Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a
representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections
of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore
how invariants – both classical cohomological invariants and recent symbol constructions – control rationality in some cases.

§ INTRODUCTION

Let $X\subset \bP^{2g+1}$ be a smooth complete intersection of two quadrics over an algebraically closed field $k$
of characteristic zero. We are particularly interested in these varieties because they have a rich birational structure, which can be completely understood in small dimensions.

In this paper, we study these varieties from the perspective of their equivariant geometry,
for regular generically free actions of finite groups.
The main problem is to distinguish such actions up to equivariant birational equivalence, and in particular, to determine which of these are linearizable, i.e., equivariantly birational to a linear action on $\bP^{2g-1}$. This shares
many similarities with the study of these varieties over nonclosed fields, but has important special features.

To address this problem, we examine various canonical constructions:

* automorphism groups and their induced actions on geometric invariants;

* the structure of varieties of linear subspaces on $X$ and associated pencils of quadric hypersurfaces;

* intermediate Jacobians and their principal homogeneous spaces.

We elaborate on constructions of Reid [28], Desale-Ramanan [14], Donagi [13],
and Bhargava-Gross-Wang [2, 31] from a functorial/moduli perspective applicable to equivariant
analysis. We also present a new connection with hyperkähler geometry (see Section <ref>),
extending Kummer-type constructions to higher dimensions; connections between Fano and hyperkähler geometry are in
the focus of many recent studies, including [16].

Recent work [9, 22, 21, 8, 23] addresses rationality questions for geometrically-rational threefolds
over nonclosed fields. Our principal theorem (Theorem <ref>) demonstrates how these results translate
into equivariant contexts: for smooth complete intersections of two quadrics in $\bP^5$, rationality is governed by the existence
of lines.

In Section <ref>, we present fundamental notions of $G$-equivariant rationality and related cohomological
invariants. We summarize key geometric structures arising from odd-dimensional complete intersections of two quadrics
in Section <ref>.  Equivariant constructions and results over nonclosed fields are developed in parallel.
The resulting principal homogeneous spaces and their embeddings are explored in
Section <ref>. Section <ref> breaks from the main narrative to make a connection with hyperkähler
manifolds in small dimensions. The rest of the paper focuses on rationality problems. Two key constructions are presented
in Section <ref>. Section <ref> relates existence of fixed points to the analogous questions on
rational points over function fields.
We close with detailed analysis of the three-dimensional case in Section <ref>, highlighting both generic behavior and the special properties
of examples with large automorphism groups. This brings into sharp relief
the similarities and differences between the equivariant geometry and geometry
over nonclosed fields.

Acknowledgments: The first author was partially supported by Simons Foundation Award 546235 and
NSF grant 1701659, the second author by NSF grant 2000099.

§ ACTIONS AND INVARIANTS

In this section, the base field $k$ is algebraically closed of characteristic zero.

Let $G$ be a finite group and $X$ a smooth projective variety with regular $G$-action; we call such varieties $G$-varieties. We say that
$G$-varieties $X,Y$ are $G$-birational if there exists a $G$-equivariant birational morphism $X\to Y$;
stable $G$-birationality means $G$-birationality of
$X\times \bP^n$ and $Y\times \bP^m$, with trivial $G$-actions on the second factors. Of particular interest are cases
when $Y=\bP^n$ is projective space, with linear $G$-action.

§.§ Linear actions

A linear action on $\bP^n$ may have at least three different meanings:

* strictly linear action: the projectivization of a linear representation $G \ra \GL(W \oplus 1)$ where
$W=\bA^n$ and $1$ is the trivial representation;

* linear action: the projectivization of a linear representation $G \ra \GL(V)$ where $V=\bA^{n+1}$;

* projectively linear action: a representation $G \ra \PGL(V)$ where $V=\bA^{n+1}$.

We record a few obvious facts:

* strictly linear actions admit fixed points;

* if $L$ is a one-dimensional representation of $G$ then the representations
$V$ and $V\otimes L$ give rise to the same projective linear actions;

* using the exact sequence

$$1 \rightarrow \mu_{n+1} \rightarrow \SL_{n+1} \rightarrow \PGL_{n+1} \rightarrow 1

any projective linear action lifts to a linear action for a central extension

$$1 \rightarrow \mu_{n+1} \rightarrow \widetilde{G} \rightarrow G \rightarrow 1

and the exact sequence

$$1 \rightarrow \bG_m \rightarrow \GL_{n+1} \rightarrow \PGL_{n+1} \rightarrow 1 $$

lifts any projective linear action to a representation on a central extension

$$ 1 \rightarrow \bG_m \rightarrow \widehat{G} \rightarrow G \rightarrow 1;$$

* given a projective representation $\rho:G \ra \PGL_{n+1}$, the resulting cohomology class

\begin{equation} \label{alpha}
\alpha(\rho) \in \rH^2(G,\bG_m)[n+1]
\end{equation}

measures the obstruction to lifting $\rho$ to a linear representation of $G$.

One last observation: Suppose we are given a projective linear action of $G$ with a fixed point
$p \in \bP^n$. We obtain a lift

$$\widetilde{G} \rightarrow \SL(V)$$

and the preimage of $p$ gives a one-dimensional subspace $L\subset V$. The tensor product $V\otimes L^{-1}$
is also a representation of $\widetilde{G}$ on which $\mu_{n+1}$ acts trivially, thus descends to a
representation of $G$. Thus we find:

A projective representation with fixed point is strictly linear and the class $\alpha$ vanishes.

If there is an equivariant embedding

$$X \hookrightarrow \bP^N$$

such that $\bP^N$ admits no fixed points then the same holds true for $X$.

§.§ Notions of rationality

We say that the $G$-action on $X$ is strictly linear, linear, or projectively linear if $X$ admits a
$G$-equivariant birational map to $\bP^n,n=\dim(X),$ with the same property.  (We think of these as equivariant analogs of
rationality over nonclosed fields.) When $G$ is abelian, the existence of a fixed point is a birational invariant of smooth projective $G$-varieties <cit.>. Thus for abelian actions, the existence of a fixed point is a necessary condition for strict linearity.

The classification of $G$-varieties has been essentially settled in dimension two [12],
but is largely open in higher dimensions. The birational classification of finite group actions on projective space
remains a challenging and interesting problem [25].

Note also that two strictly linear and generically-free actions of $G$ on projective spaces
are tautologically stably $G$-birational but need not be $G$-birational.
The first assertion follows from a more general statement:

Suppose $X$ and $Y$ are smooth varieties
with generically-free $G$-actions. If there exist $G$-equvariant vector bundles $E\ra X$ and
$F\ra Y$ such that $E$ and $F$ are equivariantly birational then $X$ and $Y$ are equivariantly stably

This is a corollary of the `No-name Lemma' <cit.>: $E$ is $G$-equivariantly
birational to $\bA^{\operatorname{rank}(E)} \times X$ over $X$, where the affine factor has trivial $G$-action.
One can find examples of $X$ and $Y$ that are not $G$-birational using the invariants of  [30].

As corollary, the followings notions of $G$-equivariant stable birational equivalence coincide

* $X \times \bA^m$ and $Y\times \bA^n$ are $G$-equivariantly birational, where the affine spaces
have trivial $G$-actions;

* $X \times V$ and $Y\times W$ are $G$-equivariantly birational, where $V$ and $W$ are linear
representations of $G$;

* $X\times \bP^m$ and $Y \times \bP^n$ are $G$-equivariantly birational, where the actions on the projective spaces
are strictly linear or admit a fixed point.

The last statement follows from Proposition <ref>.

§.§ Picard and Brauer groups

We continue to assume that $G$ is a finite group acting regularly on a smooth projective variety $X$.

We refer the reader to [7] for background on line bundles and group actions.
A $G$-linearized line bundle $L\rightarrow X$ consists of a line bundle $L$ over $X$ and an action of $G$ on $L$, compatible with the action on $X$,
such that the induced action on the fibers is linear.  It follows that $\Gamma(X,L)$ is naturally a representation of $G$. Conversely, if $X\hookrightarrow \bP^n$ is $G$-equivariant with $G$ acting linearly on $\bP^n$ then $L=\cO_{\bP^n}(-1)|X$ has a natural linearization. The equivariant Picard group $\Pic_G(X)$ parametrizes $G$-linearized line bundles on $X$, up to equivariant isomorphism. We have an exact sequence <cit.>

$$0 \rightarrow {\cO(X)^\times}^G \rightarrow \cO(X)^\times \rightarrow G^\vee \rightarrow \Pic_G(X) \rightarrow \Pic(X) \rightarrow \Pic(X\times G).$$

Here $\cO(X)^\times$ denotes units and $G^\vee=\Hom(G,\bG_m)$.

Assume that $X$ is irreducible so that $\cO(X)^\times=k^\times$. Then we can refine the exact sequence to <cit.>

$$0 \rightarrow G^\vee \rightarrow \Pic_G(X) \rightarrow \Pic(X)^G \rightarrow \rH^2(G,k^\times).$$

Given a $G$-equivariant $X \hookrightarrow \bP^n$ with $G$ acting projectively on $\bP^n$, the final coboundary morphism
applied to $\cO_{\bP^n}(-1)|X$ vanishes precisely when the class $\alpha=0$ (see (<ref>).
In particular, for $L\in \Pic(X)^G$ some power $L^{\otimes N},N>0$ admits a linearization.

Let $\Br_G(X)$ denote the equivariant Brauer group of $X$, following Fröhlich and Wall [17, 18].  It parametrizes Azumaya algebras
$\cA \rightarrow X$ with $G$-action, compatible with the action on $X$ and linear over the fibers. We mod out by those of the form
$\End(\cE)$, where $\cE$ is a $G$-equivariant locally-free sheaf on $X$.  It may be computed with a Hochschild-Serre type spectral sequence
cf. <cit.>, with graded pieces

\begin{align*}
\operatorname{coker} \bigl( \Pic(X)^G &\rightarrow \rH^2(G,k^\times)\bigr) \\
\ker \bigl(\rH^1(G,\Pic(X)) &\rightarrow \rH^3(G,k^\times) \bigr) \\
\ker \Bigl( \ker \bigl( \rH^0(G,\Br(X)) \rightarrow \rH^2(G,\Pic(X)) \bigr)  & \rightarrow  \operatorname{coker} \bigl(\rH^1(G,\Pic(X)) \rightarrow \rH^3(G,k^\times) \bigr) \Bigr).
\end{align*}

Let $V$ be a finite-dimensional linear representation of $G$ over $k$. Then the homomorphism

\begin{equation} \label{eqn:pull}
\rH^2(G,\bG_m) = \Br_G(\mathrm{point}) \rightarrow \Br_G(\bP(V)),
\end{equation}

induced by the structure morphism, is an isomorphism.

Suppose that $\rho:G \rightarrow \PGL(V)$ is a projective representation.
Then the kernel of (<ref>) contains $\alpha(\rho)$.

Let $X$ be a smooth projective $G$-variety and $E\ra X$ a $G$-equivariant vector bundle.
Then the induced homomorphism

$$\Br_G(X) \rightarrow \Br_G(\bP(E))$$

is an isomorphism.

The first statement of the proposition follows from applying the Hochschild-Serre formalism to $\bP(V)$.
For the second, note that projection

$$\bP(V) \times \bP(V) \rightarrow \bP(V)$$

admits the diagonal section so Proposition <ref> gives the vanishing of $\alpha(\rho)$ on
pullback to $\bP(V)$.  The corollary follows by computing the étale Leray spectral sequence for $\bP(E)\rightarrow X$,
using Proposition <ref>.

$\Br_G$ is an equivariant stable birational invariant of smooth projective $G$-varieties.

The stable birational invariance of the Brauer group is well-known <cit.>.

We first prove birational invariance.
By weak factorization, it suffices to prove the assertion for a blow up

$$\Bl_Z(X) \rightarrow X$$

where $Z\subset X$ is a smooth and irreducible as a $G$-variety, i.e., $G$ permutes the connected components transitively.
In this situation, we observe that

* The exceptional divisor $E$ is $G$-invariant and irreducible, thus the bottom graded piece is unchanged.

* The connected components of $E$ generate a permutation module for $G$, with trivial $\rH^1$, thus the
middle graded piece is unchanged.

* We have $\Br(\Bl_Z(X))=\Br(X)$ thus the top graded piece is unchanged.

For the stable case, observe that Proposition <ref> gives

$$\Br_G(X \times \bP^n) = \Br_G(X)$$

provided the $G$-action on $\bP^n$ is linear.

Let $X$ be a smooth projective variety with $G$-action.  Assume there is a $G$-equivariant embedding

$$X \hookrightarrow \bP^n$$

where the action of $G$ on $\bP^n$ is not linear. Then $X$ is not equivariantly birational to projective space
with a linear $G$-action.

Indeed, the kernel of $\Br_G(\mathrm{point}) \rightarrow \Br_G(X)$ is a $G$-birational invariant, equal to zero for projective spaces with
linear $G$-actions.

Much of this extends to a nonclosed field $k$, except for the interpretation of
$\Br_G(\Spec(k))$ as the group cohomology $\rH^2(G,\bG_m)$. Moreover, we
would have to keep track of the Galois actions on $\mu_n$ and $\widetilde{G}$.

§ ALGEBRAIC GEOMETRY OF PENCILS OF QUADRICS

Here we assume either

* all objects are defined over a field $k$ of characteristic zero;

* the ground field is algebraically closed of characteristic zero and all objects
are equivariant for the action of a finite group $G$.

Consider a smooth complete intersection of two quadrics

$$X= \{ Q_1=Q_2=0\} \subset \bP^{2g+1}.$$

We write

$$\cQ=\Bl_X(\bP^{2g+1})=\{t_1Q_1+t_2Q_2=0\} \subset \bP^{2g+1} \times \bP^1_{t_1,t_2}$$

for the corresponding pencil and

$$q:\cQ \ra \bP^1$$

for the projection onto the second factor, a fibration in quadric hypersurfaces.

In the equivariant context, if $\bP^{2g+1}$ comes with a linear $G$-representation then $\bP^1$ has one as well.
Given a projective representation $\rho:G \ra \PGL_{2g+2}$, the existence of an $X\subset \bP^{2g+1}$ equivariant under
$\rho$ implies $4[\alpha(\rho)]=0$. Indeed, $\wedge^2(\Sym^2(\rho))$ admits a fixed point – the pencil – so Proposition <ref> gives

$$\alpha(\wedge^2(\Sym^2 (\rho)))=4\alpha(\rho)=0.$$

We recall fundamental results from [28] and [13].
The singular members of the pencil $\cQ$ are given by the degeneracy locus

$$B=\{ \det(t_1Q_1+t_2Q_2)=0\} \subset \bP^1,$$

which consists of $2g+2$ distinct geometric points $b_1,\ldots,b_{2g+2}$, each corresponding to a nodal
fiber $\cQ_{b_i}$.

Let $F_g(q) \ra \bP^1$ denote the relative variety of maximal isotropic subspaces with Stein factorization

$$F_g(q) \ra C \ra \bP^1,$$

where the first arrow is smooth.  The double cover $C\ra \bP^1$
encodes the two connected components of the variety of maximal isotropic subspaces.
The pullback of the point class on $\bP^1$ to $C$ is written $g^1_2$.
We have natural bijections between

* the branch points $b_1,\ldots,b_{2g+2}$ of $C\rightarrow \bP^1$;

* the nodes of members of the pencil $\cQ_t$.

The points $b_i$ generate a subgroup of $\Pic(C)$ presented as follows:

* $2b_i=2b_j=g^1_2$ for all $i,j$;

* $b_1+\cdots+b_{2g+2}=(g+1)g^1_2$.

The elements $b_i-b_j$ generate $J(C)[2]$, the two-torsion of the Jacobian of $C$.

The relative variety of maximal isotropic subspaces $F_g(q)$ is isomorphic to the variety parametrizing
$(g-1)$-dimensional quadric hypersurfaces contained in $X$, which is stratified by rank

$$K_0(X) \subset K_1(X) \cdots \subset K_g(X)=F_g(q).$$

Consider the variety $F_{g-1}(X) \subset \Gr(g,2g+2)$ parametrizing linear subspaces contained in $X$. We have:

* $F_{g-1}(X)$ is a principal homogeneous space over the Jacobian $J(C)$.

* There is a correspondence

\begin{array}{ccc}
\widetilde{K_1(X)} & \ra & K_1(X) \\
\downarrow &  & \\
F_{g-1}(X) \times F_{g-1}(X)  & &
\end{array}

where the horizontal arrow is the double cover reflecting the support of elements of $K_1(X)$.

* The correspondence induces a morphism

\begin{equation} \label{morphism:Wang}
\Sym^2(F_{g-1}(X)) \rightarrow \Pic^1(C)
\end{equation}

taking $K_1(X)$ to $C \subset \Pic^1(C)$, with fibers
Kummer varieties singular along the $2^{2g}$ points of $K_0(X)$ over each
point of $C$. Work of X. Wang [31] establishes the formula

\begin{equation}
2[F_{g-1}(X)]=[\Pic^1(C)], \label{eqn:Wang}
\end{equation}

as principal homogenous spaces over the Jacobian $J(C)$ of $C$,
i.e., as elements of the Weil-Châtelet group of $J(C)$.

* $K_0(X)$ may be interpreted as a
$J(C)[2]$-principal homogeneous space over $C$.
We have $2^{2g}(2g+2)$ distinguished points of $K_0(X)$ corresponding to the elements lying
over the Weierstrass points of $C$.

* The automorphisms of $X$ act faithfully on $K_0(X)$ but automorphisms lifting the hyperelliptic
involution of $C$ fix the $2^{2g}(2g+2)$ distinguished points.

We summarize the implications of the discussion above for the automorphisms:

We have an extension

$$1 \rightarrow J(C)[2] \rightarrow \Aut(X) \rightarrow \Aut(X,C) \rightarrow 1,

where $\Aut(X,C)$ is the image of $\Aut(X) \rightarrow \Aut(C)$. Moreover,

* $\Aut(X,C)$ contains the hyperelliptic involution $\iota$,

* $\Aut(X,C)/\left<\iota\right> \subset \Aut(\bP^1)$ acts via permutation on the $2g+2$ branch points of  the cover
$C\rightarrow \bP^1$,

* the induced action of $\Aut(X,C)$ on $J(C)[2]$ is induced by this permutation action.

Suppose we diagonalize the forms

$$Q_1= \sum_{i=1}^{2g+2} x_i^2 , \quad Q_2 = \sum_{i=1}^{2g+2} \lambda_i x_i^2.$$

The combinations $Q_2-\lambda_i Q_1$ correspond to the $b_i$. The $2$-elementary extension

\begin{equation}
1 \rightarrow J(C)[2] \rightarrow H \rightarrow \left< \iota\right>\rightarrow 1 \label{eqn:2extension}
\end{equation}

acts diagonally via $\pm 1$, with the image in the quotient encoding the determinant.

§ PRINCIPAL HOMOGENEOUS SPACES FOR THE JACOBIAN

We maintain the assumptions of Section <ref>.

§.§ Abstracting the principal homogeneous space

Let $C\rightarrow \bP^1$ be a hyperelliptic curve of genus $g$; in the equivariant context, we assume that the $G$-action on $C$ lifts to a linear
action on $\bP^1$.  The hyperelliptic involution $\iota$
induces an involution

\begin{equation} \label{eq:iota}
\begin{array}{rcl}
\iota: \Pic^1(C)& \rightarrow &\Pic^1(C) \\
D & \mapsto & g^1_2  - D.
\end{array}
\end{equation}

Now suppose that $F$ is a square root of $\Pic^1(C)$, a $J(C)$ principal homogeneous space satisfying
Wang's relation (<ref>). The automorphisms of $F$ include translations by $J(C)[2]$ and transformations

$$x \mapsto \iota(x)+\tau,  \quad  \tau \in \Pic^1(C), 2\tau = g^1_2,$$

encoded by the extension (<ref>).

Given $C$, does there exists an $X\subset \bP^{2g+1}$ whose associated pencil yields $C$?
A necessary condition is the existence of a $J(C)$-principal homogeneous space $F$ satisfying Wang's relation (<ref>).
However, this is not the only way such a variety may arise. Consider
a Brauer-Severi variety $P \subset \bP^{\binom{2g+3}{2}-1}$, realized geometrically as a $2$-Veronese
reimbedding of $\bP^{2g+1}$, and a pencil of hyperplane sections of $P$. The base locus $X$ is geometrically
a complete intersection of two quadrics in $\bP^{2g+1}$ and gives rise to auxiliary varieties $F_{g-1}(X)$ and $C$
as above. However $X$ is generally not embeddable in $\bP^{2g+1}$; in equivariant terms, a $G$-action

$$G\times X \rightarrow X$$

may not linearize to $\bP^{2g+1}$.
This is often the only obstruction:

Let $C\rightarrow \bP^1$ be a hyperelliptic curve of genus $g$.

In the equivariant context, we assume the group acts with a fixed point outside the branch locus.

A square root $F\rightarrow \Pic^1(C)$, as $J(C)$ principal homogeneous spaces, exists if and only if there exists codimension-two linear section

$$X \subset P\subset \bP^{\binom{2g+3}{2}-1}$$

where $P$ is a form of $\bP^{2g+1}$ realized as a Veronese variety.
Thus there exists

$$X \subset \bP^{2g+1},$$

with the group acting linearly in $\bP^{2g+1}$ in the equivariant context, if and only if we may choose $P$ such that $[P]=0 \in \rH^2(\bG_m)$.

The cohomology group is $\rH^2(G,\bG_m)$ in the $G$-equivariant context and $\Br(k)$ over a nonclosed field $k$.

We follow [2]; the case of nonclosed fields is a corollary of their results.
Given $p\in \bP^1$ ($k$-rational or $G$-fixed) that is not a branch point of $C\ra \bP^1$, write $p',p''\in C$ for the points over $p$.
We have

\begin{equation} 0 \ra T \rightarrow J_m(C) \rightarrow J(C) \rightarrow 0 \label{eq:GJextension}
\end{equation}

where the first term

$$T = (\mathbf{R}_{\{p',p''\}/\{p\}}\bG_m) / \bG_m.$$

For $p\in \bP^1$, the middle term
is the generalized Jacobian of $C$ with respect to $\{p',p''\}$. Taking two-torsion gives

\begin{equation} 0 \ra \mu_2 \ra J_m(C)[2] \ra J(C)[2]\rightarrow 0. \label{gjacmod2}
\end{equation}

Suppose that $L$ is an étale algebra of degree $2g+2$ over $k$
associated with the branch points of $C\rightarrow \bP^1$.  We have <cit.> identifications

\begin{align}
J_m(C)[2] &\Leftrightarrow (\mathbf{R}_{L/k} \mu_2)_{N=1} \\
J(C)[2] & \Leftrightarrow (\mathbf{R}_{L/k} \mu_2)_{N=1}/\mu_2
\end{align}

where $N:\mathbf{R}_{L/k} \ra \mu_2$ is the norm map from the restriction of scalars.
These act linearly and projective linearly on $\bP^{2g+1}$ respectively.
The existence of $F$ is controlled by

$$0 \rightarrow J(C)[2] \rightarrow J(C)[4] \stackrel{\times 2}{\ra} J(C)[2] \rightarrow 0;$$

given $[\Pic^1(C)] \in \rH^1(J(C)[2])$, the obstruction to a square root $[F] \in \rH^1(J(C)[4])$ sits in

$$\rH^2(J(C)[2])=\rH^2((\mathbf{R}_{L/k} \mu_2)_{N=1}/\mu_2),$$

i.e., the Steenrod square of $[\Pic^1(C)]$.
The vanishing means $\bP^{2g+1}$ descends to a Brauer-Severi variety $P$.
Moreover, the obstruction to producing $X\subset \bP^{2g+1}$ is controlled by <cit.>

$$0 \rightarrow J_m(C)[2] \rightarrow J_m(C)[4] \stackrel{\times 2}{\ra} J_m(C)[2] \rightarrow 0;$$

once $F$ exists, this is controlled via (<ref>) by a class

\alpha \in \operatorname{coker}(\rH^1(J(C)[2])\rightarrow \rH^2(\mu_2),

where $\alpha\equiv [P]$ by the identifications.

What does this argument yield (in the equivariant context) when there is no fixed point?
Assume first that the cocycle in $\rH^2(G,J(C)[2])$ vanishes.
Write $U\subset \bP^1$ for the complement of the branch points, with the induced $G$-action. The
generalized Jacobian $J_m(C)$ may still be defined over $U$ using (<ref>) with

$$T=(\mathbf{R}_{C\times_{\bP^1} U/U}\bG_m)/\bG_m.$$

We obtain

\xymatrix{
\cX  \ar@{^{(}->}[r] \ar[rd] & \cP \ar[d] \\
& U

where the vertical arrow is a Brauer-Severi fibration of relative dimension $2g+1$. If the cocycle in $\rH^2(G,J_m(C)[2])$ vanishes
then the vertical arrow is a linear $\bP^{2g+1}$ fibration.

§.§ Projective geometry

Assume first that the base field is algebraically closed.
Recalling the imbedding $F_{g-1}(X) \subset \Gr(g,2g+2)$, we have

$$\cO_{\Gr(g,2g+2)}(1)|F_{g-1}(X) = \cO_{J(C)}(4\Theta),$$

where $\Theta$ is the class of a theta divisor.
Note however that the corresponding embedding is not linearly normal as

$$4^g = \dim \Gamma(\cO_{J(C)}(4\Theta) > \dim \Gamma(\cO_{\Gr(g,2g+2)}(1) = \binom{2g+2}{g}, \quad g>1.$$

For small $g$, we have

\begin{array}{r|cc}
g & 4^g & \binom{2g+2}{g} \\
\hline
1 & 4  & 4 \\
2 & 16 & 15 \\
3 & 64 & 56 \\
4 & 256 & 210
\end{array}

We explain the reason for this discrepancy. Suppose that $(J,\Theta)$ is a principally polarized abelian variety and
$L$ is a line bundle on $J$ representing $\Theta$. For each $n\in \bN$, the Heisenberg extension associated with

$$1 \rightarrow \bG_m \rightarrow \cG(L^n) \rightarrow J[n] \rightarrow 1$$

acts on the space of global sections $\Gamma(L^n)$. Recall that the extension data is given by the commutator

$$J[n] \times J[n] \rightarrow \mu_n \subset \bG_m$$

associated with the polarization form. Suppose that $n=4$ and realize $J[2] \subset J[4]$ in the standard way; the
commutator pairing for $4\Theta$ is isotropic on $J[2]$, i.e., we may regard

$$\mu_2 \times J[2] \subset \cG(L^4)$$

as an abelian subgroup. Thus it is reasonable to diagonalize the theta functions for this group. Indeed, we have
Let $\vartheta \in \Gamma(J,L)$ denote a generator and $\tau_x^*\vartheta$ its translate under $x\in J$. Then the

$$\{2^*\vartheta_x : x\in J[2] \}$$

form a basis for $\Gamma(J,L^4)$, naturally indexed the $2$-torsion elements of $J$. (Here $2:J\ra J$ is multiplication by two.)

Assume that either $k$ is nonclosed or that all the varieties and constructions are $G$ equivariant.
For our application, we use the squaring map

$$F_{g-1}(X) \rightarrow \Pic^1(C)$$

introduced in (<ref>); thus Proposition <ref> applies.

Consider the canonical theta divisor $\vartheta=\Sym^{g-1}(C) \subset \Pic^{g-1}(C)$.
We analyze the translates of $\vartheta$ by elements in

$$\left<b_1,\ldots,b_{2g+2}\right> \subset \Pic(C)$$

contained in $\Pic^1(C)$.
These have the structure of a principal homogeneous space for $J(C)[2]$.
The Galois action on $\left<b_1,\ldots,b_{2g+2}\right>$
factors through the permutation representation on the branch points.

Suppose first that $g$ is even; here the principal homogeneous space is trivial with distinguished divisor

$$\vartheta - \frac{g-2}{2} g^1_2,$$

i.e., the canonical theta divisor translated by the $g^1_2$. Recall that $J(C)[2]$ corresponds to even partitions

$$S \sqcup S^c = \{b_1,\ldots,b_{2g+2} \}, \quad |S|=2j.$$

These are counted via the combinatorial identity for even $g$

$$4^g = \binom{2g+2}{g} + \sum_{j=0}^{g/2-1} \binom{2g+2}{2j}.$$

The sections of

correspond to translates associated with sums of $g$ branch points $b_1,\ldots,b_{2g+2}$.

For even $g$, any square root $F$ of $\Pic^1(C)$ admits a distinguished polarization of type $4\Theta$.
However, given a projective representation $\rho:G \ra \PGL(V)$ note that

$$\alpha(\wedge^g \rho) = g \alpha(\rho).$$

Thus two-torsion $\alpha(V) \in \rH^2(G,\bG_m)$ vanishes on passage from $X\subset \bP(V)$ to $F_{g-1}(X)\subset \bP(\wedge^g V)$.

Now take $g$ odd.  The odd-degree divisors in $\left<b_1,\ldots,b_{2g+2}\right>$ – a principal homogeneous space of $J(C)[2]$ – correspond to
odd partitions

$$S \sqcup S^c =\{b_1,\ldots,b_{2g+2} \}, \quad |S|=2j-1.$$

These index translates of $\vartheta$ pulling back to our desired polarization on $F_1(X)$.
We have the combinatorial identity for odd $g$

$$4^g = \binom{2g+2}{g} + \sum_{j=1}^{(g-1)/2} \binom{2g+2}{2j-1}.$$

Again, the sections correspond to translates associated with sums of $g$ branch points.

For odd $g$, any square root $F$ of $\Pic^1(C)$ has a (Galois or $G$)
invariant divisor class: the pull back of the divisors

$$\vartheta - \sum_{j\in J} b_j - \frac{g-2j-1}{2}g^1_2  \in \Pic^1(C), \quad |J|=2j-1,$$

to $F$. However, there may be an obstruction to the existence of a line bundle (defined over $k$
or linearized for $G$) on $X$ realizing this class.  Nonzero two-torsion $\alpha(V) \in \rH^2(G,\bG_m)$ remains nonzero on passage from $X\subset \bP(V)$ to $F_{g-1}(X)\subset \bP(\wedge^g V)$.

§ LAGRANGIAN INTERPRETATION IN DIMENSION THREE

The relationship between hyperkähler manifolds $K$ and Fano varieties arising as Lagrangian submanifolds
is rich and intricate. Lagrangian $\bP^n \subset K$ can often be characterized via intersection properties of the
Hodge lattice of $K$ [3, 20]. Further subtle constructions have been studied in <cit.>.
For example, cubic fourfolds arise as Lagrangian submanifold of a $K$ of dimension eight.
Here we offer an elementary example in this spirit, elaborating on the constructions introduced above.

§.§ The basic construction

Assume that the ground field is algebraically closed.

Let $X \subset \bP^5$ denote a smooth complete intersection of two quadrics,
$F_1(X)$ its variety of lines,
and $\Alb(F_1(X))$ the associated principally polarized abelian surface.
The variety of conics on $X$ fibers over a genus two curve $C$, parametrizing
connected components of the varieties of maximal isotropic subspaces in
the quadric hypersurfaces cutting out $X$.
We may interpret $\Alb(F_1(X))\simeq J(C)$.

Consider the Hilbert scheme $F_1(X)^{[4]}$ and the natural map

F_1(X)^{[4]} & \ra & J(C) \\
(\ell_1,\ell_2,\ell_3,\ell_4) & \mapsto & \ell_1+\ell_2+\ell_3+\ell_4 - h^2
\end{array}

i.e., $0$ is identified with cycles of lines obtained as
codimension-two linear sections of $X$.
The preimage of $0$ is a subvariety

$$K(X) \subset F_1(X)^{[4]},$$

a twist of the generalized Kummer sixfold $K_3(J(C))$ associated with $J(C)$.

Consider the incidence variety

$$Z=\{[x,\ell]:x \in \ell \} \subset X \times F_1(X)$$

and the projection $\pi:Z \ra X$. This is generically finite
of degree four.
The projection $\pi$ is flat over $X$.

since $Z$ and $X$ are both smooth and projective it suffices
to show that $\pi$ is equidimensional.  But if a point
$x\in X$ were contained in a positive dimensional family of
lines then either

* $X$ admits a ruled hyperplane section through $x$ – a cone
over an elliptic quartic curve – which would force $X$ to
be singular at $x$;

* $X$ admits a ruled surface of degree $<4$ through $x$,
violating the Lefschetz hyperplane theorem.

Since $X$ is assumed to be smooth, we conclude the flatness of $\pi$.

As a corollary, we obtain

There is an imbedding

\begin{array}{rcl}
j: X & \hookrightarrow & K(X) \\
x & \mapsto & \pi^{-1}(x)
\end{array}

realizing $X$ as a holomorphic Lagrangian subvariety of $K(X)$.

§.§ Numerical invariants

Assume that $X$ is general in the sense that  $\NS(J(C))$ is of rank one, generated by $[\Theta]$.
Then the Néron-Severi group of $K_3(J(C))$ has rank two and is
generated by

* $\theta$ – subschemes with support along a theta

* $e$ – where the nonreduced subschemes have
class $2e$.

The restriction homomorphism

$$j^*:\NS(K(X)) \ra \NS(X) \simeq \bZ h$$

is given by

$$j^*(\theta)=5h, \quad j^*(e)=4h.$$

The class $\theta$ on $J(C)=F_1(X)$ may be realized as the
locus $W_{\ell}\subset F_1(X)$ of lines incident
to a fixed line $\ell\subset X$, which sweeps out a divisor on $X$.
This divisor is the exceptional divisor of the projection

$$\pi_{\ell}: X \stackrel{\sim}{\dashrightarrow} \bP^3;$$

the center of the inverse map is a quintic space curve $C$ of genus two
so the exceptional locus has degree $20$. This yields the
first equation.

The locus of nonreduced subschemes on $K(X)$ restricts to the branch locus
of $\pi:Z\ra X$. Restricting to a line $\ell\subset X$, we see that

$$\pi^{-1}(\ell) = \ell \sqcup W_{\ell}$$

where the latter component has genus two and is realized as a degree-three
cover of $\ell$. Such a cover has eight branch points, so the branch locus
has class $8h$ and we get the second equation.

§.§ Reversing the construction

For the moment, let $A$ be an abelian surface over $k$, not necessarily
principally polarized.

* The group $A[4]$ acts on $A$ via translation.

The extension

$$ 0 \ra A[4] \ra G \ra \mu_2  \ra 0,$$

where $\mu_2$ acts on $A[4]$ via $\pm 1$,
acts on $A$ is well.

* Consider the addition map

$$\alpha:A^{[4]} \ra A$$

and the induced action of $G$ on $A^{[4]}$.
Note that $A[4]$ acts on the fibers and the
action of $\mu_2$ commutes with addition.
Thus $G$ acts on $K_3(A)$ as well.

Note that $G$ admits a distinguished $2$-elementary subgroup

H = A[2] \times \mu_2.

Now assume $A=J(C)$ and consider the various Lagrangian threefolds in $K_3(J(C))$.
The subgroup $H\simeq (\bZ/2\bZ)^5$ stabilizes each, acting via automorphisms.
The full group $G$ gives an orbit of $16$ components, with transitive action of

G/H\simeq A[4]/A[2]\simeq A[2].

Assume that the base field $k$ is arbitrary, of characteristic zero.
Let $C$ be a smooth projective curve of genus two over $k$ and $F$ a principal homogeneous space
for $J(C)$ such that $2[F]=[\Pic^1(C)]$.  Let $K \subset F^{[4]}$ denote the generalized Kummer variety lying
over the divisor $g^1_2 \in \Pic^2(C)$.  We may realize $F$ as the image of a cocycle for $J(C)[4]$;
the $16$ conjugate Lagrangian threefolds naturally form a principal homogeneous space
for $J(C)[4]/J(C)[2]\simeq J(C)[2]$. However, even when this is trivial there is no guarantee that the corresponding
$X$ can be defined over $k$ – the obstruction to descent is discussed in Section <ref>.

§ RATIONALITY CONSTRUCTIONS

§.§ Simple rational parametrizations

Fix a line $\ell \subset X$ and consider the projection

$$\pi_{\ell}:X \stackrel{\sim}{\dashrightarrow} \bP^{2g-1},$$

a birational map, resolved by blowing up $\ell$.
The inverse map

$$\bP^{2g-1} \rightarrow \bP^{2g+1}$$

is obtained as follows: Consider a matrix

$$\left( \begin{matrix} l_{00} & l_{01} \\
l_{10} & l_{11} \\
q_0  & q_1 \end{matrix} \right),

where the $l_{ij}$ are linear and and $q_i$ are quadratic.  The
$2\times 2$ minors generate an ideal $I_Z$, where $Z$ is the base locus
of $\pi_{\ell}^{-1}$.  Cubic forms in $I_Z$ yield the linear series inducing this
map. The kernel of the matrix gives a rational map

$$\phi: Z \dashrightarrow \bP^1,$$

which is regular for $g=2,3$. The generic fiber of $\phi$ is a quadric hypersurface
in $\bP^{2g-3}$; we may interpret this as
$\ell^{\perp}/\ell$, understood as a subquotient of the generic fiber of $q:\cQ \rightarrow \bP^1$, i.e.,
the generic fibers of $\phi$ and $q$ are equivalent in the Witt ring of $k(\bP^1)$.

Much more can be said when $g=2$; we refer the reader to the corresponding case in Section <ref>.

For the $g=3$ case, the fibration

$$\phi:Z \rightarrow \bP^1$$

is a quadric surface fibration with discriminant curve $C$. Specifically, the relative variety of lines factors

F_1(\phi) \stackrel{\varpi}{\rightarrow} C \rightarrow \bP^1,

where $\varpi$ is a smooth conic fibration.
The following conditions are equivalent:

* $\phi$ admits a section;

* $\varpi$ admits a section;

* $q:\cQ \ra \bP^1$ admits an isotropic plane containing $\ell \times \bP^1$.

These are elementary properties of quadratic forms.
The Amer-Brumer Theorem <cit.> gives a fourth condition:

* there exists a plane $\bP^2 \subset X$.

The fibration $\varpi$ admits a section if and only if we can express

$$F_1(\phi) = \bP(\cE)$$

for some rank-two vector bundle $\cE \ra C$. Desale-Ramanan [14] use such constructions to analyze
rank-two vector bundles of odd degree on hyperelliptic curves, relating automorphisms of $X$ to natural
tensor and duality operations on the vector bundles.

Over nonclosed fields, e.g. $k=\bC(s)$, and for $g\ge 2$ (resp. $g\ge 3$),
it is possible for $X\subset \bP^{2g+1}$ (resp. $F_1(X)\subset \Gr(2,2g+2)$)
to admit a rational point even when $C$ admits no divisors of odd degree.

Consider a hyperelliptic curve $C_0 \rightarrow \bP^1$ represented as a double cover branched over
$(g+1)$-orbits for an involution on $\bP^1$.  For example, if the involution is

$$[1,t] \mapsto [1,-t]$$

we could take the branch points as the roots $\pm \lambda_1, \ldots, \pm \lambda_{g+1}$ of
$\prod_{i=1}^{g+1} (t^2-a_i)$ with the $a_i$ distinct and nonzero.  Write

X_0= \{\sum_{i=1}^{2g+2} x_i^2 = \sum_{i=1}^{g+1} \lambda_i (x_{2i-1}^2-x_{2i}^2) =0 \},

with involution given by

$$(x_1,x_2,x_3,x_4,\ldots,x_{2g+1},x_{2g+2}) \mapsto (x_2,x_1,x_4,x_3, \ldots x_{2g+2},x_{2g+1});$$

the associated hyperelliptic curve $C_0$ has the desired branch locus.

Choose a quadratic extension $L/k$ and let $X$ and $C$ denote the associated quadratic twists of
$X_0$ and $C_0$.  By construction, $C$ admits no cycles of odd degree. However, $X$ is geometrically
rationally connected. The same holds for $F_1(X)$ provided $g\ge 3$
– indeed, it has ample anticanonical class and thus is geometrically rationally connected. Thus both varieties
have $k$-rational points by the Graber-Harris-Starr Theorem.

Note that <cit.> implies that $F_{g-1}(X)$ admits no $k$-rational points.

§.§ Stable rationality constructions

Fix a $(g-1)$-dimensional subspace $L \subset X$ and look at the projection from $L$:

$$\pi: X \dashrightarrow \bP^{g+1}$$

with generic fiber a projective space $\bP^{g-2}$.
We analyze the structure of this bundle.

Suppose that $L=\{x_0=\ldots=x_{g+1}\}$ so that the induced map on linear spaces factors through

\bP(\cO_{\bP^{g+1}}(-1) \oplus \cO_{\bP^{g+1}}^g)

so that $y_0,\ldots,y_g$ are the trivializing sections of the factors and linear series to $\bP^{2g+1}$ is given by

y_0,\ldots,y_g, zx_0, \ldots, zx_{g+1}.

The proper transform of $X$ has equations

A(x_i;y_j) + z Q(x_j) = B(x_i;y_j) + z R(x_j)=0,

where $A$ and $B$ are bilinear and $Q$ and $R$ are quadratic.  Eliminate $z$ to get the relation

z = -A/Q = -B/R

which gives relation

AQ-BR = F_0y_0+\cdots +F_{g-1}y_{g-1}=0

which is cubic in $x_i$ and linear in $y_j$.
This is the formula for the $\bP^{g-2}$ bundle in the product

\bP^{g+1}_{x_i} \times \bP^{g-1}_{y_j}.

Look at the locus $C$ where the morphism

\Bl_L(X) \rightarrow \bP^{g+1}

fails to be flat. The $F_j$ are linear combinations of $Q$ and $R$ with linear coefficients. The locus $C$ is the residual intersection to the locus

Z = \{Q=R=0\}

in the intersection of cubics

\{F_0=\cdots=F_{g-1}=0\}.

The contribution of $Z$ to the intersection is $3^g-(2g+1)$ whence $C$ has degree $2g+1$.
The curve $C$ is hyperelliptic, embedded via a generic $(2g+1)$-degree polarization $D$.

This is a computation with Fulton's excess intersection formula, encoded by the exact sequence

0 \rightarrow N_{Z/\bP^{g+1}}\simeq \cO_Z(2)^2 \rightarrow \cO_Z(3)^g \rightarrow Q \rightarrow 0,

where the equivalence of $Z$ equals $c_{g-2}(Q)$. Note that

(1+3ht)^g / (1+2ht)^2 = 1 + (3g-4)ht + \cdots + \frac{3^g - 2g -1}{4} h^{g-2}t^{g-2}$$

using the identity

$$\sum_{j+k=0}^{g-2} \binom{g}{k} 3^k  (j+1) (-2)^j = \frac{3^g - 2g -1}{4}.$$

Fixing $(C,D)$, what are the constructions that arise? We analyze the induced morphism

\gamma: F_{(g-1)}(X) &\rightarrow& \Pic^{2g-1}(C) \\
L & \mapsto & D.
\end{array}

following [13] and [2].
First, translation by two-torsion in $J(C)[2]$ corresponds to an automorphism of $X$ acting trivially on cohomology.
Thus we have

$$\gamma([L]+\tau) = \gamma([L]), \quad \tau \in J(C)[2].$$

Automorphisms with determinant $-1$ may be presented in the form

$$ p \mapsto -p+ \beta, \quad \beta \in \Pic^1(C), 2\beta = g^1_2.$$

These act by $-1$ on middle cohomology whence

$$\gamma(-[L]+\beta) = (2g+1)g^1_2 - D.$$

Moreover, Wang's formula $2[F_1(X)]=[\Pic^1(C)]$ [31] implies these are the only possible relations intertwining
$\gamma$ for a generic curve $C$.

To summarize our discussion:

For each hyperelliptic curve $C$ of genus $g$ and unordered pair of divisors

$$D,D' \in \Pic^{2g+1}(C), \quad D+D' = (2g+1)g^1_2,$$

we obtain a group of stable birational equivalences of $\bP^{g+1}$ parametrized by the

$$(\bZ/2\bZ)^{2g+1} \simeq \left<b_1,\ldots,b_{2g+2}\right>/\left<g^1_2\right> \subset \Pic(C)/\left<g^1_2\right>.$$

When $g=2$, we obtain birational equivalences of $\bP^3$. This is the subgroup of the Cremona group on
$\bP^3$ associated with an orbit of $F_1(X)$ under the diagonalizable automorphisms of $X$.  If $\ell \subset X$
is a line and $h \in \Aut(X)$ is a diagonalizable automorphism then the associated birational map is

$$\bP^3 \stackrel{\pi_{\ell}}{\dashleftarrow} X \stackrel{h}{\rightarrow} X \stackrel{\pi_{\ell}}{\dashrightarrow} \bP^3.$$

§ REDUCTION TO NONCLOSED FIELDS

We are interested in translating rationality criteria for geometrically rational threefolds over nonclosed fields to the equivariant context.
For example, the existence of points and subvarieties of prescribed type sometimes suffice to characterize rationality over nonclosed fields. We hope that the corresponding criteria for $G$-equivariant rationality are valid for varieties with $G$-action, where $G$ is a finite group.  We focus on situations where principal homogeneous spaces over abelian varieties control rationality.

Let $G$ be a finite group acting generically freely from the right on $P$,
an abelian variety. We do not assume that $G$ has a fixed point on $P$.
We write $\Alb(P)$ for the Albanese of $P$, the $G$-equivariant
abelian variety parametrizing the group of translations on $P$.
In other words, $P$ is a $G$-equvariant principal homogeneous space
for $\Alb(P)$.

Fix a base curve $B$, smooth and projective of genus $g$ over $k$.
Let $f:C \ra B$ be a Galois $G$ morphism, i.e., $C$ is smooth and
irreducible and admits a left $G$-action such that $k(C)/k(B)$
is Galois with group $G$. Consider the induced

$$P \times C \rightarrow C$$

and where the left-hand-side has induced left $G$-action

$$\gamma\cdot (p,c) = (p\gamma^{-1},\gamma c)$$

with quotient $P\times_G C$.
Consider the induced

$$\pi_f: P\times_G C \ra B$$

whose fibers, away from the branch locus of $f$, are geometrically isomorphic
to $P$.

Assume that $g > \dim(P)+1$ and $B$ is of general moduli.

Suppose that for every $f$ as above, $\pi_f$ has a section.
Then $G$ acts with a fixed point on $P$.

The existence of a section for $\pi_f$ is equivalent to a $G$-equivariant

$$\phi_f:C \rightarrow P$$

which factors

$$C \hookrightarrow \Pic^1(C) \ra P.$$

We recall the main result of Boggi-Looijenga <cit.>:
Let $C$ be a very general curve with a faithful action of a group $G$,
such that the quotient $C_G=G\backslash C$ has genus at least three.
Then the homomorphism

$$\bQ G \rightarrow \End_{\bQ}(J(C))$$

is an isomorphism. Here $J(C)$ is the Jacobian of $C$ and $\End$ denotes
its endomorphisms as an abelian variety.

There do exist examples over $k$, even when $k$ happens to be countable.
Indeed, the proof of Boggi-Looijenga gives the following <cit.>: Suppose that
$\cC \rightarrow Y$ is a family of curves with $G$-actions, over an
irreducible base $Y$, that dominates a component of the moduli space of
such objects. (We continue to assume that the quotient has genus at least three.) Then

$$\bQ G \rightarrow \End_{\bQ}(J(C_{\overline{k(y)}}))$$

is an isomorphism, i.e., there are no extra endomorphisms.
A standard argument using the Hilbert irreducibility theorem
and the $\ell$-adic description of endomorphisms (see [15], for example) shows that
$Y$ admits points over some finite extension with the general endomorphism

Their proof – and our argument – uses a refinement of the Riemann-Hurwitz
theorem due to Chevalley-Weil <cit.> that computes the multiplicities
of representations of $G$ in the cohomology of $C$ in terms of the branching
data of $C \ra C_G$. It implies that the multiplicity of an
irreducible representation of $G$ is at least $g(C_G)-1$.

Our assumption – that $g(C_g)=g > \dim(P)+1$ – shows that the
simple factors arising in the decomposition of $\End_{\bQ}(J(C))$ have dimension
greater than $\dim(P)$. It follows that there is no nontrivial
$G$-equivariant homomorphism

$$J(C) \rightarrow \Alb(P).$$

This forces $\phi_f$ to be constant, which gives the triviality.

§ APPLICATIONS IN DIMENSION THREE

In this section, we present parametrizations arising from the existence of $G$-invariant linear subspaces

$$L \subset X \subset \bP^{2g+1}.$$

The geometry here is both simpler and richer than the geometry over nonclosed fields. There are more possible Galois actions
than actions via automorphisms; not every subgroup of $\fS_{2g+2}$ arises as the automorphisms of a configuration of
$2g+2$ points. On the other hand, if one can find a linear subspace

$$L \simeq \bP^r \subset X$$

defined over $k$, one automatically has subspaces $\bP^s \subset X$ for all $s\le r$. This is not the case in the equivariant
context, as the underlying representation may be irreducible.

Throughout this section, $k$ is algebraically closed of characteristic zero.

§.§ Review of surface case

We review the classification of $G$-actions on smooth intersections of two quadrics in dimension $2$.
The general strategy for attacking this question uses the $G$-equivariant minimal model program; the most systematic
description may be found in [12].

Let $X \subset \bP^4$ be a smooth quartic del Pezzo surface with a generically free action of a finite group $G$.
There are $16$ lines on $X$, which are permuted by $G$.  If there exists a $G$-equivariant collection of
disjoint lines, it may be blown down to obtain a del Pezzo surface of larger degree. del Pezzo surfaces of
degree $7,8,$ or $9$ are equivariantly birational to $\bP^2$ or $\bP^1 \times \bP^1$. In degrees $5$ and $6$ there
are actions not birational to actions on homogeneous spaces as above; see Section 8 of [12] for more details.

For our purposes – to illustrate the aspects common to all dimensions – we focus on examples with generic automorphism
group. Any quartic del Pezzo surface may be written in diagonal form

$$x_0^2+x_1^2+x_2^2+x_3^2+x_4^2 = a_0 x_0^2 + a_1 x_1^2 + a_2 x_2^2 + a_3x_3^2 + a_4x_4^2=0$$

which admits a diagonal action of $H=\mu_2^5/\mu_2$.  Generically, these are only automorphisms.
Hence we focus on subgroups of

H\simeq (\bZ/2\bZ)^4

acting via sign changes on coordinates of diagonal quadrics defining $X$. Consider involutions $\iota \in H$:

§.§ (1,1,1,1,-1):

Here $X$ is a double cover of the quadric surface in $\bP^3$

$$(a_4-a_0)x_0^2 + (a_4-a_1)x_1^2 + (a_4-a_2)x_2^2 + (a_4-a_3)x_3^2$$

branched over the elliptic curve $E$ cut out by

This surface has no fixed lines and has nontrivial cohomology $\rH^1(G,\Pic(X))$
(from the curve $E$, cf. [6]) and thus is not $G$-rational or even stably rational. It is clearly minimal as the orbits of lines consist of eight pairs meeting at points. (These are rulings of the quadric surface tangent to $E$.)
Any $G$-action on a quartic del Pezzo surface containing this involution is $G$-irrational.

§.§ (1,1,1,-1,-1):
Here we have four fixed points given by $\{x_3=x_4=0\}$ and
eight orbits of disjoint lines. Thus $X$ is birational to a sextic del Pezzo surface, admitting three conic bundle structures.
One of these must be fixed under the involution,
thus $X$ is equivariantly birational to $\bP^1 \times \bP^1$ with linear action.

§.§ Reduction to the case of nonclosed fields

We turn to thee-dimensional cases. As a corollary of results in Section <ref>, we have:

Let $X \subset \bP^5$ be a
smooth complete intersection of two quadrics with  generically free action of a finite group $G$.
Then $X$ is $G$-equvariantly birational to $\bP^3$, with $G$ acting projective linearly,
if and only if there is a $G$-invariant line on $X$.

The action of $G$ on $X$ admits a canonical linearization on $\Gamma(\omega_X^{-1})=\Gamma(\cO_X(2))$.
It follows that $G$ acts projectively on $\Gamma(\cO_X(1))$ and thus on $\bP^5$.  A central extension

$$1 \rightarrow \mu_2 \rightarrow \widetilde{G} \rightarrow G \rightarrow 1$$

acts linearly on $\bP^5$.

An invariant line corresponds to a two-dimensional $\widetilde{G}$-invariant subspace. The complementary subspace
induces a projection

\pi_{\ell}: X \stackrel{\sim}{\dashrightarrow} \bP^3,

where $\bP^3$ admits a linear action of $\widetilde{G}$ and a projective action of $G$.

We turn to the converse statement: Suppose there is a $G$-equivariant birational map

$$X \stackrel{\sim}{\dashrightarrow} \bP^3$$

with $G$ acting projective linearly.

Let $B$ be a curve of large genus satisfying the conditions of Proposition <ref>.
Construct an isotrivial family $\cX \rightarrow B$, with generic fiber isomorphic to $X$, split over a $G$ covering of $B$.
It is birational over $B$ to a fibration $\cP \rightarrow B$ that is generically a $\bP^3$-bundle. The Tsen-Lang
Theorem implies it is birational over $B$ to $\bP^3 \times B$.
The variety of lines $F_1(\cX/B)$ is a principal homogeneous space over an abelian surface, over a dense open subset
of $B$.  The main result of [21] implies that $F_1(\cX/B)\rightarrow B$ admits a rational section.
Proposition <ref> implies that the associated action of $G$ on $F_1(X)$ necessarily admits a fixed point,
which yields a line $\ell \subset X$ invariant under the $G$-action.

Let $X$ have a $G$-action as above; we do not assume the existence of a fixed point or invariant line.
Nevertheless, we can construct the isotrivial fibration

$$\cX \ra B$$

which always admits a section by the Tsen-Lang (or Graber-Harris-Starr) Theorem.
Thus Proposition <ref>, stated for abelian varieties, is not valid for other classes of varieties such as rationally-connected varieties.

§.§ Relations with the Burnside formalism

The Burnside group formalism is presented in [24].
See <cit.> for refinements via Grothendieck classes
of varieties, and Section 6 of that paper for applications to cubic fourfolds.

Let $G$ be a finite group acting generically freely on $X \subset \bP^5$,
a smooth complete intersection of two quadrics. Suppose there exist

* an element $g\neq 1$ fixing a hyperplane section
$S\subset X$;

* a subgroup $\left<g\right> \subset H \subset G$ acting on $S$.

Assume that $S$ is not $H$-birational to either

* $\bP(V)$, a projective linear representation of $H$; or

* $\bP(E)$ where $E$ is an $H$-equivariant rank-two vector bundle over a curve.

Then $X$ is not $G$-birational to $\bP^3$ with a projectively linear $G$-action.

The surface case is addressed in <cit.>.

Suppose we have a $G$-birational map

$$\rho: \bP^3 \stackrel{\sim}{\dashrightarrow} X$$

with $G$ acting projective linearly on the source. By weak factorization, we may assume it is a composition of blowups and blowdowns along smooth centers.

Consider the center $Z \rightarrow \bP^3$ blown up to obtain $S$; it is irreducible, fixed by $g$, and has an action of $H$.  Since $S \dashrightarrow Z$ is birational, we conclude that $S$ is birational to a divisor in $\bP^3$ with nontrivial stabilizer.
The locus in $\bP^3$ with nontrivial stabilizers is a union of linear subspaces, and we are in the first case.
If $Z$ is a point then $S$ is birational to the projectivization of the tangent space at that point; again, we are in the first case.
Finally, suppose $Z$ is a curve; then $S$ is birational to the projectivization of the normal bundle to $Z$, putting us in the second case.

We turn to other representative examples. We follow Avilov [1], who enumerated actions of finite groups on three-dimensional
complete intersections or two quadrics known to be equivariantly
birational to projective space, a quadric hypersurface, or a Mori fiber space.

Suppose that $G$ admits a subgroup $H\simeq C_2 \times C_2$ acting on $X$ via the diagonal matrices

$$\operatorname{diag}(1,1,1,1,\pm 1, \pm 1).$$

which fixes a del Pezzo surface $S$. The residual action on $S$ was considered in Section <ref>.
Since there is nontrivial cohomology the two possibilities from Proposition <ref> are precluded.
It follows that $X$ is not equivariantly rational for any group containing $H$.

We emphasize that Theorem <ref> gives a stronger conclusion:
If $X$ is $G$-birational to $\bP^3$ then $G$ acts on $F_1(X)$ with fixed points.
In particular, every element of $G$ has fixed points so $G$ has no elements conjugate to
$\pm \operatorname{diag}(1,1,1,1,-1,-1)$.
Indeed, these correspond to translates by two-torsion, which act freely.
For instance, if $G=\left<(1,1,1,1,-1,-1)\right>$ the fixed locus
is an elliptic curve and there are no Burnside
invariants available.

It would be interesting to have a general theory – in the context of $G$-equivariant birational geometry –
encompassing both Burnside/symbol-type invariants and obstructions arising from Chow-theoretic
principal homogeneous spaces for intermediate Jacobians.

§.§ Rational complete intersections need not have points

For noncyclic groups, the existence of a $G$-invariant line does not give the existence of a point fixed by $G$.
There are examples of $G$-rational $X\subset \bP^5$ with no fixed points.

Let $C$ denote the complex curve

Its automorphism group contains $G$,  a central extension

$$1 \rightarrow \mu_2=\left<\iota\right> \rightarrow G \rightarrow D_{12} \rightarrow 1$$

of the dihedral group $D_{12}$ of order twelve.
We write

$$G=\left<\sigma,\tau: \sigma^6=\tau^2=1, \tau \sigma \tau^{-1} \sigma = \iota \right>$$

with action

\sigma\cdot (x,y) = (\zeta x, y), \quad \tau \cdot (x,y) = (y/x^3,1/x), \quad \zeta=e^{2\pi i /6},

where $\iota$ is the hyperelliptic involution.  The induced action on the global sections $\Gamma(\omega_C)$ is

$$\sigma \mapsto \left( \begin{matrix} \zeta & 0 \\
0 & \zeta^2 \end{matrix} \right), \quad
\tau \mapsto \left( \begin{matrix} 0 & -1 \\
-1 & 0 \end{matrix} \right), \quad
\iota \mapsto \left( \begin{matrix} -1 & 0 \\
0 & -1 \end{matrix} \right).$$

Actually, $\Aut(C)=G$; see
<cit.> for more information.

Consider the quotient of $C$ under the unique cyclic subgroup of order three $\left<\sigma^2\right>$,
with invariants and equation

$$y,z=x^3 \quad y^2=z^2+1.$$

Let $\rho:C \rightarrow R$ denote the corresponding degree-three morphism. The induced action on $R$
is generically free via the dihedral group of order eight.

The double cover and $\rho$ give a morphism

$$C \hookrightarrow \bP^1 \times R.$$

While $R$ is isomorphic to $\bP^1$ as a variety, the action of $G$ on $R$ is only projectively linear. (The subgroup
$\left<\tau,\sigma\right>$ acts on $R$ as a Klein four-group, with no fixed points; such actions are not linearizable.)
However, we do have a central extension

$$1 \rightarrow \mu_2 \rightarrow \widetilde{G} \rightarrow G \rightarrow 1$$

and $\widetilde{G}$-representations $V=\Gamma(\omega_C)$ and $W$ such that

$$C \hookrightarrow \bP(V) \times \bP(W) \hookrightarrow \bP(V\otimes W)\simeq \bP^3$$

realizing our curve as a $(2,3)$ divisor.  The linear series of cubic forms vanishing on $C$ gives a birational map

\bP^3 \stackrel{\sim}{\dashrightarrow} X \subset \bP^5,

where $X$ is a smooth complete intersection of two quadrics. This blows up $C$ and blows down
$\bP(V)\times \bP(W)$ via projection to the first factor. The map $X\stackrel{\sim}{\dashrightarrow} \bP^3$
is projection from a $G$-stable line $\ell \simeq \bP(V)$.

We claim that $X$ has no $G$-fixed points. We know that $\bP(V)$ has no fixed points so it suffices
to check that $\bP^3=\bP(V\otimes W)$ has no fixed points. Looking at $\sigma$ acting on $V$,
any fixed points would necessarily lie on

$$[1,0]\times W \text{ or }[0,1]\times W.$$

However, we have already seen that the dihedral group of order eight acts on $R$ without fixed points.

This analysis also shows that the $G$-action on $X$ does not linearize to the ambient $\bP^5$.
This means that $\Br_G(\mathrm{point}) \ra \Br_G(X)$ is not an isomorphism. Its kernel contains
$0\neq \alpha(G,W) \in \rH^2(G,\mu_2)$.  This illustrates the general obstruction analysis
in Section <ref>.

This answers a question of Avilov <cit.>, who asked where $X$ – Case 2(ii) of his Theorem 1– fits in the equivariant
birational classification.

§.§ Another special example

We return to another example highlighted by Avilov – Case 2(iv) of <cit.> .

We consider the case where $G$ fits into an exact sequence

$$1 \ra C_2^5 \ra G \ra \fS_4 \ra 1.$$

The associated binary sextic form is

The hyperelliptic curve

$$C = \{U^2 = T_0T_1(T_0^4-T_1^4) \}$$

has automorphism group

$$\Aut(C) = \left< \sigma, \tau \right>,

$$\sigma(T_0,T_1,U) = (e^{3\pi i /4}T_0, e^{\pi i /4}T_1,U) \quad
\tau(T_0,T_1,U) = (\frac{1}{\sqrt{2}}(T_1-T_0), \frac{1}{\sqrt{2}}(T_0+T_1),U).$$

The resulting group has relations

$$\sigma^4=\iota, \tau^2=\iota^2=1, \iota \tau = \tau \iota, (\sigma \tau)^3=\iota,$$

where $\iota$ is the hyperelliptic involution. It sits in a central extension

$$1 \rightarrow \mu_2=\left<\iota\right> \rightarrow \Aut(C) \rightarrow \fS_4 \rightarrow 1.$$

This curve appears as

We analyze the induced action on the branch points, using coordinate $T_0/T_1$:

$$b_1=\{ 0 \}, b_2= \{\infty\}, b_3 = \{1 \}, b_4=\{i\}, b_5 = \{-1\}, b_6=\{-i\}.$$

The generators act via permutations

$$\sigma \mapsto (3456), \quad \tau \mapsto (13)(25)(46)$$

and the hyperelliptic involution acts trivially.
Consider the induced action on

$$\Pic^1(C) \supset C$$

as in (<ref>).
Note that $\iota$ fixes only the solutions to $L^2=g^1_2$, the $16$ points (cf. Section <ref>)

None of these is simultaneous fixed by $\sigma$ and $\tau$, so $\Pic^1(C)$ admits no fixed point for $\Aut(C)$.

As for the complete intersection, we may take equations

X=\{Q_1=Q_2=0\} \subset \bP^5,

$$Q_1 = x_0^2 + x_1^2 + i x_2^2 - x_3^2 - i x_4^2, \quad
Q_2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2.$$

Suppose that $G$ acts on $X$ so that the induced homomorphism

$$G \rightarrow \Aut(C)$$

is surjective. Then $X$ is not $G$-equivariantly birational to

This partly answers a question in <cit.>; Avilov asked whether these are equivariantly birational to $\bP^3$.

By Theorem <ref>, if $X$ is birational to $\bP^3$ then $F_1(X)$ admits a fixed point. But then
$\Pic^1(C)$ would admit one as well via the squaring map $F_1(X) \ra \Pic^1(C)$. This would contradict the computation above.

The argument works under the weaker hypothesis that the image contains the $2$-Sylow subgroup

$$\left<\sigma, \tau\sigma^2\tau \right> \subset \Aut(C)$$

$\tau \sigma^2 \tau \mapsto (12)(46).$

We may have rationality when smaller groups act. We restrict attention to the automorphism of order eight acting on coordinates by

$$\gamma=\left( \begin{matrix}
\alpha & 0 & 0 & 0 &  0 & 0 \\
0      & 0 & 0 & 0 & -1 & 0 \\
0      & 1 & 0 & 0 &  0 & 0 \\
0      & 0 & 1 & 0 &  0 & 0 \\
0      & 0 & 0 & 1 &  0 & 0 \\
0      & 0 & 0 & 0 &  0 & 1
\end{matrix} \right), \quad \alpha=e^{3\pi i/4}.$$

This is chosen so that $\gamma\cdot Q_2=Q_2$ and

$$\gamma\cdot Q_1 = \alpha^2 x_0^2 + x_2^2 + i x_3^2 - x_4^2 - i x_1^2 = -i Q_1.$$

Thus we may interpret $\gamma$ as a lift of $\sigma^{-1}$.

What are the fixed points?
The action on the underlying space via the contragredient representation is:

$$\left( \begin{matrix}
\alpha^7 & 0 & 0 & 0 &  0 & 0 \\
0      & 0 & 0 & 0 & -1 & 0 \\
0      & 1 & 0 & 0 &  0 & 0 \\
0      & 0 & 1 & 0 &  0 & 0 \\
0      & 0 & 0 & 1 &  0 & 0 \\
0      & 0 & 0 & 0 &  0 & 1
\end{matrix} \right),$$

with eigenvectors

\begin{align*}
1: \ &  [0,0,0,0,0,1] \not \in X \\
\alpha: \ & p_1:=[0,1,\alpha^7,\alpha^6,\alpha^5,0]\in X \\
\alpha^3: \ & [0,1,\alpha^5,\alpha^2, \alpha^7,0] \not \in X \\
\alpha^5: \ & p_2:=[0,1,\alpha^3,\alpha^6,\alpha,0] \in X \\
\alpha^7: \ & [1,0,0,0,0,0], [0,1,\alpha,\alpha^2,\alpha^3,0]\not \in X.
\end{align*}

The last eigenspace is isotropic for $\{Q_2=0\}$ and thus meets $X$ in
two points $p_3,p_4:=[\pm 2i, 1,\alpha,\alpha^2,\alpha^3,0].$
The span of $p_1$ and $p_2$ is also isotropic for $\{Q_2=0\}$.

The lines $\ell(p_2,p_3)$ and $\ell(p_2,p_4)$ are invariant under the action.

Note however that

\begin{matrix}
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 &  0 & 0 \\
0 & 0 & -1 & 0 &  0 & 0 \\
0 & 0 & 0 & -1 &  0 & 0 \\
0 & 0 & 0 & 0 &  -1 & 0 \\
0 & 0 & 0 & 0 &  0 & 1
\end{matrix}
\right)

acts on the variety of lines on $X$ with $16$ fixed points, i.e., the lines on the
quartic del Pezzo surface $X \cap \{x_5=0\}$.

This example is instructive in that the group
actions are not associated with Weyl groups:

* $W(D_6)$ is  realized as the signed permutation
matrices with an even number of $(-1)$ signs,
a semidirect product

W(D_6) \simeq \fS_6 \ltimes C_2^5,

where $C_2^5$ is the diagonal matrices;

* the semidirect product

$$\fS_6 \ltimes C_2^6/\text{Diagonal},$$

interpreted as a Weyl group for the disconnected projective orthogonal group.

These are not the same; the Weyl group has a nontrivial central element

whereas the latter group has no central elements.

However, neither of these actions coincide with our
situation! Both lack an element $\gamma$ of order eight,
sitting over a four-cycle of $\fS_6$, with $\gamma^4 \in H'$
(the $2$-elementary group of diagonal automorphisms) having nontrivial determinant.
The automorphism group $\Aut(X)$ is NOT a semidirect
product of $\Aut(D\subset P^1)$ and $H'$, which
admits NO element $\gamma$ mapping to

\left(
\begin{matrix}
1 & 0 \\
0 & -i
\end{matrix}\right)

and whose fourth power is diagonal with entries $(-1,-1,-1,-1,-1,1)$.
Indeed, the candidate in the Weyl group

$$\tilde{\gamma} = \left(
\begin{matrix}
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 &  0 & 0 \\
0 & 0 & 1 & 0 &  0 & 0 \\
0 & 0 & 0 & 1 &  0 & 0 \\
0 & 0 & 0 & 0 &  0 & 1
\end{matrix}
\right)$$

$$\tilde{\gamma}^4 = \left(
\begin{matrix}
1& 0 & 0 & 0 & 0 & 0 \\
0 & -1& 0 & 0 & 0 & 0 \\
0 & 0 & -1& 0 & 0 & 0 \\
0 & 0 & 0 & -1& 0 & 0 \\
0 & 0 & 0 & 0 &-1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{matrix}
\right)

which has the wrong diagonal entries.

The rationality of these specific examples may be deduced in terms of the group actions:

Let $\gamma$ be an automorphism of order eight on $X$ such that
$\gamma^4$ fixes a smooth hyperplane section $S\subset X$ and
the induced action on $S$ is by an element acting on the degeneracy locus (of both $X$ and $S$) as a four-cycle. Then $S$ admits a line invariant under $\gamma$.

The fixed locus of $H'$ corresponds to a del Pezzo $S$ of degree 4
with action associated with the upper $5\times 5$ matrix.
Its lines correspond to weights in the $D_5$ root system:

-e_i - e_j - e_k - e_l - e_m,
e_i + e_j - e_k - e_l - e_m,
e_i + e_j + e_k + e_l - e_m.

With respect to the standard basis of the Picard group $\{L,E_1,E_2,E_3,E_4,E_5\}$
projected into $\Pic(S)/\bZ K_S$ we have

$$L-E_i \mapsto e_i,  \quad 2L-E_j-E_k-E_l-E_m \mapsto -e_i.$$

The induced automorphism has order four. The only possible elements of $W(D_5)$ of order four fix a line.
For example, the signed permutations

$$\left( \begin{matrix}
1 & 0 & 0 & 0 & 0 \\
0  & 0 & 0 & 0 & -1 \\
0   & 1 & 0 & 0 & 0 \\
0  & 0  & 1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0
\end{matrix}
\right),
\left( \begin{matrix}
1 & 0 & 0 & 0 & 0 \\
0  & 0 & 0 & 0 & 1 \\
0   & 1 & 0 & 0 & 0 \\
0  & 0  & 1 & 0 & 0 \\
0. & 0 & 0 & 1 & 0
\end{matrix}\right),
\left( \begin{matrix}
1 & 0 & 0 & 0 & 0 \\
0  & 0 & 0 & 0 & -1 \\
0   & -1 & 0 & 0 & 0 \\
0  & 0  & -1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0
\end{matrix}\right)

are all conjugate in $W(D_5)$ so it suffices to consider the
first. This acts on $\Pic(S)$ by

\begin{align*}
L &\mapsto 2L-E_1-E_3 -E_4 \\
E_1 & \mapsto L-E_3-E_4 \\
E_2 & \mapsto L - E_1 - E_4 \\
E_3 & \mapsto L - E_1 - E_3 \\
E_4 & \mapsto E_2 \\
E_5 & \mapsto  E_5
\end{align*}

which leaves the lines $E_5$ and $2L-E_1-E_2-E_3-E_4-E_5$

The assumption that the induced permutation of the degeneracy
locus is a four-cycle is essential. The element

$$\gamma_1=\left( \begin{matrix}
1 & 0 & 0 & 0 & 0 \\
0  & 0 & 0 & -1 & 0 \\
0   & 0 & 0 & 0 & -1 \\
0  & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0
\end{matrix}\right)

acts via

\begin{align*}
L &\mapsto 2L-E_1-E_4 -E_5 \\
E_1 & \mapsto L-E_4-E_5 \\
E_2 & \mapsto L - E_1 - E_5 \\
E_3 & \mapsto L - E_1 - E_4 \\
E_4 & \mapsto E_3 \\
E_5 & \mapsto  E_2
\end{align*}

with orbits

\begin{align*}
E_1 \mapsto L-E_4-E_5 \mapsto& 2L-E_1-E_2-E_3-E_4-E_5 \mapsto L-E_2-E_3 \\
E_2 \mapsto L-E_1-E_5 \mapsto& L-E_1-E_2 \mapsto E_5 \\
E_3 \mapsto L-E_1-E_4 \mapsto& L-E_1-E_3 \mapsto E_4 \\
L-E_2-E_4  \mapsto L-E_3-E_4 \mapsto& L-E_3-E_5 \mapsto L-E_2-E_5
\end{align*}

However, $\gamma_1$ maps to a product of two transposition in $\fS_6$.

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The following version of the `No-name Lemma' shows that $G$-equivariant stable birational equivalence is quite

Let $X$ and $Y$ be a smooth projective varieties with generically-free $G$-action. Suppose they admit $G$-equivariant vector bundles
$E$ and $F$ such that $\bP(E)$ and $\bP(F)$ are $G$-equivariantly birational. Then $X$ and $Y$ are $G$-equivariantly
stably birational.

STABLE BIRATIONAL EQUIVALENCE AND NONAME LEMMA

§ EQUIVARIANT QUADRICS

Let $V$ be a finite-dimension representation of a finite group $G$.

An equivariant nondegenerate quadratic form is a homomorphism

$$Q:\Sym^2(V)\ra L$$

to a one-dimensional representation, such that the induced
$V \ra V^* \times L$ is an isomorphism. A neutral quadratic form is one of the form

$$W \oplus (W^* \otimes L),$$

where $W$ is a representation, $L$ is one-dimensional, and the quadratic form $Q$ s the duality pairing between $W$ and $W^*$.
It follows that $W$ and $W^*\otimes L$ are isotropic for $Q$.

Suppose we are given a equivariant nondegenerate quadratic form

$$Q: V \times V \rightarrow L$$

and an isotropic subrepresentation $W \subset V$. Then there exists a distinguished subrepresentation

$$W^* \otimes L \subset V$$

such that

$$W \oplus W^*\otimes L \subset V$$

is neutral.
Suppose that $(V_1,Q_1)$ and $(V_2,Q_2)$ are equivariant nondegenerate quadratic forms with values in $L$
such that

$$V_1 \oplus_{\perp} (W \oplus W^*\otimes L) \simeq V_2  \oplus_{\perp} (W \oplus W^*\otimes L).$$

Then $(V_1,Q_1)$ and $(V_2,Q_2)$ are equivariantly isomorphic.

Motivation was to formulate an equivariant analog of the Amer-Brumer theorem:

The converse is the following variation on the Amer-Brumer theorem:
Let $k$ be a field of characteristic zero, $Q_1,Q_2\in k[x_0,\ldots,x_n]$ quadrics,
$\cQ=\{t_1Q_1+t_2Q_2=0\} \subset \bP^n \times \bP^1$ the corresponding pencil,and
$X=\{Q_1=Q_2\} \subset \bP^n$ its base locus.
Suppose we are given a linear subspace $L\subset X$ and the corresponding isotropic subspace $L\times \bP^1 \subset \cQ$.
If there exists an isotropic subspace

$$L \times \bP^1 \subset \Pi \subset \cQ$$

of relative dimension $r$ then there exist subspaces

$$L' \subset \bP^r \subset X, \quad \dim(L')=\dim(L).$$