# Equivariant birational types and derived categories

Christian Böhning  Mathematics Institute
Zeeman Building
University of Warwick
Coventry CV4 7AL
UK<EMAIL_ADDRESS>,  Hans-Christian Graf von Bothmer  Fachbereich
Mathematik
Bundesstraße 55
20146 Hamburg
Germany<EMAIL_ADDRESS>and  Yuri Tschinkel
Courant Institute
New York University
New York, NY 10012
USA<EMAIL_ADDRESS>Simons Foundation
160 Fifth Avenue
New York, NY 10010
USA

###### Abstract.

We investigate equivariant birational geometry of rational surfaces and
threefolds from the perspective of derived categories.

##  1\. Introduction

Let $X$ be a smooth projective variety over an algebraically closed field $k$,
of characteristic zero. Assume that $X$ is equipped with a regular,
generically free, action of a finite group $G$. A major topic in birational
geometry is to understand equivariant birational types, e.g., to decide
whether or not $X$ is

  * •

(projectively) linearizable, i.e., equivariantly birational to projective
space, with a (projectively) linear action of $G$, or

  * •

stably (projectively) linearizable, i.e., (projectively) linearizable after
taking a product with $\mathbb{P}^{m}$, for some $m$, with trivial action on
the second factor.

One of the motivations is the analogy of this theory with birational geometry
over nonclosed ground fields and, in particular, with the central problem of
(stable) rationality over such fields, where the role of $G$ is taken by the
absolute Galois group of the ground field, acting on geometric objects.

Various tools have been developed to distinguish equivariant birational types,
e.g., cohomology, derived categories, and more recently, equivariant Burnside
groups (see [HT23]). In this note, we investigate the interactions between
different perspectives on the (stable) linearizability problem. We focus on
low-dimensional examples, in particular, Del Pezzo surfaces, rational Fano
threefolds and fourfolds. We explore the compatibility of group actions with
standard (stable) rationality constructions and conjectures, and produce new
examples of stably linearizable but nonlinearizable actions.

In detail, in Section 2, we discuss basic notions of equivariant birational
geometry, classical invariants of $G$-actions on varieties, as well as the
recently developed Burnside formalism [KT22b]. We present applications of a
multilinear algebra construction, Proposition 4, to exhibit new examples of
nonlinearizable but stably linearizable actions, e.g., we show in Example 7
that, for $G=\mathfrak{A}_{5}$, the $G$-birationally rigid, and thus not
linearizable, quintic Del Pezzo threefold is stably linearizable.

In Section 3, we study exceptional sequences in derived categories, in
presence of $G$-actions, and their connections with classical invariants. In
Section 4, we prove

######  Theorem.

A smooth projective rational $G$-surface that is linearizable has a full
$G$-equivariant exceptional sequence.

The proof relies on the classification of finite subgroups in the Cremona
group of [DI09], and subsequent developments in equivariant geometry of
rational surfaces. Over nonclosed fields the situation was investigated in
[AB18] and [BD21], in particular, we view this theorem as an analog of [AB18,
Corollary 1]. However, we also give an example, in the equivariant context,
where the analog of [AB18, Theorem 1] fails.

In Section 5, we turn to Fano threefolds. For quintic Del Pezzo threefolds,
with give examples of nonlinearizable actions of finite groups $G$ with
derived categories admitting full exceptional sequences of $G$-linearized
objects. This disproves the equivariant analog of the well-known conjecture
that a smooth projective variety with a full exceptional sequence over the
ground field should be rational. The corresponding $G$-actions are stably
linearizable.

For Fano threefolds of genus 7, we show that there are nonlinearizable actions
in presence of $G$-invariant semiorthogonal decompositions, with pieces
equivalent, as $G$-categories, to derived categories of $G$-varieties of
codimension $\geq 2$.

Acknowledgments: For the purpose of open access, the first author has applied
a Creative Commons Attribution (CC-BY) licence to any Author Accepted
Manuscript version arising from this submission. The third author was
partially supported by NSF grant 2301983.

##  2\. Equivariant geometry

### Terminology

Throughout, $G$ is a finite group. We consider generically free regular
actions of $G$ on irreducible algebraic varieties over $k$, an algebraically
closed field of characteristic zero, and refer to such varieties as
$G$-varieties. We write

$X\sim_{G}Y$

if the $G$-varieties $X,Y$ are equivariantly birational, and introduce
subcategories of the category of $G$-varieties:

  * •

$\mathfrak{L}\mathrm{in}$ \- $G$-linearizable,

  * •

$\mathfrak{P}\mathfrak{L}\mathrm{in}$ \- projectively $G$-linearizable,

  * •

$\mathfrak{S}\mathfrak{L}\mathrm{in}$ \- stably $G$-linearizable.

A basic result in $G$-birational geometry is equivariant resolution of
singularities and weak factorization: $G$-birational varieties are related via
blowups and blowdowns, with centers in smooth $G$-stable subvarieties.
Moreover, after a sequence of such blowups, one can reach a standard model
$\tilde{X}\to X$ such that on $\tilde{X}$ all stabilizers are abelian, and
$G$-orbits of divisors with nontrivial stabilizers are smooth, i.e., for every
such $D$ and $g\in G$, the intersection $(D\cdot g)\cap D$ is either all of
$D$ or empty.

### Classical invariants

The $G$-action on $X$ induces actions on cohomology groups, and in particular
on the Picard group $\operatorname{Pic}(X)$. There is an exact sequence, see,
e.g., [KT22a, §3]

$\textstyle{\mathrm{Pic}(X,G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Pic}(X)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\!\\!\delta_{2}}$$\textstyle{\mathrm{H}^{2}(G,k^{\times})}$$\textstyle{\mathrm{Br}([X/G])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(G,\mathrm{Pic}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,\,\delta_{3}}$$\textstyle{\mathrm{H}^{3}(G,k^{\times}),}$

where:

  * •

$\mathrm{Pic}(X,G)$ is the group of isomorphism classes of $G$-linearized line
bundles, and $\mathrm{Pic}(X)^{G}$ the group of $G$-invariant line bundles

  * •

$[X/G]$ is the quotient stack, $\mathrm{Br}([X/G])$ its Brauer group, and

Both $\delta_{2}$ and $\delta_{3}$ are zero when there are $G$-fixed points;
these give sections of the map $[X/G]\to BG$. Other frequently studied
obstructions to (stable) linearizabilty are:

  * •

$\mathrm{Am}(X,H)$, the Amitsur invariant, i.e., the image of

$\delta_{2}:\operatorname{Pic}(X)^{H}\to\mathrm{H}^{2}(H,k^{\times}),\quad
H\subseteq G,$

  * •

(H1):
$\mathrm{H}^{1}(H,\operatorname{Pic}(X))=\mathrm{H}^{1}(H,\operatorname{Pic}(X)^{\vee})=0$,
$H\subseteq G$,

  * •

(SP): $\operatorname{Pic}(X)$ is a stable $G$-permutation module.

If $X\in\mathfrak{P}\mathfrak{L}\mathrm{in}$ or $\mathfrak{L}\mathrm{in}$,
then (H1) and (SP) hold; when $X\in\mathfrak{L}\mathrm{in}$ then the Amitsur
invariant vanishes.

### Burnside formalism

Let $G$ be a finite group, acting on $X$, a standard model for the action. On
such a model one computes the class of the $G$-action in the Burnside group:

(2.1) $[X\righttoleftarrow
G]=\sum_{F,H}(H,Y\mathrel{\reflectbox{$\righttoleftarrow$}}k(F),\beta)\in\operatorname{Burn}_{n}(G),\,n=\dim(X),$

as a sum of symbols, recording ($G$-orbits of) irreducible subvarieties
$F\subset X$ with nontrivial generic stabilizer $H$, together with the induced
action of a subgroup $Y\subseteq Z_{G}(H)/H$ on the function field $k(F)$ and
the collection $\beta$ of weights of $H$ in the normal bundle to $F$ (all
defined up to conjugation in $G$). In particular, this sum contains the
trivial summand

$(1,G\mathrel{\reflectbox{$\righttoleftarrow$}}k(X),()),$

The symbols are subject to explicit relations so that the class (2.1) is an
equivariant birational invariant (see [KT22b], [HKT21] for definitions and
examples). The trivial summand does not participate in relations; we say that
the $G$-action on $X$ has trivial Burnside class if

$[X\righttoleftarrow
G]=(1,G\mathrel{\reflectbox{$\righttoleftarrow$}}k(X),())$

in $\operatorname{Burn}_{n}(G)$. Incompressible divisorial symbols (modulo
conjugation relation), in the terminology of [KT22c, Definition 3.3],
generate, freely, a direct summand of $\operatorname{Burn}_{n}(G)$; in many
situations, it suffices to compare their contribution to $[X\righttoleftarrow
G]$ to distinguish $G$-actions up to equivariant birationality, see [TYZ23,
Section 3.6].

The paper [KT22c] provides an algorithm for the computation of
$[\mathbb{P}(V)\righttoleftarrow G]$ for linear and projective linear actions
of a finite group $G$; this algorithm has been implemented in Magma, see
[TYZ23]. While the formalism and the computations can be involved,
incompressible divisorial symbols allow to quickly show nonlinearizability of
some actions. Indirectly, they also lead to constraints on possible actions:

######  Example 1.

Let $X\subset\mathbb{P}^{n}$ be a prime (smooth) Fano threefold of index 1, in
its anticanonical embedding. Let $\sigma\in\operatorname{PGL}_{n+1}$ be an
involution preserving a hyperplane. Does $\sigma$ preserve $X$?

If so, we would have $X^{\sigma}=S$, a surface, yielding a symbol

$(\langle\sigma\rangle,1\mathrel{\reflectbox{$\righttoleftarrow$}}k(S),(1))\in\operatorname{Burn}_{3}(C_{2}).$

Generically, $S$ would be a K3 surface, the symbol incompressible, and thus
the action not linearizable.

On the other hand, consider smooth Fano threefolds $X=X_{22}$ of genus 12. We
know that $G$-actions on $X$ are linearizable, if there is a (sufficiently
general) fixed point; in the arithmetic setup this is discussed in [KP23,
Theorem 5.17].

This tension can be reconciled, in fact, $X$ cannot carry such involutions. We
sketch an argument: According to Mukai, cf. [Sch01], $X$ can be constructed as
follows: start with a $7$-dimensional vector space $V$, a $3$-dimensional
vector space $U$, and a linear map $\eta\colon\wedge^{2}(V)\to U^{*}$. Dually,
this arises from a linear map $\eta^{*}\colon U\to\wedge^{2}(V^{*})$. Consider
the Grassmannian $\mathrm{Gr}(3,V)\subset\mathbb{P}(\wedge^{3}(V))$, and for
the universal subbundle $\mathcal{U}$ notice that
$\mathrm{H}^{0}(\mathrm{Gr}(3,V),\mathcal{U}^{*})=\wedge^{2}(V^{*})$. The
zeros of the sections in $U$ on $\mathrm{Gr}(3,V)$ yield $X$. Equivalently, we
have a linear map

$\wedge^{3}(V)\to V\otimes U^{*},$

induced by wedging elements in $\wedge^{3}(V)$ with elements in $U$, and the
kernel $K$ is a $14$-dimensional subspace of $\wedge^{3}(V)$ such that

$X=\mathrm{Gr}(3,V)\cap\mathbb{P}(K).$

We can view $\eta^{*}$ as an element in $\wedge^{2}(V^{*})\otimes U^{*}$,
i.e., a skew-symmetric $7\times 7$-matrix with entries in $U^{*}$. The
$6\times 6$ Pfaffians of this matrix define an Artinian Gorenstein module of
codimension $3$ over $k[U^{*}]$, with dual socle generator a quartic $F$, see
[Sch01, Theorem 2.6 and its proof]. Conversely, the datum of this quartic or
Artinian Gorenstein module allows to reconstruct the skew-symmetric $7\times
7$-matrix with entries in $U^{*}$ by considering the middle map in the
Buchsbaum-Eisenbud resolution of the module.

By [Sch01, Theorem 6.1], the Scorza quartic $S_{F}$ covariantly associated to
$F$ is isomorphic to the Hilbert scheme of lines $F_{1}(X)$ in $X$, and by
[DM22, Appendix, Claim A.1.1., (by Prokhorov)], the automorphisms of $X$ embed
injectively into those of $F_{1}(X)$.

Now assume that $G=C_{2}$ is acting faithfully on $X$, through the
$G$-representations $U$ and $V$ and an equivariant map $\eta$, as described
above. Then $G$ embeds into the automorphisms of the quartic $C$, and acts
faithfully on $U$ with weights $(1,1,0)$ or $(0,0,1)$. Then [Sch01, Theorem
2.6] implies that $V$ can be recovered as the kernel of the linear map

$\mathrm{Sym}^{3}(U)\to U$

given by contracting with the equation of the $C_{2}$-invariant quartic $F$.
This map is equivariant (possibly after tensoring the target $U$ with a sign).
After that, we recover $K$ as the kernel of the $G$-equivariant map

$\wedge^{3}(V)\to V\otimes U^{*},$

above. Working through all sign combinations, one verifies that $G$ cannot act
on $K$ with all weights but one equal to each other.

######  Example 2.

Let $X\subset\mathbb{P}^{5}$ be a smooth cubic fourfold, with an action of
$G=C_{m}$, with weights $(0,0,0,0,0,b)$; note that only $m=2,3$ are possible,
under this assumption. Assume that the divisor $D\subset X$, given by the
vanishing of the last coordinate, is smooth. Then
$X\notin\mathfrak{P}\mathfrak{L}\mathrm{in}$.

Indeed, the corresponding symbol
$(C_{m},1\mathrel{\reflectbox{$\righttoleftarrow$}}k(D),(b))$ is
incompressible, since $D$ is not birational to $S\times\mathbb{P}^{1}$, for
any surface $S$, and does not appear in classes of linear actions, (see
[TYZ23, Corollary 6.1]).

For $m=2$, [Mar22, Theorem 1.2(2)] shows that a very general $X$ carrying such
an involution does not have an associated K3 surface and is expected to be
nonrational; and, in particular, the action would not be linearizable. The
same argument applies for $m=3$.

### Pfaffians and Grassmannians

In [BvBT23, Section 7], we have used a construction from multilinear algebra,
the Pfaffian construction, to exhibit nonlinearizable but stably linearizable
actions of finite groups on rational varieties, e.g., rational cubic
fourfolds. The starting point is a Pfaffian variety

$X:=\mathrm{Pf}(W)\cap\mathbb{P}(L),$

where $V$ is a vector space of dimension $n=2m$ and $L\subset\wedge^{2}(W)$ a
linear subspace of dimension $n$. Then there is a diagram

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{P}(\mathcal{K}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X}$$\textstyle{\mathbb{P}(W^{*})}$

where $\mathcal{K}_{X}$ is a vector bundle of rank 2 and $q$ is birational. In
presence of group actions, choosing a $G$-representation $W$ and a
subrepresentation $L$, one obtains, under suitable genericity assumptions, an
equivariant birationality:

$X\times\mathbb{P}^{1}\sim_{G}\mathbb{P}(W^{*}),$

with trivial action on the second factor.

######  Example 3.

Let $G=C_{5}\rtimes\mathfrak{D}_{15}\rtimes C_{3}$, GapID(450,24). It acts
generically freely on the singular (toric) cubic fourfold
$X\subset\mathbb{P}^{5}$ with equation

$x_{1}x_{3}x_{5}+x_{2}x_{4}x_{6}=0,$

a degeneration of the Pfaffian cubic considered in [BvBT23, Example 14]. The
$G$-action on $X$ is not linearizable, as $G$ does not have faithful
representations of dimension $<6$. The Pfaffian construction applies: by
[BvBT23, Corollary 13], the $G$-action on $X\times\mathbb{P}^{1}$, with
trivial action on the second factor, is linearizable.

Here, we present another such construction, applicable to subvarieties of
Grassmannians. Linear sections of Grassmannians admit tautological stable
rationality constructions, that we now describe: Let $W$ be an $n$-dimensional
vector space over $k$ and $\operatorname{Gr}(2,W)$ the Grassmannian of planes
in $W$. Let $V\subset\wedge^{2}(W)$ be a linear subspace of codimension $r$.
Put

$X:=\operatorname{Gr}(2,W)\cap\mathbb{P}(V)$

and consider the diagram

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{P}(\mathcal{U}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X}$$\textstyle{\mathbb{P}(W),}$

where $\mathcal{U}_{X}$ is the restriction of the universal vector bundle over
$\operatorname{Gr}(2,W)$ to $X$. This yields a stable rationality
construction, as both $p$ and $q$ are vector bundles, in the indicated range
of dimensions.

######  Proposition 4.

Let $k$ be an algebraically closed field of characteristic zero and $G$ a
finite group. Let $W$ be an $n$-dimensional representation of $G$ over $k$ and
$V\subset\wedge^{2}(W)$ a subrepresentation of codimension $r\leq n-2$ such
that the $G$-actions on $\mathbb{P}(W)$ and $\mathbb{P}(V)$ are generically
free. Assume that

  * ($\ast$)

$X$ is irreducible of dimension $\dim(\operatorname{Gr}(2,n))-r=2(n-2)-r$.

Then

$X\times\mathbb{P}^{1}\sim_{G}\mathbb{P}(W)\times\mathbb{P}^{n-2-r},$

with trivial actions on the second factors.

###### Proof.

By the No-name Lemma,
$X\times\mathbb{P}^{1}\sim_{G}\mathbb{P}(\mathcal{U}_{X})$. Note that each
fiber of $q$ is nonempty: indeed, the fiber over $[w]\in\mathbb{P}(W)$ is
$\mathbb{P}(w\wedge W)\cap\mathbb{P}(V)$, which has dimension $\geq n-2-r\geq
0$, the last inequality by the assumption $r\leq n-2$. By assumption $(\ast)$,
it follows that for generic $w\in W$, one has

$\dim(\mathbb{P}(w\wedge W)\cap\mathbb{P}(V))=n-2-r.$

Thus $\mathbb{P}(\mathcal{U}_{X})$, which is irreducible since $X$ is, is
generically the projectivization of a $G$-vector bundle over $\mathbb{P}(W)$
via $q$. Another application of the No-name Lemma yields the result. ∎

######  Remark 5.

If we drop assumption $(\ast)$, but keep assuming $r\leq n-2$, then the
construction of Proposition 4 still yields stable linearizability for the
unique component of $X$ such that the restriction of
$\mathbb{P}(\mathcal{U}_{X})$ to it dominates $\mathbb{P}(W)$. But proving
nonlinearizability of such a component of $X$ is usually difficult, unless we
assume a condition similar to $(\ast)$, a priori.

This construction works also over nonclosed fields. However, there one does
not gain new insights: by [Xu12, Theorem 2.2.1], if $r\leq n-2$ and $X$ is
smooth then $X$ is already rational over $k$. The proof uses the same diagram,
restricted to a codimension one linear subspace $\Pi$ in $\mathbb{P}(W)$,
exhibiting $X$ as birational to a vector bundle over $\Pi$, thus rational over
$k$. In presence of group actions, this can fail, e.g., if $W$ does not admit
a subrepresentation of codimension one! This yields many examples of
nonlinearizable but stably linearizable actions.

######  Example 6.

Let $G=\mathfrak{S}_{5}$, and $W:=W_{5}$ its 5-dimensional representation. We
have a decomposition

$\wedge^{2}(W)=W_{6}\oplus W_{4},$

as representations. When $V=W_{6}$ is the 6-dimensional subrepresentation,

$S:=\operatorname{Gr}(2,W)\cap\mathbb{P}(V)$

is the del Pezzo surface of degree 5. It is easy to see that the induced
$G$-action on $S$ is not linearizable, indeed, $\mathfrak{S}_{5}$ does not
admit a linear action on $\mathbb{P}^{2}$. Even the restriction to
$\mathfrak{A}_{5}\subset\mathfrak{S}_{5}$ is not linearizable, see, e.g.,
[CS16, Theorem 6.6.1].

Note that the assumptions of Proposition 4 are not fulfilled, we have $n=5$
and $r=4$, rather than $r\leq 3$. Nevertheless, by [Pro10, Proposition 4.7],
$S\times\mathbb{P}^{1}$ is $\mathfrak{S}_{5}$-equivariantly birational to the
Segre cubic threefold, with the action of the nonstandard
$\mathfrak{S}_{5}\subset\mathfrak{S}_{6}$, which is linearizable. An
alternative proof of stable linearizability of $S$, using the equivariant
torsor formalism, is in [HT23, Proposition 20].

######  Example 7.

We modify the previous example, considering $G=\mathfrak{A}_{5}$. Then

$\wedge^{2}(W)=W_{3}\oplus W_{3}^{\prime}\oplus W_{4},$

and we put $V:=W_{3}\oplus W_{4}$. Then

$X:=\operatorname{Gr}(2,W)\cap\mathbb{P}(V)$

is a smooth threefold [CS16, Lemma 7.1.1], the quintic Del Pezzo threefold.
One of the main results of [CS16] is that $X$ is $G$-birationally rigid.

Here, the construction of Proposition 4 applies, and we obtain

$X\times\mathbb{P}^{1}\sim_{G}\mathbb{P}(W).$

Thus, $X\notin\mathfrak{L}\mathrm{in}$ but
$X\in\mathfrak{S}\mathfrak{L}\mathrm{in}$.

To check the condition $(\ast)$ in Proposition 4 and the smoothness of $X$
(independently of [CS16]), one can proceed as follows: we view
$\mathfrak{A}_{5}$ as a subgroup of $\mathrm{PSL}_{2}$ and proceed in terms of
$\mathrm{PSL}_{2}$-representations, as in [CS16, Section 7]. Put
$W=\mathrm{Sym}^{4}(k^{2})$ and consider the decomposition

(2.2) $\wedge^{2}(W)=\mathrm{Sym}^{2}(k^{2})\oplus\mathrm{Sym}^{6}(k^{2}).$

Let

$\mathbf{H}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\mathbf{X}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\mathbf{Y}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$

be the standard basis of the Lie algebra of $\mathrm{SL}_{2}$ satisfying

$[\mathbf{H},\mathbf{X}]=2\mathbf{X},\>[\mathbf{H},\mathbf{Y}]=-2\mathbf{Y},\>[\mathbf{X},\mathbf{Y}]=\mathbf{H}.$

Let $w_{4}$ be a highest weight vector in $W$ (subscripts in the sequel
indicate the weight). Thus $\mathbf{X}(w_{4})=0$, and

$w_{4},\>w_{2}=\mathbf{Y}(w_{4}),\>w_{0}=\mathbf{Y}^{2}(w_{4}),\>w_{-2}=\mathbf{Y}^{3}(w_{4}),\>w_{-4}=\mathbf{Y}^{4}(w_{4})$

form a basis for $W$; using the commutation relations inductively gives

$\mathbf{X}(w_{4})=0,\\!\mathbf{X}(w_{2})=4w_{4},\\!\mathbf{X}(w_{0})=6w_{2},\\!\mathbf{X}(w_{-2})=6w_{0},\\!\mathbf{X}(w_{-4})=4w_{-2}.$

To find a highest weight vector in the subrepresentation
$\mathrm{Sym}^{2}(k^{2})$ of $\wedge^{2}(W)$ one is looking for a linear
combination of $w_{4}\wedge w_{-2}$ and $w_{2}\wedge w_{0}$ annihilated by
$\mathbf{X}$. These are thus multiples of

$x_{2}:=3w_{2}\wedge w_{0}-2w_{4}\wedge w_{-2},$

and applying $\mathbf{Y}$ and $\mathbf{Y}^{2}$ to $x_{2}$ we obtain a basis
for $\mathrm{Sym}^{2}(k^{2})$ as a submodule of $\wedge^{2}(W)$ as

$\displaystyle x_{0}:=\mathbf{Y}(x_{2})=\mathbf{Y}\left(3w_{2}\wedge
w_{0}-2w_{4}\wedge w_{-2}\right)$ $\displaystyle=3(\mathbf{Y}(w_{2})\wedge
w_{0}+w_{2}\wedge\mathbf{Y}(w_{0}))-2(\mathbf{Y}(w_{4})\wedge
w_{-2}+w_{4}\wedge\mathbf{Y}(w_{-2}))$ $\displaystyle=3(w_{0}\wedge
w_{0}+w_{2}\wedge w_{-2})-2(w_{2}\wedge w_{-2}+w_{4}\wedge w_{-4})$
$\displaystyle=w_{2}\wedge w_{-2}-2w_{4}\wedge w_{-4}$

and

$\displaystyle x_{-2}:=\mathbf{Y}\left(w_{2}\wedge w_{-2}-2w_{4}\wedge
w_{-4}\right)$ $\displaystyle=w_{0}\wedge w_{-2}-w_{2}\wedge w_{-4}.$

Similarly, one can find a basis of
$\mathrm{Sym}^{6}(k^{2})\subset\wedge^{2}(W)$ by applying $\mathbf{Y}$
successively to the highest weight vector $w_{4}\wedge w_{2}$ in that copy of
$\mathrm{Sym}^{6}(k^{2})$.

We have thus explicitly identified both $\mathrm{Sym}^{2}(k^{2})$ and
$\mathrm{Sym}^{6}(k^{2})$ with $\mathrm{PGL}_{2}$-subrepresentations of
$\wedge^{2}(W)$, and can check that

$X=\mathrm{Gr}(2,W)\cap\mathbb{P}(\mathrm{Sym}^{6}(k^{2}))$

is irreducible, smooth of the expected dimension $3$ by computer algebra. The
necessary checks were performed using
Macaulay2111warwick.ac.uk/fac/sci/maths/people/staff/boehning/m2filesequivariantderived.

######  Example 8.

Let $G=C_{9}\rtimes C_{6}$, G:=SmallGroup(54,6). Its smallest faithful
representation has dimension 6, in particular, $G$ does not admit a linear
action on $\mathbb{P}^{4}$. Let $W$ be its unique irreducible representation
of dimension 6, it has character

$(6,0,-3,0,0,0,0,0,0,0).$

We have a decomposition:

(2.3) $\wedge^{2}(W)=V_{1}\oplus V_{1}^{\prime}\oplus
V_{1}^{\prime\prime}\oplus V_{2}\oplus V_{2}^{\prime}\oplus
V_{2}^{\prime\prime}\oplus V_{6}$

into irreducible representations. Choose a suitable subrepresentation

$V:=V_{1}\oplus V_{2}\oplus V_{2}^{\prime}\oplus V_{6},$

more precisely, that with respective characters, for $\zeta=\zeta_{3}$,

$\displaystyle{\tt X.6}=$
$\displaystyle(1,-1,1,\zeta^{2},\zeta,-\zeta,-\zeta^{2},\zeta,1,\zeta^{2}),$
$\displaystyle{\tt X.7}=$ $\displaystyle(2,0,2,2,2,0,0,-1,-1,-1),$
$\displaystyle{\tt X.8}=$
$\displaystyle(2,0,2,2\zeta,2\zeta^{2},0,0,-\zeta^{2},-1,-\zeta),$
$\displaystyle{\tt X.10}=$ $\displaystyle(6,0,-3,0,0,0,0,0,0,0).$

The complement decomposes as

$\displaystyle{\tt X.2}=$ $\displaystyle(1,-1,1,1,1,-1,-1,1,1,1),$
$\displaystyle{\tt X.3}=$
$\displaystyle(1,-1,1,\zeta,\zeta^{2},-\zeta^{2},-\zeta,\zeta^{2},1,\zeta),$
$\displaystyle{\tt X.9}=$
$\displaystyle(2,0,2,2\zeta^{2},2\zeta,0,0,-\zeta,-1,-\zeta^{2}).$

Then, according to magma,

$X:=\operatorname{Gr}(2,W)\cap\mathbb{P}(V)$

is a smooth and irreducible variety of dimension 4 and degree 14. Note that
choosing a different 11-dimensional subrepresentation $V$ also yields
irreducible fourfolds of degree 14, but some of these are singular. Thus the
construction of Proposition 4 applies, and we have
$X\notin\mathfrak{L}\mathrm{in}$ and
$X\in\mathfrak{S}\mathfrak{L}\mathrm{in}$.

Let $G=C_{3}^{3}\rtimes\mathfrak{S}_{3}$, SmallGroup(162,19). Its smallest
faithful representation has dimension 6, in particular, $G$ does not admit a
linear action on $\mathbb{P}^{4}$. Let $W$ be an irreducible
$G$-representation with character

$(6,0,-3,0,0,3,-3,0,0,0,0,0,0).$

We have a decomposition

$\wedge^{2}(W)=V_{1}\oplus V_{2}\oplus V_{3}\oplus V_{3}^{\prime}\oplus V_{6}$

into irreducible representations. We choose

$V:=V_{2}\oplus V_{3}\oplus V_{6}.$

Then

$X:=\operatorname{Gr}(2,W)\cap\mathbb{P}(V)$

is irreducible (singular) of dimension 4 as can be checked by computer
algebra222warwick.ac.uk/fac/sci/maths/people/staff/boehning/m2filesequivariantderived.
Therefore this construction satisfies the hypotheses of Proposition 4, thus
$X\notin\mathfrak{L}\mathrm{in}$ but
$X\in\mathfrak{S}\mathfrak{L}\mathrm{in}$.

##  3\. Derived categories

### Terminology

Let $X$ be a smooth projective variety (over an algebraically closed field $k$
of characteristic zero) and $\mathsf{D}^{b}(X)$ its derived category of
coherent sheaves. We use freely the following terms; see, e.g., [BvBT23,
Section 2], or [Kuz16, Section 1 and 2] for definitions and references:

  * •

admissible subcategories of $\mathsf{D}^{b}(X)$,

  * •

exceptional objects,

  * •

(full) exceptional sequences,

  * •

(maximal) semiorthogonal decompositions.

###  $G$-actions on categories

Let $G$ be an algebraic group, not necessarily finite. Let $X$ be a smooth
projective $G$-variety, i.e., a smooth projective variety with a generically
free, regular, action of $G$. In [BvBT23, Proposition 3] it was remarked that
the fundamental reconstruction theorem by Bondal and Orlov [BO01] admits the
following equivariant version:

######  Proposition 9.

Suppose $X$ and $Y$ are smooth projective $G$-varieties over $k$, $X$ is Fano,
and

$\Phi\colon\mathsf{D}^{b}(X)\simeq\mathsf{D}^{b}(Y)$

is an equivalence as $k$-linear triangulated categories together with the
induced $G$-actions. Then $X$ and $Y$ are isomorphic as $G$-varieties, i.e.,
there exists a $G$-equivariant isomorphism

$X\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Y.$

In practice, this general theorem is not very useful since the derived
category contains too much information; in the context of rationality
problems, the focus is on trying to extract information about the variety from
more accessible data, such as a piece, or several pieces, in a semiorthogonal
decomposition of $\mathsf{D}^{b}(X)$.

We will explore the extent to which these considerations apply in the
equivariant context. We investigate, in several representative geometric
examples, the effects of $G$-equivariant birationalities on

  * •

the existence of full exceptional sequences in $\mathsf{D}^{b}(X)$ that are
compatible with $G$-actions, and

  * •

derived Hom-spaces between objects in $\mathsf{D}^{b}(X)$.

###  $G$-actions and exceptional sequences

######  Definition 10.

An object $E\in\mathsf{D}^{b}(X)$ is called _$G$ -invariant_ if $g^{*}E$ is
isomorphic to $E$, for all $g\in G$. It is called _$G$ -linearized_ if it is
equipped with a $G$-linearization, i.e. a system of isomorphisms

$\lambda_{g}\colon E\to g^{*}E,\quad\forall g\in G,$

satisfying the compatibility condition

$\lambda_{1}=\mathrm{id}_{E},\quad\lambda_{gh}=h^{*}(\lambda_{g})\circ\lambda_{h}.$

Several notions of compatibility of exceptional sequences with $G$-actions
have been studied; we follow [CT20, Definition 2.1].

######  Definition 11.

Let $X$ be a smooth projective $G$-variety and

${\bf E}:=(E_{1},\dots,E_{n})$

a full exceptional sequence in $\mathsf{D}^{b}(X)$.

  1. (1)

${\bf E}$ is _$G$ -invariant_ if for every $r\in\\{1,\dots,n\\}$ and every
$g\in G$, there is an $s$ such that $g^{*}E_{r}\simeq E_{s}$.

  2. (2)

${\bf E}$ is $G$-_equivariant_ if it is $G$-invariant and, for all $r$,
$E_{r}$ is isomorphic to a $G_{r}$-linearized object in $\mathsf{D}^{b}(X)$,
where $G_{r}\subseteq G$ is the stabilizer of the isomorphism class of
$E_{r}$.

  3. (3)

${\bf E}$ is $G$-_linearized_ if it is $G$-equivariant and, for all $r$,
$G_{r}=G$, i.e., each $E_{r}$ is a $G$-linearized object.

######  Example 12.

Consider $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$, and the full exceptional
sequence in $\mathsf{D}^{b}(X)$ from [Kap88]

${\bf E}=(\mathcal{O}(-1,-1),\mathcal{O},\mathcal{O}(1,0),\mathcal{O}(0,1)).$

Then

  * •

${\bf E}$ is $H$-linearized, if we view $X$ as
$\mathbb{P}(V)\times\mathbb{P}(V)$ with its natural diagonal $H$-action, where
$H$ is a finite group admitting a two-dimensional faithful linear
representation $V$ such that $H$ acts generically freely on $\mathbb{P}(V)$.

  * •

${\bf E}$ is $G$-equivariant, but not $G$-linearized, if we let
$G=\mathbb{Z}/2\times H$ act on $X=\mathbb{P}(V)\times\mathbb{P}(V)$, with the
first factor $\mathbb{Z}/2$ in $G$ switching the rulings,

  * •

${\bf E}$ is $G$-invariant, but not $G$-equivariant, if we let
$H\simeq\mathbb{Z}/2\times\mathbb{Z}/2$ act on $\mathbb{P}^{1}$ via the two-
dimensional faithful irreducible representation of its Schur cover
$\mathfrak{D}_{8}$ instead, and then let $G=\mathbb{Z}/2\times H$ act on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, again with $\mathbb{Z}/2$ switching the
factors and $H$ acting diagonally.

######  Example 13.

The moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with
$n$ marked points has a full $\mathfrak{S}_{n}$-equivariant exceptional
sequence, where $\mathfrak{S}_{n}$ is the symmetric group permuting the marked
points, by the main result of [CT20].

The following observation will be useful in applications.

######  Lemma 14.

Let $X$ be a smooth projective $G$-variety, and

${\bf E}:=(E_{1},\dots,E_{n})$

a $G$-invariant exceptional sequence consisting of line bundles. If $X$ is
$G$-linearizable, then ${\bf E}$ is a $G$-equivariant exceptional sequence.

###### Proof.

If $X$ is $G$-linearizable, $\mathrm{Am}(X,G)$ is trivial. The same holds for
$\mathrm{Am}(X,H)$, for any $H\subseteq G$, since $X$ is also
$H$-linearizable. Therefore, under the assumptions of the Lemma, any line
bundle on $X$ is $H$-linearized for every subgroup $H$ that leaves this line
bundle invariant. ∎

### Connections with classical invariants

######  Proposition 15.

Let $G$ be a finite group and $X$ a smooth projective $G$-variety admitting a
$G$-linearized full exceptional sequence. Then

$\mathrm{Pic}(X,G)\twoheadrightarrow\mathrm{Pic}(X)^{G},$

in particular,

$\mathrm{Am}(X,G)=0.$

###### Proof.

Taking the first Chern class gives a well-defined homomorphism

$c_{1}\colon\mathsf{D}^{b}(X)\to\mathrm{Pic}(X).$

If $\mathsf{D}^{b}(X)$ is generated by an exceptional sequence ${\bf
E}=(E_{1},\dots,E_{n})$ of $G$-linearized objects, then every class in
$\mathrm{Pic}(X)$ is a $\mathbb{Z}$-linear combination of the $c_{1}(E_{r})$,
which are $G$-linearized. Indeed, the first Chern class of a $G$-linearized
complex is $G$-linearized; it is the alternating sum of the Chern classes of
the cohomology sheaves which are $G$-linearized, so we just need to show
invariance of the Chern class for a $G$-linearized sheaf. Such a sheaf always
has a finite locally free resolution by $G$-linearized vector bundles since
there exists a $G$-linearized ample sheaf on $X$ and the statement is true on
projective space. ∎

######  Remark 16.

Let $G$ be a finite group and $X$ a smooth projective $G$-variety with ample
anticanonical class. By Proposition 9, the derived category
$\mathsf{D}^{b}(X)$ determines $X$, as a $G$-variety. In particular, we can
extract the $G$-action on $\operatorname{Pic}(X)$, and determine whether or
not it satisfies (H1) or (SP). Concretely, we have

$\operatorname{Aut}(\mathsf{D}^{b}(X))=(\operatorname{Pic}(X)\times\mathbb{Z})\rtimes\operatorname{Aut}(X),$

and the derived automorphisms acting trivially on point objects can be
identified with $\operatorname{Pic}(X)$.

In the following sections we investigate connections between existence of full
exceptional sequences with various compatibility properties with the
$G$-action, and (stable) linearizability of $X$. It turns out that
$G$-linearizability often implies the existence of a full equivariant
exceptional sequence, provided such sequences exist in the non-equivariant
setting, as for Del Pezzo surfaces.

##  4\. Del Pezzo surfaces

### Terminology

By the Minimal Model Program, every rational surface is birational to a conic
bundle over $\mathbb{P}^{1}$ or a Del Pezzo surface, i.e., a smooth projective
surface $X$ with ample anticanonical class $-K_{X}$; we let

$d=d(X)=(-K_{X})^{2}$

be its degree. The same holds over nonclosed field, and in presence of group
actions.

Here and below conic bundle means that $X$ is smooth and all fibers of
$f\colon X\to\mathbb{P}^{1}$ are isomorphic to reduced conics in
$\mathbb{P}^{2}$. We recall the terminology of [DI09, Pro15]: a conic bundle
$f\colon X\to\mathbb{P}^{1}$ is called exceptional if for some positive
integer $g$ the number of degenerate fibers equals $2g+2$ and there are two
disjoint sections $C_{1}$ and $C_{2}$ with $C_{1}^{2}=C_{2}^{2}=-(g+1)$.
Exceptional conic bundles can be constructed explicitly, see [DI09, §5.2].

### Nonlinearizable actions

In this section, $G$ is a finite group. The following nonlinearizability
results for $G$-conic bundles are probably known to experts in birational
rigidity; here, we rely on the Burnside formalism.

######  Lemma 17.

Let $X\to\mathbb{P}^{1}$ be a relatively minimal $G$-conic bundle with
$K_{X}^{2}=1$. Then $X$ is not linearizable.

###### Proof.

If $X$ fails (H1), then then $X\notin\mathfrak{L}\mathrm{in}$. If $X$
satisfies (H1), then the classification in [Pro15, §8], Theorem 8.3, shows
that $G$ must be the binary dihedral group $\widetilde{\mathfrak{D}}_{5}$, a
nontrivial central $C_{2}$-extension of $\mathfrak{D}_{5}$. Such $X$, with the
$G$-action, are given by an explicit construction [Pro15, Construction 8.4].
In particular, there is a distinguished involution $\tau\in G$, generating the
center of $G$ and fixing a smooth rational curve $C$. The residual action of
$\mathfrak{D}_{5}=G/\langle\tau\rangle$ is generically free on $C$. Applying
the Burnside formalism to this situation, we find a unique, incompressible,
symbol

(4.1)
$(C_{2},\mathfrak{D}_{5}\mathrel{\reflectbox{$\righttoleftarrow$}}k(\mathbb{P}^{1}),(1)),$

contributing to the class

$[X\righttoleftarrow G]\in\operatorname{Burn}_{2}(G).$

A generically free linear action of $\widetilde{\mathfrak{D}}_{5}$ on
$\mathbb{P}^{2}$ necessarily arises from a representation $V=V_{1}\oplus
V_{2}$, where $V_{1}$ is 1-dimensional representation which is nontrivial on
$\tau$ and $V_{2}$ is a 2-dimensional representation which is trivial on
$\tau$. Then $G$ fixes the point $p_{0}:=[1:0:0]\in\mathbb{P}^{2}$ and
stabilizes the line given by $x_{0}=0$. Passing to a standard model, we
observe, as in a similar situation in [HKT21, Section 7.6], that the linear
action contributes two symbols (4.1), one from the exceptional divisor of the
blowup of $p_{0}$ and the other from $\mathbb{P}^{1}=\mathbb{P}(V_{2})$. It
follows that

$[X\righttoleftarrow G]\neq[\mathbb{P}^{2}\righttoleftarrow G]$

in $\operatorname{Burn}_{2}(G)$, and the $G$-action on $X$ is not
linearizable. ∎

######  Lemma 18.

Let $X$ be a minimal rational $G$-surface that is an exceptional conic bundle
with $K_{X}^{2}=2$ and $g=2$. Then $X$ is not linearizable.

###### Proof.

If $X$ fails (H1) then $X$ is not linearizable. The other cases have been
classified in [Pro15, §8]: consider the representation

$\varrho\colon G\to\mathrm{Aut}(\mathrm{Pic}(X)),$

its kernel $\mathrm{ker}(\varrho)$, and the exact sequence

(4.2) $1\to G_{F}\to G\to G_{B}\to 1,$

where $G_{F}\subset G$ is the largest subgroup acting trivially on the base
$B=\mathbb{P}^{1}$. By [Pro15, Theorem 8.3], we have
$\mathrm{ker}(\varrho)\neq\\{1\\}$; by [Pro15, Theorem 8.6(2)] it is cyclic,
whereas $G_{B}\simeq\mathfrak{D}_{n}$, with $n\geq 3$, or
$G_{B}\simeq\mathfrak{S}_{4}$. The table in [Pro15, Section 8.7] shows that
$G_{B}=\mathfrak{S}_{4}$, and [Pro15, Thm. 8.6] shows that
$G_{F}=\mathrm{ker}(\varrho)=C_{m}$, a nontrivial cyclic group of order $m$.

Write $C_{m}=C_{2^{r}}\times C_{m^{\prime}}$ with $\gcd(m^{\prime},2)=1$, and
consider a $2$-Sylow subgroup $G_{2}$ of $G$ that contains $C_{2^{r}}$. Then
$G_{2}$ has order $2^{r}\times 8$ and sits in an extension

$1\to C_{2^{r}}\to G_{2}\to\bar{G}_{2}\to 1$

where $\bar{G}_{2}$ is a subgroup of $\mathfrak{S}_{4}$ of order $8$, hence
equal to $\mathfrak{D}_{4}$. Since the order of a group is divisible by the
degree of any of its irreducible representations, every $3$-dimensional
representations $V$ of $G_{2}$ has to decompose into irreducible summands of
degrees $1,1,1$ or $1,2$. Only the latter can be generically free. Thus, we
may assume that $V$ is of the form

$V=V_{1}\oplus V_{2}$

with $V_{i}$ irreducible of dimension $i$. Here $V_{1}=k_{\chi}$ is a
representation of $G_{2}$ by some character $\chi$, and we can assume that
$V_{1}$ is trivial, and $V_{2}$ is a faithful $G_{2}$-representation. A
standard model for $G_{2}$-action is the blowup
$\widetilde{\mathbb{P}(V)}\to\mathbb{P}(V)$ of the $G_{2}$-fixed point
$p_{0}=[1:0:0]$, see [HKT21, Section 7.2]. The only incompressible divisorial
symbols might arise from the exceptional divisor, respectively, the preimage
of the projectivization
$\mathbb{P}^{1}=\mathbb{P}(V_{2})\subset\mathbb{P}(V)$. The corresponding
symbols are

$(C,G_{2}/C\mathrel{\reflectbox{$\righttoleftarrow$}}k(\mathbb{P}^{1}),(\chi)),\quad(C,G_{2}/C\mathrel{\reflectbox{$\righttoleftarrow$}}k(\mathbb{P}^{1}),(\bar{\chi})),$

where $C\subset G_{2}$ is a cyclic group and $\chi$ is a primitive character
of $C$. Their sum in $\operatorname{Burn}_{2}(G_{2})$ cannot equal to

$(C_{2^{r}},\mathfrak{K}_{4}\mathrel{\reflectbox{$\righttoleftarrow$}}k(\mathbb{P}^{1}),(\psi))$

with $\psi$ some primitive character of $C_{2^{r}}$, for any choices of
$C,\chi$. Thus

$[X\righttoleftarrow G_{2}]\neq[\mathbb{P}^{2}\righttoleftarrow G_{2}]$

in $\operatorname{Burn}_{2}(G_{2})$, for any generically free linear action of
$G_{2}$ on $\mathbb{P}^{2}$. ∎

### Linearization and derived categories

We consider rational $G$-surfaces and investigate which pieces and properties
of the $G$-category $\mathsf{D}^{b}(X)$ are sensitive to geometric, and in
particular, $G$-birational, characteristics of the $G$-action on $X$.

######  Lemma 19.

Let $X$ be a rational $G$-surface $X$ admitting a full $G$-invariant
exceptional sequence. Then the $G$-action on $\operatorname{Pic}(X)$ satisfies
(H1) and (SP).

###### Proof.

The classes of the terms of the sequence in the Grothendieck
$\mathrm{K}$-group $\mathrm{K}_{0}(X)$ form a $\mathbb{Z}$-basis that is
permuted by $G$. Thus $\mathrm{K}_{0}(X)$ is a permutation module, and since

$\mathrm{K}_{0}(X)\simeq\mathbb{Z}\oplus\mathrm{Pic}(X)\oplus\mathbb{Z}$

as $G$-modules, with trivial $G$-action on the two summands $\mathbb{Z}$, we
obtain the claim. ∎

Incidentally, assuming $X$ is a minimal $G$-Del Pezzo surface, Theorem 1.2 of
[Pro15] shows that (H1) is equivalent to the fact that $G$ does not fix a
curve of positive genus, and also equivalent to the condition $K_{X}^{2}\geq
5$ or $X$ being a special quartic Del Pezzo surface with a very special
action, described in [Pro15, Thm. 1.2, (iii), (b)].

######  Lemma 20.

Let $X=\mathbb{P}^{2}$ with a projectively linear but nonlinear action of a
finite group $G$. Then $X$ does admit a full $G$-invariant exceptional
sequence, but no full $G$-equivariant exceptional sequence.

###### Proof.

The exceptional sequence $(\mathcal{O},\mathcal{O}(1),\mathcal{O}(2))$ is
$G$-invariant. From [KO94] it is known that every exceptional object in
$\mathsf{D}^{b}(X)$ is, up to shift, a vector bundle. Thus assume that
$(\mathcal{E}_{1},\mathcal{E}_{2},\mathcal{E}_{3})$ is a $G$-equivariant full
exceptional sequence consisting of vector bundles. Since the map

$\mathsf{D}^{b}(X)\to\mathrm{K}_{0}(X)$

is $G$-equivariant and the action on $\mathrm{K}_{0}(X)$ is trivial in this
case, we see that every element in $G$ fixes the isomorphism class of each
$\mathcal{E}_{i}$ (because the images of
$\mathcal{E}_{1},\mathcal{E}_{2},\mathcal{E}_{3}$ form a $\mathbb{Z}$-basis of
$\mathrm{K}_{0}(X)$). If we compose $\mathsf{D}^{b}(X)\to\mathrm{K}_{0}(X)$
with the first Chern class map, we get a surjective map to $\mathrm{Pic}(X)$.
In other words, the top exterior powers of the $\mathcal{E}_{i}$ generate the
Picard group, hence at least one of them has to be isomorphic to
$\mathcal{O}_{\mathbb{P}^{2}}(r)$ for some odd integer $r$. Thus
$\mathcal{O}_{\mathbb{P}^{2}}(1)$ is also $G$-linearized, contradicting our
assumption that the action is nonlinearizable. ∎

### Examples with $\operatorname{Aut}(X)$-equivariant exceptional sequences

We present examples of rational surfaces $X$ such that $\mathsf{D}^{b}(X)$
admits a full $\mathrm{Aut}(X)$-equivariant exceptional sequence but $X$ is
not lineariable, for some $G\subseteq\operatorname{Aut}(X)$.

DP6: Let $X$ be a Del Pezzo surface of degree $6$. Then $X$ has full
$\mathrm{Aut}(X)$-_invariant_ exceptional sequence. Indeed, recall that there
is an exact sequence

$0\to T\to\mathrm{Aut}(X)\to W_{X}\to 0,$

where

$W_{X}\simeq\mathbb{Z}/2\times\mathfrak{S}_{3},\quad\mathrm{Aut}(X)\simeq
N(T)\rtimes\mathbb{Z}/2,$

and $T$ is the maximal torus of $\operatorname{PGL}_{3}$, the quotient of
$(k^{\times})^{3}$ by the diagonal subgroup $k^{\times}$, $N(T)$ its
normalizer. A generator of $\mathbb{Z}/2$ in
$W_{X}=\mathbb{Z}/2\times\mathfrak{S}_{3}$ can be identified with the lift of
the standard Cremona involution on $\mathbb{P}^{2}$ and $\mathfrak{S}_{3}$ is
realized as the group of permutations of the points

$p_{1}=(1:0:0),\quad p_{2}=(0:1:0),\quad p_{3}=(0:0:1)$

that are blown up to obtain $X$. There is always a full invariant exceptional
collection for the entire automorphism group of $X$. Indeed, $X$ has the
following (three block) exceptional sequence:

$\displaystyle\mathcal{O}_{X},\quad\mathcal{O}_{X}(H),\quad\mathcal{O}_{X}(2H-E_{1}-E_{2}-E_{3}),$
$\displaystyle\quad\mathcal{O}_{X}(2H-E_{1}-E_{2}),\quad\mathcal{O}_{X}(2H-E_{2}-E_{3}),\quad\mathcal{O}_{X}(2H-E_{1}-E_{3}),$

where $H$ is the pullback of a hyperplane class and $E_{i}$ the exceptional
divisors. The Cremona involution $\sigma$ acts as

$H\mapsto 2H-E_{1}-E_{2}-E_{3},\quad E_{i}\mapsto
H-E_{j}-E_{k},\quad\\{i,j,k\\}=\\{1,2,3\\},$

whereas $\mathfrak{S}_{3}$ permutes the $E_{i}$ and fixes $H$, and $T$ fixes
$H,E_{i}$. However, this sequence is not always an equivariant exceptional
sequence (for example, the normalizer of a maximal torus in
$\operatorname{PGL}_{3}$ is in the stabilizer of $\mathcal{O}_{X}(H)$, but the
line bundle is not linearized since the action does not lift to a linear
action on $k^{3}$).

DP5: Let $X$ be a Del Pezzo surface of degree $5$. We have

$\operatorname{Aut}(X)\simeq\mathfrak{S}_{5}.$

By [CT20, Theorem 1.2 and Example 1.3], $X$ has a full
$\mathfrak{S}_{5}$-equivariant exceptional collection. However, $X$ is
$G$-superrigid for $G=\mathfrak{A}_{5}$ [DI09].

### A special DP4

By [Pro15, Theorem 1.2], there is a unique minimal $G$-Del Pezzo surface $X$
of degree $\leq 4$ that satisfies (H1). It is a Del Pezzo surface of degree 4,
an interection of two quadrics in $\mathbb{P}^{4}$

(4.3) $x_{1}^{2}+\zeta
x_{2}^{2}+\zeta^{2}x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+\zeta^{2}x_{2}^{2}+\zeta
x_{3}^{2}+x_{5}^{2}=0,$

with $\zeta=\zeta_{3}$ a primitive cube root of unity, and
$G=\mathbb{Z}/3\rtimes\mathbb{Z}/4$, with generators

$\displaystyle\gamma\colon(x_{1},x_{2},x_{3},x_{4},x_{5})$
$\displaystyle\mapsto(x_{2},x_{3},x_{1},\zeta x_{4},\zeta^{2}x_{5}),$
$\displaystyle\beta^{\prime}\colon(x_{1},x_{2},x_{3},x_{4},x_{5})$
$\displaystyle\mapsto(x_{1},x_{3},x_{2},-x_{5},x_{4}).$

######  Theorem 21.

The derived category $\mathsf{D}^{b}(X)$ of the minimal $G$-Del Pezzo surface
$X$ given by (4.3) does not admit a full $G$-invariant exceptional sequence.

###### Proof.

Arguing by contradiction, we assume that such a sequence

$(\mathcal{E}_{1},\dots,\mathcal{E}_{8})$

exists. The group $G$ acts on the terms of the sequence by permutations,
decomposing the set of terms into $G$-orbits, each of which is again an
exceptional sequence. Let

$(\mathcal{F}_{1},\dots,\mathcal{F}_{r})$

be one of the orbits. Consider the classes $v_{i}$ of the $\mathcal{F}_{i}$ in
the Grothendieck group

$\mathrm{K}_{0}(X)\simeq\mathbb{Z}\oplus\mathrm{Pic}(X)\oplus\mathbb{Z}\simeq\mathbb{Z}^{8}.$

Let $\chi(-,-)$ be the Euler bilinear pairing on $\mathrm{K}_{0}(X)$ and
$v:=v_{r}=[\mathcal{F}_{r}]$. Since $(v_{1},\dots,v_{r})$ is a numerically
exceptional sequence with respect to the Euler pairing, we have

$(\ast)\quad\quad\chi\bigl{(}v,g(v)\bigr{)}=\left\\{\begin{matrix}0&\text{if
$g(v)\neq v$,}\\\ 1&\text{if $g(v)=v$,}\end{matrix}\right.$

for all $g\in G$. These are quadratic equations for the coefficients of $v$.
Let $H\subseteq G$ be the subgroup fixing $v$. If $H=G$, then $r=1$ and
$v_{r}=v_{1}$ is $G$-invariant. If $H=1$, then $r=12$, a contradiction. Let us
now assume that $H$ is a nontrivial proper subgroup of $G$. There are six such
subgroups and they are all cyclic. Let
$\mathrm{K}_{0}(X)^{H}\simeq\mathbb{Z}^{r}$ be the space of $H$-invariants and
consider the ideal $I_{H}$ generated by the conditions $(\ast)$ in
$\mathbb{Z}[s_{1},\dots,s_{r}]$. One can show that $I_{H}=(1)$ mod $3$ for all
such $H$, e.g., with
Macaulay2333warwick.ac.uk/fac/sci/maths/people/staff/boehning/m2filesequivariantderived.
Hence only $H=G$ is possible. Since this is the case for all $G$-orbits, we
obtain that all classes $[E_{i}]\in\mathrm{K}_{0}(X)$ are $G$-invariant.
However, they also form a $\mathbb{Z}$-basis of $\mathrm{K}_{0}(X)$. But
$\mathrm{K}_{0}(X)^{G}\neq\mathrm{K}_{0}(X)$.

More precisely, we proceed as follows. Since $X$ is a DP4, there are $16$
lines on $X$. To determine these lines explicitly consider the rank $2$ skew
matrix

$L={\tiny\begin{pmatrix}0&1&-1&1&1\\\ -1&0&1&\zeta^{2}&\zeta\\\
1&-1&0&\zeta&\zeta^{2}\\\ -1&-\zeta^{2}&-\zeta&0&\zeta-\zeta^{2}\\\
-1&-\zeta&-\zeta^{2}&-\zeta+\zeta^{2}&0\end{pmatrix}}\in\mathrm{Gr}(2,W)\subset\mathbb{P}(\wedge^{2}(W)),$

with $\dim(W)=5,$ and a diagonal matrix $D$ with entries $\pm 1$ on the
diagonal. Then we check that $DLD^{t}$ represents a line on $X$. This gives
the $16$ lines on $X$ which are permuted by $\beta$ and $\gamma$. Observe that
the line represented by $L$ is $\gamma$-invariant.

We now choose $6$ lines whose classes are a basis of $\operatorname{Pic}(X)$
as follows. There are precisely $5$ lines $L_{1},\dots,L_{5}$ that intersect
the line represented by $L$. Two of these lines are $\gamma$-invariant.
Without loss of generality, we can assume these are $L_{1}$ and $L_{2}$.
Finally there is a unique fourth $\gamma$-invariant line $L_{6}$.

Then $L_{1},\dots,L_{6}$ form a basis of $\operatorname{Pic}(X)$: indeed their
intersection matrix can be computed as

$\left(\begin{smallmatrix}{-1}&0&0&0&0&1\\\ 0&{-1}&0&0&0&1\\\
0&0&{-1}&0&0&0\\\ 0&0&0&{-1}&0&0\\\ 0&0&0&0&{-1}&0\\\
1&1&0&0&0&{-1}\end{smallmatrix}\right)$

We can now compute the representation of the other lines in this basis by
considering their intersections with the lines in the given basis. We find
representations

$B=\left(\begin{matrix}1&1&{-1}&{-1}&{-1}&2\\\ 0&0&0&0&0&1\\\
1&0&0&{-1}&0&1\\\ 1&0&{-1}&0&0&1\\\ 1&0&0&0&{-1}&1\\\
1&0&0&0&0&0\end{matrix}\right)\quad\text{and}\quad
C=\left(\begin{matrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&0&0&1&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&1\end{matrix}\right)$

of $\beta$ and $\gamma$, respectively. Moreover, the canonical class $K_{X}$
is determined by the fact that the intersection number of $-K_{X}$ with all
lines is equal to $1$. One finds:

$-K_{X}=2L_{1}+2L_{2}-L_{3}-L_{4}-L_{5}+3L_{6},$

which is also equal to the sum of the four $\gamma$-invariant lines
$L,L_{1},L_{2},L_{6}$.

Following [BGvBS14, Section 3], we work out the Euler pairing explicitly. The
Chern character

$\displaystyle\mathrm{ch}\colon\mathrm{K}_{0}(X)$
$\displaystyle\to\mathrm{CH}^{*}(X)_{\mathbb{Q}}$ $\displaystyle[\mathcal{E}]$
$\displaystyle\mapsto\mathrm{rk}(E)+c_{1}(\mathcal{E})+\frac{c_{1}(\mathcal{E})^{2}-2c_{2}(\mathcal{E})}{2}$

is an injective ring homomorphism with values in the sublattice

$\Lambda:=\left\\{x+y_{1}l_{1}+y_{2}l_{2}+\dots+y_{6}l_{6}+\frac{1}{2}zp\right\\}\simeq\mathbb{Z}^{8}\subset\mathrm{CH}^{*}(X)_{\mathbb{Q}},$

where $(x,y_{1},y_{2},\dots,y_{6},z)\in\mathbb{Z}^{8}$, $p$ is the class of a
point, and $l_{i}:=c_{1}(\mathcal{O}(L_{i}))$. We set $v=(x,y,z)$, where

$y=y_{1}l_{1}+y_{2}l_{2}+\dots+y_{6}l_{6}.$

Thus $\Lambda$ is generated by $\mathrm{CH}^{0}(X)\simeq\mathbb{Z}$,
$\mathrm{CH}^{1}(X)\simeq\mathrm{Pic}(X)\simeq\mathbb{Z}^{6}$ and
$\frac{1}{2}\mathrm{CH}^{2}(X)$, where $\mathrm{CH}^{2}(X)\simeq\mathbb{Z}$ is
generated by the Chern character of the skyscraper sheaf of a point $p$, which
is just the class of $p$ in the Chow ring. Its image is an index $2$
sublattice $\mathrm{ch}(\mathrm{K}_{0}(X))\subset\Lambda$: indeed,
$\frac{1}{2}p$ is not in $\mathrm{ch}(\mathrm{K}_{0}(X))$ since for the Euler
pairing $\chi$

$\chi(\mathcal{O}_{X},\mathcal{O}_{p})=1$

and $\chi$ takes integral values on $\mathrm{ch}(\mathrm{K}_{0}(X))$. The
class of $\frac{1}{2}p$ generates the quotient
$\Lambda/\mathrm{ch}(\mathrm{K}_{0}(X))$. By Riemann-Roch,

$\chi(X,\mathcal{E})=\deg\left(\mathrm{ch}(\mathcal{E}).\mathrm{td}(\mathcal{T}_{X})\right)_{2},$

where

$\mathrm{td}(\mathcal{T}_{X})=1-\frac{1}{2}K_{X}+\frac{1}{12}(K_{X}^{2}+c_{2})=1-\frac{1}{2}K_{X}+p.$

The subscript $2$ in the second to last formula means that one only considers
the top-dimensional component. Hence in terms of $v=(x,y,z)$,

$\chi(X,\mathcal{E})=x-\frac{1}{2}y.K_{X}+\frac{1}{2}z\,.$

If $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are bundles, then

$\chi(\mathcal{E}_{1},\mathcal{E}_{2})=\chi(X,\mathcal{E}_{1}^{\vee}\otimes\mathcal{E}_{2})$

and

$\displaystyle\mathrm{ch}(\mathcal{E}_{1}^{\vee}\otimes\mathcal{E}_{2})$
$\displaystyle=\mathrm{ch}(\mathcal{E}_{1}^{\vee}).\mathrm{ch}(\mathcal{E}_{2})=(x_{1}-y_{1}+\frac{1}{2}z_{1})(x_{2}+y_{2}+\frac{1}{2}z_{2})$
$\displaystyle=x_{1}x_{2}+(x_{1}y_{2}-x_{2}y_{1})+\frac{1}{2}(x_{1}z_{2}+x_{2}z_{1}-2y_{1}y_{2}),$

whence

$\chi(\mathcal{E}_{1},\mathcal{E}_{2})=x_{1}x_{2}-\frac{1}{2}(x_{1}y_{2}-x_{2}y_{1}).K_{X}+\frac{1}{2}(x_{1}z_{2}+x_{2}z_{1}-2y_{1}y_{2})\,.$

We work out the Euler pairing $\chi$ on the lattice $\Lambda$ in the above
$\mathbb{Z}$-basis:

$\left(\begin{smallmatrix}1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
{-\frac{1}{2}}&1&0&0&0&0&{-1}&0\\\ {-\frac{1}{2}}&0&1&0&0&0&{-1}&0\\\
{-\frac{1}{2}}&0&0&1&0&0&0&0\\\ {-\frac{1}{2}}&0&0&0&1&0&0&0\\\
{-\frac{1}{2}}&0&0&0&0&1&0&0\\\ {-\frac{1}{2}}&{-1}&{-1}&0&0&0&1&0\\\
\frac{1}{2}&0&0&0&0&0&0&0\end{smallmatrix}\right)$

We now show that equations $(\ast)$ cannot be solved even in the larger
lattice $\Lambda$. Namely, consider the subgroup $H\subset G$ generated by
$\beta$. The invariants of $\beta$ in $\Lambda$ are

$v=({z}_{1},{-2\,{z}_{3}},{-2\,{z}_{3}},-{z}_{2}-{z}_{3},{z}_{2}+3\,{z}_{3},{z}_{3},{-3\,{z}_{3}},{z}_{4})$

for $z_{i}\in\mathbb{Z}$. Now

$\displaystyle 1$
$\displaystyle=\chi\bigl{(}v,v\bigr{)}={z}_{1}^{2}+2\,{z}_{2}^{2}+8\,{z}_{2}{z}_{3}+4\,{z}_{3}^{2}+{z}_{1}{z}_{4},$
$\displaystyle 0$
$\displaystyle=\chi\bigl{(}v,vC\bigr{)}={z}_{1}^{2}-{z}_{2}^{2}-4\,{z}_{2}{z}_{3}-8\,{z}_{3}^{2}+{z}_{1}{z}_{4}.$

Subtracting the second equation from the first we obtain

$1=3\,{z}_{2}^{2}+12\,{z}_{2}{z}_{3}+12\,{z}_{3}^{2}$

which has no solution modulo $3$.

The same computation can be done for all nontrivial proper subgroups of $G$. ∎

### Implications

Figure 1 shows relations between the different notions for a minimal $G$-Del
Pezzo surface $X$ with $\mathrm{rk}\,\mathrm{Pic}(X)^{G}=1$.

$X$ has a full
---
exceptional sequence
of $G$-linearized objects
45 $X=\mathbb{P}^{2}\in\mathfrak{L}\mathrm{in}$
---
$X\in\mathfrak{L}\mathrm{in}$
---
1 $X$ has a full equivariant exceptional sequence
---
2 $X$ has a full invariant exceptional sequence
---
3 $X$ satisfies (H1)
---

Figure 1.

  * •

The implications 1\- 4 are strict, see below for references to proofs.

  * •

3

is proven in Lemma 19, whereas 2, 4 are immediate from the definitions once 5
is proven.

  * •

1

is not reversible, e.g., for $X$ a DP6.

  * •

5

follows from Proposition 15 since $X$ then has Picard rank $1$, and this also
shows that 4 is not reversible.

  * •

2

is not reversible, by Lemma 20.

  * •

3

is not reversible, by Theorem 21.

The main result, which requires a longer argument, is the implication 1. We
will prove this more generally whenever $X$ is a smooth rational $G$-surface.

######  Theorem 22.

A smooth projective rational $G$-surface that is linearizable has a full
$G$-equivariant exceptional sequence.

The proof will occupy the remainder of this section. It is based on a detailed
analysis of actions, following [DI09] and [Pro15].

###### Proof.

We assume that $X$ is linearizable.

Step 1. We reduce to $G$-minimal surfaces: Indeed, consider a blowup
$\tilde{X}\to X$ in a $G$-invariant set of points. By Orlov’s blowup formula
[Orl92], if $X$ admits a full $G$-equivariant exceptional sequence, then so
does $\tilde{X}$. The stabilizer $G_{x}\subseteq G$ of a point $x\in X$ acts
linearly on the tangent bundle of $X$ at $x$; hence the $G_{x}$-action on the
sheaves $\mathcal{O}_{E}(r)$ (where $E$ is the exceptional divisor over $x$)
is linearized.

Step 2. By [DI09, Thm. 3.8], a minimal rational $G$-surface $X$ either admits
a structure of a $G$-conic bundle over $\mathbb{P}^{1}$ with
$\mathrm{Pic}(X)^{G}\simeq\mathbb{Z}^{2}$ or $X$ is isomorphic to a Del Pezzo
surface with $\mathrm{Pic}(X)^{G}\simeq\mathbb{Z}$. We proceed via
classification in [DI09, Section 8], depending on the possible values of
$d=K_{X}^{2}$.

Step 3.

Case $d\leq 0$: $X$ is a rigid $G$-conic bundle with $8-d$ singular fibres,
and in particular, $X\notin\mathfrak{L}\mathrm{in}$.

Case $d=1$: $X$ is a rigid $G$-Del Pezzo surface, thus
$X\notin\mathfrak{L}\mathrm{in}$, or a $G$-conic bundle, treated in Lemma 17.

Case $d=2$: $X$ is a rigid $G$-Del Pezzo surface, thus
$X\notin\mathfrak{L}\mathrm{in}$, or a $G$-conic bundle. If the conic bundle
is not exceptional, it is rigid; exceptional conic bundles with $g=2$ are
treated in Lemma 18.

Case $d=3$: $X$ is either a minimal $G$-Del Pezzo surface that is rigid, thus
$X\notin\mathfrak{L}\mathrm{in}$; or a minimal $G$-conic bundle, in which case
$G$ contains three commuting involutions two of which have fixed point curves
of genus $2$, yielding the (H1)-obstruction to linearizability, contradicting
the assumption.

Case $d=4$: $X$ can be a minimal $G$-Del Pezzo surface. If $X^{G}=\emptyset$,
$X$ is either rigid or superrigid, hence $X\notin\mathfrak{L}\mathrm{in}$. If
$X^{G}\neq\emptyset$, then $X$ is $G$-birational to a minimal conic bundle
with $d=3$ and we conclude as in the previous case.

If $X$ is a minimal $G$-conic bundle, then either $X$ is an exceptional conic
bundle with $g=1$: assuming that $X$ is linearizable, [Pro15, Theorem 8.3]
implies that the kernel of

$\varrho\colon G\to\mathrm{Aut}(\mathrm{Pic}(X))$

is non-trivial, since otherwise $K_{X}^{2}$ has to be odd. Then [DI09,
Classification in §8.1] implies that no elementary transformation is possible
and $X$ is not $G$-birational to any Del Pezzo surface, hence
$X\notin\mathfrak{L}\mathrm{in}$.

Secondly, $X$ can also be a $G$-Del Pezzo surface with two sections with self-
intersection $-1$ intersecting at one point. In this case, $X$ is obtained by
regularizing a de Jonquières involution; since such a de Jonquières involution
is not conjugate to a projective involution, $X\notin\mathfrak{L}\mathrm{in}$.

Case $d=5$: has been considered above.

Case $d=6$: $X$ always has a full $G$-invariant exceptional sequence. If
$X\in\mathfrak{L}\mathrm{in}$, Lemma 14 applies.

Case $d=8$: If $X=\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$, then it
has the full exceptional sequence, see [Kap88],

${\bf E}=(\mathcal{O}(-1,-1),\mathcal{O},\mathcal{O}(1,0),\mathcal{O}(0,1)),$

which is invariant under the full automorphism group

$\mathrm{Aut}(X)=\mathrm{PGL}_{2}(\mathbb{C})\wr C_{2}.$

If $X$ is linearizable for a subgroup $G\subset\operatorname{Aut}(X)$, then
Lemma 14 applies. In that case, every $G$-invariant full exceptional sequence
is a $G$-equivariant full exceptional sequence.

When $X=\mathbb{F}_{n}$ with $n\geq 2$, we apply Proposition 23.

Case $d=9$: $X=\mathbb{P}^{2}$, and there is nothing to show. ∎

######  Proposition 23.

Let $X$ be a $G$-Hirzebruch surface $\mathbb{F}_{n}$, $n\geq 2$, that is
$G$-linearizable. Then $X$ admits a full $G$-equivariant exceptional sequence.

###### Proof.

If $X=\mathbb{F}_{n}$, $n\geq 2$, [DI09, Theorem 4.10] shows that any finite
subgroup $G\subset\mathrm{Aut}(X)$ is contained in
$\mathrm{GL}_{2}(k)/\mu_{n}$, which is embedded into $\mathrm{Aut}(X)$ as
follows: view $\mathbb{F}_{n}$ as the quotient
$(\mathbb{A}^{2}\backslash\\{0\\})^{2}/\mathbb{G}_{m}^{2}$, acting by

$\displaystyle\mathbb{G}_{m}^{2}\times(\mathbb{A}^{2}\backslash\\{0\\})^{2}$
$\displaystyle\to(\mathbb{A}^{2}\backslash\\{0\\})^{2}$
$\displaystyle\bigl{(}(\lambda,\mu),(x_{0},x_{1},y_{0},y_{1})\bigr{)}$
$\displaystyle\mapsto(\lambda\mu^{-n}x_{0},\lambda x_{1},\mu y_{0},\mu
y_{1}),$

and with projection

$\displaystyle\pi\colon\mathbb{F}_{n}$ $\displaystyle\to\mathbb{P}^{1}$
$\displaystyle(x_{0},x_{1},y_{0},y_{1})$ $\displaystyle\mapsto(y_{0}:y_{1}),$

identifying

$\mathbb{F}_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(n)\oplus\mathcal{O}_{\mathbb{P}^{1}}),$

as a $\mathbb{P}^{1}$-bundle. Letting $A=(a_{ij})\in\mathrm{GL}_{2}(k)$ act on
the $y$-coordinates

$A\cdot(x_{0},x_{1},y_{0},y_{1})=(x_{0},x_{1},a_{11}y_{0}+a_{12}y_{1},a_{21}y_{0}+a_{22}y_{1}),$

we obtain an action of $\mathrm{GL}_{2}(k)$ on $\mathbb{F}_{n}$; clearly
$\mu_{n}\subset\mathrm{GL}_{2}(k)$ acts trivially on $\mathbb{F}_{n}$, and we
get an induced action of $\mathrm{GL}_{2}(k)/\mu_{n}$. Actually the full
automorphism group of $\mathbb{F}_{n}$ is a semidirect product of
$\mathrm{GL}_{2}(k)/\mu_{n}$ by a normal subgroup $k^{n+1}$, thought of as the
space of binary forms of degree $n$ with its natural action of
$\mathrm{GL}_{2}(k)/\mu_{n}$, because $\mathbb{F}_{n}$ can also be realized as
the blowup of the weighted projective space $\mathbb{P}(1,1,n)$ at its
singular point.

The group $\mathrm{GL}_{2}(k)/\mu_{n}$ is a central product of $k^{\times}$,
embedded diagonally, and $\mathrm{SL}_{2}(k)$, intersecting in the subgroup
generated by $-\mathrm{id}$, for $n$ odd. Then every term in the exceptional
sequence (using the relative version of Beilinson’s theorem as in [Orl92])

$(\ast)\quad\quad(\mathcal{O}_{\mathbb{P}^{1}},\mathcal{O}_{\mathbb{P}^{1}}(1),\pi^{*}(\mathcal{O}_{\mathbb{P}^{1}}),\pi^{*}(\mathcal{O}_{\mathbb{P}^{1}}(1))\otimes\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)),$

where
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}(n)\oplus\mathcal{O}_{\mathbb{P}^{1}}$
and $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is the relative hyperplane
bundle on
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(n)\oplus\mathcal{O}_{\mathbb{P}^{1}})\to\mathbb{P}^{1}$,
is invariant under $\mathrm{GL}_{2}(k)/\mu_{n}$. We conclude by Lemma 14. ∎

##  5\. Threefolds

There is a wealth of results concerning linearizability of $G$-actions on
rational threefolds, in the context of birational rigidity. In absense of this
property, only few examples are known. Of particular interest are threefolds
without obvious obstructions, such as nontriviality of the Amitsur invariant
or failure of (H1).

In the arithmetic context, rationality over nonclosed fields of smooth
geometrically rational Fano threefolds, e.g., those of Picard number one, has
been investigated in [HT21a], [HT21b], [BW23], [KP23]:

  * (1)

$V_{5}$,

  * (2)

$\mathbb{P}^{3}$, $Q_{3}$, $X_{12}$, $X_{22}$,

  * (3)

$V_{4}$, $X_{16}$, $X_{18}$

(we use standard notation for the threefolds from those papers). As is well-
known, $V_{5}$, a Del Pezzo threefold of degree 5, is always rational. The
rationality of forms of varieties in group (2) is controlled by the existence
of rational points. In addition to this condition, rationality of varieties in
group (3), i.e., forms of complete intersections of two quadrics, Fano
threefolds of degree 16, 18, requires the existence, over the ground field, of
lines, twisted cubics, or conics, respectively. The papers [KP23], [Kuz22] put
this into the framework of derived categories, investigating the
semiorthogonal decompositions in this context. In particular, the only cases
where the semiorthogonal decomposition does not involve Brauer classes from
the base, are $V_{5}$ and $X_{12}$, by [Kuz22, Theorem 1.1 and Theorem 1.3].

We turn to the equivariant setting and linearizability questions. The case of
quadrics is already involved [TYZ23, Section 9]:

  * •

existence of fixed points is not necessary for linearizability, when $G$ is
nonabelian,

  * •

there are cases, when linearizability is obstructed by the Burnside formalism,
but stable linearizability is open,

  * •

there are cases with no visible obstructions, but resistant to all attempts to
linearize the action.

As we are interested in situations where no obstructions are visible in the
derived category, we focus on $V_{5}$ and $X_{12}$.

### Quintic Del Pezzo threefolds

As in Example 7, let $W$ be a faithful 5-dimensional representation of a
finite group $G$, such that $\wedge^{2}(W)$ contains a faithful 7-dimensional
subrepresentation $V$, so that

$X=\operatorname{Gr}(2,W)\cap\mathbb{P}(V),$

is a smooth threefold, with generically free action of $G$, a quintic Del
Pezzo threefold. The restriction $\mathcal{U}_{X}$ of the universal rank-2
subbundle $\mathcal{U}$ over $\operatorname{Gr}(2,W)$ to $X$ is naturally
$G$-linearized. Orlov [Orl91] showed that there is a full exceptional sequence
in $\mathsf{D}^{b}(X)$ of the form

$\langle(W\otimes\mathcal{O})/\mathcal{U}\otimes\mathcal{O}(-1),\mathcal{U},\mathcal{O},\mathcal{O}(1)\rangle.$

This is a sequence of $G$-linearized objects.

There is a distinguished such threefold, with $G=\mathfrak{A}_{5}$-action,
considered in Example 7. It is:

  * •

$G$-birationally rigid, and thus $X\notin\mathfrak{P}\mathfrak{L}\mathrm{in}$,
and

  * •

$X\in\mathfrak{S}\mathfrak{L}\mathrm{in}$.

On the other hand, recall that there is a longstanding conjecture in the
context of derived categories relating the rationality of a smooth projective
variety (over an algebraically closed field of charactertistic zero) to the
existence of a full exceptional sequence in $\mathsf{D}^{b}(X)$. The above
example contradicts the most suggestive analog of this conjecture in the
equivariant context. Note that in the arithmetic context, every form of a
quintic Del Pezzo threefold is rational, see, e.g., [KP23, Theorem 1.1].

### Fano threefolds of genus 7

We follow the discussion in [Muk92, Muk95] and its summary in [Kuz05] and
[PS99]. Consider a ten-dimensional complex vector space $V$ with a non-
degenerate symmetric bilinear form on it, and denote by $\mathrm{Spin}_{10}$
the associated spinor group with $16$-dimensional half-spinor representations
$\mathrm{S}^{\pm}V$. Consider the Lagrangian Grassmannian of $5$-dimensional
isotropic subspaces of $V$: it has two connected components
$\mathrm{LGr}_{+}(V)$ and $\mathrm{LGr}_{-}(V)$ that can be identified with
the closed orbits of the group $\mathrm{Spin}_{10}$ in
$\mathbb{P}(\mathrm{S}^{+}V)$ and $\mathbb{P}(\mathrm{S}^{-}V)$. The
representations $\mathrm{S}^{+}$ and $\mathrm{S}^{-}$ are dual to each other.
Choose a pair of subspaces and their orthogonal subspaces (subscripts denote
dimensions)

$A_{8}\subset A_{9}\subset\mathrm{S}^{+}V,\quad B_{7}\subset
B_{8}\subset\mathrm{S}^{-}V.$

Let

$\displaystyle X:=$
$\displaystyle\mathrm{LGr}_{+}(V)\cap\mathbb{P}(A_{9})\subset\mathbb{P}(\mathrm{S}^{+}V),$
$\displaystyle S:=$
$\displaystyle\mathrm{LGr}_{+}(V)\cap\mathbb{P}(A_{8})\subset\mathbb{P}(\mathrm{S}^{+}V)$

and

$\displaystyle C^{\vee}:=$
$\displaystyle\mathrm{LGr}_{-}(V)\cap\mathbb{P}(B_{7})\subset\mathbb{P}(\mathrm{S}^{-}V),$
$\displaystyle S^{\vee}:=$
$\displaystyle\mathrm{LGr}_{-}(V)\cap\mathbb{P}(B_{8})\subset\mathbb{P}(\mathrm{S}^{-}V).$

It is known that $X$ is smooth if and only if $C^{\vee}$ is smooth, and $S$ is
smooth if and only if $S^{\vee}$ is smooth, which we will now assume. Then
$X=X_{12}$ is an index $1$ degree $12$ genus $7$ Fano threefold with a smooth
K3 hyperplane section $S$ (a polarized K3 surface of degree $12$); all such
pairs $(X,S)$ are obtained via the above linear algebra construction by
[Muk92, Muk95]. Moreover, $S^{\vee}$ is also a K3 surface of degree $12$ and
$C^{\vee}$ is a canonically embedded curve of genus $7$. Note that restricting
the universal bundle $\mathcal{U}$ from $\mathrm{Gr}(5,V)$, we obtain rank $5$
vector bundles $\mathcal{U}_{+},\mathcal{U}_{-}$ on $\mathrm{LGr}_{+}(V)$ and
$\mathrm{LGr}_{-}(V)$.

It is known that the Sarkisov link with center a general point $x\in X$ gives
a birational map to a quintic Del Pezzo threefold, hence $X$ is rational
[KP23, Theorem 5.17 (i)], [PS99].

We now pivot to the equivariant setup, following [Pro12, Example 2.11].
Consider $G=\mathrm{SL}_{2}(\mathbb{F}_{8})$ and let $U$ be a 9-dimensional
irreducible representation of $G$. There is a unique $G$-invariant quadric
$Q\subset\mathbb{P}(U)$, with generically free $G$-action. Note that, quite
generally, the spinor varieties for $\mathrm{Spin}_{2n-1}$ and
$\mathrm{Spin}_{2n}$ are isomorphic, indeed, projectively equivalent as
subvarieties of projective space $\mathbb{P}^{N}$, $N=2^{n-1}-1$, in their
spinor embeddings. So we can also think of $\mathrm{LGr}_{+}(V)$ as well as
$\mathrm{LGr}_{-}(V)$ as the spinor variety for $\mathrm{Spin}_{9}$,
parametrizing projective spaces of dimension $3$ on a smooth quadric in
$\mathbb{P}^{8}$. Hence $G$ acts on $\mathrm{LGr}_{+}(V)$ (and
$\mathrm{LGr}_{-}(V)$). The embedding of these into $\mathbb{P}(S^{+}V)$ and
$\mathbb{P}(S^{-}V)$ is given by the positive generator of the Picard group,
eight times of which is the anticanonical bundle. According to [Pro12, Example
2.11], the group $G$ acts in $\mathbb{P}^{15}$ with invariant projective
subspaces of dimensions $8$ and $6$ such that we get an action of $G$ on $X$
and $C^{\vee}$.

The group $G$ contains the Frobenius group $\mathfrak{F}_{8}$, which does not
act on $\mathbb{P}^{3}$, so that the $G$-action on $X$ is not linearizable.

The derived category of $X$ is described in [Kuz05, Theorem 4.4] and [Kuz22,
Theorem 5.15]. It has a semiorthogonal decomposition

$\mathsf{D}^{b}(X)=\langle\mathcal{U}_{+},\mathcal{O}_{X},\mathsf{D}^{b}(C^{\vee})\rangle.$

The proof uses the interpretation of $C^{\vee}$ as the moduli space of stable
rank $2$ vector bundles on $X$ with $c_{1}=1,c_{2}=5$ given in [IM04]. Indeed,
define

$\mathcal{A}_{X}:=\,^{\perp}\langle\mathcal{U}_{+},\mathcal{O}_{X}\rangle.$

Kuznetsov constructs a fully faithful Fourier-Mukai functor

$\Phi_{\mathcal{E}}\colon\mathsf{D}^{b}(C^{\vee})\to\mathcal{A}_{X}\subset\mathsf{D}^{b}(X)$

from the universal bundle $\mathcal{E}$ on $X\times C^{\vee}$, and this is an
equivalence of categories.

In our context, we need to check that $\Phi_{\mathcal{E}}$ is a morphism of
$G$-categories, in the sense of, e.g., [BO23, Section 2]. We do this by
showing that the $G$-category structure on $\mathrm{D}^{b}(C^{\vee})$, given
by the geometric action of $G$ on $C^{\vee}$, and the $G$-category structure
on $\mathcal{A}_{X}$ as the left orthogonal to the exceptional sequence of
$G$-linearized objects $\mathcal{U}_{+},\mathcal{O}_{X}$, coincide. This
follows directly from the fact that the Fourier-Mukai kernel bundle
$\mathcal{E}$ is a $G$-linearized vector bundle on $X\times C^{\vee}$. The
easiest way to see this is to use the explicit description of $\mathcal{E}$ in
[Kuz05, Constructions in §2 and Corollary 2.5]. Indeed, denoting by
$\mathcal{U}_{+}^{X}$ and $\mathcal{U}_{-}^{C^{\vee}}$ the pullbacks of the
tautological subbundles from $\mathrm{LGr}_{+}(V)$ and $\mathrm{LGr}_{-}(V)$
to

$X\times C^{\vee}\subset\mathrm{LGr}_{+}(V)\times\mathrm{LGr}_{-}(V),$

the bundle $\mathcal{E}$ is cokernel of the morphism

$\xi\colon\mathcal{U}_{-}^{C^{\vee}}\hookrightarrow
V\otimes\mathcal{O}_{X\times C^{\vee}}\simeq V^{*}\otimes\mathcal{O}_{X\times
C^{\vee}}\to(\mathcal{U}_{+}^{X})^{\vee},$

where the isomorphism in the middle is given by the quadratic form and the
other maps are the canonical inclusion and surjection.

In summary, the genus 7 $G$-Fano threefold $X$ furnishes another example with
a nonlinearizable action, where all pieces in a semiorthogonal decomposition
of the derived category are “geometric”, i.e., equivalent as $G$-categories to
derived categories of $G$-varieties (of dimension $\leq 1$). Thus the pieces
of these decompositions fail to detect the nonlinearizability of $X$.

## References

  * [AB18] A. Auel and M. Bernardara.  Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields.  Proc. Lond. Math. Soc. (3), 117(1):1–64, 2018.
  * [BD21] M. Bernardara and S. Durighetto.  A categorical invariant for geometrically rational surfaces with a conic bundle structure.  In Rationality of varieties, volume 342 of Progr. Math., pages 113–128. Birkhäuser/Springer, Cham, [2021] ©2021.
  * [BGvBS14] Chr. Böhning, H.-Chr. Graf von Bothmer, and P. Sosna.  On the Jordan-Hölder property for geometric derived categories.  Adv. Math., 256:479–492, 2014.
  * [BO01] A. Bondal and D. O. Orlov.  Reconstruction of a variety from the derived category and groups of autoequivalences.  Compositio Math., 125(3):327–344, 2001.
  * [BO23] T. Beckmann and G. Oberdieck.  On equivariant derived categories.  Eur. J. Math., 9(2):39, 2023.  Id/No 36.
  * [BvBT23] Chr. Böhning, H.-Chr. Graf von Bothmer, and Yu. Tschinkel.  Equivariant birational geometry of cubic fourfolds and derived categories, 2023.  arXiv:2303.17678.
  * [BW23] O. Benoist and O. Wittenberg.  Intermediate Jacobians and rationality over arbitrary fields.  Ann. Sci. Éc. Norm. Supér. (4), 56(4):1029–1084, 2023.
  * [CS16] I. Cheltsov and C. Shramov.  Cremona groups and the icosahedron.  Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.
  * [CT20] A.-M. Castravet and J. Tevelev.  Derived category of moduli of pointed curves – II, 2020.  arXiv:2002.02889.
  * [DI09] I. V. Dolgachev and V. A. Iskovskikh.  Finite subgroups of the plane Cremona group.  In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, volume 269 of Progr. Math., pages 443–548. Birkhäuser Boston, Boston, MA, 2009.
  * [DM22] Th. Dedieu and L. Manivel.  On the automorphisms of Mukai varieties.  Math. Z., 300(4):3577–3621, 2022.
  * [HKT21] B. Hassett, A. Kresch, and Yu. Tschinkel.  Symbols and equivariant birational geometry in small dimensions.  In Rationality of varieties, volume 342 of Progr. Math., pages 201–236. Birkhäuser/Springer, Cham, [2021] ©2021.
  * [HT21a] B. Hassett and Yu. Tschinkel.  Rationality of complete intersections of two quadrics over nonclosed fields.  Enseign. Math., 67(1-2):1–44, 2021.  With an appendix by J.-L. Colliot-Thélène.
  * [HT21b] B. Hassett and Yu. Tschinkel.  Rationality of Fano threefolds of degree 18 over non-closed fields.  In Rationality of varieties, volume 342 of Progr. Math., pages 237–247. Birkhäuser/Springer, Cham, [2021] ©2021.
  * [HT23] B. Hassett and Yu. Tschinkel.  Torsors and stable equivariant birational geometry.  Nagoya Math. J., 250:275–297, 2023.
  * [IM04] A. Iliev and D. Markushevich.  Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7.  Adv. Geom., 4(3):287–318, 2004.
  * [Kap88] M. M. Kapranov.  On the derived categories of coherent sheaves on some homogeneous spaces.  Invent. Math., 92(3):479–508, 1988.
  * [KO94] S. A. Kuleshov and D. O. Orlov.  Exceptional sheaves on Del Pezzo surfaces.  Izv. Ross. Akad. Nauk Ser. Mat., 58(3):53–87, 1994.
  * [KP23] A. Kuznetsov and Yu. Prokhorov.  Rationality of Fano threefolds over non-closed fields.  Amer. J. Math., 145(2):335–411, 2023.
  * [KT22a] A. Kresch and Yu. Tschinkel.  Cohomology of finite subgroups of the plane Cremona group, 2022.  arXiv:2203.01876.
  * [KT22b] A. Kresch and Yu. Tschinkel.  Equivariant birational types and Burnside volume.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 23(2):1013–1052, 2022\.
  * [KT22c] A. Kresch and Yu. Tschinkel.  Equivariant Burnside groups and representation theory.  Selecta Math. (N.S.), 28(4):Paper No. 81, 39, 2022.
  * [Kuz05] A. Kuznetsov.  Derived categories of the Fano threefolds $V_{12}$.  Mat. Zametki, 78(4):579–594, 2005.
  * [Kuz16] A. Kuznetsov.  Derived categories view on rationality problems.  In Rationality problems in algebraic geometry, volume 2172 of Lecture Notes in Math., pages 67–104. Springer, Cham, 2016.
  * [Kuz22] A. Kuznetsov.  Derived categories of families of Fano threefolds, 2022.  arXiv:2202.12345.
  * [Mar22] L. Marquand.  Cubic fourfolds with an involution, 2022.  to appear in Trans. Amer. Math. Soc., arXiv: 2202.13213.
  * [Muk92] Sh. Mukai.  Curves and symmetric spaces.  Proc. Japan Acad. Ser. A Math. Sci., 68(1):7–10, 1992.
  * [Muk95] Sh. Mukai.  Curves and symmetric spaces. I.  Amer. J. Math., 117(6):1627–1644, 1995.
  * [Orl91] D. O. Orlov.  Exceptional set of vector bundles on the variety $V_{5}$.  Vestnik Moskov. Univ. Ser. I Mat. Mekh., (5):69–71, 1991.
  * [Orl92] D. O. Orlov.  Projective bundles, monoidal transformations, and derived categories of coherent sheaves.  Izv. Ross. Akad. Nauk Ser. Mat., 56(4):852–862, 1992.
  * [Pro10] Yu. Prokhorov.  Fields of invariants of finite linear groups.  In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 245–273. Birkhäuser Boston, Boston, MA, 2010.
  * [Pro12] Yu. Prokhorov.  Simple finite subgroups of the Cremona group of rank 3.  J. Algebraic Geom., 21(3):563–600, 2012.
  * [Pro15] Yu. Prokhorov.  On stable conjugacy of finite subgroups of the plane Cremona group, II.  Michigan Math. J., 64(2):293–318, 2015.
  * [PS99] A. N. Parshin and I. R. Shafarevich, editors.  Algebraic geometry. V, volume 47 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 1999.  Fano varieties, A translation of Algebraic geometry. 5 (Russian), Ross. Akad. Nauk, Vseross. Inst. Nauchn. i Tekhn. Inform., Moscow.
  * [Sch01] F.-O. Schreyer.  Geometry and algebra of prime Fano 3-folds of genus 12.  Compositio Math., 127(3):297–319, 2001.
  * [TYZ23] Yu. Tschinkel, K. Yang, and Zh. Zhang.  Equivariant birational geometry of linear actions, 2023.  arXiv:2302.02296.
  * [Xu12] F. Xu.  On the smooth linear section of the Grassmannian $Gr(2,n)$, 2012\.  https://hdl.handle.net/1911/70498.