# Automorphisms of unstable ${\mathbb{P}}^{1}$-bundles

János Kollár<EMAIL_ADDRESS>
###### Abstract.

Let $P\to X$ be a ${\mathbb{P}}^{1}$-bundle over a variety $X$. The aim of
this note is to understand all connected, algebraic groups

$\operatorname{Aut}^{\circ}(P)\subset
G\subset\operatorname{Bir}(X\times{\mathbb{P}}^{1}).$

We get a quite complete answer if $\operatorname{Aut}^{\circ}(X)$ is a
maximal, connected, algebraic subgroup of $\operatorname{Bir}(X)$, and $P$ is
sufficiently unstable.

This gives examples of connected, algebraic subgroups of
$\operatorname{Bir}({\mathbb{P}}^{4})$ that are not contained in any maximal
one.

Enriques described all connected, algebraic subgroups of the 2-dimensional
Cremona group $\operatorname{Bir}({\mathbb{P}}^{2})$ in 1893. In dimension 3,
Umemura classified all such subgroups, in a series of papers completed in
[Ume88]. They are all contained in a maximal one. Much less is known about
algebraic subgroups of $\operatorname{Bir}({\mathbb{P}}^{n})$ for $n\geq 4$.

Fong and Zikas gave examples of connected, algebraic subgroups of
$\operatorname{Bir}(C\times{\mathbb{P}}^{n})$ that are not contained in any
maximal one, where $C$ is a non-rational curve [Fon24, FZ23]. Fanelli, Floris
and Zimmermann constructed similar subgroups in
$\operatorname{Bir}({\mathbb{P}}^{n})$ for $n\geq 5$, answering a question of
Blanc [BFT23]; see [FFZ23]. The proof in [FFZ23] relies on the existence of a
stably rational but nonrational conic bundle 3-fold $X$ [BCTSSD85], a careful
investigation of the birational geometry of unstable
${\mathbb{P}}^{1}$-bundles over $X$, and a study of unipotent group actions.
For background and previous results, see [BFT22, BFT23] for birational group
actions, and [Sar80, Cor95, KM98, KSC04] for minimal models and birational
maps.

In this note we follow [Fon24, FZ23, FFZ23], but, instead of using birational
geometry, we focus entirely on the action of the unipotent radical of an
algebraic group. This leads to examples in
$\operatorname{Bir}({\mathbb{P}}^{4})$ as well.

As another consequence, our results apply over algebraically closed fields of
arbitrary characteristic.

The main results are Theorems 12 and 20. Here we state a special case that is
easy to formulate, yet leads to many interesting subgroups of
$\operatorname{Bir}({\mathbb{P}}^{4})$.

######  Theorem 1.

Let $X$ be a smooth, projective variety such that
$\operatorname{Aut}^{\circ}(X)$ is a maximal, connected, algebraic subgroup of
$\operatorname{Bir}(X)$. Let $L$ be an
$\operatorname{Aut}^{\circ}(X)$-linearized, very ample line bundle on $X$, and

$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}\subset
G\subset\operatorname{Bir}(X\times{\mathbb{P}}^{1})$

a connected, algebraic subgroup. Assume that
$(X,L)\not\cong\bigl{(}{\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}$.

Then there is an $\operatorname{Aut}^{\circ}(X)$-invariant, effective divisor
$D$ on $X$ such that

$G\subset\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}\bigl{(}{\mathcal{O}}_{X}{\oplus}L(D)\bigr{)}\bigr{)}.$

Note that [Fon24, FZ23, FFZ23] consider cases for which
$\operatorname{Aut}^{\circ}(X)=\\{1\\}$, and use a Noether-Fano type argument
to show that $G$ preserves the negative section of
${\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)$. It is quite likely that one
can prove Theorem 1 using the methods of [Fon24, FZ23, FFZ23], at least in
characteristic 0.

For Corollaries 2–4 we use the notation and assumptions of Theorem 1, and the
description of
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}$
given in Paragraph 18.

######  Corollary 2.

$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}$
is a maximal, connected, algebraic subgroup of
$\operatorname{Bir}(X\times{\mathbb{P}}^{1})$ iff
$H^{0}(X,L)=H^{0}\bigl{(}X,L(D)\bigr{)}$ for every
$\operatorname{Aut}^{\circ}(X)$-invariant divisor $D$ on $X$. ∎

######  Corollary 3.

$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}$
is a maximal, connected, algebraic subgroup of
$\operatorname{Bir}(X\times{\mathbb{P}}^{1})$ for every very ample $L$ iff
there are no $\operatorname{Aut}^{\circ}(X)$-invariant divisors on $X$.

Proof. One direction follows from Corollary 2. Conversely, if $D\neq 0$ is an
$\operatorname{Aut}^{\circ}(X)$-invariant divisor then there is a very ample
$L$ such that $h^{0}(X,L)<h^{0}\bigl{(}X,L(D)\bigr{)}$. Then
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}\subsetneq\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L(D))\bigr{)}$∎

If there are few $\operatorname{Aut}^{\circ}(X)$-orbits, quite likely this can
also be proved using the Noether-Fano method [KSC04], but the general case is
not clear to me.

######  Corollary 4.

$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}$
is not contained in any maximal, connected, algebraic subgroup of
$\operatorname{Bir}(X\times{\mathbb{P}}^{1})$, if there is an
$\operatorname{Aut}^{\circ}(X)$-invariant divisor $D^{\prime}$ such that
$h^{0}\bigl{(}X,{\mathcal{O}}_{X}(D^{\prime})\bigr{)}>1$.

Proof. By Theorem 1, if
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}\subset
G$, then
$G\subset\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L(D))\bigr{)}$
for some $\operatorname{Aut}^{\circ}(X)$-invariant, effective divisor $D$.

Since $h^{0}\bigl{(}X,{\mathcal{O}}_{X}(D^{\prime})\bigr{)}>1$, we see that
$h^{0}\bigl{(}X,L(D+D^{\prime})\bigr{)}>h^{0}(X,L(D))$. Thus
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L(D))\bigr{)}$
is contained in the strictly bigger group
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L(D+D^{\prime}))\bigr{)}$.
∎

Using Umemura’s classification of maximal, connected, algebraic subgroups of
$\operatorname{Bir}({\mathbb{C}}{\mathbb{P}}^{3})$, there are 3 cases over
${\mathbb{C}}$ to which Corollary 4 applies. [BFT23] gives an algebraic proof
of the classification, which also works in characteristics $p>5$ (though some
parts are not yet fully written down).

######  Example 5.

Let $X$ be a Fano 3-fold with an action $\mathrm{PGL}_{2}\curvearrowright X$
with a dense orbit. The complement is an ample divisor $D$. Thus
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}\bigl{(}{\mathcal{O}}_{X}{\oplus}{\mathcal{O}}_{X}(nD)\bigr{)}\bigr{)}$
is not contained in any maximal, connected, algebraic subgroup of
$\operatorname{Bir}({\mathbb{P}}^{4})$ for $n\geq 1$.

There are only 2 such examples, both described in [MU83]. One is now called
the Mukai-Umemura 3-fold $X\subset{\mathbb{P}}^{21}$. The other is the unique
$X\subset{\mathbb{P}}^{6}$ whose general hyperplane section is a degree 5 Del
Pezzo surface.

######  Example 6.

Let $g(u_{0},u_{1})$ be a homogeneous form without multiple roots and

$Q_{g}:=(x_{0}^{2}-x_{1}x_{2}-g(u_{0},u_{1})x_{3}^{2}=0)\subset{\mathbb{P}}^{3}_{x}\times{\mathbb{P}}^{1}_{u}.$

Together with a certain $\mathrm{PGL}_{2}$-action on the
${\mathbb{P}}^{3}_{x}$-factor as in [BFT23, 4.4.4], these are called Umemura
quadric fibrations.

We again get that
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{Q_{g}}({\mathcal{O}}_{Q_{g}}{\oplus}L)\bigr{)}$
is not contained in any maximal, connected, algebraic subgroup of
$\operatorname{Bir}({\mathbb{P}}^{4})$. The higher dimensional Umemura
fibrations studied in [FZ24] give similar examples in
$\operatorname{Bir}({\mathbb{P}}^{n})$ for $n>4$.

The subtle example of [FFZ23] can also be treated using Corollary 4.

######  Example 7.

Let $X$ be a non-rational but rationally connected 3-fold. Then $\\{1\\}$ is
the maximal, connected, algebraic subgroup of $\operatorname{Bir}(X)$. We get
that
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}$
is not contained in any maximal, connected, algebraic subgroup of
$\operatorname{Bir}(X\times{\mathbb{P}}^{1})$.

As noted in [FFZ23], there is such an $X$ for which
$X\times{\mathbb{P}}^{2}\stackrel{{\scriptstyle
bir}}{{\sim}}{\mathbb{P}}^{5}$; see [BCTSSD85]. [FFZ23] uses this to obtain
subgroups

$G\cong\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\bigr{)}\times\mathrm{PGL}_{n-3}\subset\operatorname{Bir}({\mathbb{P}}^{n})$

that are not contained in any maximal algebraic subgroup of
$\operatorname{Bir}({\mathbb{P}}^{n})$ for $n\geq 5$.

The construction in [FFZ23] uses a non-ample line bundle; these can be treated
using Theorem 12.

The next example shows that the assumption
$(X,L)\not\cong\bigl{(}{\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}$
is necessary in Theorem 1. (Strictly speaking,
${\mathcal{O}}_{{\mathbb{P}}^{n}}(1)$ does not have a
$\mathrm{PGL}_{n+1}$-linearization for $n>0$. However, is has an
$\mathrm{SL}_{n+1}$-linearization, which is enough for Theorem 1.)

######  Example 8.

Let $X={\mathbb{P}}^{n}$ and $L={\mathcal{O}}_{{\mathbb{P}}^{n}}(1)$. Then
${\mathbb{P}}_{{\mathbb{P}}^{n}}\bigl{(}{\mathcal{O}}_{{\mathbb{P}}^{n}}{\oplus}{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}$
is birational to ${\mathbb{P}}^{n+1}$, hence

$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{{\mathbb{P}}^{n}}\bigl{(}{\mathcal{O}}_{{\mathbb{P}}^{n}}{\oplus}{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}\bigr{)}\subset\operatorname{Aut}^{\circ}({\mathbb{P}}^{n+1})\cong\mathrm{PGL}_{n+2}.$

By contrast,
$\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{{\mathbb{P}}^{n}}\bigl{(}{\mathcal{O}}_{{\mathbb{P}}^{n}}{\oplus}{\mathcal{O}}_{{\mathbb{P}}^{n}}(r)\bigr{)}\bigr{)}$
is a maximal, connected subgroup of $\operatorname{Bir}({\mathbb{P}}^{n+1})$
for $r\geq 2$ by Corollary 2 (though this is also easy to see directly).

##  1\. Groups acting on ruled varieties

The idea behind the proof could be summarized as follows.

######  Principle 9.

Let $G\subset\operatorname{Bir}(Y)$ be an algebraic subgroup. If a normal,
unipotent subgroup $U\subset G$ acts on $Y$ with small orbits, then every
$G\subset\bar{G}\subset\operatorname{Bir}(Y)$ is likely to have a normal,
unipotent subgroup $\bar{U}\subset\bar{G}$ that acts on $Y$ with the same
orbits.

Theorem 12 illustrates this when the orbits are 1 dimensional.

######  Definition 10.

Let $\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}\subset\operatorname{Bir}(X\times{\mathbb{P}}^{1})$ denote the
subgroup of those birational maps $\phi:X\times{\mathbb{P}}^{1}\dasharrow
X\times{\mathbb{P}}^{1}$ that commute with projection to $X$ and map
$X\times\\{\infty\\}$ birationally to $X\times\\{\infty\\}$. Thus there is a
birational map $\phi_{X}:X\dasharrow X$ and a commutative diagram

$\begin{array}[]{ccc}X\times{\mathbb{P}}^{1}&\stackrel{{\scriptstyle\phi}}{{\dasharrow}}&X\times{\mathbb{P}}^{1}\\\
\pi\downarrow\hphantom{\pi}&&\hphantom{\pi}\downarrow\pi\\\
X&\stackrel{{\scriptstyle\phi_{X}}}{{\dasharrow}}&X,\end{array}$

where $\pi$ is the first coordinate projection.

Note that $\phi\mapsto\phi_{X}$ is a group homomorphism; its kernel is denoted
by $\operatorname{Bir}_{X}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\bigr{)}$.
Thus we have a group extension

$1\to\operatorname{Bir}_{X}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\bigr{)}\to\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}\to\operatorname{Bir}(X)\to 1.$

######  Definition 11.

Given any connected, algebraic, subgroup
$G\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}$, let
$G^{v}:=G\cap\operatorname{Bir}_{X}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\bigr{)}$
denote its vertical subgroup, and $G^{h}:=G/G^{v}\subset\operatorname{Bir}(X)$
its horizontal quotient. Thus we have a group extension

$1\to G^{v}\to G\to G^{h}\to 1.$

Note that $G^{v}$ is a solvable, normal subgroup of $G$. Let $U^{v}_{G}$
denote its unique maximal, connected, unipotent subgroup. It is a normal
subgroup of $G$. Let $U_{G}\subset G$ denote a maximal, connected, unipotent
subgroup; it is unique up to conjugation. Note that $U^{v}_{G}\subset U_{G}$,
and $U^{h}_{G}:=U_{G}/U^{v}_{G}\subset G^{h}$ is a maximal, connected,
unipotent subgroup. (Connectedness is automatic in characteristic 0.)

The following is the main result about subgroups of
$\operatorname{Bir}(X\times{\mathbb{P}}^{1})$.

######  Theorem 12.

Let $X$ be a normal, projective variety and
$G\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}$ a connected, algebraic subgroup. Assume that

  1. (12.1)

$G^{h}\subset\operatorname{Bir}(X)$ is a maximal, connected, algebraic
subgroup,

  2. (12.2)

$\dim U^{v}_{G}\geq 2$, and

  3. (12.3)

for every connected, algebraic subgroup $\\{1\\}\neq W\subset U^{h}_{G}$, the
fixed point set of the $W$-action by conjugation on $U^{v}_{G}$ has
codimension $\geq 2$ in $U^{v}_{G}$.

Then every connected, algebraic subgroup

$G\subset\bar{G}\subset\operatorname{Bir}(X\times{\mathbb{P}}^{1})$

is contained in $\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}$. Furthermore

  1. (12.4)

$\bar{G}^{h}=G^{h}$, so $U^{h}_{\bar{G}}=U^{h}_{G}$,

  2. (12.5)

$G^{v}\subset\bar{G}^{v}$, and

  3. (12.6)

$U^{v}_{G}\subset U^{v}_{\bar{G}}$.

In particular, $\bar{G}$ also satisfies (12.1–3).

Note. (12.3) implies (12.2) whenever $\\{1\\}\neq U^{h}_{G}$.

Proof. We may assume that $G\neq\bar{G}$.

First we show that there is an intermediate group $G\subsetneq
G_{1}\subset\bar{G},$ such that
$G_{1}\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}$.

To see this, we apply the purely group theoretic Lemma 16 with $R:=U^{v}_{G}$.
Assumption (12.3) says that the alternative (16.2) does not happen.

Thus there is a subgroup $\\{1\\}\neq R^{\prime}\subset U^{v}_{G}$ whose
(connected) normalizer $N_{\bar{G}}(R^{\prime})^{\circ}$ is not contained in
$G$.

A key point is that generically $R^{\prime}$ and $U^{v}_{G}$ have the same
orbits. In particular, the closures of the 1-dimensional $R^{\prime}$-orbits
are still the fibers of the projection $\pi:X\times{\mathbb{P}}^{1}\to X$.
Thus $N_{\bar{G}}(R^{\prime})^{\circ}$ birationally preserves
$\pi:X\times{\mathbb{P}}^{1}\to X$. The section $X\times\\{\infty\\}$ is the
fixed point set of the $R^{\prime}$-action, so this is also preserved. Thus

$N_{\bar{G}}(R^{\prime})^{\circ}\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}.$

We can then take $G_{1}:=\langle
G,N_{\bar{G}}(R^{\prime})^{\circ}\rangle\subset\bar{G}$.

Now $G_{1}$ clearly satisfies $G^{v}\subset G_{1}^{v}$, $U^{v}_{G}\subset
U^{v}_{G_{1}}$, and $G^{h}\subset G_{1}^{h}\subset\operatorname{Bir}(X)$.
Since $G^{h}$ is a maximal, connected, algebraic subgroup, we must have
$G^{h}=G_{1}^{h}$. Therefore $G_{1}$ also satisfies (12.1–3), and induction on
$\dim\bar{G}-\dim G$ completes the proof. ∎

######  Remark 13.

If we do not assume that $G^{h}$ is maximal, the above proof still gives a
$G\subsetneq G_{1}\subset\bar{G}$ that is a subgroup of
$\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to X\bigr{)}$.
However, if $U_{G}\neq U_{G_{1}}$ then (12.3) may not hold for $G_{1}$, and
the induction may break down.

##  2\. Lemmas on algebraic groups

######  Notation 14.

For an algebraic group $G$, we let $G^{\circ}$ denote the maximal, connected
subgroup. $B_{G}\subset G$ denotes a Borel subgroup, $U_{G}\subset B_{G}$ a
maximal, connected, unipotent subgroup, and $R_{G}\subset U_{G}$ the unipotent
radical of $G$. $Z(\ )$ denotes the center, and $N_{G}(\ )$ the normalizer in
$G$.

######  Lemma 15.

Let $G\subsetneq G^{\prime}$ be connected, algebraic groups. Then there is a
subgroup $V\subset U_{G}$ of codimension $\leq 1$ such that
$N_{G^{\prime}}\bigl{(}Z(V)\bigr{)}^{\circ}\not\subset G$.

Proof. Let $U_{G}\subset U_{G^{\prime}}$ be maximal unipotent subgroups.

If $U_{G}\neq U_{G^{\prime}}$ then let $U_{G}\subset N\subset U_{G^{\prime}}$
be the normalizer of $U_{G}$ in $U_{G^{\prime}}$. Note that $U_{G}\neq N$
since $U_{G^{\prime}}$ is nilpotent, and $N\not\subset G$ since $U_{G}$ is a
maximal unipotent subgroup of $G$. Then $N\subset
N_{G^{\prime}}(Z(U_{G}))^{\circ}$, and we can take $V:=U_{G}$.

Otherwise $U_{G}=U_{G^{\prime}}$. For every simple root of $G^{\prime}$ there
is a minimal parabolic subgroup $B_{G^{\prime}}\subsetneq P_{i}\subset
G^{\prime}$; let $V_{i}\subset U_{G}$ be its unipotent radical. Then
$P_{i}/V_{i}$ is either $\mathrm{SL}_{2}$ or $\mathrm{PSL}_{2}$. Therefore
$V_{i}\subset U_{G^{\prime}}=U_{G}$ has codimension 1 and $P_{i}\subset
N_{G^{\prime}}(V_{i})$.

The $P_{i}$ generate $G^{\prime}$, so $P_{i}\not\subset G$ for some $i_{0}$.
Then $V:=V_{i_{0}}$ works. ∎

######  Lemma 16.

Let $G\subsetneq G^{\prime}$ be connected, algebraic groups, $U_{G}\subset G$
a maximal, connected, unipotent subgroup and $\\{1\\}\neq R\subset U_{G}$ a
connected, normal subgroup. Then one of the following holds.

  1. (16.1)

There is a connected subgroup $\\{1\\}\neq R^{\prime}\subset R$ such that
$N_{G^{\prime}}(R^{\prime})^{\circ}\not\subset G$.

  2. (16.2)

There is a connected subgroup $\\{1\\}\neq W\subset U_{G}/R$ such that the
$W$-action by conjugation on $R$ is trivial on a subgroup $V_{R}\subset R$ of
codimension $\leq 1$ in $R$.

Proof. By (15) there is a subgroup $V\subset U_{G}$ of codimension $\leq 1$
such that $N_{G^{\prime}}\bigl{(}Z(V)\bigr{)}^{\circ}\not\subset G$. We are
done if $Z(V)\subset R$. Otherwise

$W:=\langle Z(V),R\rangle/R\cong Z(V)/\bigl{(}Z(V)\cap R\bigr{)}$

acts trivially on $V_{R}:=V\cap R$, which has codimension $\leq 1$. ∎

##  3\. Regularizing group actions

First recall some results on automorphisms of vector bundles; see for example
[Mar71] or [FFZ23, Sec.3].

######  17Automorphisms of vector bundles.

Let $C$ be a smooth, projective curve and $E$ a rank 2 vector bundle over $C$.
Up to twisting with a line bundle, we have one of the following cases.

  1. (17.1)

$E$ is stable, and then $\operatorname{Aut}_{C}(E)\cong{\mathbb{G}}_{m}$.

  2. (17.2)

$E\cong{\mathcal{O}}_{C}{\oplus}{\mathcal{O}}_{C}$, and then
$\operatorname{Aut}_{C}(E)\cong\mathrm{GL}_{2}$.

  3. (17.3)

There is an extension $0\to L\to E\to{\mathcal{O}}_{C}\to 0$, where $\deg
L\geq 0$ and the extension is non-split if $L\cong{\mathcal{O}}_{C}$. Then the
unipotent radical of $\operatorname{Aut}(E)$ is naturally isomorphic to
$H^{0}(C,L)$, and

    1. (17.3.a)

$\operatorname{Aut}_{C}(E)/H^{0}(C,L)\cong{\mathbb{G}}_{m}$ if the extension
is non-split, and

    2. (17.3.b)

$\operatorname{Aut}_{C}(E)/H^{0}(C,L)\cong{\mathbb{G}}_{m}^{2}$ if the
extension is split.

Let now $\bar{X}$ be a normal, projective variety and $X\subset\bar{X}$ an
open subset such that $\bar{X}\setminus X$ has codimension $\geq 2$. Let $E$
be a rank 2 vector bundle over $X$ such that $\operatorname{Aut}(E)$ contains
a unipotent subgroup of dimension $\geq 2$. Working on general curve sections
and using (17.1–3) we get that, after twisting with a line bundle and possibly
shrinking $X$, there is a unique extension

$None$ $0\to L\to E\to{\mathcal{O}}_{X}\to 0$

such that $h^{0}(X,L)\geq 2$. Furthermore

  1. (17.5)

the extension (17.4) is $\operatorname{Aut}_{X}(E)$ invariant,

  2. (17.6)

the unipotent radical of $\operatorname{Aut}_{X}(E)$ is naturally isomorphic
to $H^{0}(X,L)$,

  3. (17.7)

$\operatorname{Aut}_{X}(E)/H^{0}(X,L)\cong{\mathbb{G}}_{m}$ if (17.4) is non-
split, and

  4. (17.8)

$\operatorname{Aut}_{X}(E)/H^{0}(X,L)\cong{\mathbb{G}}_{m}^{2}$ if (17.4) is
split.

######  18Automorphisms of unstable ${\mathbb{P}}^{1}$-bundles.

Continuing with the notation of (17), let $\pi:P:={\mathbb{P}}_{X}(E)\to X$ be
the corresponding ${\mathbb{P}}^{1}$-bundle. Then $\operatorname{Aut}_{X}(P)$
is the quotient if $\operatorname{Aut}_{X}(E)$ by the scalars. Thus we get
that, if $\operatorname{Aut}_{X}(P)$ contains a unipotent subgroup of
dimension $\geq 2$, then

  1. (18.1)

there is a unique $\operatorname{Aut}_{X}(P)$ invariant section $D\subset P$,

  2. (18.2)

the unipotent radical of $\operatorname{Aut}_{X}(P)$ is naturally isomorphic
to $H^{0}(X,L)$,

  3. (18.3)

$\operatorname{Aut}_{X}(P)=H^{0}(X,L)$ if (17.4) is non-split,

  4. (18.4)

$\operatorname{Aut}_{X}(P)/H^{0}(X,L)\cong{\mathbb{G}}_{m}$ if (17.4) is
split, and

  5. (18.5)

the splittings of $E\to{\mathcal{O}}_{X}$ correspond to the splittings of
$\operatorname{Aut}_{X}(P)\to{\mathbb{G}}_{m}$.

Note that the natural way to get a vector bundle on $X$ is to take
$\pi_{*}{\mathcal{O}}_{P}(D)$. This sits in the exact sequence

$None$
$0\to{\mathcal{O}}_{X}=\pi_{*}{\mathcal{O}}_{P}\to\pi_{*}{\mathcal{O}}_{P}(D)\to{\mathcal{O}}_{P}(D)|_{D}\to
0.$

Thus $\pi_{*}{\mathcal{O}}_{P}(D)\cong E\otimes L^{-1}$.

Next we show a converse to these observations.

######  Proposition 19.

Let $\bar{X}$ be a normal, projective variety with smooth locus
$X\subset\bar{X}$, and
$G\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}$ an algebraic subgroup. Assume that
$G^{h}\subset\operatorname{Aut}(\bar{X})=\operatorname{Aut}(X)$. Then there is
a $G^{h}$-invariant open subset $X^{\circ}\subset X$, and a
${\mathbb{P}}^{1}$-bundle $\pi^{\circ}:P^{\circ}\to X^{\circ}$ with a rational
section $D^{\circ}\subset P^{\circ}$, such that

$G\subset\operatorname{Aut}\bigl{(}(P^{\circ},D^{\circ})\to
X^{\circ}\bigr{)}.$

Moreover $X\setminus X^{\circ}$ can be chosen to have codimension $\geq 3$ in
$X$.

######  Theorem 20.

Using the notation and assumptions of (19), assume in addition that there is a
${\mathbb{G}}_{m}\cong T\subset G^{v}/U^{v}_{G}$ and $\dim U^{v}_{G}\geq 1$.
Then there is a unique ${\mathbb{P}}^{1}$-bundle $\pi^{m}:P^{m}\to X$ with a
$G$-invariant section $D^{m}\subset P^{m}$, such that

  1. (20.1)

the set of fibers with trivial $U^{v}_{G}$-action has codimension $\geq 2$,
and

  2. (20.2)

for every other $\pi^{\prime}:(P^{\prime},D^{\prime})\to X^{\prime}$
satisfying $G\subset\operatorname{Aut}\bigl{(}(P^{\prime},D^{\prime})\to
X^{\prime}\bigr{)}$, there is a natural $G$-equivariant

$\pi^{\prime}_{*}{\mathcal{O}}_{P^{\prime}}(D^{\prime})\lhook\joinrel\to\pi^{m}_{*}{\mathcal{O}}_{P^{m}}(D^{m}).$

Proof of (19) and (20). By [Wei55] the $G$-action is regularized on some
$Y_{1}\stackrel{{\scriptstyle bir}}{{\sim}}\bar{X}\times{\mathbb{P}}^{1}$. Let
$Y_{2}\subset\bar{X}\times Y_{1}$ be the closure of the graph of the rational
map $Y_{1}\dasharrow\bar{X}\times{\mathbb{P}}^{1}\to\bar{X}$. Then $G$ acts
regularly on $Y_{2}$. After normalization we get a $G$-equivariant
$Y_{3}\to\bar{X}$, where $Y_{3}$ is normal. It is a ${\mathbb{P}}^{1}$-bundle
over a $G^{h}$-invariant, dense open subset $X_{3}\subset X$.

Let $x_{i}\in X\setminus X_{3}$ be the generic points that have codimension 1
in $X$. Working over the localization ${\mathcal{O}}_{x_{i},X}$, (23) gives a
$G$-invariant ${\mathbb{P}}^{1}$-bundle extending $Y_{3}\to X_{3}$ over some
open neighborhood of $x_{i}$. Gluing these to $Y_{3}\to X_{3}$ we get a
$G$-equivariant ${\mathbb{P}}^{1}$-bundle $\pi_{4}:Y_{4}\to X_{4}$ with a
rational section $D_{4}\subset Y_{4}$, where $X\setminus X_{4}$ has
codimension $\geq 2$ in $X$.

Let $F$ denote the push-forward of $(\pi_{4})_{*}{\mathcal{O}}_{Y_{4}}(D_{4})$
to $X$. Let $X^{\circ}\subset X$ be the largest open subset where $F$ is
locally free. Then $X\setminus X^{\circ}$ has codimension $\geq 3$ in $X$.

Set $P^{\circ}:={\mathbb{P}}_{X^{\circ}}\bigl{(}F|_{X^{\circ}}\bigr{)}$. The
$G$-action extends to a regular $G$-action on $P^{\circ}$ by by the Matsusaka-
Mumford lemma; see [Kol23, 11.39] for the version that we need. This completes
the proof of (19).

In (20) the $T$-action on $(\pi_{4})_{*}{\mathcal{O}}_{Y_{4}}(D_{4})$ produces
a splitting
$(\pi_{4})_{*}{\mathcal{O}}_{Y_{4}}(D_{4})\cong{\mathcal{O}}_{X_{4}}\oplus
L^{-1}_{4},$ and $U^{v}_{G}$ is naturally identified with a subspace
$W_{G}\subset H^{0}(X,L_{4})$. Let $L_{5}\subset L_{4}$ be the invertible
subsheaf generated by the sections in $W_{G}$ at codimension 1 points. Then
$L_{5}$ extends to a line bundle $L$ on $X$, and $G$ also acts on
$E:={\mathcal{O}}_{X}\oplus L^{-1}$. Its projectivization gives the reguired
${\mathbb{P}}^{1}$-bundle for (20).

Uniqueness can be checked on curves. We can thus work with a rank 2 free
module $E_{R}:=Re_{1}\oplus Re_{2}$ over a DVR $R$ with maximal ideal $(t)$
and quotient field $K$, where the $T$ action is $(e_{1},e_{2})\mapsto(\lambda
e_{1},\lambda^{-1}e_{2})$, and the elements of the unipotent subgroup act as

$(r_{1}e_{1},r_{2}e_{2})\mapsto\bigl{(}r_{1}e_{1},(r_{2}+ur_{1})e_{2}\bigr{)}\quad\mbox{where}\quad
u\in R.$

The unipotent action is nontrivial on the central fiber iff not all the $u$
are divisible by $t$.

Every other $T$-equivariant extension of $E_{K}$ is of the form
$E^{*}_{R}:=t^{c_{1}}Re_{1}\oplus t^{c_{2}}Re_{2}$. In the basis
$f_{i}:=t^{c_{i}}e_{i}$ the unipotent action becomes

$\bigl{(}r_{1}f_{1},r_{2}f_{2}\bigr{)}\mapsto\bigl{(}r_{1}e_{1},(r_{2}+ut^{c_{1}-c_{2}}r_{1})f_{2}\bigr{)}.$

This is a regular action iff $c_{1}\geq c_{2}$ and it is nontrivial on the
central fiber iff $c_{1}=c_{2}$. In this case the projectivizations of $E_{R}$
and of $E^{*}_{R}$ are naturally isomorphic. ∎

######  Remark 21.

If $\bar{X}$ is factorial, then $P^{m}$ in (20) extends to a
${\mathbb{P}}^{1}$-bundle. If $\bar{X}$ is ${\mathbb{Q}}$-factorial, then $L$
extends to a ${\mathbb{Q}}$-line bundle $\bar{L}$, and $P^{m}$ extends to an
orbifold ${\mathbb{P}}^{1}$-bundle

$\operatorname{Proj}_{\bar{X}}\Bigl{(}\oplus_{r\geq
0}\bigl{(}\oplus_{i=0}^{r}\bar{L}^{[i]}\bigr{)}\Bigr{)}.$

######  Example 22.

The codimension $3$ subset $X\setminus X^{\circ}$ is usually unavoidable in
(19), as shown by the following example taken from [AW98]. Consider the second
coordinate projection

$\pi:Y:=(\textstyle{\sum}_{i}x_{i}y_{i}=0)\subset{\mathbb{P}}^{2}_{y}\times{\mathbb{A}}^{3}_{x}\to{\mathbb{A}}^{3}_{x}.$

The fiber over $(0,0,0)$ is ${\mathbb{P}}^{2}$, all other fibers are
${\mathbb{P}}^{1}$. We can chose $D:=(y_{0}=0)$ as a rational section. Note
that $Y$ is smooth and the projection $\pi$ is $\mathrm{GL}_{3}$-equivariant
for the action
$\bigl{(}{\mathbf{x}},{\mathbf{y}}\bigr{)}\mapsto\bigl{(}A^{t}{\mathbf{x}},A^{-1}{\mathbf{y}}\bigr{)}.$
(This shows that [FFZ23, Lem.2.2] is not quite true.)

######  Lemma 23.

Let $k$ be a field, $(t\in T)$ the spectrum of an excellent DVR over $k$, and
$\pi:P\to T$ a projective morphism with general fiber ${\mathbb{P}}^{1}$, and
$D\subset P$ a section. Let $B$ be the group of $k$-automorphisms of
$\pi:(P,D)\to T$.

Then $\pi:(P,D)\to T$ is $B$-equivariantly birational to a trivial bundle
$\pi^{\prime}:({\mathbb{P}}^{1}\times T,\\{\infty\\}\times T)\to T$.

Proof. We can resolve the singularities by normalizing and blowing up the
singular set, hence the $B$ action the lifts. Thus we may assume that $P$ is
regular. Note that $(-1)$-curves in the central fiber are disjoint from each
other, we can thus contract all $(-1)$-curves disjoint from $D$ in a
$B$-equivariant way. After such contractions we get a minimal conic bundle
with a section, hence a ${\mathbb{P}}^{1}$-bundle. (The residue field of $T$
may be imperfect, see [Tan18] for a proof in these cases.) ∎

######  Example 24.

The non-uniqueness without the ${\mathbb{G}}_{m}$-action is illustrated by the
following, at the vector bundle level.

Let $R$ be a DVR with maximal ideal $(t)$. Consider the rank 2 free module
$E:=Re_{1}{\oplus}Re_{2}$ with an endomorphism given by
$\phi(e_{1})=e_{2},\phi(e_{2})=0$. The $\phi$-action is nontrivial on $E/tE$.
Another extension of $Re_{1}$ by $Re_{2}$ is given by
$E^{\prime}:=R(e_{1}+t^{-1}e_{2}){\oplus}Re_{2}$. The $\phi$-action is again
nontrivial on $E^{\prime}/tE^{\prime}$.

##  4\. Proof of Theorem 1

######  25Proof of Theorem 1.

First we aim to apply (12). Note that (12.1) is assumed in Theorem 1. There is
nothing to prove if $X$ is a point, otherwise (12.2) also holds. We check in
(26) that (12.3) is satisfied. Thus we get that

$G\subset\operatorname{Bir}\bigl{(}X\times({\mathbb{P}}^{1},\infty)\to
X\bigr{)}.$

Now we can apply (20) to get that

$G\subset\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}\bigl{(}{\mathcal{O}}_{X}{\oplus}L_{G}\bigr{)}\bigr{)}$

for some line bundle $L_{G}$.

Set
$G_{0}:=\operatorname{Aut}^{\circ}\bigl{(}{\mathbb{P}}_{X}\bigl{(}{\mathcal{O}}_{X}{\oplus}L\bigr{)}\bigr{)}$.
Since $L$ is generated by its global sections,
${\mathbb{P}}_{X}({\mathcal{O}}_{X}{\oplus}L)\to X$ is isomorphic to the
regularization of $G_{0}$ that is denoted by $P^{m}$ in (20.2). The inclusion
in (20.2) induces a $G$-equivariant $L_{G}^{-1}\lhook\joinrel\to L^{-1}$, thus
$L_{G}=L(D)$ for some effective $G^{h}$-invariant divisor $D$. ∎

The following accounts for the exceptional case
$(X,L)\not\cong\bigl{(}{\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}$
in Theorem 1.

######  Lemma 26.

Let $X$ be a normal, projective variety with a nontrivial
${\mathbb{G}}_{a}$-action. Let $L$ be a very ample,
${\mathbb{G}}_{a}$-linearized line bundle. Assue that the induced
${\mathbb{G}}_{a}$-action is trivial on a codimension 1 subspace $W\subset
H^{0}(X,L)$.

Then $X$ is a cone, and the ${\mathbb{G}}_{a}$-action is along its generators.

If $X$ is smooth of dimension $n$, then
$(X,L)\cong\bigl{(}{\mathbb{P}}^{n},{\mathcal{O}}_{{\mathbb{P}}^{n}}(1)\bigr{)}$.

Proof. Let $C\subset X$ be the closure of a ${\mathbb{G}}_{a}$-orbit and $c\in
C$ the fixed point. $L$ gives an imbedding
$X\lhook\joinrel\to{\mathbb{P}}^{N}$; let $\langle
C\rangle\subset{\mathbb{P}}^{N}$ be the linear span of $C$. Let
$\pi:\bar{C}\to C$ be the normalization. Then

$\dim\operatorname{im}\bigl{[}H^{0}(X,L)\to
H^{0}(\bar{C},\pi^{*}L)\bigr{]}=1+\dim\langle C\rangle.$

The ${\mathbb{G}}_{a}$-action on
$H^{0}\bigl{(}{\mathbb{P}}^{1},{\mathcal{O}}_{{\mathbb{P}}^{1}}(d)\bigr{)}$ is
trivial only on a 1-dimensional subspace. Thus the image of $W$ in
$H^{0}(\bar{C},\pi^{*}L)$ has dimension 1 and codimension 1. Therefore
$\dim\langle C\rangle=1$, thus $C$ is a line, and the image of $W$ consists of
the sections that vanish at $c$.

Given 2 orbits $C_{i}\ni c_{i}$, if $c_{1}\neq c_{2}$ then vanishing at both
gives a subset of codimension $\geq 2$. Thus all ${\mathbb{G}}_{a}$-orbit
closures are lines, and all these lines pass trough the same point. So $X$ is
a cone.

The only smooth cone is a linear subspace. ∎

######  Acknowledgments.

The main impetus came from the lecture On Algebraic Subgroups of the Cremona
Group of Floris at the Conference on Higher Dimensional Geometry (May, 2024)
at the Simons Center for Mathematics & Physical Sciences.

In the lecture Floris explained the main steps very clearly, but omitted the
technical details. So, when I was trying to think of a proof after the talk, I
was free to start on a somewhat different path.

I thank E. Floris and P. Fong for helpful comments. Partial financial support
was provided by the NSF under grant number DMS-1901855.

## References

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  * [BCTSSD85] Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, _Variétés stablement rationnelles non rationnelles_ , Ann. of Math. (2) 121 (1985), no. 2, 283–318. MR MR86m:14009
  * [BFT22] Jérémy Blanc, Andrea Fanelli, and Ronan Terpereau, _Automorphisms of $\mathbb{P}^{1}$-bundles over rational surfaces_, Épijournal Géom. Algébrique 6 (2022), Art. 23, 47. MR 4545346
  * [BFT23] by same author, _Connected algebraic groups acting on three-dimensional Mori fibrations_ , Int. Math. Res. Not. IMRN (2023), no. 2, 1572–1689. MR 4537333
  * [Cor95] Alessio Corti, _Factoring birational maps of threefolds after Sarkisov_ , J. Algebraic Geom. 4 (1995), no. 2, 223–254. MR 1311348
  * [FFZ23] Andrea Fanelli, Enrica Floris, and Susanna Zimmermann, _Maximal subgroups in the Cremona group_ , 2023.
  * [Fon24] Pascal Fong, _Connected algebraic groups acting on algebraic surfaces_ , Ann. Inst. Fourier (Grenoble) 74 (2024), no. 2, 545–587.
  * [FZ23] Pascal Fong and Sokratis Zikas, _Connected algebraic subgroups of groups of birational transformations not contained in a maximal one_ , C. R. Math. Acad. Sci. Paris 361 (2023), 313–322. MR 4538552
  * [FZ24] Enrica Floris and Sokratis Zikas, _Umemura quadric fibrations and maximal subgroups of $\mathrm{Cr}_{n}(\mathbb{C})$_, 2024.
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  * [Kol23] János Kollár, _Families of varieties of general type_ , Cambridge Tracts in Mathematics, vol. 231, Cambridge University Press, Cambridge, 2023, With the collaboration of Klaus Altmann and Sándor Kovács. MR 4566297
  * [KSC04] János Kollár, Karen E. Smith, and Alessio Corti, _Rational and nearly rational varieties_ , Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004.
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  * [Sar80] V. G. Sarkisov, _Birational automorphisms of conic bundles_ , Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974. MR 587343
  * [Tan18] Hiromu Tanaka, _Minimal model program for excellent surfaces_ , Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 345–376. MR 3795482
  * [Ume88] Hiroshi Umemura, _Minimal rational threefolds. II_ , Nagoya Math. J. 110 (1988), 15–80. MR 945907
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Princeton University, Princeton NJ 08544-1000,


Distributed Rate-Splitting Multiple Access for Multilayer Satellite Communications

Yunnuo Xu, Longfei Yin, Student Member, IEEE, Yijie Mao, Member, IEEE,
Wonjae Shin, Senior Member, IEEE and Bruno Clerckx, Fellow, IEEE

Y. Xu and L. Yin are with the Communications and Signal Processing Group,
Department of Electrical and Electronic Engineering, Imperial College
London, London SW7 2AZ, U.K. (email: yunnuo.xu19, longfei.yin17@imperial.ac.uk).
Y. Mao is with the School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China (email: maoyj@shanghaitech.edu.cn).
W. Shin is with the School of Electrical Engineering, Korea University, Seoul 02841, South Korea (email: wjshin@korea.ac.kr).
B. Clerckx is with the Department of Electrical and Electronic Engineering at Imperial College London, London SW7 2AZ, UK (email: b.clerckx@imperial.ac.uk).


11institutetext: Chalmers $|$ University of Gothenburg, Gothenburg, Sweden
and Volvo Cars, Gothenburg, Sweden
<EMAIL_ADDRESS>

Authors’ Instructions

# Understanding Work Practices of Autonomous Agile Teams: A Social-
Psychological Review

Lucas Gren

###### Abstract

The purpose of this paper is to suggest additional aspects of social
psychology that could help when making sense of autonomous agile teams. To
make use of well-tested theories in social psychology and instead see how they
replicated and differ in the autonomous agile team context would avoid
reinventing the wheel. This was done, as an initial step, through looking at
some very common agile practices and relate them to existing findings in
social-psychological research. The two theories found that I argue could be
more applied to the software engineering context are social identity theory
and group socialization theory. The results show that literature provides
social-psychological reasons for the popularity of some agile practices, but
that scientific studies are needed to gather empirical evidence on these
under-researched topics. Understanding deeper psychological theories could
provide a better understanding of the psychological processes when building
autonomous agile team, which could then lead to better predictability and
intervention in relation to human factors.

###### Keywords:

programming, social psychology, agile practices, teams

##  1 Introduction

The importance of understanding team autonomy has increased in the last
decades due to agile development processes [1]. There have been studies on the
barriers of self-organization in agile teams [2], the emerging roles of self-
organizing agile teams and how these roles enable agility [3], and the role of
senior management [4] to just mention a few. Some authors, like Moe et al.
[5], suggest using theories from social psychology to better understand team
autonomy. They refer to studies on self-organization in psychology but here
are many more theories in that field that would make sense to use in the
software development context. The two theories from social psychology, namely
Social Identity Theory and Group Socialization Theory where selected from a
textbook on social psychology [6]. I do not consider these theories more
important than others, they were only selected based on the vast number of
studies on them in the last decades. Therefore, they seem to be quite robust
and relevant to investigate in the software development context. Some results
might replicate, but others might not.

In order to theoretically analyze these two theories from social psychology
and their connection to agile teams, I use the study by So [7] in which the
author divide common agile practices into _core_ , _technical_ , _team
interaction_ , and _customer interaction_ practices. Due to saving space, I
selected only the _core_ and _team interaction_ practices (five in total) for
this paper. I do not consider these practices to cover all aspects of agility
nor to be the most important agile practices, however, the practices chosen
are widely used in industry [8].

I will describe the general agile work practices and connect these to existing
social, management, and organizational psychology findings, but I will start
by presenting the two important psychological theories that are in focus.

##  2 Important Psychological Theories

I have chosen to focus on two popular theories in social psychology on which
none or very few studies exist in software engineering research. The first one
is group socialization, which can be defined as the “dynamic relationship
between the group and its members that describes the passage of members
through a group in terms of commitment and of changing roles” [6]. The idea is
that a new team members will go through a certain set of phases through the
group’s lifespan. The group as a whole will evaluate a new member first by
assessing how much a potential new member can contribute to the group’s goal-
fulfillment. The individual will also assess how much the group can fulfill
their personal needs. Step two is commitment, which takes the outcome from the
evaluation as input and is an assessment of both parties’ beliefs about the
rewadingness (i.e. the quality of being rewarding) of the relationship (and
other alternative ones). The last phase is role transition in which the
commitment reaches a critical level and the relationship thereby changes.
These three phases are continuously depending on the result of the assessment.
The individual goes through five phases of group socialization: (1)
Investigation, (2) Socialization, (3) Maintenance, (4) Resocialization, and
(5) Remembrance. These phases have transition steps in-between that are: (1)
Entry, (2) Acceptance, (3) Divergence, and (4) Exit [9].

Group socialization is separate from stages that the whole group goes through
together. The most famous and used model of group development was introduced
in 1965 by Tuckman [10]. He integrated many theories and research findings
into a model with the four stages Forming, Storming, Norming, and Performing.
The forming stage is when the group is new and need to set the stage and
figure our what it is supposed to do and who can do what. Storming is a
conflict stage where people now feel safe to question the other team members,
which is needed to figure out good group goals and good strategies of work.
The Norming phase is when the group starts to set group norms of their
collaboration and know how to organize to be productive, and the final stage,
Performing, is when the team can focus the most of being productive because
they have created a system of good collaboration and effective conflict
resolution techniques. These phases are similar to the ones suggested by
Agazarian and Gantt [11] in systems centered theory, and what Wheelan [12]
also suggested as an integrated model of group development.

Another well-researched approach to explaining many group phenomena is the
social identity theory (see e.g. Hewstone et al. [13] or Hogg [6]). Not only
has the theory gained empirical evidence in social psychology research but
also in quite recent research in social neuroscience (see e.g. van Bavel and
Cunningham [14]). To understand that theory, we first need to understand the
concepts on which it is based. Social categorization is the classification of
people into different social groups, which is a deeply-rooted human trait, and
a person’s social identity is the part of the self that is derived from the
various memberships we have in social groups. Social identity theory is,
therefore, the theory of group membership and intergroup relations based on
self-categorization, social comparison and a self-definition regarding in-
group111A group that an individual is a member of [6] properties (i.e. a
prototype222Cognitive representation of the typical/ideal defining features of
a category [6]). Self-categorization is how we categorize ourselves and
thereby construct a social identity [15]. According to the minimal group
paradigm [16], even explicitly random group assignments trigger discriminatory
behavior against an out-group333A group that an individual is not a member of
[6]. The idea is that a successful intergroup bias creates or protects (high)
in-group status, which provides a positive social identity (which in turn
satisfies group-members’ need for positive self-esteem444Feelings about and
evaluations of oneself [6]). Researchers have successfully explained how
groups gain positive self-esteem through intergroup bias but have been less
successful when explaining intergroup bias motives due to threats or depressed
self-esteem [13]. However, Hogg and Williams [15] suggest that competition for
positive social identity characterizes intergroup behavior.

##  3 Agile Practices and Social Psychology

###  3.1 Iterative Development – A core practice of agile development

Delivering in short iterations has high face validity, but when broken down,
these ideas include a diversity of competences and dynamics needed by the
agile team to deliver value in such short iterations. In more general
management research, there has been more thorough research on which general
work practices contribute to performance (see e.g. Combs et al. [17]) and to
successfully implement iterative development, the team must have a high degree
and maturity of, for example, staffing, decentralized decision-making, and
communication [18]. So to understand the dynamics of iterative development, we
should consider these confounding factors before we, as researchers, jump to
conclusions about other found effects.

###  3.2 Iteration Planning – A teamwork practice

Obtaining empowered and motivated individuals that have the needed support to
solve any given task together with high levels of trust, are all aspects known
to be necessary [19] but are not always in place [20]. Creating a shared
vision has also been shown in research to be a key for success since the
beginning of the 1990s and is one of the main components of transformational
leadership [21]. A shared vision is necessary since the team needs an overall
goal to break down when planning the upcoming iteration. Regarding the
importance of simplicity in agile is somewhat connected to the concept of
reducing waste in lean manufacturing, together with the continued avoidance of
doing unnecessary activities in the project (or process) life-cycle [22]. To
plan in such a way, the team must know the members’ real competences and
abilities, which also implies maturity in the development process and that the
members of the group are committed and fully integrated into the group. With
such prerequisites, understanding the group socialization process then becomes
paramount when understanding how teams plan in short iterations.

###  3.3 Stand-up meetings – A teamwork practice

Developers, but also business people and testers, should be on the same team
and collaboratively work together through the whole project life-cycle (i.e.
having cross-functional teams). When connecting the popularity of having
cross-functional teams in the modern workplace (see e.g. Denison et al. [23])
to social identity theory, it becomes clear that it, in fact, decreases
intergroup bias. Having these various organizational functions share their
chores and issues often, would be expected to increase cohesion and
understanding of the whole project through shared mental models, which have
also gained initial empirical support [24]. Having social identity theory and
intergroup bias as factors in software engineering research would then
probably increase the explained variance.

###  3.4 Retrospectives – A teamwork practice

The idea of a retrospective meeting is that the team should reflect on
possible improvement points about their teamwork at the end of each iteration
[25]. More generally, such reflective meetings are often called _team
debriefs_ , and have been shown with scientific rigor to increase
effectiveness [26]. McHugh et al. [19] found that these types of meeting need
work and careful guidance to function in their intended way also in software
development. In a recent longitudinal study, Lehtinen et al. [27] showed that,
initially, newly formed teams focus more on task progress and task outcome
and, as the teams mature, they focus to a larger extent on process and
cooperation. Such findings also relate the “agility” of a team to group
socialization and group development since members of the group will behave
differently depending on how well integrated they are in the team [9], meaning
that a well-integrated individual will be more likely to perform
retrospectives in the way they are intended. If the socialization process is
not a part of understanding the dynamics of retrospective meetings, studies
will have difficulty explaining and predicting patterns of behavior.

###  3.5 Co-location – A teamwork practice

Having the team co-located in the same room with requirements as sticky notes
on physical boards have been promoted by the agile community in order to,
again, increase the velocity of the development in a rapidly changing
environment. Many cases have been reported where the communication challenges
of distributed teams have been satisfactory dealt with using modern technology
and slightly different practices (see e.g. Berczuk [28]). Another study showed
that both agile and traditional projects have the same issues regarding co-
location [29]. All-in-all, every social aspect of building relationships will
become more cumbersome with distance and implies that more effort is needed to
mitigate these challenges [30]. Since the social problems are amplified with
distance, failing to understand their influence in distributed agile teams
will have even larger negative effects on teamwork. And since agile processes
are dependent on the team as a working unit, understanding the social aspects
of both distributed and co-located teams are a key to building effective agile
teams.

##  4 Discussion and Implications

As we have seen in this review, there is a lot of overlap between existing
knowledge of, and research on, the workplace in general and the agile
practices. A few internal organizational examples being decreasing inter-group
bias through cross-functional teams [23], striving towards self-organization
of teams in order to increase responsiveness to change [18], creating
organizational citizenship behavior through shared visions [21], empowerment
and trust [31], and removing _waste_ in the process [22]. All these aspect are
of interest to agile software engineering researchers when trying to
understand the development of software using agile teams because these
theories might add explanatory power to the observed behaviors.

However, the theory could be seen as complex and hard to grasp for people
without any behavioral science education, which means researchers must first
run experiments to gather empirical evidence in order to eventually build a
theory of “agility,” and then provide scientifically founded and validated
guidelines to practitioners. One large hurdle of achieving this, though, is
the fact that an overwhelming majority of human factors research in software
engineering is conducted by software engineering researchers interested in
psychology and not psychology researchers interested in software engineering,
which often means that the research findings have little depth and offer
little new insights from a psychological perspective. I will not cite any
studies here due to the fact that such studies were conducted with the best of
intentions and do have high value in that they have highlighted the importance
of looking at psychological factors in the software engineering domain, which
was not the case at all before.

Social identity theory could be utterly useful when navigating through the
added complexity of the different social relationship surfacing in an agile
project. Hogg and Williams [32] explicitly suggest a set of propositions for
how social identity and self-categorization relate to the organizational
context. One of their propositions is that changes in which out-groups the in-
group compares itself to, will change the view of the group’s own identity,
including the properties of the ideal member (i.e. the prototype). Another
proposition is that harmonious relations between different subgroups of the
organization is best kept by recognizing both the subgroups (e.g. Quality
Assurance Engineer, Software Developers, Software Tester, etc.) and other
organizational constellations, including the teams and the company as a whole.
This means that the cross-functional agile teams must recognize both the value
of the team as a whole but also the different roles and make distinctions
between them. All these aspects should be part of agile team measurements in
the future in order to fully make sense of the agile team context.

When looking at the descriptions of the agile practices overall, many of the
internal practices seem to assume full group-membership seen from a group
socialization perspective [9]. They also assume the entire work-group to be
mature from a developmental perspective [10]. In order to fully understand the
social-psychological components of the team-based workplace in general, and
the agile context in particular, we also need to investigate the temporal
perspective of the interplay between group development, group socialization,
and the agile approach to projects by setting up autonomous teams.

As have also been shown in this short review, the prescribed behavior in these
agile practices are well-founded in social psychology, which provides social-
psychological reasons for their popularity. The reason for this is that if the
agile practices enable mechanisms known to work well for people in other
contexts, it is likely that they would also be appropriate in some variation
in the agile context. One example is the decrease of intergroup bias by having
cross-functional teams. Therefore, I argue for that these theories should be
applied more to the study of autonomous agile teams. In the review by [33],
they also call for more theory-based research since the current status of the
field mostly comprises method-specific case studies, which is particularly the
case in software engineering studies on human factors. In this present study,
I have explained some social-psychological underpinnings in relation to five
common agile practices, which contributes to founding agile practices in more
general social psychology theories. An understanding of such underpinnings can
help abstract the essentials of agile software development as opposed to other
approaches, but also guide researchers in conducting experiments using theory
from social psychology in the software development context. Many of the agile
principles are far from new in relation to human knowledge of work-groups.
However, what might be considered as having gotten a stronger acceptance is
the implementation of being responsive to change. The reasons for not relating
agile software development to any existing science outside of software
engineering might be due to lack of research knowledge from practitioners, but
it might also reflect the difficulty of interdisciplinary research.

##  5 Conclusion

Without understanding the psychology of groups, agile maturity survey findings
are hard to use in order to improve one’s own practices. Relating agile
practices to deeper psychological theories, like in this study, could instead
provide a deeper understanding of the psychological processes of implementing
autonomous agile teams.

## References

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