Automorphism group schemes of bielliptic and quasi-bielliptic surfaces

Bielliptic and quasi-bielliptic surfaces form one of the four classes of minimal smooth projective surfaces of Kodaira dimension $0$. In this article, we determine the automorphism schemes of these surfaces over algebraically closed fields of arbitrary characteristic, generalizing work of Bennett and Miranda over the complex numbers; we also find some cases that are missing from the classification of automorphism groups of bielliptic surfaces in characteristic $0$.


Introduction
We are working over an algebraically closed field k of characteristic p ≥ 0. Bielliptic and quasi-bielliptic surfaces form one of the four types of minimal smooth projective surfaces of Kodaira dimension 0. Each bielliptic surface X is a quotient π : E × C → (E × C)/G = X, where E and C are elliptic curves and G ⊆ E is a finite subgroup scheme of E that acts faithfully on C via α : G → Aut C . Moreover, the image of α is not entirely contained in the group of translations C. This latter condition guarantees that X is not an Abelian surface. All possible combinations of E, C, G and α have been determined: if p = 0 by Bagnera and de Franchis in [BdF10], and if p 0 by Bombieri and Mumford in [BM77].
Similarly, quasi-bielliptic surfaces, which exist if and only if p ∈ {2, 3}, are obtained by replacing C by a cuspidal plane cubic curve and by imposing on α the condition that the cusp of C is not a fixed point of the group scheme α(G). As in the bielliptic case, it is possible to determine all combinations of E, C, G and α. We refer the reader to [BM76], but note that not all cases listed there actually occur (see Remark 5.12 and Remark 5.13).
Bielliptic and quasi-bielliptic surfaces come with two natural fibrations: one of them is the Albanese map f E : X → E/G =: E , which is quasi-elliptic if X is quasi-bielliptic, and elliptic if X is bielliptic. All closed fibers of f E are isomorphic to C, since this holds after pulling back along the faithfully flat morphism E → E/G. The second fibration f C : X → C/α(G) =: C P 1 is always elliptic, but has multiple fibers.
The purpose of this article is to determine the automorphism group scheme Aut X of X. If p = 0, this has been carried out by Bennett and Miranda in [BM90]. By Proposition 3.1, the actions of the centralizers C Aut E (G) and C Aut C (α(G)) on the first and second factor of E × C, respectively, descend to X and we consider them as subgroup schemes of Aut X via these actions. Then, the following theorem is the key result of this article.   Table 3. Automorphism group schemes of quasi-bielliptic surfaces in characteristic 2 Remark 1.4. Let us explain the meaning of the stars and daggers in Table 1. We denote by O ∈ E the neutral element with respect to the group law on E: • Stars: If p 2, 3 and j(E) = 1728, then every automorphism h E of (E, O) of order 4 fixes a unique cyclic subgroup of E of order 2. Similarly, if p 2, 3 and j(E) = 0, then every automorphism h E of (E, O) of order 3 fixes a unique cyclic subgroup of E of order 3. A star after a j-invariant in Table 1 denotes that the translation subgroup of G or α(G) coincides with this cyclic subgroup. By Lemma 5.1, this implies that h E is in the corresponding centralizer. We note that such special 2 and 3-torsion points do not exist if p = 2, 3, because (E, O) has more automorphisms in these characteristics. • Daggers: A dagger after j(E) denotes that the special 2 or 3-torsion points described above maps to a translation in α(G). In these cases, the automorphism (h E , h C ), where h E is an automorphism of order 4 or 3 of (E, O) and h C is translation by a suitable 4 or 3-torsion point, respectively, normalizes the G-action on E × C and hence descends to X. Since (h E , h C ) does not centralize the G-action, it induces a non-trivial element of M. See the proof of Proposition 5.5 for a precise description of the automorphism (h E , h C ) in these cases.
These cases seem to be missing from [BM90], since they were not listed in [BM90, Table 1.1], which is why [BM90, Table 3.2] differs from our Table 1.
Remark 1.5. In the quasi-bielliptic case in characteristic 2, the action of G on E × C sometimes depends on a parameter λ ∈ k and so does Aut X . For an explicit description of λ, see Section 5.2.2. The parameter λ should be thought of as a replacement for the j-invariant of the curve C.
Recall that the space H 0 (X, T X ) is the tangent space of Aut X at the identity. Since E (Aut • X ) red , Aut X is smooth if and only if h 0 (X, T X ) = 1. A careful inspection of Tables 1, 2, and 3, and of the orders of the canonical bundle ω X determined in [BM77] and [BM76] shows the following. Corollary 1.6. Let X be a bielliptic or quasi-bielliptic surface. Then, the following hold: (1) h 0 (X, T X ) ≤ 3.
(3) h 0 (X, T X ) = 1 if and only if ω X O X if and only if Aut X is smooth.

Acknowledgements
I would like to thank Daniel Boada de Narváez, Christian Liedtke, and Claudia Stadlmayr for helpful comments on a first version of this article and Curtis Bennett and Rick Miranda for interesting discussions. I am indebted to Azniv Kasparian and Gregory Sankaran for pointing out a mistake in an earlier version of this article. I am grateful to the anonymous referee for thorough comments and very helpful suggestions that helped me to improve the exposition and to fix inaccuracies. Finally, I would like to thank the Department of Mathematics at the University of Utah for its hospitality while this article was written.

Notation and generalities on automorphism group schemes
Let π : Y → X be a morphism of proper varieties over an algebraically closed field k. There are several k-group schemes of automorphisms associated to π. We follow the notation of [Bri18, Section 2.4], which we recall for the convenience of the reader. Throughout, T is an arbitrary k-scheme.
• The automorphism group scheme Aut X of X is the k-group scheme whose group of T -valued points , Aut X is a group scheme locally of finite type over k. The identity component of Aut X is denoted by Aut • X . • The automorphism group scheme Aut π of the morphism π is the k-group scheme such that Aut π (T ) consists of pairs (g, h) ∈ Aut Y (T ) × Aut X (T ) making the diagram hence Aut π is representable by a group scheme locally of finite type over k. • The group scheme Aut π comes with projections to Aut Y and Aut X . If π is faithfully flat, then the first projection Aut π → Aut Y is a closed immersion and we will use this to consider Aut π as a subgroup scheme of Aut Y . We denote the second projection by π * : Aut π → Aut X . • The automorphism group scheme Aut Y /X of Y over X is the k-group scheme whose group of T -valued points Aut Y /X (T ) consists of automorphisms g ∈ Aut Y (T ) such that π T • g = π T . By definition, there is an exact sequence realizing Aut Y /X as a subgroup scheme of Aut π : The centralizer C Aut Y (G) of G in Aut Y is the group scheme whose T -valued points satisfy the stronger condition h T • g • (h T ) −1 = g instead. By [ABD + 66, Exposé VIB, Proposition 6.2 (iv)], both N Aut Y (G) and C Aut Y (G) are closed subgroup schemes of Aut Y .
Caution 2.1. The notation Aut Y /X is also a standard notation for the group functor on the category of X-schemes that associates to an X-scheme Z the automorphism group of Y × X Z over Z. Since these relative automorphism group functors do not occur in this article, we decided to use the notation introduced above instead of more cumbersome, albeit more precise, notation such as Aut Y /X/k .

Automorphism group schemes of quotients
In this section, we study Sequence (1) in the case where π : Y → X is a finite quotient.
Proposition 3.1. If G is a finite group scheme acting freely on a proper variety Y such that the geometric quotient π : Y → Y /G =: X exists as a scheme, then we have Aut Y /X = G and Aut π = N Aut Y (G) as subgroup schemes of Aut Y . In particular, Sequence (1) becomes where the T -valued points of the latter are Hom(Y × T , G) and G is embedded as G = Hom(Spec k, G). Since Y is a proper variety and taking global sections commutes with flat base change, , which is what we had to show. Next, we show Aut π = N Aut Y (G). For this, let h ∈ Aut Y (T ) be an automorphism of Y T . Then, h ∈ Aut π (T ) if and only if there is h ∈ Aut X (T ) such that the following diagram commutes Comparing degrees, it is easy to check that the geometric quotient of Y T by the induced free action of G coincides with π T , so the morphism π : Y → X is a universal geometric quotient of Y , hence also a universal categorical quotient by [ Example 3.2. Contrary to the situation for abstract groups, Proposition 3.1 typically fails if Y is a non-proper variety or the action of G is not free. Indeed, consider any infinitesimal subgroup scheme G ⊆ PGL 2 of length p. The k-linear Frobenius F : Y := P 1 → P 1 =: X is the geometric quotient for the action of G on P 1 and Aut Y /X = PGL 2 [F] is the kernel of Frobenius on PGL 2 . Moreover, we have Aut F = PGL 2 . Thus, Aut Y /X and Aut F are strictly bigger than G and N PGL 2 (G) even though F is a G-torsor over an open subscheme of X.
Even though Example 3.2 shows that Proposition 3.1 fails for non-free actions on curves, we can at least describe the k-rational points in Sequence (1) if the quotient is smooth. Proof. We can consider the four groups as subgroups of the group Aut k (k(D)) of k-linear field automorphisms of k(D) via the injective restriction map Aut(D) → Aut k (k(D)). We have a tower of field extensions G(k) is purely inseparable and k(D) G(k) ⊆ k(D) is a Galois extension with Galois group G(k). An elementary calculation shows that N Aut k (k(D)) (G(k)) is the subgroup of Aut k (k(D)) of automorphisms preserving k(D) G(k) . Since D is a curve, we have k(D ) = (k(D) G(k) ) p n for some n ≥ 0, so an automorphism of k(D) preserves k(D) G(k) if and only if it preserves k(D ). Hence, N Aut k (k(D)) (G(k)) is also the group of automorphisms of k(D) preserving k(D ). On the other hand, since D is smooth and proper, Aut ϕ (k) consists precisely of those automorphisms of D which, when restricted to k(D), preserve k(D ). Hence, we have N Aut(D) (G(k)) = N Aut k (k(D)) (G(k)) ∩ Aut(D) = Aut ϕ (k), and Aut D/D (k) = Aut k(D ) (k(D)) ∩ Aut(D) = G(k), which is what we had to show.

Proof of Theorem 1.1
Throughout this section, E and C are integral curves of arithmetic genus 1, and we assume that E is smooth and C is either smooth or has a single cusp as singularity. We choose a point O ∈ E and consider E as an elliptic curve with identity element O. We fix a finite subgroup scheme G ⊆ E, and a monomorphism α : G → Aut C such that α(G) is not contained in the group of translations of C if C is smooth, and not contained in the stabilizer of the cusp if C is singular. In particular, the actions of G on E (via translations) and C (via α) give rise to a product action of G on E × C and we set X := (E × C)/G with quotient map π : E × C → X. We have the following commutative diagram with two cartesian squares: Since G acts freely on E, the map π E induces isomorphisms on the fibers of E × C → E and X × E E → E and thus, as both maps are flat, the morphism π E is an isomorphism. The following lemma shows that the automorphism group scheme of X is controlled by the fibrations f E and f C .
Lemma 4.1. There is a unique action of Aut X on C and on E such that both f E : X → E and f C : X → C are Aut X -equivariant. In particular, there are exact sequences Proof. The Aut • X -action on X descends to both E and C by Blanchard's Lemma [BSU13, Proposition 4.2.1]. Since f E and f C are the only fibrations of X and E C P 1 , it is also clear that the action of the abstract group Aut(X) descends to E and C . By [Bri18, Lemma 2.20 (ii)], this is enough to prove that the whole Aut X -action descends uniquely to the two curves E and C .
With respect to the Aut X -actions of the previous paragraph, we have Aut X = Aut f C = Aut f E , hence the short exact sequences in the statement of the lemma are special cases of Sequence (1).
The idea for the proof of Theorem 1.1 is to use the isomorphism π E to lift group scheme actions from X to E × C. By Proposition 3.1, the automorphisms of X that come from E × C are induced by the normalizer N Aut E×C (G). Therefore, before proving Theorem 1.1, we study N Aut E×C (G). For the following lemma, note that there is a natural inclusion Aut E × Aut C → Aut E×C given by letting Aut E and Aut C act on the first and second factor, respectively. In particular, we can consider C Aut E (G) × C Aut C (α(G)) and N Aut E (G) × N Aut C (α(G)) as subgroup schemes of Aut E×C .
Lemma 4.2. The normalizer N Aut E×C (G) of G in Aut E×C satisfies the following properties:  (2), so it suffices to prove the statement for T = Spec k. Let h ∈ N Aut E×C (G)(k). Since h normalizes G, it descends to X by Proposition 3.1. The induced automorphism of X preserves both f C and f E , because they are the only fibrations of X and E has genus 1, while C P 1 . Since the projections E × C → E and E × C → C coincide with the Stein factorizations of f E • π and f C • π, respectively, both projections are preserved by h. Hence, h ∈ Aut(E) × Aut(C). An automorphism of this form normalizes the G-action on E × C if and only if it normalizes the G-action on both factors and the automorphisms of G induced by the two conjugations are identified via α. This proves Claim (3).
Claim (4) follows from the N Aut E×C (G)-equivariance of π, f E , and f C , since the two projections E × C → E and E × C → C are faithfully flat.
Recall that, by Proposition 3.1, the action of N Aut E×C (G) on E × C descends to X and we denote the corresponding homomorphism by π * : N Aut E×C (G) → Aut X . After these preparations, we are ready to prove the following refined version of Theorem 1.1.
There is a short exact sequence of group schemes

M is finite and étale, and M(k) is a subquotient of the groups
This always holds if X is bielliptic.
Proof. For Claim (1), we first show that the Aut X/C -action lifts to E × C. For this, choose a general point c ∈ C and let c ∈ C be its image in C , so that π restricted to E × {c} yields an identification of E with the fiber F of f C over c . Via this identification, the morphism (f E )| F : F → E is identified with the quotient map E → E/G = E . By Lemma 4.1, the action of Aut X/C on X descends to an action on E , and we can use the restriction homomorphism Aut X/C → Aut F and the identification of F with E to get a compatible action of Aut X/C on E. Using the isomorphism π E : E × C → X × E E, we thus obtain an action of Aut X/C on E × C that lifts the action of Aut X/C on X. Hence, Aut X/C is in the image of π * and it remains to describe its preimage. By Lemma 4.2 (4), a subgroup scheme H ⊆ N Aut E×C (G) ⊆ Aut E × Aut C maps to Aut X/C via π * if and only if it maps to Aut C/C under the second projection. To prove Claim (1), we have to show that such an H in fact centralizes G. By Lemma 4.2 (2) this holds for H • , so we have to prove that H(k) centralizes α(G). Observe that H(k) is mapped to Aut C/C (k) under the second projection and Aut C/C (k) = α(G)(k) by Proposition 3.3. This, and the fact that G is abelian, implies that . This holds if G is étale, for then Aut C/C is the constant group scheme associated to α(G) by Proposition 3.3.
For Claim (3), we only have to show that the Aut X/E -action lifts to E × C, because the description of the preimage of Aut X/E under π * works as in the proof of Claim (1). Since Aut X/E acts trivially on E , we can use the trivial action of Aut X/E on E to define an action of Aut X/E on X × E E lifting the action of Aut X/E on X. Using the isomorphism π E : E × C → X × E E, we thus obtain the desired lifting.
For Claim (4), we use that the G-action on E is free. By Proposition 3.1 this implies that Next, let us prove Claim (5). By Proposition 3.1, the image of (1) and Claim (3), this image coincides with the subgroup scheme of Aut X generated by the two normal subgroup schemes Aut X/C and Aut X/E , hence it is itself normal. In particular, the quotient M and the exact sequence in Claim (3) exist. It remains to describe M.
First, consider the exact sequence In particular, M is a subquotient of Aut E ,O and hence it is finite and étale.
Then, the restriction of π to {O} × C gives an identification of C with F such that the quotient map ϕ : In the following, we use this identification to write C instead of F and ϕ instead of (f C )| F . Since Aut E ,O fixes O , the action of H on X preserves C and the morphism ϕ is H-equivariant, since f C is Aut X -equivariant by Lemma 4.1. In other words, the H-action on C factors through Aut ϕ . By Claim (1), the kernel of this action is contained in (C Aut E (G) × C Aut C (α(G)))/G, hence M is a subquotient of Aut ϕ /C Aut C (α(G)). Now, it suffices to observe that Aut ϕ (k) = N Aut(C) (α(G)(k)), which follows from Proposition 3.3.
Finally, for Claim (6), the description of M follows immediately from Lemma 4.2 (3) and Proposition 3.1. By the previous paragraph, we can lift every element of M(k) to an automorphism g ∈ Aut(X) mapping under (f C ) * to the image of Aut ϕ (k) → Aut C (k). In particular, g lifts to an automorphism h of X × C C. Since π C : E × C → E × C C is birational, we obtain a birational automorphism h of E × C. Now, if X is bielliptic, then E × C is smooth, minimal, and non-ruled hence h extends to a biregular automorphism of E × C lifting g.
Remark 4.4. We remark that if G is not étale, then the group N Aut(C) (α(G)(k)) will usually be bigger than N Aut C (α(G))(k). Only later it will turn out that M is in fact a subquotient of the smaller group N Aut C (α(G))(k)/C Aut C (α(G))(k) in every case.
Remark 4.5. In the case-by-case analysis of quasi-bielliptic surfaces in Section 5.2, we will show that the assumptions of Theorem 4.3 (6) are also satisfied for all quasi-bielliptic surfaces, hence the description of M also holds for these surfaces.
Remark 4.6. It will follow from the calculations of Section 5 that Theorem 4.3 (2) holds for all bielliptic surfaces. Indeed, the situation where X is bielliptic and G is not étale only occurs if p = 2 and G = µ 2 ×Z/2Z and in this case explicit calculations show that C Aut C (α(G)) ∩ Aut C/C = α(G), hence the existence of an isomorphism Aut X/C C Aut E (G) follows from Theorem 4.3 (1). In particular, for bielliptic surfaces, we always have Aut X/C ∩ Aut X/E = π * (G × α(G)) G.
If X is quasi-bielliptic, then it is not true in general that Aut X/C C Aut E (G). Indeed, for example if p = 3 and G = α 3 , then C → C is purely inseparable of degree 3, hence Aut C/C = Aut C [F]. Calculations (see Section 5.2.1, Case (d)) show that C Aut C (α 3 ) • α 2 3 and C Aut E (G) E Z/3Z. Hence, by Theorem 4.3 (1), Aut X/C is non-reduced while C Aut E (G) is reduced, so they cannot be isomorphic. In particular, for quasi-bielliptic surfaces, Aut X/C ∩ Aut X/E can be larger than G.
We end this section with a description of (Aut • X ) red . We are thankful to the editors for sharing an observation that allowed us to avoid forward references to Section 5 in the proof of the following proposition.   (1) G 2 normalizes G 1 .
Proof. Note that if T is a k-scheme, g ∈ G 1 (T ), and t s ∈ G 2 (T ), then we have In particular, if s = g(s) for all t s ∈ G 2 (T ), then G 1 and G 2 commute, hence (3) ⇒ (2). The implication (2) ⇒ (1) is clear, hence it remains to prove (1) ⇒ (3): if G 2 normalizes G 1 , then Equation (3) shows that t s−g(s) (O T ) = O T for all T -schemes T and t s ∈ G 2 (T ). This is only possible if s = g(s), hence G 2 (T ) ⊆ D G 1 (T ).

Bielliptic surfaces
We use the notation of Section 4 and Lemma 5.1, but assume that D is smooth. In each of the cases p 2, 3, p = 3 and p = 2, we will recall the structure of the subgroup scheme Aut D,O ⊆ Aut D . Moreover, for every commutative subgroup H ⊆ Aut D,O , we list the fixed locus D H and, if Aut D,O is non-commutative, also the centralizer and normalizer of H in Lemma 5.2, Lemma 5.6, and Lemma 5.9. All of this is well-known and elementary to check, and we refer the reader to [Sil09, Section III.10 and Appendix A] for details. Together with Lemma 5.1, it will be straightforward to calculate the groups C Aut E (G) and C Aut C (α(G)) of Theorem 1.1 and produce Table 1. We will leave the details to the reader, but we will explain how the calculations work in Example 5.3. Using Theorem 4.3 (6), we calculate M in every case. The results of the calculations of this section are summarized in Table 1. To simplify notation, we define N := N Aut(C) (α(G)(k))/(C Aut C (α(G))(k)).  Table 4.

Characteristic
. This is Case b) in the first row of Table 1 and it seems to be missing from [BM90, Table 3.2], see also Remark 1.4.
Similarly, one can calculate the centralizers of G and α(G) for all seven possibilities of G. They are listed in Table 1. As for the group N , we have the following: Lemma 5.4. The group N is as in Table 5. Proof. If α(G) does not contain translations, then N Aut C (α(G)) = C Aut C (α(G)) by Lemma 5.1 and because Aut C,O is abelian. Hence, N is trivial in these cases. If G = (Z/2Z) 2 , then conjugation by N Aut C (α(G)) fixes the unique non-trivial 2-torsion point c in α(G). By Lemma 5.2 and Lemma 5.1, this implies |N | | 2. The non-trivial element of N is induced by a 4-torsion point c of C with 2c = c.
If G = (Z/3Z) 2 , then conjugation by N Aut C (α(G)) preserves the subgroup c ⊆ α(G) generated by a non-trivial 3-torsion point c in α(G). Thus, the action of N Aut C (α(G)) descends to C := C/ c . There, it maps to the normalizer in Aut C of a subgroup G ⊆ Aut C ,O of order 3, where O is the image of O. By Lemma 5.1 and Table 4, the normalizer of G is isomorphic to Z/3Z Z/6Z, where G sits inside the second factor. Thus, N is isomorphic to a subgroup of S 3 . One can check that the involution in Aut C,O and a 3-torsion point not contained in c induce non-trivial elements of N , hence N S 3 .
Finally, if G = Z/4Z × Z/2Z, then, again, conjugation by N Aut C (α(G)) fixes the unique non-trivial 2-torsion point c in α(G). In this case, however, the involution in α(G) ∩ Aut C,O is the unique element in α(G) which is divisible by 2, hence it is also fixed by N Aut C (α(G)). Thus, by Lemma 5.1, a translation can be in N Aut C (α(G)) only if it is a translation by a 2-torsion point. The non-trivial 2-torsion point that commutes with α(G) is already contained in α(G), hence N Z/2Z is generated by one of the other two non-trivial 2-torsion points. Proof. Assume that M is non-trivial. By Theorem 4.3 (5) and Table 5, this can only happen if G ∈ {(Z/2Z) 2 , (Z/3Z) 2 , Z/4Z × Z/2Z}. Assume G = (Z/2Z) 2 . By Theorem 4.3 (5) and Table 5, we have |M| | 2. If j(E) 1728, then Aut E /((f E ) * C Aut E (G)) has odd order, hence M = {1} by Theorem 4.3 (5). If j(E) = 1728, we use Theorem 4.3 (6): by our description of the centralizers and normalizers, both N Aut E (G)/C Aut E (G) and N Aut C (α(G))/C Aut C (α(G)) are isomorphic to Z/2Z and every non-trivial element of M(k) can be represented by h = (h E , h C ), where h E ∈ Aut E,O is of order 4 and h C is translation by a non-trivial 4-torsion point such that h 2 C ∈ α(G). By Lemma 5.1 and Table 4, we have α • ad h E = ad h C • α if and only if the fixed point of h E maps via α to the unique translation in α(G). This is Case (1).
Next, assume G = (Z/3Z) 2 . Let h = (h E , h C ) be an automorphism of E × C lifting a non-trivial element of M(k). By our description of C Aut E (G) and N , we may assume that h C is either the involution in Aut C,O or translation by a 3-torsion point c α(G), and that h E ∈ Aut E,O . If h E is an involution, then ad h E fixes only the identity in G, while ad h C has more fixed points on α(G). Hence, by Theorem 4.3 (6), h does not normalize the G-action on E × C in this case, a contradiction to Proposition 3.1. Thus, we may further assume that j(E) = 0 and h E has order 3. Then, we may assume that h C is translation by c . By Lemma 5.1 and Table 4, we have α • ad h E = ad h C • α if and only if the fixed points of h E on E map to translations in α(G). This is Case (2).
Finally, assume G = Z/4Z × Z/2Z. Assume M is non-trivial and, using Theorem 4.3 (6), let h = (h E , h C ) be an automorphism mapping to a non-trivial element in M(k). We may assume that h E ∈ Aut E,O is the involution and h C is a translation by one of the 2-torsion points not contained in α(G). Observe that ad h E maps elements of order 4 in G to their inverses while ad h C maps the automorphism σ of order 4 in This contradiction shows that M = {1} in this case.

Characteristic p = 3.
By Bombieri and Mumford [BM77,p.37], the groups G leading to bielliptic surfaces X = (E × C)/G are the six groups The translation subgroup of α(G) is trivial in the first four of these cases, and isomorphic to Z/2Z in the other two cases.  Table 6. As in characteristic 2, 3, it is straightforward to calculate the centralizers of G and α(G) and they are listed in Table 1.
Lemma 5.7. The group N is as in Table 7. Proof. If α(G) does not contain translations, then a translation in Aut C normalizes α(G) if and only if it centralizes α(G) by Lemma 5.1. Thus, in these cases, N can be read off from the last two columns of Table  6. The proof of the two remaining cases is the same as for Lemma 5.4.
Proof. By Theorem 4.3 (5) and Table 7 Table 8. As before, it is straightforward to calculate the centralizers of G and α(G) and they are listed in Table 1.
Lemma 5.10. The group N is as in Table 9. Proof. If α(G) does not contain translations, then a translation in Aut C normalizes α(G) if and only if it centralizes α(G) by Lemma 5.1. Thus, in these cases, N can be read off from the last two columns of Table 8. For G = (Z/3Z) 2 , the proof is the same as for Lemma 5.4. Finally, if G = µ 2 × Z/2Z, then N Aut(C) (α(G)(k)) is generated by α(G)(k) and the unique non-trivial 2-torsion point in C(k) by the same argument as in the proof of Lemma 5.1. Translation by this 2-torsion point commutes with α(G), hence N is trivial.
Proof. By Theorem 4.3 (5) and Table 9, we may assume G ∈ {(Z/3Z) 2 , Z/4Z}. The proof for G = (Z/3Z) 2 is the same as in Proposition 5.5 with the only difference that every non-trivial 3-torsion point in E is fixed by some automorphism of order 3, so we do not have an extra condition as in Proposition 5.5.
is of order 4 and not contained in α(G) and h E is the inversion involution on E. By Lemma 4.2 (3), h normalizes the G-action on E × C and, by Proposition 3.1, induces a non-trivial element of M. Hence, we have M Z/2Z.

Quasi-bielliptic surfaces
In the case of quasi-bielliptic surfaces, E is still smooth, so the group C Aut E (G)/E can be calculated using the results of the previous section. We will thus focus on the calculation of C Aut C (α(G))/α(G) and M. We identify the smooth locus of C with A 1 = Spec k[t] and use the description of automorphisms of A 1 coming from C given in [BM76, Proposition 6].

Characteristic p = 3.
By [BM76, Proposition 6] the T -valued automorphisms of A 1 coming from C are of the form Now, let us calculate C Aut C (α(G)) and M for the surfaces in Case (a),...,(e). To this end, we take a k-scheme T and arbitrary elements g ∈ α(G)(T ) as in the above list and h ∈ Aut C (T ) as in (4). One can check that the inverse of h is given by Thus This group scheme is isomorphic to µ 3 × S 3 . Therefore, we have C Aut C (α(G))/α(G) S 3 .
To calculate M, first note that |M| | 2, since E and E are ordinary, M is a subquotient of Aut E /((f E ) * C Aut E (G)) by Theorem 4.3 (5), and Aut • E ⊆ ((f E ) * C Aut E (G)). If M is non-trivial, then it can be represented by an automorphism g ∈ Aut(X) that induces the inversion involution on E . This involution can be lifted to E, hence g lifts to an automorphism of E × C. However, by the above calculations there is no element of Aut(C) that acts as an inversion on α(G). So, Theorem 4.3 (6) shows that M is trivial.
Next, we calculate M. Using Lemma 5.1 and Lemma 5.6, one can check that Choose any automorphism h E ∈ Aut E,O of order 4. Since α 3 ⊆ E is the kernel of Frobenius, it is preserved by h E . Moreover, by Lemma 5.1 and Lemma 5.6, the centralizer of α 3 in Aut E,O has order 3, so conjugation by h E induces an automorphism of α 3 of order 4. By the calculations of the first paragraph, we have a surjection N Aut C (α(G)) → Aut α(G) G m , hence we can find an Since α 3 is the identity component of α 3 × Z/2Z, we can use the results of (d) to deduce that h normalizes α(G) if and only if c = 0 and it centralizes α(G) if and only if additionally b 2 = 1. Thus, we get C Aut C (α(G)) α 3 × Z/2Z and the normalizer of α(G) is N Aut C (α(G)) α 3 G m . In particular, C Aut C (α(G))/α(G) = {1}.
Since the automorphism g generates the group α(G)(k), the calculation of the previous paragraph also shows that N Aut(C) (α(G)(k)) = G m (k). Thus, M is isomorphic to a subquotient of G m (k) by Theorem 4.3 (5) and, in particular, the order of M is prime to 3. By the same theorem, M is also a subquotient of Aut E /((f E ) * C Aut E (G)), which is isomorphic to Z/3Z Z/4Z since C Aut E (G) E in the current case. Hence, M is a subquotient of Z/4Z. Using the same construction as in (d), one can show that M Z/4Z.
Remark 5.13. In [BM76,p. 214], Bombieri and Mumford do not give restrictions on the parameter λ ∈ k in Case (f). However, all the α 2 -actions with λ 0 described by them are conjugate, so we may assume λ ∈ {0, 1}. For more details, we refer the reader to the discussion of Case (f) below.
Remark 5.14. To see that the group scheme in Case (h) is indeed M 2 , denote the transformation in Case (h) associated to z i with z 4 i = 0 by t z i . Observe that t z 1 • t z 2 = t z 1 +z 2 +λz 2 1 z 2 2 . So, if G = Spec k[z]/z 4 is the group scheme in Case (h), then its co-multiplication is given by Consider the supersingular elliptic curve E with affine Weierstrass equation y 2 +λy = x 3 and set z = x/y, w = 1/y, so that the equation becomes z 3 = w + λw 2 . Then, the 2-torsion subscheme M 2 of E is the subscheme given by z 4 = w 2 = 0, and thus w = z 3 . By [Sil09,p.120] the co-multiplication on k[z]/z 4 induced by the group structure on E is precisely the one described above. Hence, we have G = M 2 .
For later use, we note that by [Sil09, Appendix A, Proposition 1.2], the group of automorphisms of E preserving w = z = 0 is given by the substitutions x → b 2 x + c 2 , y → y + b 2 cx + d with b 3 = 1, c 4 + λc = 0 and d 2 + λd + c 6 = 0. In particular, they act on k[z]/z 4 as In particular, if we think of the substitutions in Case (h) above as defining a homomorphism of group schemes E ⊇ M 2 α → Aut C , then precomposing α with ad h E where h E ∈ Aut E,O is as described in the previous paragraph, then α • ad h E corresponds to M 2 acting on C as t → t + (b 2 a + bca 2 ) + bλa 2 t 2 + ba 2 t 4 . Now, we are prepared to calculate C Aut C (α C ) and M in Cases (a),...,(h). As in characteristic 3, we take a k-scheme T and arbitrary elements g ∈ α(G)(T ) as in the above list and h ∈ Aut C (T ) as in (5). One can check that the inverse of h is given by
(b) Since µ 2 is the identity component of the group scheme Z/3Z×µ 2 , it suffices to calculate the normalizer of Z/3Z × µ 2 in A 4 × µ 2 , which is equal to its centralizer and both are equal to Z/3Z × µ 2 . In particular, C Aut C (α(G))/α(G) = {1}. To see that M = {1}, one can use the same arguments as in Case (a) in characteristic 3 to show that the action of M lifts to E × C. Since N Aut C (α(G)) = C Aut C (α(G)), Theorem 4.3 (6) shows that M is trivial.
(e) Since µ 4 is the identity component of µ 4 × Z/2Z, we can use the computations of (d) to immediately conclude that centralizer and normalizer of α(G) are both equal to µ 4 × Z/2Z and thus C Aut C (α(G))/α(G) = {1}. Also, M = {1} follows by the same argument as in Case (b).
Finally, let us explain how to compute M in the case λ = 0. As in the case λ = 1, we have |M| | 3. Choose an element h E ∈ Aut E,O of order 3. Since α(G) = α 2 is the kernel of Frobenius on E, it is preserved by h E . By Lemma 5.1 and Lemma 5.9, conjugation by h E induces an automorphism of α 2 of order 3. On the other hand, the conjugation action of N Aut C (α(G)) on α 2 factors through N Aut C (α(G))/C Aut C (α(G)). By the calculations of the previous paragraph and since Aut α 2 G m , we can find an automorphism h C ∈ N Aut C (α(G)) of order 9 such that α • ad h E = ad h C • α. By Lemma 4.2 (3), h = (h E , h C ) normalizes the G-action on E × C. By Proposition 3.1, h descends to X and induces a non-trivial element of M. Hence, we have M Z/3Z. Putting this together with the conditions obtained in (f), we deduce that the normalizer of α(G) is the group scheme of maps t → bt + et 4 with e 2 = 0, which is isomorphic to α 2 G m . Moreover, we see that C Aut C (α(G)) = α(G).
Since the automorphism g generates the group α(G)(k), the calculation of the previous paragraph also shows that N Aut(C) (α(G)(k)) = G m (k). Thus, M is a subquotient of G m (k) by Theorem 4.3 (5) and, in particular, the order of M is prime to 2. By the same theorem, M is also a subquotient of Aut E /((f E ) * C Aut E (G)), which is isomorphic to Q 8 Z/3Z since C Aut E (G) E in the current case. Hence, M is a subquotient of Z/3Z. Using the same construction as in (f), one can show that M = Z/3Z.
This means that h normalizes α(G) if and only if it satisfies b 3 = 1, and In fact, since d 4 = 0, we can square (9) to deduce b = 1, and since λ 0, we get d 2 = 0. Now, h centralizes α(G) if and only if additionally d = c 4 + λc 2 . (10) Squaring (10), we obtain c 8 + λ 2 c 4 = 0. Hence, the centralizer C Aut C (α(G)) of α(G) is the group scheme of maps t → t + c + (c 4 + λc 2 )t 2 + et 4 with e 2 = 0 and c 8 + λ 2 c 4 = 0, which is isomorphic to (M 2 × α 2 ) Z/2Z, and the normalizer of α(G) is the group scheme of maps t → t + c + dt 2 + et 4 with d 2 = e 2 = 0, which is isomorphic to G a α 2 2 . In particular, we have C Aut C (α(G))/α(G) α 2 × Z/2Z. To calculate M, note first that M is a subquotient of Aut E /((f E ) * C Aut E (G)) A 4 by Theorem 4.3 (5). Since E → E is purely inseparable, we can lift the action of Aut X to E × C, where it normalizes the G-action. By the previous paragraph, we have N Aut C (α(G))/C Aut C (α(G)) G a and therefore M is isomorphic to a subquotient of (Z/2Z) 2 , again by Theorem 4.3 (5). We may assume that E is given by the equation y 2 + λy = x 3 . Choose c, d ∈ k such that c 3 = λ and d 2 + λd + λ 2 = 0 and let h E,c,d be the corresponding automorphism of E as in Remark 5.14. Then, by the calculations of the previous paragraph and by Remark 5.14, α T • ad h E,c,d = ad h C,c • α T , where h C,c is a substitution h C,c : t → t + c with c 4 + λc 2 = c. Therefore the automorphisms (h E,c,d , h C,c ) of E × C descend to X. The three different values of c yield three distinct non-trivial elements of M, so M (Z/2Z) 2 .
This finishes the calculation of the groups C Aut E (G)/E, C Aut C (α(G))/α(G), M, and thus also of the full automorphism group schemes for all bielliptic and quasi-bielliptic surfaces in all characteristics. The results are summarized in Table 1, Table 2 and Table 3.