Equivariant extensions of Ga-torsors over punctured surfaces
Adrien Dubouloz Isac Hedén Takashi Kishimoto
math.AG




IMB UMR5584, CNRS, Univ. Bourgogne Franche-Comt, F-21000 Dijon,
France.

adrien.dubouloz@u-bourgogne.fr



Research Institute for Mathematical Sciences, Kyoto University, Kyoto
606-8502, Japan

Isac.Heden@kurims.kyoto-u.ac.jp



Department of Mathematics, Faculty of Science, Saitama University,
Saitama 338-8570, Japan 

tkishimo@rimath.saitama-u.ac.jp

This work was partially funded by Grant-in-Aid for JSPS Fellows Number
15F15751and Grant-in-Aid for Scientific Research of JSPS No. 15K04805.
The research was done during visits of the first and second authors
at Saitama University, and during visits of the third author at the
Institut de Mathmatiques de Bourgogne. The authors thank these institutions
for their generous supports and the excellent working conditions offered.

[2000]14R20; 14R25; 14R05; 14L30, 14D06



Motivated by the study of the structure of algebraic actions the additive
group on affine threefolds X, we consider a special class of such
varieties whose algebraic quotient morphisms X→ X//𝔾_a
restrict to principal homogeneous bundles over the complement of a
smooth point of the quotient. We establish basic general properties
of these varieties and construct families of examples illustrating
their rich geometry. In particular, we give a complete classification
of a natural subclass consisting of threefolds X endowed with proper
𝔾_a-actions, whose algebraic quotient morphisms π:X→ X//𝔾_a
are surjective with only isolated degenerate fibers, all isomorphic
to the affine plane 𝔸^2 when equipped with their reduced
structures. 


Equivariant extensions of 𝔾_a-torsors over punctured
surfaces
    Takashi Kishimoto 
    
=============================================================





§ INTRODUCTION


Algebraic actions of the complex additive group 𝔾_a=𝔾_a,ℂ
on normal complex affine surfaces S are essentially fully understood:
the ring of invariants 𝒪(S)^𝔾_a,ℂ
is a finitely generated algebra whose spectrum is a smooth affine
curve C=S//𝔾_a, and the inclusion 𝒪(S)^𝔾_a⊂𝒪(S)
defines a surjective morphism π:S→ C whose general
fibers coincide with general orbits of the action, hence are isomorphic
to the affine line 𝔸^1 on which 𝔾_a acts
by translations. The degenerate fibers of such 𝔸^1-fibrations
are known to consist of finite disjoint unions of smooth affine curves
isomorphic to 𝔸^1 when equipped with their reduced structure.
A complete description of isomorphism classes of germs of invariant
open neighborhoods of irreducible components of such fibers was established
by Fieseler <cit.>. 

In contrast, very little is known so far about the structure of 𝔾_a-actions
on complex normal affine threefolds. For such a threefold X, the
ring of invariants 𝒪(X)^𝔾_a is again finitely
generated <cit.> and the morphism π:X→ S induced
by the inclusion 𝒪(X)^𝔾_a⊂𝒪(X)
is an 𝔸^1-fibration over a normal affine surface S.
But in general, π is neither surjective nor equidimensional.
Furthermore, it can have degenerate fibers over closed subsets of
pure codimension 1 as well as of codimension 2. All of these
possible degeneration are illustrated by the following example:

The restriction of the projection pr_x,y to the smooth
threefold X={x^2(x-1)v+yu^2-x=0} in 𝔸^4 is
an 𝔸^1-fibration π:X→𝔸^2 which
coincides with the algebraic quotient morphism of the 𝔾_a-action
on X associated to the locally nilpotent derivation ∂=x^2(x-1)∂_u-2yu∂_v
of its coordinate ring. The restriction of π over the principal
open subset x^2(x-1)≠0 of 𝔸^2 is a trivial principal
𝔾_a-bundle, but the fibers of π over the points
(1,0) and (0,0) are respectively empty and isomorphic to 𝔸^2.
Furthermore, for every y_0≠0, the inverse images under π
of the points (0,y_0) and (1,y_0) are respectively isomorphic
to 𝔸^1 but with multiplicity 2, and to the disjoint
union of two reduced copies of 𝔸^1.

Partial results concerning the structure of one-dimensional degenerate
fibers of 𝔾_a-quotient 𝔸^1-fibrations
were obtained by Gurjar-Masuda-Miyanishi <cit.>. In the present
article, as a step towards the understanding of the structure of two-dimensional
degenerate fibers, we consider a particular type of non equidimensional
surjective 𝔾_a-quotient 𝔸^1-fibrations
π:X→ S which have the property that they restrict to
𝔾_a-torsors[sometimes also referred to as Zariski locally trivial principal 𝔾_a-bundles]
 over the complement of a finite set of smooth points in S. These
are simpler than the general case illustrated in the previous example
since they do not admit additional degeneration of their fibers over
curves in S passing through the given points. The local and global
study of some classes of such fibrations was initiated by the second
author <cit.>. He constructed in particular many examples of
𝔾_a-quotient 𝔸^1-fibrations on smooth
affine threefolds X with image 𝔸^2 whose restrictions
over the complement of the origin are isomorphic to the geometric
quotient SL_2→SL_2/𝔾_a
of SL_2 by the action of unitary upper triangular matrices. 

One of the simplest examples of this type is the smooth threefold
X_0⊂𝔸_x,y,p,q,r^5 defined by the equations

    X_0:    
    xr-yq     =0
    
    yp-x(q-1)     =0
    
    pr-q(q-1)     =0

and equipped with the 𝔾_a-action associated to the locally
nilpotent ℂ[x,y]-derivation x^2∂_p+xy∂_q+y^2∂_r
of its coordinate ring. The equivariant open embedding SL_2={xv-yu=1}↪ X_0
is given by (x,y,u,v)↦(x,y,xu,xv,yv). The 𝔾_a-quotient
morphism coincides with the surjective 𝔸^1-fibration
π_0:pr_x,y:X_0→𝔸^2. Its restriction
over 𝔸^2∖{(0,0)} is isomorphic to the quotient
morphism SL_2→SL_2/𝔾_a,
while its fiber over (0,0) is the smooth quadric {pr-q(q-1)=0}⊂𝔸_p,q,r^3,
isomorphic to the quotient SL_2/𝔾_m of SL_2
by the action of its diagonal torus (see Example <ref>).
A noteworthy property of this example is that the 𝔾_a-quotient
morphism π:X_0→𝔸^2 factors through a locally
trivial 𝔸^1-bundle ρ:X_0→𝔸̃^2
over the the blow-up τ:𝔸̃^2→𝔸^2
of the origin. 

It is a general fact that every irreducible component of a degenerate
fiber of pure codimension one of a 𝔾_a-quotient 𝔸^1-fibration
π:X→ S on a smooth affine threefold is an 𝔸^1-uniruled
affine surface (see Proposition <ref>).
We do not know whether every 𝔸^1-uniruled surface can
be realized as an irreducible component of the degenerate fiber of
a 𝔾_a-extension. But besides the smooth affine quadric
SL_2/𝔾_m appearing in the previous example,
the following one confirms that the affine plane 𝔸^2
can also be realized (see also Examples <ref>
and <ref> for other types of surfaces that can
be realized): Let X_1⊂𝔸_x,y,z_1z_2,w^5
be the smooth affine threefold defined by the equations 

    X_1:    
    xw-y(yz_1+1)     =0
    
    xz_2-z_1(yz_1+1)     =0
    
    z_1w-yz_2    =0,

equipped with the 𝔾_a-action associated to the locally
nilpotent ℂ[x,y]-derivation x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w
of its coordinate ring. The morphism SL_2↪ X_1
given by (x,y,u,v)↦(x,y,u,uv,yv) isequivariant open embedding.
The 𝔾_a-quotient morphism coincides with the surjective
𝔸^1-fibration π_1=pr_x,y:X_1→𝔸^2,
whose fiber over the origin is the affine plane 𝔸^2=Spec(ℂ[z_2,w])
and whose restriction over 𝔸^2∖{(0,0)} is
again isomorphic to the quotient morphism SL_2→SL_2/𝔾_a.
A special additional feature is that the 𝔾_a-action
on X_1 extending that on SL_2 is not only fixed
point free but actually proper: its geometric quotient X_1/𝔾_a
is separated. One can indeed check that X_1/𝔾_a is
isomorphic to the complement 𝔸̃^2∖{o_1}
of a point o_1 supported on the exceptional divisor E of the
blow-up 𝔸̃^2 of 𝔸^2 at the origin
(see Example <ref>). 


Relaxing the hypothesis that the 𝔸^1-fibration π:X→ S
arises as the quotient of a 𝔾_a-action on an affine
threefold X to consider the broader problem of describing the geometry
of degeneration of 𝔸^1-fibrations over irreducible closed
subsets of pure codimension two of their base, we are led to the following
more general notion:

Let (S,o) be a pair consisting of a normal separated 2-dimensional
scheme S essentially of finite type over a field k of characteristic
zero and of a closed point o contained in the smooth locus of S.
A 𝔾_a-extension of a 𝔾_a-torsor
ρ:P→ S∖{o} is a 𝔾_a-equivariant
open embedding j:P↪ X into an integral scheme X
equipped with a surjective morphism π:X→ S of finite
type and a 𝔾_a,S-action, such that the commutative diagram

    P [r,hook,"j"] [d,"ρ"']     X [d,"π"] 
     S∖{o}[r,hook]     S

is cartesian.


The examples X_0 and X_1 above provide motivation to study
the following natural classes of 𝔾_a-extensions π:X→ S
of a 𝔾_a-torsor ρ:P→ S∖{o},
which are arguably the simplest possible types of 𝔾_a-extensions
from the viewpoints of their global geometry and of the properties
of their 𝔾_a-actions:

- (Type I) Extensions for which π factors through a locally trivial
𝔸^1-bundle over the blow-up τ:S̃→ S
of the point o, the fiber π^-1(o) being then the total space
of a locally trivial 𝔸^1-bundle over the exceptional
divisor of τ. 

- (Type II) Extensions for which π^-1(o)_red is
isomorphic to the affine plane 𝔸_κ^2 over the
residue field κ of S at o, X is smooth along π^-1(o)
and the 𝔾_a,S-action on X is proper. 


The first main result of this article, Proposition <ref>
and Theorem <ref>, is a complete description of
𝔾_a-extensions of Type I together with an effective
characterization of which among them have the additional property
that the morphism π:X→ S is affine. Our second main
result, Theorem <ref>, consists of a classification
of 𝔾_a-extensions of Type II, under the additional assumption
that the morphism π:X→ S is quasi-projective. More
precisely, given a 𝔾_a-torsor ρ:P→ S∖{o}
and a 𝔾_a-extension π:X→ S with proper
𝔾_a,S-action and reduced fiber π^-1(o)_red
isomorphic to 𝔸_κ^2, we establish that the possible
geometric quotients S'=X/𝔾_a belong to a very special
class of surfaces isomorphic to open subsets of blow-ups of S with
centers over o which we fully describe in  <ref>.
We show conversely that every such surface is indeed the geometric
quotient of a 𝔾_a-extension of ρ:P→ S∖{o}
with the desired properties. 

In a second step, we tackle the question of existence of 𝔾_a-extensions
π:X→ S of Type II for which the structure morphism
π is not only quasi-projective but affine. Our method to produce
extensions with this property is inspired by the observation that
the threefolds X_0 and X_1 above are not only birational
to each other due to the property that they both contain SL_2
as open subset, but in fact that the birational morphism 

    η:X_1→ X_0,  (x,y,z_1,z_2,w)↦(x,y,p,q,r)=(x,y,xz_1,yz_1+1,w)

expresses X_1 as a 𝔾_a-equivariant affine modification
of X_0 in the sense of Kaliman and Zaidenberg <cit.>.
This suggests that extensions of Type II for which X is affine
over S could be obtained as equivariant affine modification in
a suitable generalized sense from extensions of Type I with the same
property. Using this technique, we are able to show in Theorem <ref>
that for each possible geometric quotient S' above, there exist
𝔾_a-extensions π:X→ S of ρ:P→ S∖{o}
with geometric quotient X/𝔾_a=S' such that π is
an affine morphism. 

As an application towards the initial question of the structure 𝔾_a-quotient
𝔸^1-fibrations on affine threefolds, we in particular
derive from this construction the existence of uncountably many pairwise
non isomorphic smooth affine threefolds X endowed with proper 𝔾_a-actions,
containing SL_2 as an invariant open subset with complement
𝔸^2, whose geometric quotients are smooth quasi-projective
surfaces which are not quasi-affine, and whose algebraic quotients
are all isomorphic to 𝔸^2. 

 

The scheme of the article is the following. The first section begins
with a review of general properties of 𝔾_a-extensions.
We then set up the basic tools which will be used through all the
article: locally trivial 𝔸^1-bundles with additive group
actions and equivariant affine birational morphisms between these.
In section two, we study 𝔾_a-extensions of Type I. The
last section is devoted to the classification of quasi-projective
𝔾_a-extensions of Type II. 





§ PRELIMINARIES 


In the rest of the article, the term surface refers to a normal
separated 2-dimensional scheme essentially of finite type over
a field k of characteristic zero. A punctured surface S_*=S∖{o}
is the complement of a closed point o contained in the smooth locus
of a surface S. We denote by κ the residue field of S
at o. 



We do not require that the residue field κ of S at o
is an algebraic extension of k. For instance, S can very well
be the spectrum of the local ring 𝒪_X,Z of an arbitrary
smooth k-variety X at an irreducible closed subvariety Z
of codimension two in X and o its unique closed point, in which
case the residue field κ is isomorphic to the field of rational
functions on Z. 


In this section, we first review basic geometric properties of equivariant
extensions of 𝔾_a-torsors over punctured surfaces. We
then collect various technical results on additive group actions on
affine-linear bundles of rank one and their behavior under equivariant
affine modifications. 



 §.§ Equivariant extensions of 𝔾_a-torsors 


A 𝔾_a-torsor over punctured surface S_*=S∖{o}
is an S_*-scheme ρ:P→ S_* equipped with a 𝔾_a-action
μ:𝔾_a,S_*×_S_*P→ P for which
there exists a Zariski open cover f:Y→ S_* of S_*
such that P×_S_*Y is equivariantly isomorphic to 𝔾_a,Y
acting on itself by translations. In the present article, we primarily
focus on 𝔾_a-torsors ρ:P→ S_* whose
restrictions P×_S_*U→ U∖{o} over every
Zariski open neighborhood U of o in S are nontrivial. Since
in this case the total space of P is affine over S (see e.g.
<cit.> whose proof carries over verbatim to
our more general situation), it follows that for every 𝔾_a-extension
j:P↪ X the fiber π^-1(o)⊂ X of the surjective
morphism π:X→ S has pure codimension one in X. Two
important families of examples of non trivial normal 𝔾_a-extensions
j:SL_2→ X of the 𝔾_a-torsor
ρ:SL_2→SL_2/𝔾_a≃𝔸^2∖{(0,0)},
where 𝔾_a acts on SL_2 via left multiplication
by upper triangular unipotent matrices, were constructed in <cit.>.
Various other extensions were obtained from these by performing suitable
equivariant affine modifications. One can observe that for all these
extensions, the fiber π^-1({(0,0)}) is an 𝔸^1-ruled
surface, a property which is a consequence of the following more general
fact: 

 Let ρ:P→ S_*
be a non trivial 𝔾_a-torsor over the punctured spectrum
S∖{o} of a regular local ring of dimension 2 over
an algebraically closed field k and with residue field κ(o)=k,
and let π:X→ S be a 𝔾_a-extension of
P. If X is smooth along π^-1(o), then every irreducible
component F of π^-1(o)_red is a uniruled surface.
Furthermore, if X is affine then F is 𝔸^1-uniruled,
hence 𝔸^1-ruled when it is normal. 



Since π^-1(o) has pure codimension one in X and X is
smooth along π^-1(o), every irreducible component of π^-1(o)
is a 𝔾_a-invariant Cartier divisor on X. The complement
X' in X of all but one irreducible component of π^-1(o)
is thus again a 𝔾_a-extension of P, and we may therefore
assume without loss of generality that F=π^-1(o)_red
is irreducible. Let x∈ F be a closed point in the regular locus
of F. Since F and X are smooth at x and X is connected,
there exists a curve C⊂ X, smooth at x and intersecting
F transversally at x. The image π(C) of C is a curve
on S passing through o, and the closure B of π^-1(π(C)∩ S_*)
in X is a surface containing C. Since ρ:P→ S_*
is a 𝔾_a-torsor, the restriction of π to B∩ P
is a trivial 𝔾_a-torsor over the affine curve π(C).
So π|_B:B→π(C) is an 𝔸^1-fibration.
Let ν:C̃→π(C) be the normalization of π(C).
Then π|_B lifts to an 𝔸^1-fibration θ:B̃→C̃
on the normalization B̃ of B. The fiber of θ
over every point in ν^-1(o) is a union of rational curves.
Since the normalization morphism μ:B̃→ B is
surjective, one of the irreducible components of ν^-1(o) is
mapped by μ onto a rational curve in F passing through x.
This shows that for every smooth closed point x of F, there
exists a non constant rational map h:ℙ^1 F
such that x∈ h(ℙ^1). Thus F is uniruled. If X
is in addition affine, then B and B̃ are affine surfaces,
and the fibers of the 𝔸^1-fibration θ:B̃→C̃
consist of disjoint union of curves isomorphic to 𝔸^1
when equipped with their reduced structure. This implies that F
is not only uniruled but actually 𝔸^1-uniruled. 


Let X be the smooth affine threefold
in 𝔸^2×𝔸^4=Spec(k[x,y][c,d,e,f])
defined by the equations 

    xd-y(c+1)     =0
    
    xc^2-y^2e     =0
    
    yf-c(c+1)     =0
    
    xf^2-(c+1)^2e     =0
    
    de-cf     =0

equipped with the 𝔾_a-action induced by the locally
nilpotent k[x,y]-derivation 

    xy∂_c+y^2∂_d+x(2c+1)∂_f+(2x^2f-2xye)∂_e

of its coordinate ring. The morphism j:SL_2={xv-yu=1}→ X
defined by (x,y,u,v)↦(x,y,yu,yv,xu^2,xuv) is an open embedding
of SL_2 in X as the complement of the fiber over
o=(0,0) of the projection π=pr_x,y:X→𝔸^2.
So j:SL_2→ X is an affine 𝔾_a-extension
of the 𝔾_a-torsor ρ:SL_2→SL_2/𝔾_a=𝔸^2∖{o},
for which π^-1(o) consists of the disjoint union of two copies
D_1={x=y=c=0}≃Spec(k[d,f]) and D_2={x=y=c+1=0}≃Spec(k[d,e])
of 𝔸^2. Note that the induced 𝔾_a-action
on each of these is the trivial one. 



Let X be the affine 𝔾_a-extension
constructed in the previous example and let C⊂ D_1 be any
smooth affine curve. Let τ:X̃→ X be the blow-up
of X along C, let i:X'↪X̃ be the open
immersion of the complement of the proper transform of D_1∪ D_2
in X̃ and let π'=π∘τ∘ i:X'→𝔸^2.
Since C and D_1∪ D_2 are 𝔾_a-invariant,
the 𝔾_a-action on X lifts to a 𝔾_a-action
on X̃ which restricts in turn to X'. By construction,
π' is surjective, with fiber π'^-1(o) isomorphic to
C×𝔸^1 and τ∘ i:X'→ X restricts
to an equivariant isomorphism between X'∖π'^-1(o)
and X∖π^-1(o)≃SL_2. So π':X'→𝔸^2
is a 𝔾_a-extension of the 𝔾_a-torsor ρ:SL_2→SL_2/𝔾_a=𝔸^2∖{o}. 





 §.§ Recollection on affine-linear bundles


Affine-linear bundles of rank one over a scheme are natural generalization
of 𝔾_a-torsors. To fix the notation, we briefly recall
their basic definitions and properties 

By a line bundle on a scheme S, we mean the relative spectrum p:M=Spec(Sym^·ℳ^∨)→ S
of the symmetric algebra of the dual of an invertible sheaf of 𝒪_S-module
ℳ. Such a line bundle M can be viewed as a locally
constant group scheme over S for the group law m:M×_SM→ M
whose co-morphism 

    m^♯:Sym^·ℳ^∨→Sym^·ℳ^∨⊗Sym^·ℳ^∨≃Sym^·(ℳ^∨⊕ℳ^∨)

is induced by the diagonal homomorphism ℳ^∨→ℳ^∨⊕ℳ^∨.
An M-torsor is then an S-scheme θ:W→ S
equipped with an action μ:M×_SW→ W which is
Zariski locally over S isomorphic to M acting on itself by translations. 

This is the case precisely when there exists a Zariski open cover
f:Y→ S and an 𝒪_Y-algebra isomorphism
ψ:f^*𝒜→Sym^·f^*ℳ^∨
such that over Y'=Y×_SY the automorphism p_1^*ψ∘p_2^*ψ^-1:Sym^·ℳ_Y'^∨→Sym^·ℳ_Y'^∨
of the symmetric algebra of ℳ_Y'^∨=p_2^*f^*ℳ^∨=p_1^*f^*ℳ^∨
is affine-linear, i.e. induced by an 𝒪_Y'-module
homomorphism ℳ_Y'^∨→Sym^·ℳ_Y'^∨
of the form 

    β⊕id:ℳ_Y'^∨→𝒪_Y'⊕ℳ_Y'^∨↪⊕_n≥0(ℳ_Y'^∨)^⊗ n=Sym^·ℳ_Y'^∨

for some β∈Hom_Y'(ℳ_Y'^∨,𝒪_Y')≃ H^0(Y',ℳ_Y')
which is a Čech 1-cocyle with values in ℳ for
the Zariski open cover f:Y→ S. Standard arguments show
that the isomorphism class of θ:W→ S depends only
on the class of β in the Čech cohomology group Ȟ^1(S,ℳ),
and one eventually gets a one-to-one correspondence between isomorphism
classes of M-torsors over S and elements of the cohomology group
H^1(S,M)=H^1(S,ℳ)≃Ȟ^1(S,ℳ)
with the zero element corresponding to the trivial torsor p:M→ S. 

It is classical that every locally trivial 𝔸^1-bundle
θ:W→ S over a reduced scheme S can be equipped
with the additional structure of a torsor under a uniquely determined
line bundle M on S. The existence of this additional structure
will be frequently used in the sequel, and we now quickly review its
construction (see also e.g. <cit.>). Letting
𝒜=θ_*𝒪_W, there exists by definition
a Zariski open cover f:Y→ S and a quasi-coherent 𝒪_Y-algebra
isomorphism φ:f^*𝒜→𝒪_Y[u].
Over Y'=Y×_SY equipped with the two projections p_1
and p_2 to Y, the 𝒪_Y'-algebra isomorphism
Φ=p_1^*φ∘p_2^*φ^-1
has the form 

    Φ:𝒪_Y'[u]→𝒪_Y'[u], u↦ au+b

for some a∈Γ(Y',𝒪_Y'^*) and b∈Γ(Y',𝒪_Y')
whose pull back over Y”=Y×_SY×_SY by the three projections
p_12,p_23,p_13:Y”→ Y'
satisfy the cocycle relations p_13^*a=p_23^*a·p_12^*a
and p_13^*b=p_23^*a·p_12^*b+p_23^*b
in Γ(Y”,𝒪_Y”^*) and Γ(Y”,𝒪_Y”)
respectively. The first one says that a is a Čech 1-cocycle
with values in 𝒪_S^* for the cover f:Y→ S,
which thus determines, via the isomorphism H^1(S,𝒪_S^*)≃Pic(S),
a unique invertible sheaf ℳ on S together with an
𝒪_Y-module isomorphism α:f^*ℳ^∨→𝒪_Y
such that p_1^*α∘p_2^*α^-1:𝒪_Y'→𝒪_Y'
is the multiplication by a. The second one can be equivalently
reinterpreted as the fact that β=p_2^*(^tα)(b)∈Γ(Y',ℳ_Y')
is a Čech 1-cocycle with values in ℳ for the
Zariski open cover f:Y→ S. Letting Sym^·(α):Sym^·f^*ℳ^∨→𝒪_Y[u]
be the graded 𝒪_Y-algebra isomorphism induced by α,
the isomorphism ψ=Sym^·(α^-1)∘φ:f^*𝒜→Sym^·f^*ℳ^∨
has the property that p_1^*ψ∘p_2^*ψ^-1
is affine-linear, induced by the homomorphism β⊕id:ℳ_Y'^∨→𝒪_Y'⊕ℳ_Y'^∨.
So θ:W→ S is a torsor under the line bundle M=Spec(Sym^·ℳ^∨),
with isomorphism class in H^1(S,M) equal to the cohomology class
of the cocyle β. Summing up, we obtain;

Let θ:W→ S be a locally
trivial 𝔸^1-bundle. Then there exists a unique pair
(M,g) consisting of a line bundle M on S and a class g∈ H^1(S,M)
such that θ:W→ S is an M-torsor with isomorphism
class g. 





 §.§ Additive group actions on affine-linear bundles of rank one 


Given a locally trivial 𝔸^1-bundle θ:W→ S,
which we view as an M-torsor for a line bundle M=Spec(Sym^·ℳ^∨)→ S
on S, with corresponding action μ:M×_SW→ W,
every nonzero group scheme homomorphism ξ:𝔾_a,S→ M
induces a nontrivial 𝔾_a,S-action ν=μ∘(ξ×id):𝔾_a,S×_SW→ W
on W. A nonzero group scheme homomorphism ξ:𝔾_a,S=Spec(𝒪_S[t])→ M=Spec(Sym^·ℳ^∨)
is uniquely determined by a nonzero 𝒪_S-module homomorphism
ℳ^∨→𝒪_S, equivalently by a
nonzero global section s∈Γ(S,ℳ). The following
proposition asserts conversely that every nontrivial 𝔾_a,S-action
on an M-torsor θ:W→ S uniquely arises from such
a section. 

(<cit.>) Let
θ:W→ S be a torsor under the action μ:M×_SW→ W
of a line bundle M=Spec(Sym^·ℳ^∨)→ S
on S and let ν:𝔾_a,S×_SW→ W be
a non trivial 𝔾_a,S-action on W. Then there exists
a non zero global section s∈Γ(S,ℳ) such that ν=μ∘(ξ×id)
where ξ:𝔾_a,S→ M is the group scheme homomorphism
induced by s. 



Let 𝒜=θ_*𝒪_W and let f:Y→ S
be a Zariski open cover such that there exists an 𝒪_Y-algebra
isomorphism φ:f^*𝒜→𝒪_Y[u],
and let 

    Φ=p_1^*φ∘p_2^*φ^-1:𝒪_Y'[u]→𝒪_Y'[u],   u↦ au+b

be as in (<ref>) above. Since θ:W→ S
is an M-torsor, φ also determines an 𝒪_Y-module
isomorphism α:f^*ℳ^∨→𝒪_Y
such that p_1^*α∘p_2^*α^-1:𝒪_Y'→𝒪_Y'
is the multiplication by a. The 𝔾_a,S-action ν
on W pulls back to a 𝔾_a,Y-action ν×id
on W×_S̃Y. The co-mophism η:𝒪_Y[u]→𝒪_Y[u]⊗𝒪_Y[t]
of the nontrivial 𝔾_a,Y-action φ∘(ν×id)∘(id×φ^-1)
on Spec(𝒪_Y[u]) has the form u↦ u⊗1+1⊗γ t
for some nonzero γ∈Γ(Y,𝒪_Y). Letting ℐ=γ·𝒪_Y
be the ideal sheaf generated by γ, η factors as 

    η=(id⊗ j)∘η̃:𝒪_Y[u]→𝒪_Y[u]⊗Sym^·ℐ→𝒪_Y[u]⊗𝒪_Y[t]

where η̃ is the co-morphism of an action of the line
bundle Spec(Sym^·ℐ)→ Y
on 𝔸_S^1×_SY≃ W×_SY and j:Sym^·ℐ→𝒪_Y[t]
is the homomorphism induced by the inclusion ℐ⊂𝒪_Y.
Pulling back to Y', we find that p_2^*γ=a·p_1^*γ,
which implies that ^tα(γ)∈Γ(Y,f^*ℳ)
is the pull-back f^*s to Y of a nonzero global section s∈Γ(S,ℳ).
Letting D=div_0(s) be the divisors of zeros of s,
we have ℳ^∨≃𝒪_S(-D)⊂𝒪_S
and f^*ℳ^∨≃𝒪_Y(-f^*D)⊂𝒪_Y
is equal to the ideal ℐ=γ·𝒪_Y. The
global section f^*s viewed as a homomorphism f^*ℳ^∨→𝒪_Y
coincides via these isomorphisms with the inclusion γ·𝒪_Y↪𝒪_Y.
We can thus rewrite η in the form 

    η=(id⊗Sym^·f^*s)∘η̃:𝒪_Y[u]→𝒪_Y[u]⊗Sym^·f^*ℳ^∨→𝒪_Y[u]⊗𝒪_Y[t].

By construction η̃=(φ⊗id)∘ f^*μ^♯∘φ^-1
where f^*μ^♯ is the pull-back of the co-morphism μ^♯:𝒜→𝒜⊗Sym^·ℳ^∨
of the action μ:M×_SW→ W of M on W. It
follows that the pull-back f^*ν^♯ of the co-morphism
of the action ν:𝔾_a,S× W→ W factors
as 

    f^*ν^♯=(id⊗Sym^·f^*s)∘ f^*μ^♯=f^*𝒜→ f^*𝒜⊗Sym^·f^*ℳ^∨→ f^*𝒜⊗𝒪_Y[t]

This in turn implies that ν^♯ factors as (id⊗Sym^·s)∘μ^♯:𝒜→𝒜⊗Sym^·ℳ^∨→𝒜⊗𝒪_Y[t]
as desired. 


In the setting of Proposition <ref>, letting
U⊂ S be the complement of the zero locus of s, the morphism
ξ restricts to an isomorphism of group schemes ξ|_U:𝔾_a,U→ M|_U
for which W|_U equipped with the 𝔾_a,U-action ν|_U:𝔾_a,U×_UW|_U→ W|_U
is a 𝔾_a,U-torsor. This isomorphism class in H^1(U,𝒪_U)
of this 𝔾_a,U-torsor coincides with the image of the
isomorphism class g∈ H^1(S,ℳ) of W by the composition
of the restriction homomorphism res:H^1(S,ℳ)→ H^1(U,ℳ|_U)
with the inverse of the isomorphism H^1(U,𝒪_U)→ H^1(U,ℳ|_U)
induced by s|_U. 





 §.§ 𝔾_a-equivariant affine modifications of affine-linear
bundles of rank one


Recall <cit.> that given an integral scheme X with sheaf
of rational functions 𝒦_X, an effective Cartier divisor
D on X and a closed subscheme Z⊂ X whose ideal sheaf
ℐ⊂𝒪_X contains 𝒪_X(-D),
the affine modification of X with center (ℐ,D)
is the affine X-scheme σ:X'=Spec(𝒪_X[ℐ/D])→ X
where 𝒪_X[ℐ/D] denotes the quotient of the
Rees algebra 

    𝒪_X[(ℐ⊗𝒪_X(D))]=⊕_n≥0(ℐ⊗𝒪_X(D))^nt^n⊂𝒦_X[t]

of the fractional ideal ℐ⊗𝒪_X(D)⊂𝒦_X
by the ideal generated by 1-t. In the case where X=Spec(A)
is affine, D=div(f) is principal and Z is defined by
an ideal I⊂ A containing f then X̃ is isomorphic
to the affine modification X'=Spec(A[I/f]) of X with
center (I,f) in the sense of <cit.>. 

Now let S be an integral scheme and let θ:W→ S
be a locally trivial 𝔸^1-bundle. Let C⊂ S be
an integral Cartier divisor, let D=θ^-1(C) be its inverse
image in W and let Z⊂ D be a non empty integral closed
subscheme of D on which θ restricts to an open embedding
θ|_Z:Z↪ C. Equivalently, Z is the closure
in D of the image α(U) of a rational section α:C→ D
of the locally trivial 𝔸^1-bundle θ|_D:D→ C
defined over a non empty open subset U of C. The complement
F of θ|_Z(Z) in C is a closed subset of C hence
of S. Letting i:S∖ F↪ S be the natural
open embedding, we have the following result:

 Let σ:W'→ W be the
affine modification of W with center (ℐ_Z,D). Then
the composition θ∘σ:W'→ S factors through
a locally trivial 𝔸^1-bundle θ':W'→ S∖ F
in such a way that we have a cartesian diagram 
    W' [d]_θ '[r]^σ    W [d]^θ
     S∖ F [r]^i     S.
 



The question being local with respect to a Zariski open cover of S
over which θ:W→ S becomes trivial, we can assume
without loss of generality that S=Spec(A), W=Spec(A[x]),
C=div(f) for some non zero element f∈ A. The integral
closed subscheme Z⊂ D is then defined by an ideal I of
the form (f,g) where g(x)∈ A[x] is an element whose image
in (A/f)[x] is a polynomial of degree one in t. So g(x)=a_0+a_1x+x^2fR(x)
where a_0∈ A, a_1∈ A has non zero residue class in A/f
and R(x)∈ A[x]. The condition that θ|_Z:Z→ C
is an open embedding implies further that the residue classes a_0
and a_1 of a_0 and a_1 in A/f generate
the unit ideal. The complement F of the image of θ|_Z(Z)
in C is then equal to the closed subscheme of C with defining
ideal (a_1)⊂ A/f, hence to the closed subscheme
of S with defining ideal (f,a_1)⊂ A. The algebra A[t][I/f]
is isomorphic to 

    A[x][u]/(g-fu)=A[x][u-x^2R(x)]/(a_0+a_1x-f(u-t^2R(x))≃ A[x][v]/(a_0+a_1x-fv).

One deduces from this presentation that the morphism θ∘σ:W'=Spec(A[I/f])→Spec(A)
corresponding to the inclusion A→ A[I/f] factors through
a locally trivial 𝔸^1-bundle θ':W'→ S∖ F
over the complement of F. Namely, since a_0 and
a_1 generate the unit ideal in A/f, it follows
that a_1 and f generate the unit ideal in A[x][u]/(g-fu).
So W' is covered by the two principal affine open subsets 

    W'_a_1   ≃Spec(A_a_1[x][v]/(a_0+a_1x-fv))≃Spec(A_a_1[v])≃ S_a_1×𝔸^1
    
    W'_f   ≃Spec(A_f[x][v]/(a_0+a_1x-fv))≃Spec(A_f[x])≃ S_f×𝔸^1

on which θ' restricts to the projection onto the first factor. 

With the notation above, θ:W→ S and θ':W'→ S∖ F
are torsors under the action of line bundles M=Spec(Sym^·ℳ^∨)
and M'=Spec(Sym^·ℳ'^∨)
for certain uniquely determined invertible sheaves ℳ
and ℳ' on S and S∖ F respectively. 

 (<cit.>) Let σ:W'→ W
be the affine modification of W with center (ℐ_Z,D)
as is Lemma <ref>. Then ℳ'=ℳ⊗_𝒪_S𝒪_S(-C)|_S∖ F
and the cartesian diagram of Lemma <ref> is equivariant
for the group scheme homomorphism ξ:M'→ M induced by
the homomorphism ℳ⊗_𝒪_S𝒪_S(-C)→ℳ
obtained by tensoring the inclusion 𝒪_S(-C)↪𝒪_S
by ℳ. 



Since M and M' are uniquely determined, the question is again
local with respect to a Zariski open cover of S over which θ:W→ S,
hence M, becomes trivial. We can thus assume as in the proof of
Lemma <ref> that S=Spec(A), W=Spec(A[x]),
that C=div(f) for some non zero element f∈ A and
that Z⊂ D is defined by the ideal (f,g) for some g=a_0+a_1x+fx^2R(x)∈ A[x].
Furthermore, the action of M≃𝔾_a,S=Spec(A[t])
on W≃ S×𝔸^1 is the one by translations x↦ x+t
on the second factor. Let N=Spec(Sym^·𝒪_S(C))≃Spec(Sym^·f^-1A)
where f^-1A denotes the free sub-A-module of the field of
fractions Frac(A) of A generated by f^-1. As in
the proof of Proposition <ref>, the inclusion
𝒪_S(-C)=f·𝒪_S↪𝒪_S
induces a group-scheme homomorphism ξ:N→ M whose co-morphism
ξ^♯ concides with the inclusion A[t]⊂Sym^·f^-1A=A[(f^-1t)].
The co-morphism of the corresponding action of N on W is given
by 

    A[x]→ A[x]⊗ A[f^-1t], x↦ x⊗1+1⊗ t=x⊗1+f⊗ f^-1t.

This action lifts on W'≃Spec(A[x][v]/(a_0+a_1x-fv))
to an action ν:N×_SW'→ W' whose co-morphism

    A[x][v]/(a_0+a_1x-fv)→ A[x][v]/(a_0+a_1x-fv)⊗ A[f^-1t]

is given by x↦ x⊗1+1⊗ t and v↦ v⊗1+a_1⊗ f^-1t.
By construction, the principal open subsets W'_a_1≃Spec(A_a_1[v])≃Spec(A_a_1[v/a_1])
and W'_f≃Spec(A_f[x])≃Spec(A_f[x/f])
of W' equipped with the induced actions of N|_S_a_1 and
N|_S_f respectively are equivariantly isomorphic to N|_S_a_1
and N|_S_f acting on themselves by translations. So θ':W'→ S∖ F
is an N|_S∖ F-torsor, showing that ℳ'=ℳ⊗_𝒪_S𝒪_S(-C)|_S∖ F
as desired. 




§ EXTENSIONS OF 𝔾_A-TORSORS OF TYPE I: LOCALLY TRIVIAL
BUNDLES OVER THE BLOW-UP OF A POINT


Given a surface S and a locally trivial 𝔸^1-bundle
θ:W→S̃ over the blow-up τ:S̃→ S
of a closed point o in the smooth locus of S, the restriction
of W over the complement S̃∖ E of the exceptional
divisor E of τ is a locally trivial 𝔸^1-bundle
τ∘θ:W|_S̃∖ E→S̃∖ E≃→S∖{o}.
This observation combined with the following re-interpretation of
an example constructed in <cit.> suggests that locally trivial
𝔸^1-bundles over the blow-up of closed point o in
the smooth locus of a surface S form a natural class of schemes
in which to search for nontrivial 𝔾_a-extension of 𝔾_a-bundles
over punctured surfaces. 

 Let o=V(x,y) be a global
scheme-theoretic complete intersection closed point in the smooth
locus of a surface S. Let ρ:P→ S∖{o}
and π_0:X_0→ S be the affine S-schemes with
defining sheaves of ideals (xv-yu-1) and (xr-yq,yp-x(q-1),pr-q(q-1))
in 𝒪_S[u,v] and 𝒪_S[p,q,r] respectively.
The morphism of S-schemes j_0:P→ X_0 defined by
(x,y,u,v)↦(x,y,xu,xv,yv) is an open embedding, equivariant
for the 𝔾_a,S-actions on P and X_0 associated
with the locally nilpotent 𝒪_S-derivations x∂_u+y∂_v
and x^2∂_p+xy∂_q+y^2∂_r of ρ_*𝒪_P
and (π_0)_*𝒪_X_0 respectively. It is straightforward
to check that ρ:P→ S∖{o} is a 𝔾_a,S_*-torsor
and that π_0:X_0→ S is a 𝔾_a-extension
of P whose fiber over o is isomorphic to the smooth affine quadric
{pr-q(q-1)=0}⊂𝔸_κ^3. Viewing the blow-up
S̃ of o as the closed subscheme of S×_kProj(k[u_0,u_1])
with equation xu_1-yu_0=0, the morphism of S-schemes θ:X_0→S̃
defined by 

    (x,y,p,q,r)↦((x,y),[x:y])=((x,y),[q:r])=((x,y),[p:q-1])

is a locally trivial 𝔸^1-bundle, actually a torsor under
the line bundle corresponding to the invertible sheaf 𝒪_S̃(-2E),
where E≃ℙ_κ^1 denotes the exceptional divisor
of the blow-up. 



Given a surface S and a closed point o in the smooth locus of
S, with residue field κ, we denote by τ:S̃→ S
be the blow-up of o, with exceptional divisor E≃ℙ_κ^1.
We identify S̃∖ E and S_*=S∖{o} by
the isomorphism induced by τ. For every ℓ∈ℤ,
we denote by M(ℓ)=Spec(Sym^·𝒪_S̃(-ℓ E))
the line bundle on S̃ corresponding to the invertible sheaf
𝒪_S̃(ℓ E). 


The aim of this section is to give a classification of all possible
𝔾_a-equivariant extensions of Type I of a given 𝔾_a-torsor
ρ:P→ S_*, that is 𝔾_a-extensions π:W→ S
that factor through locally trivial 𝔸^1-bundles θ:W→S̃. 



 §.§ Existence of 𝔾_a-extensions of Type I


By virtue of Propositions <ref> and <ref>,
there exists a one-to-one correspondence between 𝔾_a-equivariant
extensions of a 𝔾_a-torsor ρ:P→ S_*
that factor through a locally trivial 𝔸^1-bundle θ:W→S̃
and pairs (M,ξ) consisting of an M-torsor θ:W→S̃
for some line bundle M on S̃ and a group scheme homomorphism
ξ:𝔾_a,S̃→ M restricting to an isomorphism
over S̃∖ E, such that W equipped with the 𝔾_a,S̃-action
deduced by composition with ξ restricts on S_*=S̃∖ E
to a 𝔾_a,S_*-torsor θ|_S_*:W|_S_*→ S_*
isomorphic to ρ:P→ S_*. The condition that ξ:𝔾_a,S̃→ M
restricts to an isomorphism outside E implies that M≃ M(ℓ)
for some ℓ, which is necessarily non negative, and that ξ
is induced by the canonical global section of 𝒪_S̃(ℓ E)
with divisor ℓ E. 

 Let ρ:P→ S_* be a
𝔾_a,S_*-torsor. Then there exists an integer ℓ_0≥0
depending on P only such that for every ℓ≥ℓ_0, P
admits a 𝔾_a-extension to a uniquely determined M(ℓ)-torsor
θ_ℓ:W(P,ℓ)→S̃ equipped with the 𝔾_a,S̃-action
induced by the canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E))
with divisor ℓ E. 



The invertible sheaves 𝒪_S̃(nE), n≥0,
form an inductive system of sub-𝒪_S̃-modules
of the sheaf 𝒦_S̃ of rational function on S̃,
where for each n, the injective transition homomorphism j_n,n+1:𝒪_S̃(nE)↪𝒪_S̃((n+1)E)
is obtained by tensoring the canonical section 𝒪_S̃→𝒪_S̃(E)
with divisor E with 𝒪_S̃(nE). Let i:S_*=S̃∖ E↪S̃
be the open inclusion. Since E is a Cartier divisor, it follows
from <cit.> that i_*𝒪_S_*≃colim_n≥0𝒪_S̃(nE).
Furthermore, since E≃ℙ_κ^1 is the exceptional
divisor of τ:S̃→ S, we have 𝒪_S̃(E)|_E≃𝒪_ℙ_κ^1(-1),
and the long exact sequence of cohomology for the short exact sequence

    0→𝒪_S̃(nE)→𝒪_S̃((n+1)E)→𝒪_S̃((n+1)E)|_E→0,   n≥0,

combined with the vanishing of H^0(ℙ_κ^1,𝒪_ℙ_κ^1(-n-1))
for every n≥0 implies that the transition homomorphisms 

    H^1(j_n,n+1):H^1(S̃,𝒪_S̃(nE))→ H^1(S̃,𝒪_S̃((n+1)E)), n≥0,

are all injective. By assumption, S whence S̃ is noetherian,
and i:S_*→S̃ is an affine morphism as E is
a Cartier divisor on S̃. We thus deduce from <cit.>
and <cit.> that the canonical homomorphism

    ψ:colim_n≥0H^1(S̃,𝒪_S̃(nE))→ H^1(S_*,𝒪_S_*)

obtained as the composition of the canonical homomorphisms 

    colim_n≥0H^1(S̃,𝒪_S̃(nE))→ H^1(S̃,colim_n≥0𝒪_S̃(nE))=H^1(S̃,i_*𝒪_S_*)

and H^1(S̃,i_*𝒪_S_*)→ H^1(S_*,𝒪_S_*)
is an isomorphism. 

Let g∈ H^1(S_*,𝒪_S_*) be the isomorphism class
of the 𝔾_a,S_*-torsor ρ:P→ S_*.
If g=0, then since ψ is an isomorphism, we have ψ^-1(g)=0
and, since the homomorphisms H^1(j_n,n+1) are injective, it
follows that ψ^-1(g) is represented by the zero sequence (0)_n∈ H^1(S̃,𝒪_S̃(nE)),
n≥0 . Consequently, the only 𝔾_a-extensions of
P are the line bundles W(P,ℓ)=M(ℓ), ℓ≥0, each
equipped with the 𝔾_a,S̃-action induced by its
canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)). 

Otherwise, if g≠0, then h=ψ^-1(g)≠0, and since the
homomorphisms H^1(j_n,n+1), n≥0 are injective, it follows
that there exists a unique minimal integer ℓ_0 such that h
is represented by the sequence 

    h_n=H^1(j_n-1,n)∘⋯∘ H^1(j_ℓ_0,ℓ_0+1)(h_ℓ_0)∈ H^1(S̃,𝒪_S̃(nE)), n≥ℓ_0

for some non zero h_ℓ_0∈ H^1(S̃,𝒪_S̃(ℓ_0E)).
It then follows from Proposition <ref> that
for every ℓ≥ℓ_0, the M(ℓ)-torsor θ_ℓ:W(P,ℓ)→S̃
with isomorphism class h_ℓ equipped with the 𝔾_a,S̃-action
induced by the canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E))
is a 𝔾_a-extension of P. 

Conversely, for every 𝔾_a-extension of P into an
M(ℓ)-torsor θ:W→S̃ equipped with the
𝔾_a,S̃-action induced by the canonical global
section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)),
it follows from Proposition <ref> again that
the image of the isomorphism class h_ℓ∈ H^1(S̃,𝒪_S̃(ℓ E))
of W in H^1(S̃∖ E,𝒪_S̃(ℓ E)|_S̃∖ E)≃ H^1(S_*,𝒪_S_*)
is equal to g. Letting h∈colim_n≥0H^1(S̃,𝒪_S̃(nE))
be the element represented by the sequence 

    h_n=(H^1(j_n-1,n∘⋯∘ j_ℓ,ℓ+1)(h_ℓ))_n≥ℓ∈ H^1(S̃,𝒪_S̃(nE)), n≥ℓ

we have ψ(h)=g and since ψ is an isomorphism, we conclude
that W≃ W(P,ℓ). 




 §.§ 𝔾_a-extensions with
affine total spaces 


The extensions θ:W→S̃ we get from Proposition
<ref> are not necessarily affine over S.
In this subsection we establish a criterion for affineness which we
then use to characterize all extensions θ:W→S̃
of a 𝔾_a-torsor ρ:P→ S_* whose total
spaces W are affine over S.

 Let S=Spec(A) be an
affine surface and let o=V(x,y) be a global scheme-theoretic complete
intersection point in the smooth locus of S . Let τ:S̃→ S
be the blow-up of o with exceptional divisor E and let θ:W→S̃
be an M(ℓ)-torsor for some ℓ≥0. Then the following
hold:

a) H^1(W,𝒪_W)=0. 

b) If H^1(W,θ^*𝒪_S̃(ℓ E))=0 for
some ℓ≥2 then W is an affine scheme. 



Since o is a scheme-theoretic complete intersection, we can identify
S̃ with the closed subvariety of S×_kℙ_k^1=S×_kProj(k[t_0,t_1])
defined by the equation xt_1-yt_0=0. The restriction p:S̃→ℙ_k^1
of the projection to the second factor is an affine morphism. More
precisely, letting U_0=ℙ_k^1∖{[1:0]}≃Spec(k[z])
and U_∞=ℙ_k^1∖{[0:1]}≃Spec(k[z'])
be the standard affine open cover of ℙ_k^1, we have
p^-1(U_0)≃Spec(A[z]/(x-yz) and p^-1(U_∞)≃Spec(A[z']/(y-xz')).
The exceptional divisor E≃ℙ_κ^1 of τ:S̃→ S
is a flat quasi-section of p with local equations y=0 and x=0
in the affine charts p^-1(U_0) and p^-1(U_∞) respectively.
Every M(ℓ)-torsor θ:W→S̃ for some ℓ≥0
is isomorphic to the scheme obtained by gluing W_0=p^-1(U_0)×Spec(k[u])
with W_∞=p^-1(U_∞)×Spec(k[u']) over
U_0∩ U_∞ by an isomorphism induced by a k-algebra
isomorphism of the form 

    A[(z')^±1]/(y-xz')[u']∋(z',u')↦(z^-1,z^ℓu+p)∈ A[z^±1]/(x-yz)[u]

for some p∈ A[z^±1]/(x-yz). Since H^1(W,𝒪_W)≃Ȟ^1(W,𝒪_W)≃Ȟ^1({W_0,W_∞},𝒪_W),
it is enough in order to prove a) to check that every Čech 1-cocycle
g with value in 𝒪_W for the covering of W by
the affine open subsets W_0 and W_∞ is a coboundary.
Viewing g as an element g=g(z^±1,u)∈ A[z^±1]/(x-yz)[u],
it is enough to show that every monomial g_s=hz^ru^s where
h∈ A, r∈ℤ and s∈ℤ_≥0 is a coboundary,
which is the case if and only if there exist a(z,u)∈ A[z]/(f-gz)[u]
and b(z',u')∈ A[z']/(y-xz')[u'] such that g=b(z^-1,z^ℓu+p)-b(z,u).
If r≥0 then g∈ A[z]/(x-yz)[u] is a coboundary. We thus
assume from now on that r<0. Suppose that s>0. Then we can write
u^s=z^-ℓ s(z^ℓu+p)^s-R(u) where R∈ A[z^±1]/(x-yz)[u]
is polynomial whose degree in u is strictly less than s. Then
since r<0, 

    hz^ru^s    =hz^r-ℓ s(z^ℓu+p)^s-hz^rR(u)
        =b(z^-1,z^ℓu+p)-hz^rR(u)

where b(z',u')=h(z')^-r+ℓ s(u')^s∈ A[z']/(y-xz')[u'].
So g_s is a coboundary if and only if -hz^rR(u) is. By induction,
we only need to check that every monomial g_0=hz^r∈ A[z^±1]/(x-yz)[u]
of degree 0 in u is a coboundary. But such a cocycle is simply
the pull-back to W of a Čech 1-cocycle h_0 with value
in 𝒪_S̃ for the covering of S̃ by
the affine open subsets p^-1(U_0) and p^-1(U_∞).
Since the canonical homomorphism 

    H^1(S,𝒪_S)=H^1(S,τ_*𝒪_S̃)→ H^1(S̃,𝒪_S̃)≃Ȟ^1({p^-1(U_0),p^-1(U_∞)},𝒪_S̃)

is an isomorphism and H^1(S,𝒪_S)=0 as S is affine,
we conclude that h_0 is a coboundary, hence that g_0 is
a coboundary too. This proves a). 

Now suppose that H^1(W,θ^*𝒪_S̃(ℓ E))=0
for some ℓ≥2. Let η:V→ℙ_k^1
be a non trivial 𝒪_ℙ_k^1(-ℓ)-torsor
and consider the fiber product W×_p∘θ,ℙ_k^1,ηV:

    @C-3ex    W×_p∘θ,ℙ^1_k,η V [dl] [dr] 
     W [dr]_p∘θ       V [dl]^η
       ℙ^1_k

By virtue of <cit.>, V is an affine surface.
Since p∘θ:W→ℙ_k^1 is an affine
morphism, so is pr_V:W×_ℙ_k^1V→ V
and hence, W×_ℙ_k^1V is an affine scheme. On
the other hand, since p^*𝒪_ℙ_k^1(-1)≃𝒪_S̃(E),
the projection pr_W:W×_ℙ_k^1V→ W
is a θ^*M(ℓ)-torsor, hence is isomorphic to the trivial
one q:θ^*M(ℓ)→ W by hypothesis. So W is
isomorphic to the zero section of θ^*M(ℓ), which is a
closed subscheme of the affine scheme W×_ℙ_k^1V,
hence an affine scheme. 

We are now ready to prove the following characterization: 

 A 𝔾_a,S_*-torsor ρ:P→ S_*
admits a 𝔾_a-extension to a locally trivial 𝔸^1-bundle
whose total space is affine over S if and only if for every Zariski
open neighborhood U of o, P×_S_*U→ U_*=U∖{o}
is a non trivial 𝔾_a,U_*-torsor. 

When it exists, the corresponding locally trivial 𝔸^1-bundle
θ:W→S̃ is unique and is an M(ℓ_0)-torsor
for some ℓ_0≥2, whose restriction to E≃ℙ_κ^1
is a non trivial 𝒪_ℙ_κ^1(-ℓ_0)-torsor. 



The scheme W is affine over S if and only if its restriction
W|_E over E⊂S̃ is a nontrivial torsor. Indeed,
if W|_E is a trivial torsor then it is a line bundle over E≃ℙ_κ^1.
Its zero section is then a proper curve contained in the fiber of
π=τ∘θ:W→ S, which prevents π from
being an affine morphism. Conversely, if W|_E is nontrivial,
then it is a torsor under a uniquely determined line bundle 𝒪_ℙ_κ^1(-m)
for some m≥2 necessarily. Since by construction π restricts
over S_* to ρ:P→ S_* which is an affine morphism,
π is affine if and only if there exists an open neighborhood
U of o in S such that π^-1(U) is affine. Replacing
S by a suitable affine open neighborhood of o, we can therefore
assume without loss of generality that S=Spec(A) is affine
and that o is a scheme-theoretic complete intersection o=V(x,y)
for some elements x,y∈ A. By virtue of <cit.>
every nontrivial 𝒪_ℙ_κ^1(-m)-torsor,
m≥2 , has affine total space. The Cartier divisor D=W|_E
in W is thus an affine surface, and so H^1(D,𝒪_W((n+1)D)|_D)=0
for every n∈ℤ. By a) in Lemma <ref>,
H^1(W,𝒪_W)=0, and we deduce successively from the
long exact sequence of cohomology for the short exact sequence

    0→𝒪_W(nD)→𝒪_W((n+1)D)→𝒪_W((n+1)D)|_D→0

in the case n=0 and then n=1 that H^1(W,𝒪_W(D))=H^1(W,𝒪_W(2D))=0.
Since 𝒪_W(2D)≃θ^*𝒪_S̃(2E),
we conclude from b) in the same lemma that W is affine. 

The condition that P×_S_*U→ U_* is nontrivial
for every open neighborhood U of o is necessary for the existence
of an extension θ:W→S̃ of P for which
W|_E is a nontrivial torsor. Indeed, if there exists a Zariski
open neighborhood U of o such that the restriction of P over
U_* is the trivial 𝔾_a,U_*-torsor, then the image
in H^1(U_*,𝒪_U_*) of the isomorphism class g
of P is zero and so, arguing as in the proof of Proposition <ref>,
every 𝔾_a-extension θ:W→S̃
restricts on τ^-1(U) to the trivial M(ℓ)|_τ^-1(U)-torsor
M(ℓ)|_τ^-1(U)→τ^-1(U), hence to a trivial
torsor on E⊂τ^-1(U). 

Now suppose that ρ:P→ S_* is a 𝔾_a,S_*-torsor
with isomorphism class g∈ H^1(S_*,𝒪_S_*) such
that P×_S_*U→ U_* is non trivial for every
open neighborhood U of o. The inverse image h=ψ^-1(g)∈colim_n≥0H^1(S̃,𝒪_S̃(nE))
of g by the isomorphism (<ref>) is represented by
a sequence of nonzero elements h_n∈ H^1(S̃,𝒪_S̃(nE))
as in (<ref>) above. By the long exact sequence
of cohomology of the short exact sequence (<ref>),
the image h_n of h_n in H^1(E,𝒪_S̃(nE)|_E)≃ H^1(ℙ^1,𝒪_ℙ_κ^1(-n))
is nonzero if and only if h_n is not in the image of the injective
homomorphism H^1(j_n,n-1). Since h_n coincides
with the isomorphism class of the restriction W_n|_E of an
M(n)-torsor θ_n:W_n→S̃ with isomorphism
class h_n, we conclude that there exists a unique ℓ_0≥2
such that the restriction to E of an M(ℓ_0)-torsor θ_ℓ_0:W_ℓ_0→S̃
with isomorphism class h_ℓ_0∈ H^1(S̃,𝒪_S̃(ℓ_0E))
is a nontrivial 𝒪_ℙ_κ^1(-ℓ_0)-torsor. 




 §.§ Examples


In this subsection, we consider 𝔾_a-torsors of the punctured
affine plane. So S=𝔸^2=Spec(k[x,y]), o=(0,0)
and 𝔸_*^2=𝔸^2∖{o}. We let τ:𝔸̃^2→𝔸^2
be the blow-up of o, with exceptional divisor E≃ℙ^1
and we let i:𝔸_*^2↪𝔸̃^2
be the immersion of 𝔸_*^2 as the open subset 𝔸̃^2∖ E.
We further identify 𝔸̃^2 with the total space
f:𝔸̃^2→ℙ^1 of the line bundle
𝒪_ℙ^1(-1) in such a way that E corresponds
to the zero section of this line bundle.



  §.§.§ A simple case: homogeneous 𝔾_a-torsors 


Following <cit.>, we say that a non trivial 𝔾_a,𝔸_*^2-torsor
ρ:P→𝔸_*^2 is homogeneous if it admits
a lift of the 𝔾_m-action λ·(x,y)=(λ x,λ y)
on 𝔸_*^2 which is locally linear on the fibers of
ρ. By <cit.>, this is the case if and
only if the isomorphism class g of P in H^1(𝔸_*^2,𝒪_𝔸_*^2)
can be represented on the open covering of 𝔸_*^2 by
the principal open subsets 𝔸_x^2 and 𝔸_y^2
by a Čech 1-cocycle of the form x^-my^-np(x,y)
where m,n≥0 and p(x,y)∈ k[x,y] is a homogeneous polynomial
of degree r≤ m+n-2. Equivalently, P is isomorphic the 𝔾_a,𝔸_*^2-torsor

    ρ=pr_x,y:P_m,n,p={ x^mv-y^nu=p(x,y)}∖{x=y=0}→𝔸_*^2,

which admits an obvious lift λ·(x,y,u,v)=(λ x,λ y,λ^m-du,λ^n-dv),
where d=m+n-r, of the 𝔾_m-action on 𝔸_*^2.
Let q:𝔸_*^2→𝔸_*^2/𝔾_m=ℙ^1
be the quotient morphism of the aforementioned 𝔾_m-action
on 𝔸_*^2. Then it follows from <cit.>
that the inverse image by the canonical isomorphism 

    ⊕_k∈ℤH^1(ℙ^1,𝒪_ℙ(k))≃ H^1(ℙ^1,q_*𝒪_𝔸_*^2)→ H^1(𝔸_*^2,𝒪_𝔸_*^2)

of the isomorphism class g of such an homogeneous torsor is an
element h of H^1(ℙ^1,𝒪_ℙ(-d)).
Furthermore, the 𝔾_m-equivariant morphism ρ:P→𝔸_*^2
descends to a locally trivial 𝔸^1-bundle ρ:P/𝔾_m→ℙ^1=𝔸_*^2/𝔾_m
which is an 𝒪_ℙ^1(-d)-torsor with isomorphism
class h∈ H^1(ℙ^1,𝒪_ℙ(-d)).

Since f^*𝒪_ℙ^1(-d)≃𝒪_𝔸̃^2(dE),
the fiber product W(P,d)=𝔸̃^2×_ℙ^1P/𝔾_m
is equipped via the restriction of the first projection with the structure
of an M(d)-torsor θ:W(P,d)→𝔸̃^2
with isomorphism class f^*h∈ H^1(𝔸̃^2,𝒪_𝔸̃^2(dE)).
On the other other hand, W(P,d) is a line bundle over P/𝔾_m
via the second projection, hence is an affine threefold as P/𝔾_m
is affine. By construction, we have a commutative diagram 
    @C+2ex@R-1ex    W(P,d) @->'[d]^-θ[dd] [dr] 
     P [dd]_ρ[ur]^j[rr]        P/𝔾_m [dd]_ρ
       𝔸̃^2 [dr]^f
    𝔸^2_*[ur]^i[rr]^q      ℙ^1
in
which each square is cartesian. In other words, W(P,d) is obtained
from the 𝔾_m-torsor P→ P/𝔾_m
by “adding the zero section”. The open embedding j:P↪ W(P,d)
is equivariant for the 𝔾_a-action on W(P,d) induced
by the canonical global section of 𝒪_𝔸̃^2(dE)
with divisor dE (see Proposition <ref>).
By Theorem <ref> , θ:W(P,d)→𝔸̃^2
is the unique 𝔾_a-extension of ρ:P→𝔸_*^2
with affine total space. 

In the simplest case d=2, the unique homogeneous 𝔾_a,𝔸_*^2-torsor
is the geometric quotient SL_2→SL_2/𝔾_a
of the group SL_2 by the action of its subgroup of upper
triangular unipotent matrices equipped with the diagonal 𝔾_m-action,
and we recover Example <ref>. 



  §.§.§ General case


Here, given an arbitrary non trivial 𝔾_a-torsor ρ:P→𝔸_*^2,
we describe a procedure to explicitly determine the unique 𝔾_a-extension
θ:W→𝔸̃^2 of P with affine total
space W from a Čech 1-cocycle x^-my^-np(x,y),
where m,n≥0 and p(x,y)∈ k[x,y] is a non zero polynomial
of degree r≤ m+n-2, representing the isomorphism class g∈ H^1(𝔸_*^2,𝒪_𝔸_*^2)
of P on the open covering of 𝔸_*^2 by the principal
open subsets 𝔸_x^2 and 𝔸_y^2. 

Write p(x,y)=p_d+p_d+1+⋯+p_r where the p_i∈ k[x,y]
are the homogeneous components of p, and p_d≠0. In the
decomposition 

    H^1(𝔸_*^2,𝒪_𝔸_*^2)≃ H^1(ℙ^1,q_*𝒪_𝔸_*^2)≃⊕_s∈ℤH^1(ℙ^1,𝒪_ℙ^1(s))

a non zero homogeneous component x^-my^-np_i of x^-my^-np(x,y)
corresponds to a non zero element of H^1(ℙ^1,𝒪_ℙ^1(-m-n+i)).
On the other hand, since for every ℓ∈ℤ, 𝒪_𝔸̃^2(ℓ E)=f^*𝒪_ℙ^1(-ℓ)
and f:𝔸̃^2→ℙ^1 is the total
space of the line bundle 𝒪_ℙ^1(-1), it follows
from the projection formula that 

    H^1(𝔸̃^2,𝒪_𝔸̃^2(ℓ E))≃ H^1(ℙ^1,f_*𝒪_𝔸̃^2⊗𝒪_ℙ^1(-ℓ))≃⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ)).

The image of x^-my^-np(x,y) in ⊕_s∈ℤH^1(ℙ^1,𝒪_ℙ^1(s))
belongs to ⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ))
if and only if ℓ≥ℓ_0=m+n-d≥2. Given such an ℓ,
the image (h_t)_t≥0∈⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ))
of x^-my^-np(x,y) then defines a unique M(ℓ)-torsor
θ_ℓ:W(P,ℓ)→𝔸̃^2 whose
restriction over the complement of E is isomorphic to ρ:P→𝔸_*^2
when equipped with the action 𝔾_a-action induced by
the canonical section of 𝒪_𝔸̃^2(ℓ E)
with divisor ℓ E. On the other hand, the restriction of W|_E→ E
over E is an 𝒪_ℙ^1(-ℓ)-torsor with
isomorphism class h_0∈ H^1(ℙ^1,𝒪_ℙ^1(-ℓ)).
By definition, h_0 is non zero if and only if ℓ=ℓ_0,
and we conclude from Theorem <ref> that θ_ℓ_0:W(P,ℓ_0)→𝔸̃^2
is the unique 𝔾_a-extension of ρ:P→𝔸̃^2
with affine total space. 



§ QUASI-PROJECTIVE 𝔾_A-EXTENSIONS OF TYPE II 


In this section we consider the following subclass of extensions of
Type II of a 𝔾_a-torsor over a punctured surface.

 A 𝔾_a-extension π:X→ S
of a 𝔾_a-torsor ρ:P→ S_* over a punctured
surface S_*=S∖{o} is said to be a quasi-projective
extension of Type II if it satisfies the following properties

i) X is quasi-projective over S and the 𝔾_a,S-action
on X is proper, 

ii) X is smooth along π^-1(o) and π^-1(o)_red≃𝔸_κ^2.



 Let o=V(x,y) be a global scheme-theoretic
complete intersection closed point in the smooth locus of a surface
S and let ρ:P→ S∖{o} be the 𝔾_a-torsor
with defining sheaf of ideals (xv-yu-1)⊂𝒪_S[u,v]
as in Example <ref>. Let π_1:X_1→ S
be the affine S-scheme with defining sheaf of ideals (xw-y(yz_1+1),xz_2-z_1(yz_1+1),z_1w-yz_2)⊂𝒪_S[z_1,z_2,w].
The morphism of S-schemes j_1:P→ X_1 defined by
(x,y,u,v)↦(x,y,u,uv,yu) is an open embedding, equivariant
for the 𝔾_a,S-action on X_1 associated with the
locally nilpotent 𝒪_S-derivation x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w
of π_*𝒪_X_1. The fiber π_1^-1(o) is
isomorphic to 𝔸_κ^2=Spec(κ[z_2,w])
on which the 𝔾_a,S-action restricts to 𝔾_a,κ-action
by translations associated to the derivation ∂_z_2 of
κ[z_2,w]. It is straightforward to check that X_1 is
smooth along π_1^-1(o). We claim that the geometric quotient
of the 𝔾_a,S-action on X_1 is isomorphic to the
complement of a κ-rational point o_1 in the blow-up τ:S̃→ S
of o. Such a surface being in particular separated, the 𝔾_a,S-action
on X_1 is proper, implying that j_1:P↪ X_1
is a quasi-projective extension of P of Type II.

Indeed, let us identify S̃ with the closed subvariety of
S×_kProj(k[u_0,u_1]) with equation xu_1-yu_0=0
in such a way that τ coincides with the restriction of the first
projection. The morphism f:X_1→S̃ defined by

    (x,y,z,u,v)↦((x,y),[x:y])=((x,y),[yz_1+1:w])

is 𝔾_a-invariant and maps π_1^-1(o) dominantly
onto the exceptional divisor E≃pr_S^-1(o)≃Proj(κ[u_0,u_1])
of τ. The induced morphism

    f|_π^-1(o):π^-1(o)=Spec(κ[z_2,w])→ E,  (z_2,w)↦[1:w]

factors as the composition of the geometric quotient π_1^-1(o)→π_1^-1(o)/𝔾_a,κ≃Spec(κ[w])
with the open immersion π_1^-1(o)/𝔾_a,κ↪ E
of π_1^-1(o)/𝔾_a,κ as the complement of the
κ-rational point o_1=((0,0),[0:1])∈ E. On the other
hand, the composition 

    τ∘ f∘ j_1:P≃⟶X_1∖π_1^-1(o)→S̃∖ E≃⟶S∖{o}

coincides with the geometric quotient morphism ρ:P→ S∖{o}.
So f:X_1→S̃ factors through a surjective morphism
q:X_1→S̃∖{o_1} whose fibers all
consist of precisely one 𝔾_a-orbit. Since q is a
smooth morphism, q is a 𝔾_a-torsor which implies
that X_1/𝔾_a≃S̃∖{o_1}. 


The scheme of the classification of quasi-projective extensions of
Type II of a given 𝔾_a-torsor ρ:P→ S_*
which we give below is as follows: we first construct in <ref>
families of such extensions, in the form of 𝔾_a-torsors
q:X→ S' over quasi-projective S-schemes τ:S'→ S
such that τ^-1(o)_red is isomorphic to 𝔸_κ^1,
S' is smooth along τ^-1(o), and τ:S'∖τ^-1(o)→ S_*
is an isomorphism. We then show in  <ref>
that for quasi-projective 𝔾_a-extension π:X→ S
of Type II of a given 𝔾_a-torsor ρ:P→ S_*,
the structure morphism π:X→ S factors through a 𝔾_a-torsor
q:X→ S' over one of these S-schemes S'. In the
last subsection, we focus on the special case where π:X→ S
has the stronger property of being an affine morphism. 



 §.§ A family of 𝔾_a-extensions
over quasi-projective S-schemes


Let again (S,o) be a pair consisting of
a surface and a closed point o contained in the smooth locus of
S, with residue field κ. We let τ_1:S_1→ S
be the blow-up of o, with exceptional divisor E_1≃ℙ_κ^1.
Then for every n≥2, we let τ_n,1:S_n=S_n(o_1,…,o_n-1)→S_1
be the scheme obtained from S_1 by performing the
following sequence of blow-ups of κ-rational points: 

a) The first step τ_21:S_2(o_1)→S_1
is the blow-up of a κ-rational point o_1∈E_1
with exceptional divisor E_2≃ℙ_κ^1, 

b) Then for every 2≤ i≤ n-2, we let τ_i+1,i:S_i+1(o_1,…,o_i)→S_k(o_1,…,o_i-1)
be the blow-up of a κ-rational point o_i∈E_i,
with exceptional divisor E_i+1≃ℙ_κ^1. 

c) Finally, we let τ_n,n-1:S_n(o_1,…,o_n-1)→S_n-1(o_1,…,o_n-2)
be the blow-up of a κ-rational point o_n-1∈E_n-1
which is a smooth point of the reduced total transform of E_1
by τ_1∘⋯∘τ_n-1,n-2. 

We let E_n≃ℙ_κ^1 be the exceptional
divisor of τ_n,n-1 and we let 

    τ_n,1=τ_2,1∘⋯∘τ_n,n-1:S_n(o_1,…,o_n-1)→S_1.

The inverse image of o in S_n(o_1,…,o_n-1)
by τ_1∘τ_n,1 is a tree of κ-rational
curves in which E_n intersects the reduced proper
transform of E_1∪⋯∪E_n-1 in
S_n(o_1,…,o_n-1) transversally in a unique
κ-rational point. 



 For every κ-rational point o_1∈E_1,
we let S_1(o_1)=S_1∖{o_1}, E_1=E_1∩ S_1≃𝔸_κ^1
and we let τ_1:S_1(o_1)→ S be the restriction
of τ_1. 

For n≥2, we let S_n(o_1,…,o_n-1)=S_n(o_1,…,o_n-1)∖E_1∪⋯∪E_n-1
and E_n=S_n(o_1,…,o_n-1)∩E_n≃𝔸_κ^1.
We denote by τ_n,1:S_n(o_1,…,o_n-1)→S_1
the birational morphism induced by τ_n,1 and we
let τ_n=τ_1∘τ_n,1:S_n(o_1,…,o_n-1)→ S. 


The following lemma summarizes some basic properties of the so-constructed
S-schemes: 

 For every n≥1, the
following hold for S_n=S_n(o_1,…,o_n-1):

a) τ_n:S_n→ S is quasi-projective and restricts
to an isomorphism over S_* while τ_n^-1(o)_red=E_n, 

b) S_n is smooth along τ_n^-1(o)

c) τ_n^*:Γ(S,𝒪_S)→Γ(S_n,𝒪_S_n)
is an isomorphism.

Moreover for n≥2, the morphism τ_n,1:S_n→S_1
is affine. 



The first three properties are straightforward consequences of the
construction. For the last one, let D=E_1+∑_i=2^n-1a_iE_i
where a_i is a sequence of positive rational numbers and let
m≥1 be so that mD is a Cartier divisor on S_n.
Then a direct computation shows that the restriction of 𝒪_S_n(mD)
to τ_n,1^-1(o_1)_red=⋃_i=2^nE_i
is an ample invertible sheaf provided that the sequence (a_i)_i=2,…,n-1
decreases rapidly enough with respect to the distance of E_i
to E_1 in the dual graph of E_1∪⋯∪E_n-1.
Since τ_n,1 restricts to an isomorphism over S_1∖{o_1},
it follows from <cit.> that 𝒪_S_n(mD)
is τ_n,1-ample on S_n. Since by
definition τ_n,1 is the restriction of the projective morphism
τ_n,1:S_n→S_1
to S_n=S_n∖E_1∪⋯∪E_n-1=S_n∖Supp(D),
we conclude that τ_n,1 is an affine morphism. 


By construction, τ_1^-1(o)=E_1 in S_1(o_1), but
for n≥2, we have τ_n^-1(o)=mE_n for some integer
m≥1 which depends on the sequence of κ-rational points
o_1,…,o_n-1 blown-up to construct S_n(o_1,…,o_n-1).
For instance, it is straightforward to check that m=1 if and only
if for every i≥1, o_i∈E_i is a smooth point
of the reduced total transform of E_1 in S_i(o_1,…,o_i-1). 


The structure morphism of a 𝔾_a-torsor being affine,
hence quasi-projective, the total space of any 𝔾_a-torsor
q:X→ S_n over an S-scheme τ_n:S_n=S_n(o_1,…,o_n)→ S
is a quasi-projective S-scheme π=τ_n∘ q:X→ S
equipped with a proper 𝔾_a,S-action. Furthermore π^-1(o)_red=q^-1(E_n)≃ E_n×𝔸_κ^1≃𝔸_κ^2
and X is smooth along π^-1(o) as S_n is smooth along
E_n. On the other hand, π:X→ S is by construction
a 𝔾_a-extension of its restriction ρ:P→ S_n∖ E_n≃ S_*
over S_n∖ E_n, hence is a quasi-projective 𝔾_a-extension
of P of Type II. The following proposition shows conversely that
every 𝔾_a-torsor ρ:P→ S_* admits a
quasi-projective 𝔾_a-extension of Type II  into a 𝔾_a-torsor
q:X→ S_n.

 Let ρ:P→ S_* be
a 𝔾_a-torsor. Then for every n≥1 and every S-scheme
τ_n:S_n(o_1,…,o_n-1)→ S as in Notation
<ref> there exist a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1)
and an equivariant open embedding j:P↪ X such that
in the following diagram 
    P [hook,r,"j"] [d,"ρ"']     X [d,"q"] 
     S_n(o_1,… , o_n-1)∖ E_n [hook,r] [d,"τ_n"',"≀"]     S_n(o_1,… , o_n-1) [d,"τ_n"] 
     S_* [hook,r]     S
all
squares are cartesian. In particular, j:P↪ X is a
quasi-projective 𝔾_a-extension of P of Type II. 

    




Letting S_n=S_n(o_1,…,o_n), we have to prove that
every 𝔾_a-torsor ρ:P→ S_n∖ E_n≃ S_*
is the restriction of a 𝔾_a-torsor q:X→ S_n,
or equivalently that the restriction homomorphism H^1(S_n,𝒪_S_n)→ H^1(S_n∖ E_n,𝒪_S_n∖ E_n)
is surjective. It is enough to show that there exists a Zariski open
neighborhood U of E_n in S_n and a 𝔾_a-torsor
q:Y→ U such that Y|_U∖ E_n≃ P|_U∖ E_n.
Indeed, if so then a 𝔾_a-torsor q:X→ S_n
with the desired property is obtained by gluing P and Y over
U∖ E_n by the isomorphism Y|_U∖ E_n≃ P|_U∖ E_n.
In particular, we can replace S_n by the inverse image by τ_n:S_n→ S
of any Zariski open neighborhood of o in S. We can thus assume
from the very beginning that S=Spec(A) is affine and that
o=V(f,g) is a scheme-theoretic intersection for some f,g∈ A.
Up to replacing f and g by other generators of the maximal ideal
of o in A, we can assume that the proper transform L_1
in τ_1:S_1→ S of the curve
L=V(f)⊂ S intersects E_1 in o_1. We
denote by M_1⊂S_1 the proper transform of
the curve M=V(g)⊂ S. 

We first treat the case n=1. The open subset U_1=S_1∖ L
of S_1 is then affine and contained in S_1. Furthermore
U_1∖ E_1=S_1∖τ_1^-1(L)≃ S∖ L
is also affine. The Mayer-Vietoris long exact sequence of cohomology
of 𝒪_S_1 for the open covering of S_1 by S_1∖ E_1
and U_1 then reads 

    0→    H^0(S_1,𝒪_S_1)→ H^0(U_1,𝒪_S_1)⊕ H^0(S_1∖ E_1,𝒪_S_1)→ H^0(U_1∖ E_1,𝒪_S_n)→⋯
    ⋯→    H^1(S_1,𝒪_S_1)→ H^1(U_1,𝒪_U_1)⊕ H^1(S_1∖ E_1,𝒪_S_1)→ H^1(U_1∖ E_1,𝒪_U_1)→⋯.

Since U_1∖ E_1 is affine, H^1(U_1∖ E_1,𝒪_U_1)=0
and so, the homomorphism H^1(S_1,𝒪_S_1)→ H^1(S_1∖ E_1,𝒪_S_1)
is surjective as desired. 

In the case where n≥2, the open subset V_1=S_1∖ M_1
of S_1 is affine and it contains o_1 since M_1
intersects E_1 in a point distinct from o_1.
Since τ_n,1:S_n→S_1 is an affine
morphism by Lemma <ref>, U_n=τ_n,1^-1(U_1)
is an affine open neighborhood of E_n in S_n. By construction,
S_n is then covered by the two open subset U_n and S_n∖ E_n
which intersect along the affine open subset U_n∩ S_n∖ E_n=U_n∖ E_n=τ_n,1^-1(S_1∖τ_1^-1(M))
of S_n. The conclusion then follows from the Mayer-Vietoris long
exact sequence of cohomology of 𝒪_S_n for the open
covering of S_n by S_n∖ E_n and U_n. 




 §.§ Classification 


The following theorem shows that every quasi-projective
𝔾_a-extension of Type II of a given 𝔾_a-torsor
ρ:P→ S_* is isomorphic to one of the schemes q:X→ S_n
constructed in  <ref>. 

 Let ρ:P→ S_* be a 𝔾_a-torsor
and let 
    P [r,hook,"j"] [d,"ρ"']     X [d,"π"] 
     S_* [r,hook]     S

be a quasi-projective 𝔾_a-extension of P of Type
II. Then there exists an integer n≥1 and a scheme τ_n:S_n(o_1,…,o_n-1)→ S
such that X is a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1)≃ X/𝔾_a
and ρ:P→ S_* coincides with the restriction of q
to S_n(o_1,…,o_n-1)∖ E_n≃ S_*. 



Since the 𝔾_a,S-action on X is proper, the geometric
quotient X/𝔾_a,S exists in the form of a separated algebraic
S-space δ:X/𝔾_a,S→ S. Furthermore,
since by definition of an extension π^-1(S_*)≃ P, we
have π^-1(S_*)/𝔾_a,S≃ P/𝔾_a,S≃ S_*
and so δ restricts to an isomorphism over S_*. On the
other hand, π^-1(o)≃𝔸_κ^2 is equipped
with the induced proper 𝔾_a,κ-action, whose geometric
quotient 𝔸_κ^2/𝔾_a,κ is isomorphic
to 𝔸_κ^1. It follows from the universal property
of geometric quotient that δ^-1(o)=𝔸_κ^2/𝔾_a,κ=𝔸_κ^1. 

Since X is smooth in a neighborhood of π^-1(o), X/𝔾_a,S
is smooth in neighborhood of δ^-1(o). In particular, π^-1(o)
and δ^-1(o) are Cartier divisors on X and X/𝔾_a,S
respectively. Let τ_1:S_1→ S
be the blow-up of o. Then by the universal property of blow-ups
<cit.>,
the morphisms π:X→ S and δ:X/𝔾_a,S→ S
lift to morphisms π_1:X→S_1 and δ_1:X/𝔾_a,S→S_1
respectively, and we have a commutative diagram 
    X [r,"π_1"] [d]    S_1 [d,"τ_1"]
     X/𝔾_a [r,"δ"] [ur,"δ_1"]     S
Furthermore,
since δ:X/𝔾_a,S→ S and τ_1:S_1→ S
are separated, it follows that δ_1:X/𝔾_a,S→S_1
is separated. By construction, the image of π^-1(o)_red/𝔾_a,κ
by δ_1 is contained in E_1. 

If δ_1 is not constant on π^-1(o)_red/𝔾_a,κ
then δ_1 is a separated quasi-finite birational morphism.
Since S_1 is normal, δ_1 is thus an open
immersion by virtue of Zariski Main Theorem for algebraic spaces <cit.>.
Since π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1,
the only possibility is that S_1∖δ_1(X/𝔾_a,S)
consists of a unique κ-rational point o_1∈E_1
and δ_1:X/𝔾_a,S→ S_1(o_1)=S_1∖{o_1}
is an isomorphism. So π_1:X→ S_1(o_1) is 𝔾_a-torsor
whose restriction to S_1(o_1)∖ E_1≃ S_* coincides
with ρ:P→ S_*. 

Otherwise, if δ_1 is constant on π^-1(o)_red/𝔾_a,κ,
then its image consists of a unique κ-rational point o_1∈E_1.
The same argument as above implies that π_1:X→S_1
and δ_1:X/𝔾_a,S→S_1 lift
to a 𝔾_a,S-invariant morphism π_2:X→S_2(o_1)
and a separated morphism δ_2:X/𝔾_a,S→S_2(o_1)
to the blow-up τ_2,1:S_2(o_1)→S_1
of S_1 at o_1, with exceptional divisor E_2.
If the restriction of δ_2 to π^-1(o)_red/𝔾_a,κ
is not constant then δ_2 is an open immersion and the image
of π^-1(o)_red/𝔾_a,κ is an open
subset of E_2 isomorphic to 𝔸_κ^1.
The only possibility is that δ_2(π^-1(o)/𝔾_a,κ)=E_2∖E_1.
Indeed, otherwise S_2∖δ_2(X/𝔾_a,S)
would consist of the disjoint union of a point in E_2∖(E_1∩E_2)
and of the curve E_1∖(E_1∩E_2)
which is not closed in S_2, in contradiction to the
fact that δ_2 is an open immersion. Summing up, δ_2:X/𝔾_a,S→ S_2(o_1)=S_2(o_1)∖E_1
is an isomorphism mapping π^-1(o)_red/𝔾_a,κ
isomorphically onto E_2. So π_2:X→ S_2(o_1)
is 𝔾_a-torsor whose restriction to S_2(o_1)∖ E_2≃ S_*
coincides with ρ:P→ S_*. 

Otherwise, if δ_2 is constant on π^-1(o)_red/𝔾_a,κ,
then δ_2(π^-1(o)/𝔾_a,κ) is a κ-rational
point o_2∈E_2, and there exists a unique minimal
sequence of blow-ups τ_k+1,k:S_k+1(o_1,…,o_k)→S_k(o_1,…,o_k-1),
k=2,…,m-1 of successive κ-rational points o_k∈E_k⊂S_k(o_1,…,o_k-1),
with exceptional divisor E_k+1⊂S_k+1(o_1,…,o_k)
such that π_2:X→S_2(o_1) and δ_2:X/𝔾_a,S→S_2(o_1)
lift respectively to a 𝔾_a,S-invariant morphism π_m:X→S_m(o_1,…,o_m-1)
and a separated morphism δ_m:X/𝔾_a,S→S_m(o_1,…,o_m-1)
with the property that the restriction of δ_m to π^-1(o)_red/𝔾_a,κ
is non constant. By Zariski Main Theorem <cit.>
again, we conclude that δ_m is an open immersion, mapping
π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1
isomorphically onto an open subset of E_m≃ℙ_κ^1.
As in the previous case, the image of π^-1(o)_red/𝔾_a,κ
in E_m must be equal to the complement of the intersection
of E_m with the proper transform of E_1∪⋯∪E_m-1
in S_m(o_1,…,o_m-1) since otherwise S_m(o_1,…,o_m-1)∖δ_m(X/𝔾_a,S)
would not be closed in S_m(o_1,…,o_m-1).
Since π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1,
it follows that E_m intersects the proper transform
of E_1∪⋯∪E_m-1 in a unique
κ-rational point, implying in turn that o_m-1∈E_m-1
is a smooth κ-rational point of the reduced total transform
E_1∪⋯∪E_m-1 of E_1
in S_m-1(o_1,…,o_m-2). Summing up, 

    δ_m:X/𝔾_a,S→S_m(o_1,…,o_m-1)∖E_1∪⋯∪E_m-1

is an isomorphism with an S-scheme of the form S_m(o_1,…,o_m-1)
as constructed in <ref>, mapping π^-1(o)_red/𝔾_a,κ
isomorphically onto E_m=S_m(o_1,…,o_m-1)∩E_m.
It follows in turn that π_m:X→ S_m(o_1,…,o_m-1)
is a 𝔾_a-torsor whose restriction to S_m(o_1,…,o_m-1)∖ E_m≃ S_*
coincides with ρ:P→ S_*. This completes the proof. 




 §.§ Affine 𝔾_a-extensions of Type II


In this subsection, given a 𝔾_a-torsor ρ:P→ S_*
we consider the existence of quasi-projective 𝔾_a-extensions
of Type II 
    P [r,hook,"j"] [d,"ρ"']     X [d,"π"] 
     S_* [r,hook]     S

with the additional for which X is affine over S. As in the
case of extension to 𝔸^1-bundles over the blow-up of
o treated in  <ref>, a necessary
condition for the existence of such extensions is that the restriction
of P over every open neighborhood of the closed point o in S
is nontrivial. Indeed, if there exists an affine open neighborhood
U of o over which P is trivial, then P≃ U∖{o}×𝔸_k^1
is strictly quasi-affine, hence cannot be the complement of a Cartier
divisor π^-1(o) is any affine U-scheme X|_U. The next
theorem shows that this condition is actually sufficient:

Let ρ:P→ S_*
be a 𝔾_a-torsor such that for every open neighborhood
U of o in S, the restriction P×_S_*U→ U∖{o}
is non trivial. Then for every n≥1 and every S-scheme τ_n:S_n(o_1,…,o_n-1)→ S
as in Notation <ref> there exists a quasi-projective
𝔾_a-extension of P of Type II into the total space
of a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1)
for which π=τ_n∘ q:X→ S is an affine morphism. 


The following example illustrates the strategy of the proof given
below, which consists in constructing such affine extensions π:X→ S
by performing a well-chosen equivariant affine modification of extensions
of ρ:P→ S_* into locally trivial 𝔸^1-bundles
θ:W(P)→S̃ over the blow-up τ:S̃→ S
of the point o. 

 Let again X_0 and X_1
be the 𝔾_a-extensions of ρ:P={xv-yu=1}→ S∖{o}
considered in Example <ref> and <ref>.
Recall that X_0 and X_1 are the affine S-schemes in 𝔸_S^3
defined respectively by the equations 

    X_0:  
    xr-yq     =0
    
    yp-x(q-1)     =0
    
    pr-q(q-1)     =0
        and     X_1:  
    xw-y(yz_1+1)     =0
    
    xz_2-z_1(yz_1+1)     =0
    
    z_1w-yz_2    =0

equipped with the 𝔾_a,S-actions associated with the
locally nilpotent 𝒪_S-derivations ∂_0=x^2∂_p+xy∂_q+y^2∂_r
and ∂_1=x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w
respectively. 

The morphism π_0:X_0→ S factors through the structure
morphism θ:X_0→S̃ of a torsor under a line
bundle on the blow-up τ:S̃→ S of the origin,
with the property that the restriction of X_0 to exceptional
divisor E=ℙ_κ^1 of τ is a nontrivial torsor
under the total space of the line bundle 𝒪_ℙ_κ^1(-2).
The 𝔾_a,S-action on X_0 restricts to the trivial
one on X_0|_E=π_0^-1(o). More precisely, ∂_0
is a global section of the sheaf 𝒯_X_0⊗𝒪_X_0(-2X_0|_E)
of vector fields on X_0 that vanish at order 2 along X_0|_E.
One way to obtain from X_0 a 𝔾_a-extension π:X→ S
of ρ:P→ S∖{o} with fiber π^-1(o)_red
isomorphic to 𝔸_κ^2 and a fixed point free action
is thus to perform an equivariant affine modification which simultaneously
replaces X_0|_E by a copy of 𝔸_κ^2 and
decreases the “fixed point order of ∂_0 along X_0|_E”,
typically a modification with divisor D equal to X_0|_E
and whose center Z⊂ X_0|_E is supported by a curve isomorphic
to 𝔸_κ^1 which is mapped isomorphically onto
its image by the restriction of θ. The birational S-morphism

    η:X_1→ X_0,  (x,y,z_1,z_2,w)↦(x,y,xz_1,yz_1+1,w)

is equivariant for the 𝔾_a,S-actions on X_0 and
X_1 and corresponds to an equivariant affine modification of
this type: it restricts to an isomorphism outside the fibers of π_0
and π_1 over o, and it contracts π_1^-1(o)=Spec(κ[z_2,w])
onto the curve {p=q-1=0}⊂π_0^-1(o)={pr-q(q-1)=0}.
This curve is isomorphic to 𝔸_κ^1=Spec(κ[r])
and it is mapped by the restriction 

    θ|_π_0^-1(o):π_0^-1(o)≃{pr-q(q-1)=0}→ E=ℙ_κ^1, (p,q,r)↦[p:q-1]=[q:r]

of θ isomorphically onto the complement of the κ-rational
point [0:1]∈ℙ_κ^1. 



By virtue of Theorem <ref>, there exists a unique
integer ℓ_0≥2 such that ρ:P→ S_* is the
restriction of a torsor θ_1:W_1→S_1
under the line bundle M_1(ℓ_0)=Spec(Sym^·𝒪_S_1(-ℓ_0E_1))→S_1
whose total space W_1 is affine over S_1. We
now treat the case of S_1(o_1) and S_n(o_1,…,o_n-1),
n≥2 separately.

Given a κ-rational point o_1∈E_1, the
restriction of W_1 over E_1=E_1∖{o_1}≃𝔸_κ^1
is the trivial 𝔸^1-bundle E_1×𝔸_κ^1.
Since on the other hand the restriction θ_1|_E_1:W_1|_E_1→E_1
is a non trivial 𝒪_ℙ^1(-ℓ_0)-torsor
(see Theorem <ref>), it follows that for every
section s:E_1→ W_1|_E_1 the image Z_1 of
E_1 in W_1|_E_1 is a closed curve isomorphic
to E_1. Indeed, otherwise if Z_1 is not closed in W_1|_E_1
then its closure Z_1 would be a section of θ_1|_E_1
in contradiction with the fact that θ_1|_E_1:W_1|_E_1→E_1
is a non trivial 𝒪_ℙ^1(-ℓ_0)-torsor.
Let D_1=θ_1^-1(E_1) and let σ_1:W_1'→ W_1
be the affine modification of W_1 with center (ℐ_Z_1,D_1)
. By virtue of Lemmas <ref> and <ref>,
θ_1∘σ_1:W_1'→S_1 factors
through a torsor θ_1':W_1'→S_1∖{o_1}=S_1(o_1)
under the line bundle 

    M_1'(ℓ_0-1)=Spec(Sym^·𝒪_S_1(o_1)((-ℓ_0+1)E_1))→ S_1(o_1).

Now since E_1≃𝔸_κ^1 is affine, the restriction
of θ_1' over E_1⊂ S_1(o_1) is the trivial
M_1'(ℓ_0-1)|_E_1-torsor. Letting D_2=θ_1'^-1(E_1)
and Z_2⊂ D_2 be any section of θ'_1|_D_2:D_2→ E_1,
the affine modification σ_2:W_2'→ W_1' with
center (ℐ_Z_2,D_2) is then an M_1'(ℓ_0-2)-torsor
θ'_2:W_2'→ S_1(o_1). Iterating this construction
ℓ_0-1 times, we reach a 𝔾_a,S_1(o_1)-torsor
q=θ_ℓ_0+1':X=W'_ℓ_0+1→ S_1(o_1).
Since σ_1:W_1'→ W_1 and each σ_i:W_i'→ W_i-1',
i≥2, restricts to an isomorphism over the complement of E_1,
the restriction of q:X→ S_1(o_1) over S_1(o_1)∖ E_1≃ S_*
is isomorphic to ρ:P→ S_*. Furthermore, since the
morphisms σ_i, i=1,…,ℓ_0+1 are affine and τ_1∘θ_1:W_1→ S
is an affine morphism, it follows that 

    τ_1∘ q=τ_1∘θ_1∘σ_1∘⋯σ_ℓ_0+1:X→ S

is an affine morphism. So q:X→ S_1(o_1) is a 𝔾_a-extension
of ρ:P→ S_* with the desired property. 

Now suppose that n≥2. It follows from the construction of the
morphism τ_n,1:S_n=S_n(o_1,…,o_n-1)→S_1
given in subsection <ref> that τ_n,1^*𝒪_S_1(ℓ_0E_1)≃𝒪_S_n(mE_n)
for some m≥2. The fiber product W_n=W_1×_S_1S_n
is thus a torsor θ_n:W_n→ S_n under the line
bundle 

    M_n(m)=Spec(Sym^·𝒪_S_n(-mE_n))→ S_n

whose restriction to S_n∖ E_n≃ S_* is isomorphic
to ρ:P→ S_*. Furthermore, since τ_n,1 is
an affine morphism by virtue of Lemma <ref>,
so is the projection pr_W_1:W_n→ W_1.
Since τ_1∘θ_1:W_1→ S is
an affine morphism, we conclude that τ_n∘θ_n=τ_1∘τ_n,1∘θ_n=τ_1∘θ∘pr_W_1:W_n→ S
is an affine morphism as well. Since E_n≃𝔸_κ^1,
the restriction of θ_n over E_n is the trivial M_n(m)|_E_n-torsor.
The desired 𝔾_a,S_n-torsor q:X→ S_n
extending ρ:P→ S_* is then obtained from θ_n:W_n→ S_n
by performing a sequence of m successive affine modifications similar
to those applied in the previous case. 

In the case where S is affine, the total spaces X of the varieties
q:X→ S_n(o_1,…,o_n-1) of Theorem <ref>
are all affine. To our knowledge, these are the first instances of
smooth affine threefolds equipped with proper 𝔾_a-actions
whose geometric quotients are smooth quasi-projective surfaces which
are not quasi-affine. 


We do not know in general if under the conditions of Theorem <ref>
every quasi-projective 𝔾_a-extensions of P of Type
II into the total space of a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1)
has the property that π=τ_n∘ q:X→ S is an
affine morphism. In particular, we ask the following:

Is the total space X of a quasi-projective 𝔾_a-extension
π:X→𝔸^2 of ρ=pr_x,y:SL_2={xv-yu=1}→𝔸_*^2
of Type II always an affine variety ? 





 §.§ Examples


In the next paragraphs, we construct two countable families of quasi-projective
𝔾_a-extensions of the 𝔾_a-torsor SL_2→SL_2/𝔾_a≃𝔸^2∖{(0,0)}
of Type II with affine total spaces. As a consequence of <cit.>,
for any nontrivial 𝔾_a-torsor ρ:P→ S_*
over a local punctured surface S_*, these provide, by suitable
base changes, families of examples of 𝔾_a-extensions
of P whose total spaces are all affine over S. 



  §.§.§  A family of 𝔾_a-extensions of
SL_2 of “Type II-A” 


Let S=𝔸^2=Spec(k[x,y_0]) and let X_n⊂𝔸_S^n+2=Spec(k[x,y_0][z_1,z_2,y_1,…,y_n]),
n≥1, be the smooth threefold defined by the system of equations

    y_iy_j-y_ky_ℓ    =0   i,j,k,ℓ=0,…,n, i+j=k+ℓ
    
    z_2y_i-z_1y_i+1    =0   i=0,…,n-1
    
    xy_i+1-y_i(y_0z_1+1)     =0   i=0,…,n-1
    
    xz_2-z_1(y_0z_1+1)     =0.

The threefold X_n can be endowed with a fixed point free 𝔾_a,S-action
induced by the locally nilpotent k[x,y_0]-derivation 

    x∂_z_1+(2y_0z_1+1)∂_z_2+∑_i=1^niy_0y_i-1∂_y_i

of its coordinate ring. The scheme-theoretic fiber over o={(0,0)}
of the 𝔾_a-invariant morphism π_n=pr_x,y,:X_n→ S
is isomorphic 𝔸^2=Spec(k[z_2,y_n]), on
which the induced 𝔾_a-action is a translation induced
by the derivation ∂_z_2 of k[z_2,y_n]. On the
other hand, the morphism j:SL_2={ xv-y_0u=1}→ X_n
defined by 

    (x,y,u,v)↦(x,u,uv,y,yv,yv^2,…,yv^n)

is an equivariant open embedding of SL_2 equipped with
the 𝔾_a-action induced by the locally nilpotent derivation
x∂_u+y_0∂_v of its coordinate ring into X_n
with image equal to π^-1(𝔸^2∖{o}). So
j:SL_2↪ X_n is a quasi-projective 𝔾_a-extension
of SL_2 into the affine variety X_n, with π_n^-1(o)≃𝔸_k^2. 

The restrictions of the projection 𝔸_S^n+3→𝔸_S^n+2
onto the first n+2 variables induce a sequence of 𝔾_a-equivariant
birational morphisms σ_n+1,n:X_n+1→ X_n. The
threefolds X_n thus form a countable tower of 𝔾_a-equivariant
affine modifications of X_1. It follows from Example <ref>
that X_1 is a quasi-projective extension of SL_2
of Type II with geometric quotient isomorphic to a quasi-projective
surface of the form S_1(o_1). More generally, we have the following
result. 

For every n≥2, the morphism j:SL_2↪ X_n
is a quasi-projective 𝔾_a-extension of Type II. The
geometric quotient X_n/𝔾_a is isomorphic to a quasi-projective
surface S_n=S_n(o_1,…,o_n) as in  <ref>
for which S_n(o_1,…,o_n-1)∖ S_n
consists of a chain of n-1 smooth rational curves with self-intersection
-2, i.e. the exceptional set of the minimal resolution of a surface
singularity of type A_n-1. 



To see this, we consider the following sequence of blow-ups: the first
one τ_1:S_1→ U_0=𝔸^2
is the blow-up of the origin, with exceptional divisor E_1,
and we let U_1≃𝔸^2=Spec(k[x,w_1])
be the affine chart of S_1 on which τ_1:S_1→𝔸^2
is given by (x,w_1)↦(x,xw_1). Then we let τ_2,1:S_2(o_1)→S_1
be the blow-up of the point o_1=(0,0)∈ U_1⊂S_1
with exceptional divisor E_2, and we let U_2≃𝔸^2=Spec(k[x,w_2])
be the affine chart of S_2(o_1) on which the restriction
of τ_2,1:S_2(o_1)→S_1
coincides with the morphism U_2→ U_1, (x,w_2)↦(x,xw_2).
For every 2<m≤ n, we define by induction the blow-up 

    τ_m,m-1:S_m(o_1,…,o_m-1)→S_m-1(o_1,…,o_m-2)

of the point o_m-1=(0,0)∈ U_m-1⊂S_m-1(o_1,…,o_m-2)
with exceptional divisor E_m and we let U_m≃𝔸^2=Spec(k[x,w_m])
be the affine chart of S_m(o_1,…,o_m-1) on
which the restriction of τ_m,m-1 coincides with
the morphism U_m→ U_m-1, (x,w_m)↦(x,xw_m).
By construction, we have a commutative diagram 
    @C8exS_n(o_1,…,o_n-1) [r]^τ_n,n-1   S_n-1(o_1,…,o_n-2) [r]^-τ_n-1,n-2   ⋯   [r]^τ_2,1   S_1[r]^τ_1   𝔸^2 
     U_n [u] [r]     U_n-1[u] [r]    ⋯   [r]     U_1 [u] [r]    𝔸^2=U_0. @=[u]
  

The total transform of E_1 in S_n(o_1,…,o_n-1)
is a chain E_1∪E_2∪⋯∪E_n-1∪E_n
is a chain formed of n-1 curves with self-intersection -2 and
the curve E_n which has self-intersection -1. 



The morphism π:X_n→ S lifts to a morphism π_1:X_n→S_1
defined by 

    (x,z_1,z_2,y_0,y_1,…,y_n)↦((x,y_0),[x:y_0])=((x,y),[y_0z_1+1:y_1]).

This morphism contracts π^-1(o) onto the point o_1=((0,0),[1:0])
of the exceptional divisor E_1 of τ_1.
The induced rational map π_1:X_n U_1 is
given by 

    (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_1/y_0z_1+1)

and it contracts π^-1(o) onto the origin o_1=(0,0). So
π_1 lifts to a morphism π_2:X_n→S_2(o_1),
and with our choice of charts, the induced rational map π_2:X_n U_2
is given by 

    (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_2/(y_0z_1+1)^2)

If n=2 then the image of π^-1(o)=Spec(k[z_2,y_2])
by π_2 is equal to E_2∩ U_2 and π_2^-1(E_2∩ U_2)
is equivariantly isomorphic to (E_2∩ U_2)×Spec(k[z_2])
on which 𝔾_a acts by translations on the second factor.
So π_2:X_n→S_2(o_1) factors through
a 𝔾_a-bundle q_2:X_2→ S_2(o_1)=S_2(o_1)∖E_1
and X_2/𝔾_a≃ S_2(o_1). Otherwise, if n>2
then π_2 contracts π^-1(o) onto the point o_2=(0,0)∈E_2∩ U_2⊂S_2(o_1).
So π_2:X_n→S_2(o_1) lifts to a morphism
π_3:X_n→S_3(o_1,o_2). With our
choice of charts, for each 2<m<n, the induced rational map π_m:X_n U_m
is given by 

    (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_m/(y_0z_1+1)^m)

hence contracts π^-1(o) onto the point o_m=(0,0)∈ U_m⊂S_m(o_1,…,o_m-1).
It thus lifts to a morphism π_m:X_n→S_m(o_1,…,o_m-1).
At the last step, the image of π^-1(o)=Spec(k[z_2,y_n])
by the rational map π_n:X_n U_n induced
by π_n:X_n→S_n(o_1,…,o_n-1)
is equal to E_n∩ U_n, and we conclude as above
that π_n:X_n→S_n(o_1,…,o_n-1)
factors through a 𝔾_a-bundle 

    q_n:X_n→ S_n(o_1,…,o_n-1)=S_n(o_1,…,o_n-1)∖(E_1∪⋯∪E_n-1),

hence that X_n/𝔾_a is isomorphic to the quasi-projective
surface S_n(o_1,…,o_n-1). 




  §.§.§ A family of 𝔾_a-extensions of SL_2 of
“Type II-D”


To conclude this section, we present as an illustration of the proof
of Theorem <ref> another countable family
of quasi-projective 𝔾_a-extensions of SL_2
of Type II with  affine total spaces. 

Let again τ_1:S_1→ S=𝔸^2
be the blow-up of the origin o={(0,0)} in 𝔸^2=Spec(k[x,y])
with exceptional divisor E_1≃ℙ^1,
identified with closed subvariety of 𝔸^2×ℙ_[w_0:w_1]^1
with equation xw_1-yw_0=0 in such a way that τ coincides
with the restriction of the first projection. The second projection
identifies S_1 with the total space p:S_1→ℙ^1
of the invertible sheaf 𝒪_ℙ^1(-1). We fix
trivializations p^-1(U_∞)=Spec(k[z_∞][u_∞])
and p^-1(U_0)=Spec(k[z_0][u_0]) over the open
subsets U_∞=ℙ^1∖{[0:1]}=Spec(k[z_∞])
and U_0=ℙ^1∖{[1:0]}=Spec(k[z_0])
in such a way that the gluing of p^-1(U_∞) and p^-1(U_0)
over U_0∩ U_∞ is given by the isomorphism (z_0,u_0)↦(z_∞,u_∞)=(z_0^-1,z_0u_0).

For every n≥1, we let S_2n+3,0=Spec(k[z_0,u_0^±1]),

    S_2n+3,∞=Spec(k[z_∞,u_∞,v_∞]/(u_∞^nv_∞-z_∞^2-u_∞)),

and we let S_2n+3 be the surface obtained by gluing S_2n+3,0
and S_2n+3,∞ along the open subsets S_2n+3,0∖{z_0=0}
and S_2n+3,∞∖{z_∞=u_∞=0} by the
isomorphism 

    (z_0,u_0)↦(z_∞,u_∞,v_∞)=(z_0^-1,z_0u_0,(z_0u_0)^-n(z_0^-2+z_0u_0)).

The canonical open immersion S_2n+3,0↪ p^-1(U_0)
and the projection pr_z_∞,u_∞:S_2n+3,∞→ p^-1(U_∞)
glue to a global birational affine morphism τ_2n+3,1:S_2n+3→S_1
restricting to an isomorphism S_2n+3∖{z_∞=u_∞=0}→S_1∖E_1
where we identified the closed subset E_2n+3={z_∞=u_∞=0}≃Spec(k[v_∞])
of S_2n+3,∞ with its image in S_2n+3. We leave to the
reader to check that with the notation of  <ref>,
S_2n+3=S_2n+3(o_1,…,o_2n+2) for a surface τ_2n+3,1:S_2n+3,1(o_1,…,o_2n+2)→S_1
obtained by first blowing-up the point o_1=(0,0)∈ p^-1(U_∞)
with exceptional divisor E_2, then the point o_2=E_1∩E_2
with exceptional divisor E_3, then a point o_3∈E_3∖(E_1∪E_2)
with exceptional divisor E_4 and then a sequence of
point o_i∈E_i∖E_i-1 with
exceptional divisor E_i+1, i=5,…,2n+2 in such
a way that the total transform of E_1 in S_2n+3,1
is a tree depicted in Figure <ref>. Letting τ_2n+3=τ_1∘τ_2n+3,1:S_2n+3→𝔸^2,
we have τ_2n+3^-1(o)_red=E_2n+3≃𝔸^1
and τ_2n+3^*(o)=2E_2n+3. 



Now we let q:X_2n+3→ S_2n+3 be the 𝔾_a-bundle
defined as the gluing of the trivial 𝔾_a-bundles X_2n+3,0=S_2n+3,0×Spec(k[t_0])
and X_2n+3,∞=S_2n+3,∞×Spec(k[t_∞])
over S_2n+3,0 and S_2n+3,∞ respectively along the open
subsets X_2n+3,0∖{z_0=0} and X_2n+3,∞∖{z_∞=u_∞=0}
by the 𝔾_a-equivariant isomorphism 

    (z_0,u_0,t_0)↦(z_∞,u_∞,v_∞,t_∞)=(z_0^-1,z_0u_0,(z_0u_0)^-n(z_0^-2+z_0u_0),t_0+z_0^-1u_0^-2).

Let π_2n+3=τ_1∘τ_2n+3,1∘ q:X_2n+3→𝔸^2. 

For every n≥1, the variety X_2n+3 is affine and there exists
a 𝔾_a-equivariant open embedding j:SL_2↪ X_2n+3
which makes π_2n+3:X_2n+3→𝔸^2 a quasi-projective
𝔾_a-extension of SL_2 of Type II, with
fiber π_2n+3^-1(o) isomorphic to 𝔸^2 of multiplicity
two, and geometric quotient X_2n+3/𝔾_a≃ S_2n+3.



Let j_1:SL_2↪ W=W(SL_2,2)
be the 𝔾_a-extension of SL_2 into a locally
trivial 𝔸^1-bundle θ:W→S_1
with affine total space constructed in Example <ref>.
Recall that the image of j_1 coincides with the restriction of
θ to S_1∖E_1=𝔸^2∖{o}.
With our choice of coordinates, the open subsets W_0=θ^-1(q^-1(U_0))
and W_∞=θ^-1(q^-1(U_∞)) of W are respectively
isomorphic to p^-1(U_0)×Spec(k[w_0]) and p^-1(U_∞)×Spec(k[w_∞])
glued over U_0∩ U_∞ by the isomorphism 

    (z_0,u_0,w_0)↦(z_∞,u_∞,w_∞)=(z_0^-1,z_0u_0,z_0^2w_0+z_0).

The 𝔾_a-action on W_0 and W_∞ are given
respectively by α·(z_0,u_0,w_0)=(z_0,u_0,w_0+α u_0^2)
and α·(z_∞,u_∞,w_∞)=(z_∞,u_∞,w_∞+α u_∞^2). 

Let W'=W×_S_1S_2n+3, equipped with the natural
lift of the 𝔾_a-action on W. Since τ_2n+3,1:S_2n+3→S_1
restricts to an isomorphism over S_1∖E_1,
the composition j'=τ_2n+3,1^-1∘ j_1:SL_2→ W'
is a 𝔾_a-equivariant open embedding. Furthermore, since
W is affine and τ_2n+3,1 is an affine morphism, it follows
that W' is affine. By construction, W' is covered by the two
open subsets 

    W'_0=W×_p^-1(U_0)S_2n+3,0   ≃ S_2n+3,0×Spec(k[w_0])
    
    W'_∞=W×_p^-1(U_∞)S_2n+3,∞   ≃ S_2n+3,∞×Spec(k[w_∞]).

The local 𝔾_a-equivariant morphisms 

    β_0:X_2n+3,0=S_2n+3,0×Spec(k[t_0])→ W_0'
    β_∞:X_2n+3,∞=S_2n+3,∞×Spec(k[t_∞])→ W_∞'

 of schemes over S_2n+1,0 and S_2n+3,∞ respectively
defined by t_0↦ w_0=u_0^2t_0 and t_∞↦ w_∞=u_∞^2t_∞
glue to a global 𝔾_a-equivariant birational affine morphism
β:X_2n+3→ W', restricting to an isomorphism over
S_2n+3∖ E_2n+3≃𝔸^2∖{o}. Summing
up, X_2n+3 is affine over W' hence affine, and the composition
β^-1∘ j':SL_2↪ X_2n+3 is a
𝔾_a-equivariant open embedding which realizes π:X_2n+3→𝔸^2
as a 𝔾_a-extension of SL_2 of Type II
with affine total space. By construction, π_2n+3^-1(o)=q^-1(2E_2n+3)
is isomorphic to 𝔸^2, with multiplicity two, while the
geometric quotient X_2n+3/𝔾_a is isomorphic to S_2n+3. 


For every n≥1, the birational morphism S_2(n+1)+3,∞→ S_2n+3,∞,
(z_∞,u_∞,v_∞)↦(z_∞,u_∞,u_∞v_∞)
extends to a birational morphism S_2(n+1)+3→ S_2n+3
which lifts in turn in a unique way to a 𝔾_a-equivariant
birational morphism γ_n+1,n:X_2(n+1)+3→ X_2n+3.
So in a similar way as for the family constructed in  <ref>,
the family of threefolds X_2n+3, n≥1, form a tower of 𝔾_a-equivariant
affine modifications of the initial one X_5. 


amsplain
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