On extra-special Enriques surfaces

We refine Cossec and Dolgachev’s classification of extra-special Enriques surfaces, providing a complete and concise proof.


INTRODUCTION
Throughout the paper, X will denote an Enriques surface defined over an algebraically closed field of arbitrary characteristic p.
A half-fiber on X is a divisor F such that 2F is a fiber of a genus one fibration.A c-sequence on X is a sequence of half-fibers (F 1 , . . ., F c ) such that F i .F j = 1 − δ ij .As observed by Enriques himself for classical Enriques surfaces, and later extended to non-classical Enriques surfaces by Bombieri and Mumford [1,Theorem 3], every Enriques surface admits a 1-sequence.The maximal length of a c-sequence is 10.
It is a natural question to ask whether every c-sequence can be extended to a c ′ -sequence with c ′ > c.In this context, extra-special Enriques surfaces play a fundamental role.Definition 1.1.An Enriques surface X is called • extra-special of type Ẽ8 if the dual graph of all (−2)-curves on X is • extra-special of type D8 if the dual graph of all (−2)-curves on X is • extra-special of type Ẽ7 if the dual graph of all (−2)-curves on X is An Enriques surface is called extra-special if it is extra-special of type Ẽ8 , D8 , or Ẽ7 .
The classification of Enriques surfaces on which every 1-sequence can be extended to a 2sequence is well-known and is contained in the following theorem due to Cossec and Dolgachev.
The main result of this paper is the following theorem, whose proof is obtained by combining Theorem 3.3 and Corollary 4.3.
Theorem 1.3.If X is not extra-special, then every 2-sequence on X extends to a 3-sequence.
Remark 1.4.In their first book about Enriques surfaces, Cossec and Dolgachev proved that X admits a 3-sequence if X is not extra-special [3,Theorem 3.5.1].In a similar vein, Cossec proved that if p = 2, then every 1-sequence on X extends to a 3-sequence [2, Theorem 3.5].Note that our result is strictly stronger than both of these results, since it asserts the extendability of any given 2-sequence in all characteristics.Big parts of the proof of the result of Cossec are left to the 'patient reader'.Moreover, the proof of the result of Cossec-Dolgachev occupies 32 pages and is described as 'lengthy' in the new book on Enriques surfaces, with 'no guarantee that it is correct' (see [7,Remark 6.1.14and Theorem 6.2.6]).The main ingredient that allows us to give a substantially shorter proof of a stronger result is the observation that in a non-extendable 2-sequence one of the genus one fibrations is special with a fiber of type II * (Theorem 3.3), and that Enriques surfaces with such a fibration can be easily classified in all characteristics (Theorem 4.2).
Extra-special Enriques surfaces do exist.Salomonsson [12] gave equations for all extra-special surfaces of type Ẽ8 and Ẽ7 .An alternative construction of these surfaces as well as examples of extra-special surfaces of type D8 were given by Katsura, Kondō and Martin [9, § §10, 11, 12].
There do exist non-extendable 1 or 2-sequences on extra-special Enriques surfaces [7, Proposition 6.2.7].More precisely, on extra-special surfaces of type Ẽ8 there exists only one genus one fibration, so the only 1-sequence is non-extendable.On extra-special surfaces of type D8 there are exactly three genus one fibrations, forming two distinct 2-sequences, both non-extendable.Finally, on extra-special surfaces of type Ẽ7 there are exactly two genus one fibrations, forming a non-extendable 2-sequence.
Remark 1.5.Every extra-special surface admits a genus one fibration with a double fiber of additive type, and an extra-special surface of type Ẽ7 even admits a quasi-elliptic fibration with two double fibers [7,Proposition 6.2.7].Consequently (cf.Lemma 2.2), extra-special Enriques surfaces can exist only in characteristic 2 and they are either classical or supersingular.Additionally, extra-special surfaces of type Ẽ7 can only be classical.
We infer an immediate corollary from Remark 1.5.

Corollary 1.6. If one of the following conditions holds:
(1) p = 2, (2) p = 2 and X is ordinary, (3) p = 2 and X is not extra-special, then every c-sequence on X with c ≤ 2 extends to a 3-sequence.
Let us describe one of the geometric consequences of the extendability of 1-and 2-sequences to 3-sequences.By applying [4, Theorem 3.5.1 and the following discussion] and [5,Section 7.8.1] to a 3-sequence that extends a given 2-sequence (F 1 , F 2 ), we obtain the following strong version of the Enriques-Artin Theorem for non-extra-special, classical Enriques surfaces.
Corollary 1.7.If X is a classical Enriques surface which is not extra-special, then X is the minimal resolution of a sextic S ⊆ P 3 given by an equation of the form , where L is linear and Q is a quadratic form.Moreover, for each half-fiber F 1 (resp.2-sequence (F 1 , F 2 )) on X, there is a sextic model as above such that F 1 (resp.(F 1 , F 2 )) maps to a line (resp.a pair of lines) in the non-normal locus of S.
Extending 3-sequences to 4-sequences is a much more difficult problem, even in characteristic p = 2.The classification of non-extendable 3-sequences will be the subject of a forthcoming paper.

PRELIMINARIES
In §2.1, we collect some basic definitions and facts about genus one fibrations on Enriques surfaces.In §2.2, we introduce the notion of isotropic sequences, generalizing the notion of csequence given in the introduction.

Genus one fibrations.
Recall that an Enriques surface is a smooth and proper surface Xof Kodaira dimension 0 with b 2 (X) = 10 over an algebraically closed field of arbitrary characteristic p.An Enriques surface X is called classical if its canonical divisor is not linearly equivalent to 0. A non-classical Enriques surface X (which can only exist if p = 2) is called ordinary (or singular or a µ 2 -surface) if the absolute Frobenius morphism is bijective on H 1 (X, O X ), and supersingular (or an α 2 -surface) otherwise.
A genus one fibration on X is a fibration f : X → P 1 such that the generic fiber X η is a regular genus one curve.A genus one fibration is called elliptic if X η is smooth and quasi-elliptic otherwise.We will use Kodaira's notation for singular fibers.Singular fibers of type I n are called of multiplicative type, while all others are called of additive type.Any genus one fibration on X admits at least one half-fiber, that is a curve F of arithmetic genus one such that 2F is a fiber of the fibration.In particular, the degree of a multisection of a genus one fibration f on X is divisible by 2 and if f admits a (−2)-curve as a bisection, both f and the bisection are called special.We call a divisor primitive if its class in Num(X) spans a primitive sublattice.
Lemma 2.1 ([4, Corollary 2.2.9]).An effective divisor D on X is a half-fiber of a genus one fibration on X if and only if D is nef, primitive, and D 2 = 0.
We recall here the behaviour of genus one fibrations on Enriques surfaces.Lemma 2.2 ([4, Theorem 4.10.3]).Let f : X → P 1 be a genus one fibration on X.
• If p = 2, then f is an elliptic fibration with two half-fibers, and each of them is either non-singular or singular of multiplicative type.
• If p = 2 and X is classical, then f is an elliptic or quasi-elliptic fibration with two halffibers, and each of them is either an ordinary elliptic curve or a singular curve of additive type.• If p = 2 and X is ordinary, then f is an elliptic fibration with one half-fiber, which is either a non-singular ordinary elliptic curve or a singular curve of multiplicative type.• If p = 2 and X is supersingular, then f is an elliptic or quasi-elliptic fibration with one half-fiber, which is either a supersingular elliptic curve or a singular curve of additive type.
The symbol ∼ denotes linear equivalence, and W nod X denotes the Weyl group generated by reflections along classes of (−2)-curves on X.
).If D is an effective divisor on X with D 2 ≥ 0, then there exist non-negative integers a i and (−2)-curves R i such that where D ′ is the unique nef divisor in the W nod X -orbit of D. In particular, D 2 = D ′2 .The following bound is an immediate consequence of the Shioda-Tate formula.
Lemma 2.4.The number of irreducible curves contained in s fibers of any genus one fibration on X is at most 8 + s.

Isotropic sequences.
A c-degenerate (canonical isotropic) n-sequence on X is an n-tuple of the form where the F i are half-fibers of genus one fibrations on X and the R i,j are (−2)-curves satisfying the conditions (1) ).If c = n, we simply call the above sequence a c-sequence.
We say that a c-degenerate n-sequence extends to a c ′ -degenerate n ′ -sequence if the former is contained in the latter, disregarding the ordering.The following fundamental theorem is due to Cossec and holds also in positive characteristic (cf.[7, Proposition 6.1.7]).

NON-EXTENDABLE 2-SEQUENCES
In this section, we investigate the extendability of 2-sequences and the properties of non-extendable 2-sequences.Proposition 3.1.Let (F 1 , F 2 ) be a 2-sequence on an Enriques surface X.If there are two simple fibers G 1 ∈ |2F 1 | and G 2 ∈ |2F 2 | with a common component, then there is a half-fiber F 3 extending (F 1 , F 2 ) to a 3-sequence.
Proof.By [4, Proposition 2.6.1], the linear system Since C ′′ is nef, C ′′ .F i = 0 for at most one i and then C ′′ .F j = 1 for j = i.This implies that C ′′ is primitive, and therefore C ′′ is a half-fiber of some genus one fibration on X by Lemma 2.1.
Thus, let us exclude the possibility that C ′′ .F 1 = 0 (the case is a 2-sequence on X that does not extend to a 3-sequence, then one of the half-fibers of |2F 1 | or |2F 2 | is reducible and has at least 4 components. Proof.By Theorem 2.5, we may assume that there exist (−2)-curves R 1,1 , . . ., R 1,m , R 2,1 , . . ., R 2,n with m+n = 8 and such that . By Proposition 3.1, G 1 and G 2 cannot both be simple.Therefore, R is a component of a reducible half-fiber, which also contains all R i,j ′ with j ′ = 1.
Since either m ≥ 4 or n ≥ 4, one of the reducible half-fibers has at least 4 components.
The bound of Corollary 3.2 on the number of components is not sharp, but it is enough to prove the following theorem, which is the key result of this paper.
The fiber G 2 is necessarily simple because of Lemma 2.6.Let Λ be the sublattice of Num(X) generated by the components of G 2 and R 2,1 and let Λ ′ = Λ[F 2 ] be the sublattice of Num(X) obtained from Λ by adjoining then so are all the R 1,j .This shows that Λ has rank 10, because it contains all R i,j , the class of F 1 and the class of G 2 ≡ 2F 2 .Since Λ ′ contains all the 10 divisors of the 10-sequence and since these divisors generate a lattice of rank 10 and discriminant 9, Λ ′ has index 1 or 3 in Num(X).
By [2, Lemma 1.6.2],there exists a vector e ∈ Num(X) with e 2 = 0, e.F 1 = e.F 2 = 1, e.R i,j = 0 for all (i, j) except for e.R 2,n−1 = 1, and such that e and the components of the 10-sequence generate Num(X).(In the notation of [2], we can choose e = e 9,10 , since n ≥ 3.) Note that it suffices to show that e is contained in the sublattice Λ ′ .Indeed, if this holds, then Λ ′ = Num(X) and therefore Λ has index at most 2 in Num(X).Then, consider the basis of Λ given by G 2 , R 2,1 and all the components of G 2 except a simple one.The intersection matrix of Λ with respect to this basis is where L is a root lattice of rank 8 depending on the type of G 2 .More precisely, L is the root lattice associated to the Dynkin diagram obtained by removing a simple component from G 2 .By reducing along the first row and first column, we obtain that det(Λ) = −4 det(L).Since Λ has index at most 2 in the unimodular lattice Num(X), we have |det(Λ)| ≤ 4, and hence det(L) = 1.This forces L to be isometric to the lattice E 8 , and in turn G 2 to be of type II * .
It remains to show that e ∈ Λ ′ .Let D be an effective lift of e to Pic(X).Write D ∼ D ′ + i a i R i as in Lemma 2.3.Since D ′2 = D 2 = 0 and D ′ .F i ≤ 1 with equality for at least one i, D ′ is nef and primitive, hence equal to a half-fiber by Lemma 2.1.If D ′ is not a half-fiber of |2F 1 | or |2F 2 |, then (F 1 , F 2 , D ′ ) is a 3-sequence, contradicting our assumption that (F 1 , F 2 ) does not extend to a 3-sequence.Therefore, we can suppose that D ′ is a half-fiber of |2F k | for some k ∈ {1, 2}.Take j ∈ {1, 2} with j = k.Then, D ′ .F k = 0 and D ′ .F j = 1, so ( i a i R i ) 2 = (D − D ′ ) 2 = −2 and ( i a i R i ).F j = (D−D ′ ).F j = 0. Thus, i a i R i is a connected configuration of (−2)-curves contained in a single fiber of |2F j |.But ( i a i R i ).R 2,n−1 = (D − D ′ ).R 2,n−1 = D.R 2,n−1 = 1, since R 2,n−1 .F 1 = R 2,n−1 .F 2 = 0, hence i a i R i is contained in the fiber of |2F j | containing R 2,n−1 .If j = 1, this fiber is F 1 , whereas if j = 2 this fiber is G 2 .In both cases, we obtain that e ∈ Λ ′ .
on X that does not extend to a 3-sequence, then either |2F 1 | or |2F 2 | is a special genus one fibration with a fiber of type II * .Proof.By Corollary 3.2 we can suppose that F 1 is reducible and has at least 4 components.Let R 2,1 be the component ofF 1 meeting F 2 .Since R 2,1 is a simple component of F 1 , we can find two more components R 2,2 , R 2,3 of F 1 forming a chain with R 2,1 .Applying Theorem 2.5, we extend the 2-degenerate 5-sequence (F 1 , F 2 , . . ., F 2