A counter-example to the equivariance structure on semi-universal deformation

TL;DR

This paper presents a counter-example demonstrating that a $G$-equivariant extension does not always exist on the formal semi-universal deformation of a projective variety with an algebraic group action.

Contribution

It provides the first known counter-example to the presumed equivariance structure on semi-universal deformations of algebraic varieties.

Findings

Counter-example to $G$-equivariant extensions established

Shows limitations of equivariance in deformation theory

Highlights need for revised understanding of deformation structures

Abstract

If $X$ is a projective variety and $G$ is an algebraic group acting algebraically on $X$, we provide a counter-example to the existence of a $G$-equivariant extension on the formal semi-universal deformation of $X$.

Equations

$$X_1\to X_2\to \cdots \to X_n\to \cdots .$$

$$X_{i1}\to X_{i2}\to \cdots \to X_{in}\to \cdots .$$

$$\hat{\pi }:\hat{\mathcal{X}}\to \text{Specf}(A)$$

$$\hat{\mathcal{X}}=(X,\underleftarrow{\lim }{\mathcal{O}}_{{\mathcal{X}}_n})\text{ and }\hat{\pi }=\underleftarrow{\lim }{\pi }_n.$$

$$S_n:=\mathrm{Spec}({\mathcal{O}}_{S,s}/{\mathfrak{m}}_s^{n+1})\to S$$

$${\mathcal{O}}_{S,s}/{\mathfrak{m}}_s^{n+1}\cong {\hat{\mathcal{O}}}_{S,s}/{\hat{\mathfrak{m}}}_s^{n+1}$$

$$\hat{\pi }:\hat{\mathcal{X}}\to \text{Specf}(A)$$

$$\hat{\mathcal{X}}=(X,\underleftarrow{\lim }{\mathcal{O}}_{{\mathcal{X}}_n})\text{ and }\hat{\pi }=\underleftarrow{\lim }{\pi }_n.$$

$$\left(\begin{array}{c}X\\ Y\text{par}\end{array}\right):=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}X\\ Y\text{par}\end{array}\right),$$

$$g.f$$

$$=\left(a^2{a}_0+ac{a}_1+c^2{a}_2\right)X^2+\left(2ab{a}_0+(ad+bc)a_1+2cd{a}_2\right)XY+\left(b^2{a}_0+bd{a}_1+d^2{a}_2\right)Y^2.$$

$$g.(a_0,a_1,a_2)=\left(\begin{array}{ccc}{a}^2& ac& c^2\\ 2ab& ad+bc& 2cd\\ b^2& bd& d^2\end{array}\right)\left(\begin{array}{c}{a}_0\\ a_1\\ a_2\end{array}\right).$$

$$H:={\mathbb{C}}^3⋊\text{GL}(2,\mathbb{C}).$$

$${\mathbb{F}}_2:=\{([x:y:z],[u:v])\in {\mathbb{P}}^2\times {\mathbb{P}}^1|y{v}^2=z{u}^2\}.$$

$$\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}(2)\oplus {\mathcal{O}}_{{\mathbb{P}}^1})={\sigma }^{-1}(U)\cup {\sigma }^{-1}(U^{\prime })=(U\times {\mathbb{P}}^1)\cup (U^{\prime }\times {\mathbb{P}}^1).$$

$$\{\begin{array}{ll}v{v}^{\prime }=1& \\ y^{\prime }=y{v}^2& \end{array}.$$

$${\rho }_1:U\times {\mathbb{P}}^1$$

$$(v,[x:y])$$

$${\rho }_2:U^{\prime }\times {\mathbb{P}}^1$$

$$(v^{\prime },[x^{\prime }:y^{\prime }])$$

$${\mathbb{F}}_2=\{([x:y:z],[u:v])\in {\mathbb{P}}^2\times {\mathbb{P}}^1|y{v}^2=z{u}^2\}$$

$$g.p=\{\begin{array}{ll}\left(\left[x{u}^2+y(a_0{v}^2+a_{1}uv+a_2{u}^2):y(au+bv{)}^2:y(cu+dv{)}^2\right],\left[au+bv:cu+dv\right]\right)& \text{ if }u\ne 0\\ \left(\left[x{v}^2+z(a_0{v}^2+a_{1}uv+a_2{u}^2):z(au+bv{)}^2:z(cu+dv{)}^2\right],\left[au+bv:cu+dv\right]\right)& \text{ if }v\ne 0\end{array}.$$

$$\mathrm{Spec}(\mathbb{C}[t,v,v^{-1},y])\subset \mathrm{Spec}(\mathbb{C}[t,v,y])$$

$$\mathrm{Spec}(\mathbb{C}[t^{\prime },v^{\prime },v^{\prime -1},y^{\prime }])\subset \mathrm{Spec}(\mathbb{C}[t^{\prime },v^{\prime },y^{\prime }])$$

$$\{\begin{array}{ll}v{v}^{\prime }=1& \\ x^{\prime }=x& \\ y^{\prime }=y{v}^2-tvx& \\ t^{\prime }=t.& \end{array}$$

$$\text{Proj}(\mathbb{C}[t,v,v^{-1},x,y])\text{ and }\text{Proj}(\mathbb{C}[t^{\prime },v^{\prime },v^{\prime -1},x^{\prime },y^{\prime }]).$$

$${\pi }^{-1}(t)=\{\begin{array}{ll}& {\mathbb{F}}_2\text{ if }t=0\\ & {\mathbb{P}}^1\times {\mathbb{P}}^1\text{ otherwise}.\end{array}$$

$$\varphi :{\mathbb{C}}^{\ast }\times \mathbb{C}\times {\mathbb{P}}^1$$

$$(t,v,[x:y])$$

$${\varphi }^{\prime }:{\mathbb{C}}^{\ast }\times \mathbb{C}\times {\mathbb{P}}^1$$

$$(t^{\prime },v^{\prime },[x^{\prime }:y^{\prime }])$$

$$(t^{\prime },[v^{\prime }:1],[t^{\prime }{v}^{\prime }{y}^{\prime }+t^{\prime 2}{x}^{\prime }:y^{\prime }])$$

$$=(t^{\prime },[v^{\prime }:1],[t^{\prime }yv:y{v}^2-tvx])$$

Full text

1. 1 Formal schemes and formal deformations

2. 2 The second Hirzebruch surface and its automorphism group

3. 3 A formal semi-universal deformation of $\mathbb{F}_{2}$ and formal vector fields on it

4. 3.1 Construction of the semi-universal deformation of $\mathbb{F}_{2}$

5. 3.2 Formal vector fields on the formal semi-universal deformation of $\mathbb{F}_{2}$

6. 4 The non-existence of $G$-equivariant structure on the formal semi-universal deformation

Introduction

Let $X$ be an algebraic variety defined over a field $k$ of characteristic zero. Due to Schlessinger’s work in [5], the existence of a formal semi-universal deformation (unique up to non-canonical isomorphism), which contains all the information of small deformations of $X$, is assured provided that $H^{1}(X,\mathcal{T}_{X})$ and $H^{2}(X,\mathcal{T}_{X})$ are finite dimensional vector spaces. These conditions realise for example, if $X$ is a complete scheme over $k$ or an affine scheme with at most isolated singularities (see [6, Corollary 2.4.2]). Now, we equipe $X$ with an action of an algebraic group $G$ defined over $k$. One question arising naturally is whether there exists a formal semi-universal deformation $\pi:$ $\mathcal{X}\rightarrow S$ of $X$, on which we can provide a $G$-action extending the given one on $X$. The answer is positive in the case that $G$ satisfies some vanishing conditions on its cohomology groups, i.e. $H^{1}(G,-)=0$ and $H^{2}(G,-)=0$ for a class of $G$-modules determined by $X$. In particular, these vanishing conditions hold for linearly reductive groups (see [4]). However, we do not know if there exists a non-reductive group whose action on $X$ does not extend to the formal semi-universal deformation of $X$. Therefore, we wish to give an example which illustrates this phenomenon. More precisely, we prove that the action of the automorphism group of the second Hirzebruch surface $\mathbb{F}_{2}$ does not extend to its formal semi-universal deformation.

Our proof goes as follows. First, we find a nice presentation of $G:=\operatorname{Aut}(\mathbb{F}_{2})$. Then we construct a formal semi-universal deformation $\widehat{\mathcal{X}}$ of $\mathbb{F}_{2}$. It turns out that $G$ is non-reductive and that the Lie algebra of $G$ is a $7$-dimensional vector space. As a matter of fact, we obtain seven vector fields on $\mathbb{F}_{2}$ with Lie bracket relations induced by those in $\text{Lie}(G)$. Next, we describe the general form of formal vector fields on $\widehat{\mathcal{X}}$. Finally, we conclude the paper by means of contradiction. Suppose that the $G$-action on $\mathbb{F}_{2}$ does extend to a $G$-action on $\widehat{\mathcal{X}}$ then we also have seven formal vector fields on $\widehat{\mathcal{X}}$ whose restrictions on the central fiber are nothing but our initial ones on $\mathbb{F}_{2}$. By manipulating these vector fields with a filtration $F$ given by the vanishing order at [math], we obtain the existence of a $3$-dimensional abelian Lie subalgebra in $\mathfrak{sl}_{2}(K)\times\mathfrak{sl}_{2}(K)$, where $\mathfrak{sl}_{2}(K)$ is the special linear group and $K$ is the field of formal Laurent power series $\mathbb{C}[[t,t^{-1}]]$, which is not the case. A remark is in order. Since the semi-universal family of $\mathbb{F}_{2}$ is in fact not universal, another possible way to obtain a contradiction is to use Wavrik’s criterion (see [7, Theorem 4.1]) but the calculations are rather complicated.

Acknowledgements*.*

I would like to thank Prof. Bernd Siebert for many useful discussions. Actually, I learned the idea of using the extension of vector fields and their relations as obstructions to the extension of the group action from an unpublished paper of his. This provides a strategy to attack the problem. I am specially thankful to Prof. Julien Grivaux for his careful reading and his comments which help to improve the manuscript. Finally, I am warmly grateful to the referee whose work led to a remarkable improvement of the paper.

1. Formal schemes and formal deformations

In this section, by $k$, we always mean a field of characteristic zero. We begin by recalling the definition of formal schemes. For more details, the readers are referred to [2, Chapter III. 9].

Definition 1.1**.**

Let $X$ be a noetherian scheme and let $Y$ be a closed subscheme defined by a sheaf of ideals $\mathcal{J}.$ Then we define the formal completion of $X$ along $Y$, denoted $(\widehat{X},\mathcal{O}_{\widehat{X}})$ (sometimes just $\widehat{X}$), to be the following ringed space. We take the topological space $Y$, and on it the sheaf of rings $\mathcal{O}_{\widehat{X}}=\varprojlim\mathcal{O}_{X}/\mathcal{I}^{n}$. Here we consider each $\mathcal{O}_{X}/\mathcal{I}^{n}$ as sheaf of rings on $Y$

Remark 1.1*.*

For each $n$, let $X_{n}=(X,\mathcal{O}_{X}/\mathcal{I}^{n})$. Then we obtain a sequence of closed immersions of schemes

[TABLE]

This expression is helpful in the sequel.

Definition 1.2**.**

A noetherian formal scheme is a locally ringed space $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ which has a finite open cover $\{\mathfrak{U}_{i}\}$ such that for each $i$, the pair $(\mathfrak{U}_{i},\mathcal{O}_{\mathfrak{X}}\mid_{\mathfrak{U}_{i}})$ is isomorphic, as a locally ringed space, to the completion of some noetherian scheme $X_{i}$ along a closed subscheme $Y_{i}$. A morphism of noetherian formal schemes is a morphism as locally ringed spaces.

Example 1.1**.**

If $X$ is any noetherian scheme, and $Y$ is a closed subscheme then the formal completion $\widehat{X}$ of $X$ along $Y$ is a formal scheme.

Example 1.2**.**

For $X=\mathbb{C}^{1}=\operatorname{Spec}(\mathbb{C}[t])$ and $Y=\{0\}$, the formal scheme $\widehat{X}$ is the locally ringed space $(Y,\mathcal{O}_{\widehat{X}})$, where the structure sheaf $\mathcal{O}_{\widehat{X}}$ is $\mathbb{C}[[t]]$. We denote $\text{Specf}(\mathbb{C}[[t]]):=(Y,\mathcal{O}_{\widehat{X}})$.

Let $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ be a noetherian formal scheme. We would like to define formal vector fields on $\mathfrak{X}$. Let $\{\mathfrak{U}_{i}\}$ be a finite open cover of $\mathfrak{X}$ such that for each $i$, the pair $(\mathfrak{U}_{i},\mathcal{O}_{\mathfrak{X}}\mid_{\mathfrak{U}_{i}})$ is the formal completion $(\widehat{X_{i}},\mathcal{O}_{\widehat{X}_{i}})$ of some noetherian scheme $X_{i}$ along a closed subscheme $Y_{i}$. By Remark 1.1, for each $i$ we have a sequence of closed immersions of schemes

[TABLE]

Definition 1.3**.**

A formal vector field on a noetherian formal scheme $\mathfrak{X}$ is a sequence of vector fields $\{v_{i,n}\}$ such that

1. ($i$)

Each $v_{i,n}$ is a usual vector field on the scheme $X_{i,n}$, 2. ($ii$)

$v_{i,n}$* induces $v_{i,n-1}$ via the natural inclusion $X_{i,n-1}\rightarrow X_{i,n}$,* 3. ($iii$)

$v_{i,n}\mid_{X_{i,n}\cap X_{j,n}}=v_{j,n}\mid_{X_{j,n}\cap X_{i,n}}$.

Next, we turn to the notion of infinitesimal deformations and that of formal deformations. Let $X$ be an algebraic scheme and let $A$ be an artinian local $k$-algebra with residue field $k$. An infinitesimal deformation of $X$ is a deformation of $X$ over the scheme $\operatorname{Spec}(A)$, i.e. a commutative diagram

X$$\mathcal{X}$$\operatorname{Spec}(k)$$\operatorname{Spec}(A)$$i$$\pi

where $\pi:\;\mathcal{X}\rightarrow\operatorname{Spec}(A)$ is a flat surjective morphism of schemes.

Now, let $A$ be a complete local noetherian $k$-algebra with the unique maximal ideal $\mathfrak{m}$ and with residue $k$.

Definition 1.4**.**

A formal deformation of $X$ over $A$ is a sequence $\{\nu_{n}\}$ of infinitesimal deformations of $X$, in which $\nu_{n}$ is represented by a deformation

X$$\mathcal{X}_{n}$$\operatorname{Spec}(k)$$\operatorname{Spec}(A_{n})$$f_{n}$$\pi_{n}

where $A_{n}=A/\mathfrak{m}^{n+1}$, such that for all $n\geq 1$, $\nu_{n}$ induces $\nu_{n-1}$ by pullback under the natural inclusion $\operatorname{Spec}(A_{n-1})\rightarrow\operatorname{Spec}(A_{n})$, i.e. $\nu_{n-1}$ is also represented by the deformation

X$$\mathcal{X}_{n}\times_{\operatorname{Spec}(A_{n})}\operatorname{Spec}(A_{n-1})$$\operatorname{Spec}(k)$$\operatorname{Spec}(A_{n-1})$$f_{n-1}$$\pi_{n-1}

In the language of formal schemes, we can write $\{\nu_{n}\}$ as the morphism of formal schemes

[TABLE]

where

[TABLE]

Here, $\mathcal{O}_{\mathcal{X}_{n}}$ is the structure sheaf on $\mathcal{X}_{n}$ and $\text{Specf}(A)$ is the formal scheme obtained by completing $\operatorname{Spec}(A)$ along its closed point, which corresponds to the unique maximal ideal of $A$. The easiest way to construct formal deformations is to build out of usual ones. This leads to the definition of formal deformation associated to a given deformation. Let $X$ be a projective scheme and let $\nu$ be a deformation represented by

X$$\mathcal{X}$$\operatorname{Spec}(k)$$(S,s)$$f$$\pi

where $S=\operatorname{Spec}(B)$ for some $k$-algebra of finite type $B$ and $s$ is a $k$-rational point of $S$.

Definition 1.5**.**

The formal deformation associated to $\nu$ is defined to be the sequence of deformations $\{\nu_{n}\}$ where each $\nu_{n}$ is the pullback of $\nu$ under the natural closed embedding

[TABLE]

where $\mathfrak{m}_{s}$ is the unique maximal ideal of the local ring $\mathcal{O}_{S,s}$.

Remark 1.2*.*

Note that $\{\nu_{n}\}$ is formal because of the isomorphism

[TABLE]

for all $n$.

To end this section, we introduce a very interesting kind of (formal) deformations, namely, the kind of $G$-equivariant (formal) ones, which is of central interest of the article. Let $G$ be a $k$-algebraic group acting algebraically on a projective variety $X$ and $A$ an artinian local $k$-algebra.

Definition 1.6**.**

A $G$-equivariant infinitesimal deformation of $X$ over $\operatorname{Spec}(A)$ is a usual deformation of $X$, i.e. a commutative diagram

X$$\mathcal{X}$$\operatorname{Spec}(k)$$\operatorname{Spec}(A)$$i$$\pi

where $\mathcal{X}$ and $\operatorname{Spec}(A)$ are equipped with $G$-actions in a way that any map appearing in the above diagram is $G$-equivariant. In particular, the restriction of the $G$-action on $\mathcal{X}$ on the central fiber is nothing but the initial $G$-action on $X$.

Finally, we give the definition of $G$-equivariant formal deformations and then we show how to produce formal vector fields from $G$-equivariant formal deformations.

Definition 1.7**.**

A $G$-equivariant formal deformation of $X$ over a complete local noetherian $k$-algebra $A$ with the unique maximal ideal $\mathfrak{m}$ is a formal deformation of $X$, i.e. a sequence $\{\nu_{n}\}$ of infinitestimal deformations of $X$, in which $\nu_{n}$ is represented by a $G$-infinitesimal deformation

X$$\mathcal{X}_{n}$$\operatorname{Spec}(k)$$\operatorname{Spec}(A_{n})$$f_{n}$$\pi_{n}

where $A_{n}=A/\mathfrak{m}^{n+1}$, such that for all $n\geq 1$, the $G$-equivariant deformation $\nu_{n}$ induces the $G$-equivariant deformation $\nu_{n-1}$ by pullback under the natural inclusion $\operatorname{Spec}(A_{n-1})\rightarrow\operatorname{Spec}(A_{n})$.

As before, we can write $\{\nu_{n}\}$ as the $G$-equivariant morphism of formal schemes

[TABLE]

where

[TABLE]

Here, the $G$-equivariance of $\widehat{\pi}$ means that $\widehat{\pi}$ is an inverse limit of $G$-equivariant morphisms of schemes $\pi_{n}$. On one hand, on each $n^{\text{th}}$-infinitesimal neighborhood, $G$-actions on $\mathcal{X}_{n}$ and on $\operatorname{Spec}(A_{n})$ induce vector fields on $\mathcal{X}_{n}$ and on $\operatorname{Spec}(A_{n})$, respectively. They are related by the fact that the differential of $\pi_{n}$ always maps the former ones to the latter ones. On the other hand, these induced vector fields on $\mathcal{X}_{n}$ and on $\operatorname{Spec}(A_{n})$ are also induced by those on $\mathcal{X}_{n+1}$ and on $\operatorname{Spec}(A_{n+1})$, respectively, via the natural inclusion $\operatorname{Spec}(A_{n})\rightarrow\operatorname{Spec}(A_{n+1})$. Therefore, we obtain formal vector fields, induced by the $G$-actions, on $\widehat{\mathcal{X}}$ and on $\text{Specf}(A)$, respectively.

2. The second Hirzebruch surface and its automorphism group

For the rest of the paper, we assume that $k$ is the field of complex numbers $\mathbb{C}$. The geneneral linear group $\text{GL}(2,\mathbb{C})$ has an obvious linear action on $\mathbb{C}^{2}$. This induces an action on the $\mathbb{C}$-vector space of polynomials in two variables $\mathbb{C}[X,Y]$. Since the subspace of homogeneous polynomials of degree $2$, denoted by $\mathbb{C}[X,Y]_{2}$, is $\text{GL}(2,\mathbb{C})$-invariant then we have a $\text{GL}(2,\mathbb{C})$-action on $\mathbb{C}[X,Y]_{2}$. More precisely, for $g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\text{GL}(2,\mathbb{C})$ and $f=a_{0}X^{2}+a_{1}XY+a_{2}Y^{2}\in\mathbb{C}[X,Y]_{2}$, the action of $g$ on $f$ is given by the linear substitution

[TABLE]

i.e.

[TABLE]

Identifying $\mathbb{C}[X,Y]_{2}$ with $\mathbb{C}^{3}$, the corresponding action on $\mathbb{C}^{3}$ can be written as

[TABLE]

This action gives rise to an algebraic group $H$ which is the semi-product of $\mathbb{C}^{3}$ and $\text{GL}(2,\mathbb{C})$, i.e.

[TABLE]

This is a non-reductive linear group. Recall that an algebraic group $K$ is reductive if the greatest connected normal subgroup $R_{u}(K)$ of $K$ is trivial. In our case, $R_{u}(H)=\mathbb{C}^{3}$.

Next, we recall the definition of the second Hirzebruch surface. Let $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})$ be the projectivization of $\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}$, where $\mathcal{O}_{\mathbb{P}^{1}}$ is the structure sheaf of the projective space $\mathbb{P}^{1}$.

Definition 2.1**.**

The second Hirzebruch surface is defined to be $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})$.

Proposition 2.1**.**

The second Hirzebruch surface is isomorphic to the variety

[TABLE]

Proof.

Let $\sigma$: $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})\rightarrow\mathbb{P}^{1}$ be the canonical projection of the projectivization $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})$, let $U=\operatorname{Spec}(\mathbb{C}[v])$ and $U^{\prime}=\operatorname{Spec}(\mathbb{C}[v^{\prime}])$ such that $v^{\prime}v=1$ on $U\cap U^{\prime}$. Then $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})$ has the following presentation

[TABLE]

such that on the intersection of the affine open sets $V=\operatorname{Spec}(\mathbb{C}[v,y])\subset U\times\mathbb{P}^{1}$ and $V^{\prime}=\operatorname{Spec}(\mathbb{C}[v^{\prime},y^{\prime}])\subset U\times\mathbb{P}^{1}$, we have

[TABLE]

So, an open covering of $\mathbb{F}_{2}$ is given by the open embeddings

[TABLE]

and

[TABLE]

which glue to give an isomorphism $\rho:\;\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}})\rightarrow\mathbb{F}_{2}$. ∎

Now, the algebraic group $H$ acts on the second Hirzebruch surface

[TABLE]

in the following manner: for $p=([x:y:z],[u:v])\in\mathbb{F}_{2}$ and $g=\left((a_{0},a_{1},a_{2})^{t},\begin{pmatrix}a&b\\ c&d\end{pmatrix}\right)\in H$,

[TABLE]

The following theorem is well-known (see [1, Section 6.1]).

Theorem 2.1**.**

The group of automorphisms of $\mathbb{F}_{2}$ is exactly the quotient of $H$ by the subgroup $I$ consisting of diagonal matrices of the form $\begin{pmatrix}\mu&0\\ 0&\mu\end{pmatrix}$ where $\mu\in\mathbb{C}$ such that $\mu^{2}=1$.

3. A formal semi-universal deformation of $\mathbb{F}_{2}$ and formal vector fields on it

3.1. Construction of the semi-universal deformation of $\mathbb{F}_{2}$

We shall follow the construction given in [4, Example 1.2.2.(iii)]. Consider two copies of $\mathbb{C}\times\mathbb{C}\times\mathbb{P}^{1}$ given by $W:=\text{Proj}(\mathbb{C}[t,v,x,y])$ and $W^{\prime}:=\text{Proj}(\mathbb{C}[t^{\prime},v^{\prime},x^{\prime},y^{\prime}])$ (note that these two rings are graded with respect to $x,y$ and $x^{\prime},y^{\prime}$, respectively). Consider the affine subsets $\operatorname{Spec}(\mathbb{C}[t,v,y])\subset W$, $\operatorname{Spec}(\mathbb{C}[t^{\prime},v^{\prime},y^{\prime}])\subset W^{\prime}$ and then glue them along the open subsets

[TABLE]

and

[TABLE]

by the rules

[TABLE]

This gives a gluing of $W$ and $W^{\prime}$ along

[TABLE]

We denote the resulting scheme by $\mathcal{W}$. In other words, if we let $(t,v,[x:y])$ and $(t^{\prime},v^{\prime},[x^{\prime}:y^{\prime}])$ be the coordinates on $W=\mathbb{C}\times\mathbb{C}\times\mathbb{P}^{1}$ and on $W^{\prime}=\mathbb{C}\times\mathbb{C}\times\mathbb{P}^{1}$, respectively. Then $W$ is obtained by glue $W$ and $W^{\prime}$ according to the rules $(3.1)$. Now, let $\pi:$ $\mathcal{W}\rightarrow\mathbb{C}$ be the morphism induced by the projections.

Theorem 3.1**.**

The familly $\pi:$ $\mathcal{W}\rightarrow\mathbb{C}=\operatorname{Spec}(\mathbb{C}[t])$ is a semi-universal deformation of $\mathbb{F}_{2}$. Moreover,

[TABLE]

Proof.

The map $\pi$ is obviously surjective by construction. Since $\pi$ is locally a projection, it is a flat morphism. Moreover, by Proposition 2.1, $\mathcal{W}_{0}=\pi^{-1}(0)=\mathbb{F}_{2}.$ Then $\pi:$ $\mathcal{W}\rightarrow\mathbb{C}$ is a deformation of $\mathbb{F}_{2}$. Next, let $\mathcal{W}^{*}=\pi^{-1}(\mathbb{C}^{*})$ and $\pi^{*}:\;\mathcal{W}^{*}\rightarrow\mathbb{C}^{*}$ is the restriction of $\pi$ on $\mathbb{C}^{*}=\operatorname{Spec}(\mathbb{C}[t,t^{-1}])$. We shall prove that $\mathcal{W}^{*}$ is in fact isomorphic to $\mathbb{C}^{*}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. Indeed, consider the following open embeddings

[TABLE]

and

[TABLE]

By the gluing condition $(3.1)$, we have that

[TABLE]

Hence, the above two morphisms glue to give an isomorphism

\mathcal{W}^{*}$$\mathbb{C}^{*}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$$\mathbb{C}^{*},\cong$$\operatorname{pr}_{1}$$\pi^{*}

which means precisely that $\pi^{*}:\;\mathcal{W}^{*}\rightarrow\mathbb{C}^{*}$ is the trivial family whose fibers are all isomorphic to $\mathbb{P}_{1}\times\mathbb{P}_{1}$. In particular, for $t\in\mathbb{C}^{*}$, $\pi^{-1}(t)=(\pi^{*})^{-1}(t)=\mathbb{P}_{1}\times\mathbb{P}_{1}$.

It remains to prove that the family $\pi:$ $\mathcal{W}\rightarrow\mathbb{C}$ is actually semi-universal. One way to see it is to compute the Kodaira-Spencer map $\mathcal{K}_{\pi,0}$ of $\pi$ at [math]. This map is uniquely determined by the element $\mathcal{K}_{\pi,0}(\frac{d}{dt})$ in $H^{1}(\mathbb{F}_{2},\mathcal{T}_{\mathbb{F}_{2}})$. By definition, $\mathcal{K}_{\pi,0}(\frac{d}{dt})$ represents the first order deformation of $\mathbb{F}_{2}$, obtained by gluing $W_{0}:=\text{Proj}(\mathbb{C}[\epsilon,v,x,y])$ and $W_{0}^{\prime}:=\text{Proj}(\mathbb{C}[\epsilon,v^{\prime},x^{\prime},y^{\prime}])$ along $\text{Proj}(\mathbb{C}[\epsilon,v,v^{-1},x,y])$ and $\text{Proj}(\mathbb{C}[\epsilon,v^{\prime},v^{\prime-1},x^{\prime},y^{\prime}])$ by the rules

[TABLE]

where $\mathbb{C}[\epsilon]$ is the ring of complex dual numbers. Hence, $\mathcal{K}_{\pi,0}(\frac{d}{dt})\in H^{1}(\mathcal{U},\mathcal{T}_{\mathbb{F}_{2}})$ is the $1$-cocycle which corresponds to the vector field $\{-v\frac{\partial}{\partial y}\}$ on $W_{0}\cap W_{0}^{\prime}$, where $\mathcal{U}$ is the covering $\{W_{0},W_{0}^{\prime}\}$. By [3, Example B.11(iii)], we see that $\{-v\frac{\partial}{\partial y}\}$ is nonzero and $\dim_{\mathbb{C}}H^{1}(\mathbb{F}_{2},\mathcal{T}_{\mathbb{F}_{2}})=1$. Thus, the Kodaira-Spencer map is an isomorphism and so $\pi:$ $\mathcal{W}\rightarrow\mathbb{C}$ is semi-universal. ∎

Another useful presentation of $\mathcal{W}$ is given as follows.

Proposition 3.1**.**

The scheme $\mathcal{W}$ is isomorphic to the surface

[TABLE]

Proof.

We have an open covering of $\mathcal{X}$ given by the open embeddings

[TABLE]

and

[TABLE]

which glue to give an isomorphism $\mathcal{W}\overset{\cong}{\rightarrow}\mathcal{X}$. ∎

Remark 3.1*.*

By Proposition 2.1 and by Proposition 3.1, from now on, we use interchangeably between $\mathbb{F}_{2},\mathcal{X}$ and $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}),\mathcal{W}$, respectively.

3.2. Formal vector fields on the formal semi-universal deformation of $\mathbb{F}_{2}$

The formal deformation associated to $\mathcal{W}$, $\widehat{\pi}:\widehat{\mathcal{W}}\rightarrow\text{Specf}(\mathbb{C}[[t]])$ is a formal semi-universal deformation of $\mathbb{F}_{2}$ (here $\mathbb{C}[[t]]$ is the ring of formal power series in the variable $t$). We will give explicit descriptions of formal vector fields on $\widehat{\mathcal{W}}$. Consider the covering $\{W,W^{\prime}\}$ where $W:=\text{Proj}(\mathbb{C}[t,v,x,y])$ and $W^{\prime}:=\text{Proj}(\mathbb{C}[t^{\prime},v^{\prime},x^{\prime},y^{\prime}])$, as before. A formal vector field on $W$ is of the form

[TABLE]

where $g_{1},\alpha_{1},\beta_{1},\gamma_{1},k_{1}$ are formal power series in the variable $t$. Likewise, a formal vector field on $W^{\prime}$ is of the form

[TABLE]

where $g_{2},\alpha_{2},\beta_{2},\gamma_{2},k_{2}$ are formal power series in the variable $t^{\prime}$. Therefore, a vector field on $\mathcal{W}$ which is of the form $(3.2)$ on $W$ and of the form $(3.3)$ on $W^{\prime}$ must satisfy the relation

[TABLE]

on the overlapping open set $W\cap W^{\prime}$.

Lemma 3.1**.**

A global formal vector field on $\widehat{\mathcal{W}}$ whose restriction on $W$ is

[TABLE]

must satisfy the following

[TABLE]

where $A,B,C,a,b,c,e,k_{1}$ are formal power series in the variable $t$ with a relation

[TABLE]

Proof.

By $(3.1)$, we have

[TABLE]

Substituting $(3.7)$ into the left hand side of $(3.4)$ and equalizing, we get that

[TABLE]

which implies that

[TABLE]

where $A,B,C,a,b,c,e$ are formal power series in the variable $t$ with a relation

[TABLE]

This constraint comes from the coefficient of $\frac{1}{v^{\prime}}$ in the fourth equation in $(3.8)$. ∎

Remark 3.2*.*

If $t=0$ then $(3.5)$ becomes

[TABLE]

which agrees with Kodaira’s calculations of vector fields on $\mathcal{W}_{0}=\mathbb{F}_{2}$ (see [3, Page 75]). In particular, we have seven linearly independent vector fields on $\mathbb{F}_{2}$. If $t$ is non-zero and fixed then we have six linearly independent vector fields on the fiber $\mathcal{W}_{t}$, which is due to the existence of the relation $(3.6)$.

4. The non-existence of $G$-equivariant structure on the formal semi-universal deformation

The Lie algebra of $G:=\text{Aut}(\mathbb{F}_{2})$ is $\mathbb{C}^{3}\times M(2,\mathbb{C})$, which is evidently $7$-dimensional . A $\mathbb{C}$-basis of Lie(G) is given by the following elements

[TABLE]

Then the $G$-action gives us $7$ vector fields $E_{1}^{\prime},\ldots,E_{7}^{\prime}$ on $\mathbb{F}_{2}$ with the relations

[TABLE]

Now, we are in the position to prove the main result of this paper. Suppose that the $G$-action extends on $\widehat{\mathcal{W}}$ . This implies that we also have $7$ formal vector fields $E_{1},E_{2},E_{3},E_{4},E_{5},E_{6},E_{7}$ on $\widehat{\mathcal{W}}$ with the following Lie bracket constraints

[TABLE]

These vector fields form a Lie subalgebra, denoted by $\mathfrak{g}$, of the Lie algebra of formal vector fields on $\widehat{\mathcal{W}}$. Of course, the restriction of $E_{i}$ on the central fiber is nothing but $E_{i}^{\prime}$ ($i=1,\ldots,7$).

From the previous section, we can assume that our seven vector fields are of the form

[TABLE]

(cf. Lemma 3.1) where $A,B,C,a,b,c,e$ are formal power series in $t$ ($i=1,\ldots,7$).

Theorem 4.1**.**

The action of $G$ on $\mathbb{F}_{2}$ does not extend to the formal semi-universal deformation $\widehat{\mathcal{W}}$, where $G$ is the automorphism group of $\mathbb{F}_{2}$.

Proof.

We denote by $\mathfrak{v}$ the Lie algebra of formal vector fields in one variable $t$. Let $\delta:$ $\mathfrak{g}\rightarrow\mathfrak{v}$ be the map which sends

[TABLE]

to

[TABLE]

for $i=1,\ldots,7$. Since, the first two components $\frac{\partial}{\partial v}$ and $\frac{\partial}{\partial y}$ contribute nothing to the component $\frac{\partial}{\partial t}$ in the Lie bracket then $\delta$ is a well-defined Lie homomorphism. Set $F_{i}:=\delta(E_{i})=k_{i}(t)\frac{\partial}{\partial t}$ ($i=1,\ldots,7$). Note that the seven formal vector fields $F_{i}$ ($i=1,\ldots,7$) are nothing but those induced by the $G$-action on the base $\text{Specf}(\mathbb{C}[[t]])$ (cf. the last paragraph of Section 1). Observe also that $\mathfrak{v}$ can be equipped with a filtration $F$ given by the vanishing order at [math] and we have two well-known facts

[TABLE]

for $p,q\geq 1$. Furthermore, the vanishing order of all $k_{i}$ at [math] is at least $1$. Let $k_{i}(t)=\sum_{j=1}^{\infty}a^{i}_{j}t^{j}$ ($i=1,2,4,5$). Using the first fact and the Lie relations induced by $\delta$:

[TABLE]

we obtain $a_{1}^{1}=a_{1}^{2}=a_{1}^{4}=0$. Suppose that $k_{4}(t)$ is not identically zero, then there exists $j^{*}\geq 2$ such that $a_{j^{*}}^{4}$ is nonzero. By computing explicitly the Lie relation $[F_{4},F_{5}]=-F_{4}$ in terms of power series in $t$ and then by equalizing coefficients, we get that

[TABLE]

for all $j\geq 2$. Thus, $a_{1}^{5}=\frac{1}{j^{*}-1}$, which is clearly nonzero. A similar computation for the relation $[F_{1},F_{5}]=0$ gives

[TABLE]

for all $j\geq 2$. Hence, all $a_{j}^{1}=0$ so that $k_{1}(t)=0$. By the relation $[F_{1},F_{3}]=-2F_{4}$, we deduce that $k_{4}(t)=0$, a contradiction. Therefore, $k_{4}(t)=0$. From the relations $[F_{3},F_{4}]=F_{2}$ and $[F_{4},F_{7}]=-F_{1}$, we obtain that $k_{2}(t)=0$ and $k_{1}(t)=0$. As a sequence, $E_{1},E_{2}$, and $E_{4}$ do not have the component $\frac{\partial}{\partial t}$.

In addition, by the proof of Theorem 3.1, as a scheme over $\mathbb{C}$,

[TABLE]

Then,

[TABLE]

as a scheme over $L$, where $L:=\mathbb{C}[t,t^{-1}]$ and $\mathbb{P}^{1}_{L}$ is the $1$-projective space over $L$. Therefore, the generic fiber $\widehat{\mathcal{W}}^{*}$ of $\widehat{\mathcal{W}}$ is isomorphic to $\mathcal{W}^{*}\times_{\operatorname{Spec}(\mathbb{C}[t,t^{-1}])}\operatorname{Spec}(\mathbb{C}[[t,t^{-1}]])=\mathbb{P}^{1}_{K}\times\mathbb{P}^{1}_{K}$, as a scheme over $K$, where $K$ is the field of Laurent formal power series $\mathbb{C}[[t,t^{-1}]]$. Now, by restricting on the generic fiber of $\widehat{\mathcal{W}}$, we obtain that $E_{1},E_{2}$, and $E_{4}$ are formal vector fields on $\widehat{\mathcal{W}}^{*}$, considered as a $\mathbb{C}$-scheme. However, by the first paragraph, we have proved that there is no component $\frac{\partial}{\partial t}$ in the expression of $E_{i}$ ($i=1,2,4$). So, if we think of $E_{1}$, $E_{2}$ and $E_{4}$ as vector fields with coefficients in $K$, then they are definitely vector fields on $\widehat{\mathcal{W}}^{*}$, regarded as a scheme over $K$. Note that the Lie algebra of vector fields on $\mathbb{P}^{1}_{K}\times\mathbb{P}^{1}_{K}$ is isomorphic to $\mathfrak{sl}_{2}(K)\times\mathfrak{sl}_{2}(K)$, where $\mathfrak{sl}_{2}(K)$ is the special linear group. This means that there exists a $3$-dimensional abelian Lie subalgebra of $\mathfrak{sl}_{2}(K)\times\mathfrak{sl}_{2}(K)$. The image of that subalgebra under one of the two canonical projections of the product $\mathfrak{sl}_{2}(K)\times\mathfrak{sl}_{2}(K)$ provides a $2$-dimensional abelian Lie subalgebra in $\mathfrak{sl}_{2}(K)$. This is a contradiction since $\text{rank}(\mathfrak{sl}_{2}(K))$ is only $1$. ∎

Remark 4.1*.*

A naturally posed question is if the $G$-action extends to $\mathcal{W}_{n}$ over $\operatorname{Spec}(\mathbb{C}[t]/(t^{n+1}))$ for small value $n$. Although the above proof does not give any clue to reply to this question, the answer is yes for $n=1$. More general, if $G$ is an algebraic group acting algebraically on a projective variety $X$ and $\pi:\mathcal{X}\rightarrow S$ is the semi-universal deformation of $X$ then the $G$-action on $X$ certainly extends up to the first infinitesimal deformation $\mathcal{X}_{1}$ over $S_{1}$. This follows easily from the semi-universality of the family $\pi:\mathcal{X}\rightarrow S$. Unfortunately, our example turns out to be the worst case. More precisely, we can even show that the $G$-action on $\mathbb{F}_{2}$ can not extend to the second infinitesimal $\mathcal{W}_{2}$ over $\operatorname{Spec}(\mathbb{C}[t]/(t^{3}))$ by extending $E^{\prime}_{i}$ ($i=1,\dots,7$) together with their Lie bracket relations, order by order with respect to $t$. However, the computations are somewhat lengthy and complicated.