic}(X)@>{}>{r_{*}}>\operatorname{APic}(Z^{\prime})@>{}>{}>0\\\
\end{CD}$

The vertical maps are injective, and the one in the middle has index four,
because the intersection forms on $\operatorname{Pic}(X)$ and
$\operatorname{Pic}(Z^{\prime})\oplus\operatorname{Div}_{E}(X)$ have
discriminants $\delta=1$ and $\delta=4^{2}$, respectively. Applying the Snake
Lemma, we see that the cokernel for
$\operatorname{Pic}(Z^{\prime})\subset\operatorname{APic}(Z^{\prime})$ has
order four. It also sits inside
$\operatorname{Cl}(\mathscr{O}_{Z^{\prime},b^{\prime}})$, which is cyclic of
order four.

Thus the group $\operatorname{APic}(Z^{\prime})$ is an extension of
$\mathbb{Z}/4\mathbb{Z}$ by $\mathbb{Z}$. It remains to check that it is
torsion-free, in other words the inclusion of $L=\operatorname{Div}_{E}(X)$
inside $\operatorname{Pic}(X)$ is primitive. Consider the dual lattice
$L^{*}=\operatorname{Hom}(L,\mathbb{Z})$ and the resulting discriminant group
$L^{*}/L$, which comes with a perfect $\mathbb{Q}/\mathbb{Z}$-valued pairing.
The over-lattices $L\subset L^{\prime}$ correspond to totally isotropic
subgroup $T\subset L^{*}/L$, via $T=L^{\prime}/L$, according to [Nik80,
Section 4]. In our case, the discriminant group
$L^{*}/L=\mathbb{Z}/4\mathbb{Z}$ is cyclic, so there are no such subgroups. In
turn, $L\subset\operatorname{Pic}(X)$ is primitive, thus
$\operatorname{APic}(Z^{\prime})=\mathbb{Z}$. The generator satisfies
$4\Theta=K_{Z^{\prime}}$, and the intersection number ${}^{2}=1/4$ follows. ∎

By Artin’s classification [Art77] of rational double points in positive
characteristics, there are actually two isomorphism classes of type $D_{5}$,
which are denoted by $D_{5}^{0}$ and $D_{5}^{1}$. The former is simply-
connected, the latter not. Since $\nu\colon Z^{\prime}\rightarrow Z$ is a
finite universal homeomorphism and rational double points of type $A_{1}$ are
simply-connected, we see that our singularity
$\mathscr{O}_{Z^{\prime},b^{\prime}}$ is also simply-connected, whence
formally given by the equation $z^{2}+y^{2}z+x^{2}y=0$.

## 6\. Cones and Fano varieties

In this section we collect some facts on cones, which complement the
discussions by Grothendieck [EGA2, §8] and Kollár [Kol13, Section 3.1]. They
will be used to construct Fano threefolds with unusual torsion in the next
section.

Let $k$ be a ground field of arbitrary characteristic $p\geq 0$ and $B$ be a
proper connected scheme. Suppose that $\mathscr{E}$ is a locally free sheaf of
rank two, and consider its projectivization

$P=\mathbb{P}(\mathscr{E})=\operatorname{Proj}(\operatorname{Sym}^{\bullet}\mathscr{E}).$

Let $f\colon P\rightarrow B$ the structure morphism, whose fibers are copies
of the projective line $\mathbb{P}^{1}$. From the Formal Function Theorem, one
gets a split exact sequence

(5) $0\longrightarrow\operatorname{Pic}(B)\stackrel{{\scriptstyle
f^{*}}}{{\longrightarrow}}\operatorname{Pic}(P)\stackrel{{\scriptstyle\deg}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$

where the degree is taken fiber-wise, and the splitting is given by the
tautological sheaf $\mathscr{O}_{P}(1)$. The sections $D\subset P$ correspond
to invertible quotients $\mathscr{L}=\mathscr{E}/\mathscr{N}$, via
$D=\mathbb{P}(\mathscr{L})$ and $\mathscr{L}=f_{*}(\mathscr{O}_{P}(1)|D)$.
Each section is an effective Cartier divisor. Moreover, to simplify the
notation, for any line bundle $\mathscr{L}$ on $B$ and any section $D$ in $P$,
we denote the restriction $f^{*}(\mathscr{L})|D$ by the same symbol
$\mathscr{L}$.

######  Lemma 6.1.

With the above notation, $\mathscr{O}_{P}(D)\simeq
f^{*}(\mathscr{N}^{\otimes-1})\otimes\mathscr{O}_{P}(1)$. In particular,
$\mathscr{O}_{D}(D)=\mathscr{N}^{\otimes-1}\otimes\mathscr{L}$.

###### Proof.

Set $\mathscr{F}=\mathscr{O}_{P}(1)\otimes\mathscr{O}_{P}(-D)$. Tensoring
$0\rightarrow\mathscr{O}_{P}(-D)\rightarrow\mathscr{O}_{P}\rightarrow\mathscr{O}_{D}\rightarrow
0$ with $\mathscr{O}_{P}(1)$ and taking direct images gives $0\rightarrow
f_{*}(\mathscr{F})\rightarrow\mathscr{E}\rightarrow\mathscr{L}\rightarrow 0$.
The map on the right is the quotient map $\mathscr{E}\rightarrow\mathscr{L}$
defining the section $D\subset P$, hence $f_{*}(\mathscr{F})=\mathscr{N}$. The
invertible sheaf $\mathscr{F}$ has degree zero on each fiber of $f\colon
P\rightarrow P$. Hence $\mathscr{F}=f^{*}(\mathscr{N}^{\prime})$ for some
invertible sheaf $\mathscr{N}^{\prime}$ on $B$, by the exact sequence (5).
Finally, the Projection Formula gives
$\mathscr{N}^{\prime}=f_{*}f^{*}(\mathscr{N}^{\prime})=\mathscr{N}$. ∎

From now on, we assume that $\mathscr{E}=\mathscr{O}_{B}\oplus\mathscr{L}$ for
some ample invertible sheaf $\mathscr{L}$. Then $P=\mathbb{P}(\mathscr{E})$
contains two canonical sections, namely $A=\mathbb{P}(\mathscr{O}_{X})$ and
$E=\mathbb{P}(\mathscr{L})$, coming from the two projections
$\mathscr{E}\rightarrow\mathscr{O}_{X}$ and
$\mathscr{E}\rightarrow\mathscr{L}$, and we have

(6)
$\mathscr{O}_{A}(A)=\mathscr{L}\quad\text{and}\quad\mathscr{O}_{E}(E)=\mathscr{L}^{\otimes-1}.$

Therefore we say that $E\subset S$ is the _negative section_ and $A\subset P$
is the _positive section_. Clearly, the stable base locus for the invertible
sheaf $\mathscr{O}_{P}(A)$ is contained in $A$, and the restriction
$\mathscr{O}_{A}(A)$ is ample. According to Fujita’s result (see [Fuj83],
compare also [Ein00]), the invertible sheaf $\mathscr{O}_{P}(A)$ is semiample,
and we get a contraction

$r\colon P\longrightarrow
X=\operatorname{Proj}\bigoplusop\displaylimits_{i\geq
0}H^{0}(P,\mathscr{O}_{P}(iA))$

to some projective scheme $X$, with
$\mathscr{O}_{P}(nA)=r^{*}\mathscr{O}_{X}(n)$ for some $n\geq 0$ sufficiently
large, and $\mathscr{O}_{X}=r_{*}(\mathscr{O}_{P})$. Clearly, the connected
closed set $E\subset P$ is mapped to a closed point $x_{0}\in X$. The
_exceptional set_ for the morphism $r\colon P\rightarrow X$ is defined as the
closed set $\operatorname{Exc}(P/X)=\operatorname{Supp}({}^{1}_{P/X})$.

######  Lemma 6.2.

The exceptional set $\operatorname{Exc}(P/X)$ for the morphism $r\colon
P\rightarrow X$ coincides with the negative section
$E=\mathbb{P}(\mathscr{L})$.

###### Proof.

The exceptional set is the union of all irreducible curves $C\subset P$ that
are disjoint from the positive section $A=\mathbb{P}(\mathscr{O}_{B})$. We
have to check that each such $C$ is contained in the negative section
$E=\mathbb{P}(\mathscr{L})$. For this, we may pass to the base-change
$C^{\prime}\times_{B}P$, where $C^{\prime}\rightarrow C$ is the normalization,
and assume that $B$ is an irreducible regular curve and $C\subset P$ is a
section. Then $P$ is a regular surface, and both $E,C\subset P$ are mapped to
points in $X$. Recall that $E\subset P$ corresponds to the invertible quotient
$\mathscr{L}=\mathscr{E}/\mathscr{N}$, where $\mathscr{N}=\mathscr{O}_{B}$. By
the Hodge Index Theorem, we have
$\deg(\mathscr{L})-\deg(\mathscr{N})=E^{2}<0$, and the same holds for
$C\subset P$. Since such an invertible quotient
$\mathscr{L}=\mathscr{E}/\mathscr{N}$ is unique, $C=E$ follows. ∎

In turn, the morphism $r\colon P\smallsetminus E\rightarrow
X\smallsetminus\\{x_{0}\\}$ is an isomorphism. One also says that $X$ is the
_projective cone_ on $B$ with respect to the ample sheaf $\mathscr{L}$, with
vertex $x_{0}\in X$. By abuse of notation, we regard the positive section
$A=\mathbb{P}(\mathscr{O}_{B})$ of the $\mathbb{P}^{1}$-bundles
$P=\mathbb{P}(\mathscr{E})$ also as an ample Cartier divisor $A\subset X$.

######  Proposition 6.3.

The projective scheme $X$ has Picard scheme
$\operatorname{Pic}_{X/k}=\mathbb{Z}$, and this is generated by the ample
sheaf $\mathscr{O}_{X}(A)$.

###### Proof.

Let $\mathscr{F}$ be an invertible sheaf on $X$. Its preimage takes the form
$r^{*}(\mathscr{F})=f^{*}(\mathscr{N})\otimes\mathscr{O}_{P}(dA)$ for some
unique $\mathscr{N}\in\operatorname{Pic}(B)$ and $d\in\mathbb{Z}$. Since
$r^{*}(\mathscr{F})|E=\mathscr{O}_{E}$ and
$\mathscr{O}_{E}(dA)=\mathscr{O}_{E}$, we have $\mathscr{N}=\mathscr{O}_{D}$,
and hence $\operatorname{Pic}(X)=\mathbb{Z}$, generated by
$\mathscr{O}_{P}(A)$. The statement on the Picard scheme follows likewise, by
working with the infinitesimal extension $B\otimes_{k}k[\epsilon]$. ∎

######  Proposition 6.4.

If $B$ is Gorenstein, then $P$ is Gorenstein, and we have the Canonical Bundle
Formula

$\omega_{P}=\mathscr{O}_{P}(-2E)\otimes
f^{*}(\omega_{B}\otimes\mathscr{L}^{\otimes-1}).$

###### Proof.

The scheme $P$ must be Gorenstein by [WITO69]. In light of the exact sequence
(5), the dualizing sheaf has the form $\omega_{P}=\mathscr{O}_{P}(-2E)\otimes