Algebraic subgroups of the group of birational transformations of ruled surfaces

We classify the maximal algebraic subgroups of Bir(CxPP^1), when C is a smooth projective curve of positive genus.


Introduction
In this article, all varieties are defined over an algebraically closed field k, algebraic groups are smooth group schemes of finite type (or, equivalently, reduced group schemes of finite type), and C denotes a smooth projective curve of genus g.The main results, namely Theorem A and Corollary B, hold when the characteristic of k is different from two.When X is a projective variety, the automorphism group of X is the group of k-rational points of a group scheme (see [MO67]), and we only consider its reduced structure.
The study of algebraic subgroups of the group of birational transformations started with [Enr93], where the author classified the maximal connected algebraic subgroups of Bir(P 2 ).More recently, the maximal algebraic subgroups of Bir(P 2 ) have been classified; see [Bla09b].The purpose of this text is to study the algebraic subgroups of Bir(C × P 1 ) when g ≥ 1, which will complete the classification for surfaces of Kodaira dimension −∞.
Let G be an algebraic subgroup of Bir(C × P 1 ).The strategy is classical: first regularize the action of G, find a G-equivariant completion, and run a G-equivariant minimal model program (MMP) to embed G in the automorphism group of a G-minimal fibration.The equivariant completion from Sumihiro [Sum74,Sum75] works for linear algebraic groups; therefore, his results cannot be applied in our setting.Recently, Brion proved the existence of an equivariant completion for connected (not necessarily linear) algebraic groups acting birationally on integral varieties; see [Bri17,Corollary 3].Using his results, we find an equivariant completion for (not necessarily linear or connected) algebraic groups acting on surfaces (see Proposition 2.5).Then we re-prove the G-equivariant MMP (Proposition 2.6), which is a folklore result (see e.g.[KM98, Example 2.18]), by using only elementary arguments.We are left with studying the automorphism groups of conic bundles.Following the ideas of [Bla09b], we prove Propositions 2.17 and 3.11, which reduce the study to the cases of ruled surfaces, exceptional conic bundles and (Z/2Z) 2 -conic bundles (see Section 2.1 for definitions).The Segre invariant S(X) of a ruled surface X (see Definition 2.9) was introduced in [Mar70] and [Mar71] for the classification of ruled surfaces and their automorphisms.The ruled surfaces A 0 and A 1 in Theorem A(4), (5) are the only indecomposable P 1 -bundles over C up to C-isomorphism, when C is an elliptic curve (see Definition 2.8 and [Ati57,Theorem 11] or [Har77, Theorem V.2.15]).Combining techniques from Blanc and results of Maruyama, we prove the following theorem.and such that −2D is linearly equivalent to (Then Aut(X) fits into an exact sequence where H is the finite subgroup of Aut(C) preserving the image of the singular fibres.)(3) Aut(X), where X is a (Z/2Z) 2 -conic bundle with at least one singular fibre; (Then Aut(X) fits into an exact sequence where H is the finite subgroup of Aut(C) preserving the image of the singular fibres.)(4) Aut(X), where X is a (Z/2Z) 2 -ruled surface (consequently, S(X) > 0); (Then Aut(X) fits into an exact sequence Moreover, if g = 1, there exists a unique (Z/2Z) 2 -ruled surface over C denoted by A 1 which satisfies S(A 1 ) = 1 and for which Aut(A 1 ) fits into an exact sequence (5) Aut(A 0 ), where A 0 is the unique indecomposable ruled surface over C with Segre invariant 0 when g = 1; (Then there exists an exact sequence , where X P(O C (D) ⊕ O C ) is a non-trivial decomposable ruled surface over C with deg(D) = 0 (or, equivalently, S(X) = 0), with the additional assumption that 2D is principal if g ≥ 2.
(Then Aut(X) fits into an exact sequence where Moreover, any maximal algebraic subgroup of Bir(C × P 1 ) is conjugate to one in the list above.
By Corollary 3.7, there exist exceptional conic bundles X → C, where C is a curve of positive genus, such that Aut(X) is not a maximal algebraic subgroup of Bir(C × P 1 ).This does not happen when the base curve is rational: the automorphism group of an exceptional conic bundle over P 1 is always maximal (if the number of singular fibres is at least four, see [Bla09b, Theorem 1(2)]; else the number of singular fibres equals two, and the result follows from [Bla09b, Theorem 2(3)]).Moreover, the cases (4), ( 5) and (6) of Theorem A do not exist when the base curve is rational: the Segre invariant of a ruled surface π : S → P 1 is always non-positive (see [HM82] and [Fon21, Proposition 2.18(1)]) and equals 0 if and only if π is trivial.
From the classification of Blanc, it follows that every algebraic subgroup of Bir(P 2 ) is conjugate to a subgroup of a maximal one.This does not hold anymore for algebraic subgroups of Bir(C × P 1 ) when C has positive genus.The following corollary is an analogue of [Fon21, Theorem C] for surfaces of Kodaira dimension −∞.

Corollary B.
Let k be an algebraically closed field of characteristic different from two, and let X be a surface of Kodaira dimension −∞.Then, every algebraic subgroup of Bir(X) is contained in a maximal one if and only if X is rational.
Conventions.Unless stated otherwise, all varieties are smooth and projective, and C is a smooth projective curve.

Regularization and relative minimal fibrations
Definition 2.1.Let C be a curve.
(1) A ruled surface over C is a morphism π : S → C such that each fibre is isomorphic to P 1 .
(2) A conic bundle over C is a morphism κ : X → C such that all fibres are isomorphic to P 1 , except finitely many (possibly 0) which are called singular fibres and are transverse unions of two (−1)-curves.
(3) A conic bundle κ : X → C is an exceptional conic bundle over C if there exists an n ≥ 1 such that κ has exactly 2n singular fibres and two sections of self-intersection −n.(4) If κ : X → C is a conic bundle, we denote by Bir C (X) the subgroup of Bir(X) which consists of the elements f ∈ Bir(X) such that κf = κ.We also define 2 and each non-trivial involution in this group fixes pointwise an irreducible curve, which is a 2-to-1 cover of C ramified above an even positive number of points.(6) A (Z/2Z) 2 -ruled surface over C is a ruled surface over C which is also a (Z/2Z) 2 -conic bundle over C.
Remark 2.2.Assume that C has positive genus, and let π : X → C be a conic bundle.Let f be a smooth fibre of π and α ∈ Aut(X).Since (πα) |f : f P 1 → C is constant, it follows that α(f ) is also a smooth fibre of π.The set of singular fibres is preserved by Aut(X), and π induces a morphism of group schemes π * : Aut(X) → Aut(C).This implies that every automorphism of X preserves the conic bundle structure.
Definition 2.3.Let X be a surface and G be an algebraic subgroup of Aut(X). ( The classical approach to study algebraic subgroups uses the regularization theorem of Weil [Wei55] (see also [Zai95] or [Kra18] for modern proofs).By [Bri22, Theorem 1], the regularization of X contains a G-stable dense open subset U which is smooth and quasi-projective.Then by [Bri22, Theorem 2], U admits a G-equivariant completion by a normal projective G-variety, that we can assume smooth by a G-equivariant desingularization (see [Lip78]).
In the following lemma and proposition, we give an elementary proof of the existence of an equivariant completion for surfaces equipped with the action of an algebraic group G, not necessarily connected or linear, without using results of [Bri22].
Lemma 2.4.Let X be a surface and G be an algebraic subgroup of Bir(X) such that G • acts regularly on X. Denote by Bs(X) the set of base points of the G-action, including the infinitely near ones.Then Bs(X) is finite, and the action of G lifts to a regular action on the blowup of X at Bs(X).
For each i, we fix g i ∈ G i and get that g i G • = G i .Let Bs(g i ) be the set of base points of g i , including the infinitely near ones; this set is finite because X is smooth and projective.The subgroup G • ⊂ G is normal; it follows that every element g i ∈ G i equals gg i for some g ∈ G • .Since G • acts regularly, Bs(G i ) = Bs(g i ) for each i, and this implies that Bs(X) = i=1,...,n Bs(g i ) is also finite.Moreover, for each g ∈ G • , there exists a g ∈ G • such that g i g = gg i .Then g −1 (Bs(g i )) = Bs(g i ), so G • preserves Bs(X).Since G • is connected and Bs(X) is finite, this implies that G • acts trivially on Bs(X).
If Bs(X) is empty, the result holds.Suppose that Bs(X) ∅.Let p ∈ Bs(X) ∩ X be a proper base point and η : X p → X be the blowup of X at p. We consider the action of G on X p obtained by conjugation.As p is fixed by G • , the algebraic group G • still acts regularly on X p .We prove that each element q ∈ Bs(X p ) corresponds via η to an element of Bs(X).Let q ∈ Bs(X p ); there exist a surface Y such that q ∈ Y and a birational morphism π : Y → X p such that q is a base point of η −1 g i ηπ.Let W be a smooth projective surface with birational morphisms α : W → Y and β : W → X p such that βα −1 is a minimal resolution of η −1 g i ηπ.The following diagram is commutative: Since q is a base point of η −1 g i ηπ, it must be blown up by α.There exists a (−1)-curve C in W contracted by α to q and such that its image by Let j be such that p is a proper base point of g j .Let W be a smooth projective surface with birational morphisms α : W → X p and β : W → X such that β α −1 η −1 is a minimal resolution of g j .We obtain the following commutative diagram: There exists a (−1)-curve C in W contracted to p by ηα and such that its image by β is a curve in X.The image of C by α is C or a point of C. Since β : U → U is an isomorphism, this implies that β −1 α : W W is defined on a neighbourhood of C and sends C onto either C or a point of C. Hence αβ −1 α : W Y contracts C onto q, so q ∈ Bs(g j g i ).Let Bs(X p ) be the set of base points of η −1 Gη, including the infinitely near ones.We have shown that if q ∈ Bs(X p ), then q ∈ Bs(X).The map Bs(X p ) → Bs(X) \ {p} sending the infinitesimal base point (q, π) to (q, ηπ) is injective.Conversely, if q ∈ Bs(g i ) and q p, then η −1 (q) ∈ Bs(η −1 g i η).Therefore, Bs(X p ) Bs(X) \ {p}.Proceeding by induction, the blowup of all elements of Bs gives rise to a surface on which G acts regularly.Proposition 2.5.Let X be a surface and G be an algebraic subgroup of Bir(X).Then there exists a smooth projective surface Y with a birational map ψ : X Y such that ψGψ −1 is an algebraic subgroup of Aut(Y ).

P. Fong 6 P. Fong
Proof.Apply [Bri17, Corollary 3] to X equipped with the action of the connected component of the identity G • ⊂ G.There exists a normal projective surface Z with a birational map φ : X Z such that φG • φ −1 ⊂ Aut • (Z).By an equivariant desingularization, we can also assume that Z is smooth; see [Lip78].Let H = φGφ −1 and η : Y → Z be the blowup of Z at Bs(Z).By Lemma 2.4, the action of H lifts to a regular action on Y .Then η −1 Hη ⊂ Aut(Y ) is a closed subgroup which is an algebraic subgroup of Bir(Y ).Take ψ = η −1 φ; we get that ψGψ −1 is an algebraic subgroup of Aut(Y ).
The next result is also known, see e.g.[KM98, Example 2.18]; we re-prove it in our specific situation using elementary arguments.
Proposition 2.6.Let C be a curve of positive genus, and let X be a surface birationally equivalent to C × P 1 .Let G be an algebraic subgroup of Aut(X).If (G, X) is minimal (see Definition 2.3 ), then X is a conic bundle over C.
Proof.Since X is birational to C×P 1 , there exist a morphism κ : X → C and a birational map φ : C×P 1 X such that κφ = p 1 , where p 1 : C × P 1 → C denotes the projection onto the first factor.In particular, φ is a finite composite of blowups and contractions, and there exists a non-empty open U ⊂ C such that φ |U ×P 1 is an isomorphism.Let p ∈ C \ U ; it remains to see that κ −1 (p) either is isomorphic to P 1 or is the transverse union of two (−1)-curves.Since X is the blowup of a ruled surface S in finitely many (possibly infinitely close) points, we can write , where: • Each E i is isomorphic to P 1 .
• For all distinct i, j, E i and E j either intersect transversely at a point or are disjoint.
If n = 1, then κ −1 (p) is a smooth fibre isomorphic to P 1 .If n = 2, then E 1 and E 2 intersect transversely in one point.Because there is a contraction to the ruled surface S, either E 1 or E 2 can be contracted.Therefore, E 2 1 = E 2 2 = −1.Assume from now on that n ≥ 3. First, E 2 i < 0 for all i.The contraction of any collection of disjoint (−1)-curves permuted transitively by G is G-equivariant.Since (G, X) is minimal, there exist k, l ∈ {1, . . ., n} with k l such that E k and E l are two (−1)-curves in the same G-orbit and E k ∩ E l ∅.The image of E l by the contraction of E k has self-intersection 0, and in particular it cannot be contracted.By the assumption that n ≥ 3, we can contract other (−1)-curves in κ −1 (p), which increases the self-intersection.This contradicts the existence of a contraction of X to the ruled surface S, where f 2 = 0 for any fibre f .Therefore, we must have n ≤ 2.
Proposition 2.6 motivates the study of automorphism groups of conic bundles.The next lemma can be used as a maximality criterion for their automorphism groups.
Lemma 2.7.Let C be a curve of positive genus.Let κ : X → C and κ : X → C be conic bundles.Let G be an algebraic group acting on X and X such that (G, X) and (G, X ) are minimal, and let φ : X X be a G-equivariant birational map which is not an isomorphism.Then φ = φ n • • • φ 1 , where each φ j is the blowup of a finite G-orbit of a point which is contained in the complement of the singular fibres and does not contain two points on the same smooth fibre, followed by the contractions of the strict transforms of the fibres through the points of the G-orbit.In particular, κ and κ have the same number of singular fibres.
Proof.Take a minimal resolution of φ, i.e. a surface Z with G-equivariant birational morphisms η : Z → X and η : Z → X satisfying η = φη.Let E 1 , . . ., E m ⊂ Z be a G-orbit of (−1)-curves contracted by η , and let p i = κη(E i ).For each i, denote by E i the image of E i by η, which is contained in the fibre κ contained in the complement of the singular fibres.As E 2 i = 0 and E 2 i = −1 for each i, no distinct points of Ω lie in the same smooth fibre.Because (G, X ) is minimal, we can contract the strict transforms of the fibres, which yields a G-equivariant birational map φ 1 : X X 1 such that φ factorizes through φ 1 .By induction, we find G-equivariant birational maps φ j : X j−1 X j such that φ = φ n • • • φ 1 , where each φ j is as we wanted.
Finally, applying elementary transformations in the complement of the set of singular fibres does not change the number of singular fibres.

Generalities on ruled surfaces and their automorphisms
Definition 2.8.A ruled surface π is decomposable if it admits two disjoint sections.Else, π is indecomposable.
The following notion has been already used in [Mar70,Mar71] and more recently in [Fon21].
Definition 2.9.The Segre invariant S(S) of a ruled surface π : S → C is the integer min{σ 2 | σ section of π}.A section σ of π such that σ 2 = S(S) is called a minimal section.
Lemma 2.10.Let π : S → C be a decomposable ruled surface with S(S) = 0.If two sections are disjoint, then they are both minimal sections.
Proof.Let σ be a minimal section.Let Num(S) be the group of divisors of S, up to numerical equivalence.Then Num(S) is generated by the classes of σ and f , where σ is a minimal section and f is a fibre; see [Har77, Proposition V.2.3].Let s 1 and s 2 be disjoint sections.In particular, Lemma 2.11.Let S → C be a ruled surface such that S(S) = −n < 0. The following hold: (1) There exists a unique section of negative self-intersection, and all other sections have self-intersection at least n.
(2) Two sections are disjoint if and only if one is the (−n)-section and the other has self-intersection n.
(3) There exists a section of self-intersection n if and only if S is decomposable. Proof.
(1) By assumption, there exists a section s −n of self-intersection s 2 −n = −n < 0. Let s s −n be a section; then s is numerically equivalent to s −n + bf for some integer b and 0 ≤ s (3) Let s be a section such that s 2 = n.Then s ≡ s −n +nf and s •s −n = 0; i.e. s and s −n are disjoint sections.In particular, S is decomposable.Conversely, if S is decomposable, there exist two disjoints sections, and one of them has self-intersection n by (2).Definition 2.12.
A non-trivial decomposable ruled surface S of Segre invariant 0 admits exactly two minimal sections.In [Mar71, Theorem 2(3), (4)], a necessary and sufficient condition for such surfaces to have an automorphism permuting two minimal sections is given.We provide below a revisited version of this result, that we prove by computations in local charts.Lemma 2.13.Let C be a curve.Let π : S = P(O C (D) ⊕ O C ) → C be a decomposable P 1 -bundle, and let p 1 : C × P 1 → C be the trivial P 1 -bundle over C. Then S(S) = 0 if and only if deg(D) = 0.Moreover, if S(S) = 0 and π is not trivial, then the following hold: (1) The P 1 -bundle π has exactly two minimal sections s 1 and s 2 of self-intersection 0.
(3 We now assume that S(S) = 0 and that π is not trivial, and prove (1), ( 2), (3).The proof of (1) can be found in [Mar71, Lemma 2(2)] or [Fon21, Proposition 2.18(3.iii)].We now prove (2) and (3).Let A be a very ample divisor on C. For a large enough integer m, the divisor B = D + mA is also very ample.In particular, we can find B ∼ B and A ∼ mA such that Supp(B ) ∩ Supp(D) = ∅ and Supp(A ) ∩ Supp(D) = ∅.Let E = A − B ; then D + E ∼ 0, and there exists an as trivializing open subsets of π, and local trivializations of π such that s 1 and s 2 are, respectively, the zero and the infinity sections.The transition map of S can be written as By (1), an element of Aut C (S) either fixes pointwise s 1 and s 2 , or permutes s 1 and s 2 .If φ ∈ Aut C (S) fixes s 1 and s 2 , then it induces automorphisms φ u : U × P 1 → U × P 1 , (x, [y 0 : y 1 ]) → (x, [α u (x)y 0 : y 1 ]) and φ v : fixing s 1 and s 2 .If ι ∈ Aut C (S) permutes s 1 and s 2 , then ι induces automorphisms ι u : U × P 1 → U × P 1 , (x, [y 0 : y 1 ]) → (x, [β u (x)y 1 : y 0 ]) and ι v : In particular, div(β v ) = 2D, and 2D is a principal divisor.Conversely, if 2D is a principal divisor, there exists a β ∈ k(C) * such that div(β) = 2D.Choose β v = β and β u = f −2 β v ; the automorphisms ι u , ι v glue back to a C-automorphism ι of S of order two which permutes s 1 and s 2 .Thus, Aut C (S) G m Z/2Z if and only if 2D is principal, and ι v induces the birational map ξ : S C × P 1 given in the statement.
Finally, assume ι is a square in Bir(S).Then ξιξ −1 is a square in Bir(C × P 1 ) and has determinant −β.Since π is not trivial by assumption, it follows that D is not principal and div(β) = 2D.This implies that −β = det(ξιξ −1 ) is not a square, which gives a contradiction.
Every element of Aut P 1 (F n ) fixes pointwise a section of F n .This is not true when we consider P 1 -bundles over a non-rational curve C, as we have seen in Lemma 2.13(2).The following lemma shows that it is the only exception up to conjugation.Lemma 2.14.Let C be a curve.Let π : S → C be a ruled surface, and let p 1 : C × P 1 → C be the trivial P 1 -bundle and f ∈ Aut C (S).Then f satisfies one of the following: (1) The morphism f fixes pointwise a section of π.
(2) The morphism f does not fix any section of π, and there exists a birational map ξ : S C × P 1 such that π = p 1 ξ and ξf ξ −1 = 0 β 1 0 , with div(β) = 2D for some divisor D which is not principal.Moreover, if f satisfies (2), then f is not a square in Bir C (S).
Proof.First we deal with the case S(S) ≤ 0. If S(S) < 0, or S(S) = 0 and S is indecomposable, then S has a unique minimal section which is Aut C (S)-invariant (see Lemma 2.11(1) and [Mar71, Lemma 2(1.ii)], or [Fon21, Proposition 2.18(3.ii)]).If S is trivial, then Aut C (S) = PGL(2, k) and every element fixes pointwise a section.In particular, f satisfies condition (1).Else S(S) = 0, S is decomposable, and S is not trivial.Then S = P(O C (D) ⊕ O C ) for some divisor D of degree 0, and by Lemma 2.13, Aut C (S) G m or G m Z/2Z.In particular, the automorphism f either fixes the two minimal sections of S and satisfies (1), or permutes them and satisfies (2).
Assume S(S) > 0. Then S is indecomposable, and Aut C (S) is finite; see [Mar71, Theorem 2(1)].Let s be a section of S. If f (s) = s, we are done.Else s and f (s) intersect in finitely many points which are fixed by f .Blow up these points and contract the strict transforms of their fibres, and repeat the process until the strict transforms of the sections are disjoint.This yields an f -equivariant birational map φ : S S with S decomposable.By [Fon21, Proposition 2.18(1)], it follows that S(S ) ≤ 0.Moreover, the strict transforms of s and f (s) by φ are disjoint and permuted by φf φ −1 ; hence S(S ) = 0 (see Lemma 2.11(2)).Then Lemma 2.13 implies that f satisfies (2).Since φf φ −1 is a not a square in Bir C (S) by Lemma 2.13(2), it also follows that f is also not a square in Bir C (S).

Reduction of cases
The following lemma is an analogue of [Bla09a, Lemma 6.1(1) ⇔ (2)] for not necessarily rational conic bundles.The proof is slightly more difficult, due to case (2) of Lemma 2.14, which does not exist in the rational case.
Lemma 2.15.Let C be a curve.Let κ : X → C be a conic bundle with at least one singular fibre, and let f ∈ Aut C (X) permute the irreducible components of at least one singular fibre.Then f has order two.
Proof.Let η : X → S be the contraction of one irreducible component in each singular fibre.The automorphism f 2 preserves all the irreducible components of the singular fibres; hence η is f 2 -equivariant.Let g = ηf 2 η −1 ∈ Aut C (S), which is a square in Bir C (S); then g fixes pointwise a section (see Lemma 2.14).Let s inv be a g-invariant section of S, and s inv its strict transform by η which is f 2 -invariant.As f exchanges the irreducible components of at least one singular fibre, the section s inv is not f -invariant.The sections s inv and f (s inv ) meet a general fibre in two points which are exchanged by the action of f .Thus f has order two.
Lemma 2.16.Let C be a curve.Let κ : X → C be a conic bundle with at least one singular fibre, such that its two irreducible components are exchanged by an element ρ ∈ Aut(X).Let G be the normal subgroup of Aut C (X) which leaves invariant each irreducible component of the singular fibres.The following hold: (1) If G fixes a section σ 1 of κ and G is not trivial, then κ is an exceptional conic bundle.
(2) If there exists a contraction η : X → S such that S(S) ≤ 0 and S is indecomposable, then G is trivial. Proof.
(1) The subgroup G ⊂ Aut(X) is normal; hence ρGρ −1 = G, and the section σ 2 = ρσ 1 σ 1 is also G-invariant.Let η : X → S be the contraction of one irreducible component in each singular fibre of κ, namely the one intersecting σ 2 ; then it is a G-equivariant birational morphism.Let H = ηGη −1 ⊂ Aut C (S), which is not trivial.The images of σ 1 and σ 2 by η are H-invariant sections s 1 and s 2 of S. Assume that s 1 and s 2 intersect.Choose another section s 3 .Apply elementary transformations centred on {s i ∩ s j | i, j ∈ {1, 2, 3}, i j}, and repeat until the strict transforms of s 1 , s 2 , s 3 are disjoint.This yields an H-equivariant birational map ψ : S C × P 1 .The group ψHψ −1 is an algebraic subgroup of PGL(2, k) which fixes the strict transforms of s 1 , s 2 and the basepoints of ψ −1 .The basepoints of ψ −1 coming from the contraction of the strict transforms of the fibres passing through the intersections of s 1 and s 2 are outside of the strict transforms of s 1 and s 2 .Then H is conjugate to a subgroup of PGL(2, k) fixing three distinct points on P 1 , which implies that H is trivial and gives a contradiction.Therefore, s 1 and s 2 are disjoint P. Fong 10 P. Fong sections of S, and it follows that S is decomposable and S(S) ≤ 0 by [Fon21, Proposition 2.18(1)].Thus σ 1 and σ 2 are also disjoint sections of X which pass through different irreducible components in each singular fibre.
(2) If S(S) ≤ 0 and π is indecomposable, then S has a unique minimal section which is Aut(S)-invariant, see [Mar71, Lemma 2(1)(i), (ii)] (or [Fon21, Proposition 2.18(2), (3.ii)]), and its strict transform by η is a G-invariant section of κ.If G is not trivial, it follows from (1) that X is an exceptional conic bundle.This implies that S admits two disjoint sections, which gives a contradiction.
The key result of this section is the following proposition, analogue of [Bla09b, Lemma 4.3.5],which will be useful to reduce to the study of automorphism groups of ruled surfaces, exceptional conic bundles and (Z/2Z) 2 -conic bundles.
Proposition 2.17.Assume that char(k) 2. Let C be a curve.Let κ : X → C be a conic bundle with at least one singular fibre, such that its two irreducible components are exchanged by an element of Aut(X).Let G be the normal subgroup of Aut C (X) which leaves invariant every irreducible component of the singular fibres.If G is not trivial and if there exists a contraction η : X → S with S a decomposable P 1 -bundle over C, then κ is an exceptional conic bundle.Else, Aut C (X) is isomorphic to (Z/2Z) r for some r ∈ {0, 1, 2}.
Proof.If G is trivial, then every element of Aut C (X) is an involution.This implies that Aut C (X) is a finite subgroup of PGL(2, k(C)), and the statement follows.Assume that G is not trivial, and let η : X → S be a contraction, where π : S → C is a P 1 -bundle.Then η is G-equivariant, and H = ηGη −1 ⊂ Aut C (S) is not trivial.Three cases arise: (1) First assume that S(S) < 0. Then S admits a unique minimal section, and its strict transform by η is a G-invariant section of κ.By Lemma 2.16, S is decomposable, and κ is an exceptional conic bundle.
(2) Assume that S(S) = 0.If a section of S of self-intersection 0 passes through at least one of the points blown up by η, its strict transform is a section s of X of negative self-intersection.Contracting in each fibre the irreducible component not intersecting s gives a birational morphism η : X → S with S(S ) < 0, reducing to the previous case.We now assume that no section of S of self-intersection 0 passes through any point blown up by η.Firstly, π : S → C is not a trivial bundle, as otherwise sections of S of self-intersection 0 would cover S. From Lemma 2.16(2), S is decomposable.Moreover, by Lemma 2.13(1), there exist exactly two disjoint sections s 1 , s 2 of S of self-intersection 0. Furthermore, ηGη −1 is a non-trivial subgroup of Aut C (S), isomorphic to G m or G m Z/2Z (see Lemma 2.13), that fixes the basepoints of η −1 not lying on s 1 or s 2 .We now prove that no non-trivial element of ηGη −1 can lie in G m : take a trivializing open subset U ⊆ C of π : S → C containing the image of a basepoint, and take an isomorphism π −1 (U ) U × P 1 sending s 1 , s 2 onto the zero and infinity sections; then the action of G m on U × P 1 is (x, [u : v]) → (x, [αu : v]), and thus no non-trivial element of G m fixes any point outside of s 1 , s 2 .Then ηGη −1 ∩ G m = {1}, and G has order two.By Lemma 2.15, every element of Aut C (X) is an involution.As Aut C (X) is a finite subgroup of PGL(2, k(C)), this implies that Aut C (X) (Z/2Z) r for some r ∈ {0, 1, 2}.

Infinite increasing sequence of automorphism groups
We first prove the following lemma, which is a generalization of [Fon21, Theorem A].The proof works essentially the same, based on an explicit automorphism of ruled surfaces computed in [Mar71].
Lemma 3.1.Let C be a curve of positive genus and π : S → C be a ruled surface such that S(S) < 0. Then there exists an infinite family {S i , φ i } i≥1 , where the π i : S i → C are ruled surfaces and the φ i : S S i are Aut(S)-equivariant birational maps, such that • is an infinite increasing sequence of algebraic subgroups of Bir(C × P 1 ).In particular, Aut(S) is not a maximal algebraic subgroup of Bir(C × P 1 ).
Proof.Since S(S) < 0, there exists a unique negative section (see Lemma 2.11(1)), which is Aut(S)-invariant.From [Mar71, Lemmas 6 and 7], the morphism of algebraic groups π * : Aut(S) → Aut(C) has finite image.Let p be a point on the minimal section; its orbit by the Aut(S)-action is a finite subset of the minimal section.The blowup of the orbit of p followed by the contractions of the strict transforms of the fibres defines an Aut(S)-equivariant birational map η 1 : S S 1 with S(S 1 ) < S(S).Repeating this process gives rise to a family of ruled surfaces {π i : S i → C} i≥1 with an infinite sequence We will see that this sequence is not stationary.
Take n large, and let z = π(p).By a choice of trivialization π −1 n (U ) U × P 1 , we can assume that q = (z, [0 : 1]) ∈ U × P 1 is a basepoint of η −1 n : S n S n−1 .Let V be a vector bundle of rank two over C such that P(V ) = S n , and let L ⊂ V be the line subbundle associated to the minimal section in S n .Let L = det(V ) −1 ⊗ L 2 ; it follows from [Fon21, Corollary 2.16] that deg(L) = −S(S n ).Since n is chosen large, we can assume that S(S n ) < 0 is small enough such that h 1 (C, L) = h 1 (C, L ⊗ O C (z) −1 ) = 0.By the Riemann-Roch theorem, we get that h 0 (C, L ⊗ O C (z) −1 ) < h 0 (C, L); i.e. z is not a basepoint of the complete linear system |L|.Therefore, there exists a γ ∈ H 0 (C, L) such that γ(z) 0.
Let (U i ) i be trivializing open subsets of π n ; the automorphisms ) such that f γ does not fix q and Aut(S n−1 ) η −1 n Aut(S n )η n .We have proved that the sequence ( †) is not stationary.Removing in the sequence the groups which are not strictly bigger than the previous term and renaming the elements accordingly yields the increasing sequence of the statement.Remark 3.2.Notice that the proof of Lemma 3.1 implies [Fon21, Theorem A].Let γ ∈ Γ (C, det(V ) −1 ⊗ L 2 ) be as above.For any t ∈ G a , the automorphisms glue into a C-automorphism f tγ .In particular, each automorphism f γ belongs to the connected component of the identity.Restricting the infinite chain ( †) to the connected components, one gets that )) for all n.

Exceptional conic bundles
The following lemma is a generalization of [Bla09b, Lemma 4.3.1]for exceptional conic bundles which are not necessarily rational.Lemma 3.3.Let C be a curve, and let κ : X → C be a conic bundle with 2n ≥ 0 singular fibres.The following assertions are equivalent: (1) The bundle π is exceptional.
(2) There exist exactly two sections s 1 , s 2 of self-intersection −n, which are disjoint and intersect different irreducible components of each singular fibre.
(3) There exists a birational morphism η n : X → S, where S is a decomposable P 1 -bundle over C with S(S) = −n, which consists in the blowup of 2n points on a section of self-intersection n in S. (4) There exists a birational morphism η 0 : X → S, where S is a decomposable P 1 -bundle over C with S(S) = 0, which consists in the blowup of 2n points such that no two points are in the same fibre, n are chosen on a section of self-intersection 0, and the other n are chosen on another section of self-intersection 0. Proof.
(1) =⇒ (2), (3) Assume κ is exceptional, and let s 1 , s 2 be sections of self-intersection −n.Contracting in each singular fibre the irreducible component which does not meet s 1 yields a birational morphism η n : X → S, where S is a ruled surface over C. Denote by s 1 and s 2 the images of s 1 and s 2 by η n ; then s 2 1 = −n and s 2 2 ≤ n.The case s 2 2 < n cannot happen (see Lemma 2.11(1)), and the equality implies s 1 and s 2 pass through different irreducible components of each singular fibre.Then the sections s 1 and s 2 are disjoint (see Lemma 2.11(2)), and S is decomposable (see Lemma 2.11(3)).Assume there exists a third section s 3 of self-intersection −n on X .By the same argument, s 3 has to pass through different irreducible components than s 1 and s 2 .Since each singular fibre contains exactly two irreducible components, this is not possible.
(2) =⇒ (4) Contract in n singular fibres the irreducible components meeting s 1 , and contract in the other singular fibres the irreducible components meeting s 2 .This defines a birational morphism η 0 : X → S such that the images of s 1 and s 2 by η 0 are disjoint sections of S of self-intersection 0. In particular, S is decomposable, and by Lemma 2.11(1), S(S) = 0.
(2), (3), ( 4) =⇒ (1) The implication (2) =⇒ (1) is trivial.The strict transforms by η n of the sections of S of self-intersection n and −n are two sections of κ of self-intersection −n.The strict transforms by η 0 of the two sections of S of self-intersection 0 are two sections of κ of self-intersection −n.This proves that (3) and (4) imply (1).
In [Bla09b, Lemma 4.3.3(1)], it is proven that Aut P 1 (X) G m Z/2Z when X is an exceptional conic bundle over P 1 , which implies that Aut(X) is maximal.We see below that automorphism groups of exceptional conic bundles over a non-rational curve do not always contain an involution permuting the two (−n)-sections (see Proposition 3.5) and are not always maximal (see Lemma 3.4).Lemma 3.4.Let C be a curve of positive genus, and let κ : X → C be an exceptional conic bundle.If Aut C (X) contains a non-trivial involution permuting the irreducible components of the singular fibres, then Aut C (X) G m Z/2Z and Aut(X) is maximal.Else, Aut C (X) G m , and Aut(X) can be embedded in a infinite increasing sequence of algebraic subgroups of Bir(C × P 1 ).
Proof.Denote by s 1 , s 2 the two (−n)-sections of κ.Let G be the subgroup of Aut C (X) which leaves invariant the irreducible components of each singular fibre, and let η 0 : X → S be a birational morphism which contracts an irreducible component in each singular fibre and is such that S(S) = 0 (see Lemma 3.3(4)).Let π : S → C be the morphism such that κ = πη 0 .Let s 1 , s 2 be, respectively, the images of s 1 and s 2 by η 0 .Then η 0 is G-equivariant, and η 0 Gη −1 0 is an algebraic subgroup of Aut C (S) which leaves invariant s 1 and s 2 .If π is trivial, then η 0 Gη −1 0 is an algebraic subgroup of PGL(2, k) fixing at least two points on a fibre, and thus is contained in G m ⊂ PGL(2, k).If π is not trivial, then η 0 Gη −1 0 is an algebraic subgroup of G m or G m Z/2Z by Lemma 2.13.Since the element of order two in Z/2Z permutes s 1 and s 2 , it follows that η 0 Gη −1 0 is also contained in G m in the second case.Hence η 0 Gη −1 0 is contained in a subgroup of Aut C (S) isomorphic to G m .Conversely, every element of G m fixes s 1 , s 2 (the images of s 1 , s 2 by η 0 ); hence η 0 Gη −1 0 = G m .The exceptional conic bundle κ has exactly two (−n)-sections, which are left invariant or are permuted by the elements of Aut(X).Assume that Aut C (X) contains an element ι which permutes the two (−n)-sections of κ (or, equivalently, the irreducible components of each singular fibre, by Lemma 3.3(2)).The automorphism ι acts on a general fibre by permuting two points, which implies that ι is an involution.If f ∈ Aut C (X) permutes the two (−n)-sections, then ιf does not; i.e. ιf ∈ G.This implies that Aut C (X) = G ι .Moreover, there is no ι-equivariant contraction from X, and all Aut(X)-orbits in the complement of the singular fibres are infinite or contain two points on a smooth fibre.Hence there is no Aut(X)-equivariant birational map from X (see Lemma 2.7), and Aut(X) is maximal.
If Aut C (X) does not contain an element ι as above, then Aut C (X) = G G m .Since there exist exactly two (−n)-sections, an element of Aut(X) either fixes them or permutes them.The contraction of Aut(X)orbits of (−1)-curves yields an Aut(X)-equivariant birational morphism η n : X → S, where S(S) < 0 (see Lemma 3.3(3)).Then use Lemma 3.1 to conclude.

P. Fong 14 P. Fong
C-birational map φ of S, and η −1 φη ∈ Aut C (X) is an involution permuting the irreducible components of the singular fibres of κ.Conversely, assume there exists an involution ψ ∈ Aut C (X) permuting the irreducible components of the singular fibres.Then ηψη −1 ∈ Bir(S) acts trivially on C and permutes s 1 and s 2 ; hence there exists an f ∈ k(C) such that the restriction of ηψη −1 to π −1 (U ) yields a birational map Since the set of basepoints of ηψη −1 is exactly Z ∪ P , the rational function f |U has zeros in π(Z) and poles in π(P ).Computing in local charts, one can check that ν q (f ) = 1 if q ∈ π(Z) and ν q (f ) = −1 if q ∈ π(P ).Conjugating as before by the transition maps of π gives ) Let m be the number of singular fibres of κ, and denote by ι ∈ Aut C (X) a non-trivial involution permuting the irreducible components of each singular fibre.Without loss of generality, we can replace s 2 with ι(s 1 ), and it follows that s 2 1 = s 2 2 .Contracting the irreducible components intersecting s 1 in each singular fibre of κ gives a birational morphism η : X → S, where S is decomposable with two disjoint sections s 1 and s 2 .Then S(S) ≤ 0 (see [Mar70, Corollary 1.17] or [Fon21, Proposition 2.18(1)]), and S(S) 0 by the definition of η and by Lemma 2.10 as S → C is decomposable by (1).This implies that s 2 1 = −S(S) and s 2 2 = S(S) (see Lemma 2.11(2)).On the other hand, we have In particular, η corresponds to a birational map η n as in Lemma 3.3(3), and κ is an exceptional conic bundle over C.
Combining Lemma 3.4 and Proposition 3.5, we get the main result of this section.Proposition 3.6.Let C be a curve of positive genus.Let κ : X → C be an exceptional conic bundle with two (−n)-sections s 1 and s 2 .The contraction of an irreducible component in each singular fibre gives a birational morphism η : X → S, where π : In particular, κ = πη.Denote by s 1 , s 2 the images of s 1 , s 2 by η, and by Z ⊂ s 1 , P ⊂ s 2 the sets of basepoints of η −1 .The algebraic group Aut(X) is maximal if and only if −2D is linearly equivalent to and in this case, Aut(X) fits into an exact sequence of algebraic groups where H denotes the subgroup of Aut(C) which fixes the finite subset π(Z ∪ P ).Else, Aut(X) is not maximal and can be embedded in an infinite increasing sequence of algebraic subgroups of Bir(C × P 1 ).
Proof.The structure morphism κ induces a morphism of algebraic groups κ * : Aut(X) → Aut(C), and an element in the image of κ * must preserves π(Z ∪ P ), which is the set of points in C having singular fibres.The rest of the statement follows from Lemma 3.4 and Proposition 3.5.Corollary 3.7.Let C be a curve of positive genus.Then there exist exceptional conic bundles X → C such that Aut(X) is not a maximal algebraic subgroup of Bir(C × P 1 ).
Proof.Let X be an exceptional conic bundle over C which is not the blowup of a decomposable ruled surface π : By Proposition 3.6, Aut(X) is not a maximal algebraic subgroup of Bir(C × P 1 ).
In Proposition 3.6, the morphism κ * : Aut(X) → H can be surjective: it is always the case either if C = P 1 (see [Bla09b, Lemma 4.3.3(1)]),or if C is a curve of genus g ≥ 2 with a trivial automorphism group.We give an example where this surjectivity fails.
Example 3.8.Let C be an elliptic curve over C with neutral element p 0 .Choose a 4-torsion point p 1 ∈ C such that p 2 = 2p 1 p 0 , and denote by ∆ = {p 0 , p 1 , p 2 , p 3 } the subgroup generated by p 1 .Define the ruled surface π : [Fon21,Proposition 2.15]).By Lemma 2.11(1), it follows that σ is the unique section of π with negative self-intersection, and therefore S(S) = −2.Let s 1 , s 2 be two disjoint sections with s 2 1 = −2 and s 2 2 = 2. Denote by η : X → S the blowup of s 2 ∩ π −1 (∆), and by s 1 , s 2 the strict transforms of s 1 and s 2 by η.Then κ = πη is a conic bundle.Moreover, Let f ∈ Aut(C) be the translation x → x + p 1 which preserves ∆; i.e. f ∈ H. Denote by H the subgroup of Aut(X) which fixes s 1 and s 2 .Notice that η is H-equivariant and the following diagram is commutative: Assume that f ∈ Aut(C) lifts to an element of Aut(X); then it can also be lifted in H (if the lifting permutes s 1 and s 2 , compose it with the non-trivial involution to get an element in H), and a fortiori can be lifted in Aut(S).This is not the case because f * (D) = p 0 + p 3 is not linearly equivalent to D .
Lemma 3.9 (cf.[Bla09b, Lemma 4.4.1]).Let C be a curve.Every element of order two in PGL(2, k(C)) is conjugate to an element of the form σ f = 0 f 1 0 , where f ∈ k(C) * .Moreover, σ f and σ g are conjugate if and only if f /g is a square in k(C) * .
Proof.Let σ ∈ PGL(2, k(C)) be an element of order two, and let v ∈ P 1 be such that σ (v) v. Since σ is of order two, there exists an f ∈ k(C) * such that the matrix of σ with respect to the basis {v, σ (v)} is σ f = 0 f 1 0 .Let f , g ∈ k(C) * .Assume σ f and σ g are conjugate.Let σ f , σ g be their respective representatives in GL(2, k(C)) having 1 as the lower left coefficient.Then there exist λ ∈ k(C), P ∈ GL(2, k(C)) such that P σ f P −1 = λ σ g .Taking the determinant in the last equality gives f /g = λ 2 .Conversely, assume that f /g = λ 2 for some λ ∈ k(C) * .Then where  (1) ) is also trivial.

In particular, s(h) = ( f (h), h). For all h
) is trivial, f is conjugate to the trivial cocycle; this implies that s and s c are conjugate up to an element of PGL(2, k(C)).Thus G and H are conjugate.
(2) Let σ be the element of order two of G .From Lemma 3.9, we can assume up to conjugation that σ = 0 f 1 0 for some f ∈ k(C) * which is not a square (by assumption, the determinant of σ is not trivial).We denote by N σ the normalizer of σ .By Lemma 3.10, All elements of the form a −bf b −a have order two in PGL(2, k(C)): the goal is to find one of them which with σ generates a subgroup V isomorphic to (Z/2Z) 2 and normalized by G.
The H-equivariant exact sequence Therefore, ρ is conjugate to the trivial cocycle; i.e. there exists a τ ∈ N The subgroup generated by σ and (T , 1) is normalized by G and is isomorphic to (Z/2Z) 2 .
Under the assumptions of Proposition 3.11, the automorphism group of a conic bundle X such that Aut C (X) Z/2Z is not maximal.Below, we see that a conic bundle X such that Aut C (X) (Z/2Z) 2 is always a (Z/2Z) 2 -conic bundle and has a maximal automorphism group (Lemmas 3.15 and 3.16).Lemma 3.12.Assume that char(k) 2. Let C be a curve, and let κ : X → C be a conic bundle having at least one singular fibre.Suppose there exists a non-trivial involution f ∈ Aut C (X) fixing pointwise two sections s 1 and s 2 .Then in each singular fibre, the sections s 1 and s 2 pass through different irreducible components.
Proof.Assume there is a singular fibre κ −1 (p) where s 1 and s 2 pass through the same irreducible component.Since f fixes pointwise s 1 and s 2 , the contraction of the other irreducible component gives an f -equivariant birational morphism η : κ −1 (U ) → U × P 1 , where U is an open neighbourhood of p.The C-automorphism ηf η −1 has order two, which implies that there exist a, b, c ∈ O C (U ) such that ( ) ηf η −1 : (x, [u : v]) → (x, [au + bv : cu − av]).
P. Fong 20 P. Fong (3) If S(S) = 0 and S is indecomposable, then Aut(S) is maximal if and only if g = 1.If g ≥ 2, then Aut(S) can be embedded in an infinite increasing sequence of algebraic subgroups of Bir(C × P 1 ).(4) If S(S) > 0, then Aut(S) is maximal if and only if S is a (Z/2Z) 2 -ruled surface. Proof.
(1) If S is trivial, each Aut(S)-orbit contains at least a fibre.Hence Aut(S) is maximal by Lemma 2.7.
Assume that g ≥ 2. If 2D is principal, then Aut C (S) G m Z/2Z (see Lemma 2.13(2)), and the group Aut C (S) acts on a fibre with two orbits: one is isomorphic to G m , and the other one is made of two points exchanged by the involution.From Lemma 2.7, Aut(S) is a maximal algebraic subgroup.If 2D is not principal, Aut(S) G m (see Lemma 2.13(3)).Take a point in a minimal section; its Aut(S)-orbit is finite and contains at most one point in each fibre.The blowup of this orbit followed by the contraction of the strict transforms of the fibres gives an Aut(S)-equivariant birational map S S , where S is a ruled surface with S(S ) < 0. Then apply Lemma 3.1.
(3) From [Fon21, Proposition 2.18(3.ii)],S has a unique minimal section which is Aut(S)-invariant.If g ≥ 2, take a point on this minimal section and blow up its orbit (which consists of finitely many points on the minimal section), then contract the strict transforms of the fibres.This defines an Aut(S)-equivariant map S S with S(S ) < 0. Then apply Lemma 3.1.If g = 1, then S A 0 and π * : Aut • (S) → Aut • (C) is surjective; see [Fon21,Proposition 3.6].In particular, there is no Aut(S)-orbit of dimension 0 and no Aut(S)-equivariant map.Thus Aut(S) is maximal.
By Lemma 3.15, each non-trivial element of Aut C (S) has a non-trivial determinant.If s = 0, then Aut(S) is conjugate to a finite subgroup of Aut(C) Aut(C × P 1 ) by Proposition 3.11(1).If s = 1, then by Proposition 3.11(2), Aut(S) normalizes a group V (Z/2Z) 2 containing Aut C (S); i.e. there exists a finite subgroup G ⊂ Bir(C × P 1 ) containing Aut(S) such that V is the kernel of the action of G on C. In particular, Aut(S) G. Therefore, we get that Aut(S) is not maximal if s ∈ {0, 1}.If s = 2, then S is a (Z/2Z) 2 -ruled surface.Conversely, the automorphism group of a (Z/2Z) 2 -ruled surface is maximal by Lemma 3.16.

Examples of (Z/2Z) 2 -conic bundles
If C is an elliptic curve, the Atiyah bundle A 1 is the only (Z/2Z) 2 -ruled surface.We give below examples of (Z/2Z) 2 -conic bundles over any curve C of genus g ≥ 2. If X is a (Z/2Z) 2 -conic bundle over P 1 with at least one singular fibre, then every element of order two in Aut P 1 (X) acts non-trivially on Pic(X) by permuting the irreducible components of a singular fibre (by [Bla09b,Lemma 4.3.5].If there exists an element of Aut P 1 (X) \ {1} acting trivially on Pic(X), then X → P 1 is an exceptional conic bundle, and thus not a (Z/2Z) 2 -conic bundle by [Bla09b, Lemmas 4.3.3(1)and 4.3.5]).The following example also shows that this does not hold anymore when C has positive genus.
Example 3.18.Assume that char(k) 2. Let C be a curve of genus g ≥ 1 and D be a non-principal divisor such that 2D is principal.Let S be the decomposable ruled surface P(O C (D) ⊕ O C ). From Lemma 2.13, Aut C (S) = G m Z/2Z, and the element σ of order two that generates Z/2Z is conjugate to 0 f 1 0 for some f ∈ k(C) * such that div(f ) = 2D.Since D is not principal, f is not a square.In particular, det(σ ) 1 and σ fixes pointwise an irreducible curve birational to a 2-to-1 cover of C and ramified above an even positive number of points (see Lemma 3.13(2)).
The matrix τ = a −bf b −a of order two has determinant −a 2 + b 2 f = −N (a + b f ), where N : k(C)[ f ] → k(C) is the norm, which is surjective (see [Ser68, Propositions X.10 and X.11]).Choose a and b such that det(τ) has a pole with odd multiplicity at a point where f is regular.Then det(τ) 1 and det(σ τ) 1.

Proofs of the results
Proof of Theorem A. Each algebraic group in the list is a maximal algebraic subgroup of Bir(C × P 1 ) by Lemma 3.16 and Propositions 3.6 and 3.17.Conversely, let G be a maximal algebraic subgroup of Bir(C ×P 1 ), where C is a curve of genus g ≥ 1.Using the regularization theorem (Proposition 2.5) and the G-equivariant MMP (Proposition 2.6), it follows that G is conjugate to Aut(X) for some conic bundle κ : X → C. If κ has no singular fibre, directly apply Proposition 3.17.Else κ has at least one singular fibre.If there is no element of Aut(X) permuting two irreducible components of a singular fibre, then there exists an Aut(X)-equivariant contraction X → S, where S is a ruled surface; apply Proposition 3.17 to conclude.Else, apply Proposition 2.17 with Proposition 3.11; it follows that either X is an exceptional conic bundle, or Aut(X) is conjugate to a subgroup of Aut(C × P 1 ) or Aut(X ), where X is a (Z/2Z) 2 -conic bundle.To conclude, apply Proposition 3.6 for the case of exceptional conic bundles, and apply Lemma 3.16 for the case of (Z/2Z) 2 -conic bundles.Finally, the exact sequences of (4) in the case g = 1 and (5) are taken from [Mar71, Theorem 3].
Proof of Corollary B. From [Bla09b, Theorem 1], every algebraic subgroup of Bir(P 2 ) is included in a maximal one.From Theorem A, every maximal algebraic subgroups of Bir(C × P 1 ) has dimension at most four.By Remark 3.2, there exist algebraic subgroups of arbitrary large dimension, and they cannot be subgroups of the maximal ones.
s 1 and s 2 are both different from s −n , then b 1 ≥ n and b 2 ≥ n by (1), and this contradicts the equality 0 = −n + b 1 + b 2 .Then we can assume that s 1 = s −n and b 1
the group homomorphism a bf b a → [a + b f ], and Z/2Z is generated by the diagonal involution.The action of Z/2Z on N • σ sends a bf b a onto a −bf −b a .Proof.Since σ has order two, the normalizer of σ equals the centralizer of σ .Then it is a straightforward computation in PGL(2, k(C)) to check that matrices commuting with σ are of the form a bf b a or a −bf b −a for some a, b ∈ k(C), and . the action of Z/2Z on N • σ sends a bf b a onto a −bf −b a .The following key proposition is an analogue of [Bla09b, Proposition 5.2.2] for non-rational ruled surfaces.We prove it by copying, mutatis mutandis, the proof of [Bla09b, Proposition 5.2.2].Proposition 3.11 (cf.[Bla09b, Proposition 5.2.2]).Assume that char(k) 2. Let C be a curve of positive genus, and let G be a finite subgroup of Bir(C × P 1 ) = PGL(2, k(C)) Aut(C).Denote by G ⊂ G and H ⊂ Aut(C) the kernel and the image of the action of G on the base of the fibration.Then the following hold: and Z/2Z acts on N • σ by sending a bf b a onto a −bf −b a .
Assume that char(k) 2. Let C be a curve.Let σ is generated by an involution with a non-trivial determinant, then G normalizes a group V ⊂ PGL(2, k(C)) isomorphic to (Z/2Z) 2 and containing G .