ON MODULI SPACES OF POLARIZED ENRIQUES SURFACES ANDREAS LEOPOLD KNUTSEN

arXiv:2001.10769v2 [math.AG] 17 Mar 2020

ANDREAS LEOPOLD KNUTSEN

Abstract. We prove that, for anyg≥2, the ´etale double coverρg :Eg→Ebgfrom the moduli spaceEg of complex polarized genus gEnriques surfaces to the moduli space

b

Eg of numerically polarized genus gEnriques surfaces is disconnected precisely over irreducible components ofEbg parametrizing 2-divisible classes, answering a question of Gritsenko and Hulek [13]. We characterize all irreducible components of Eg in terms of a new invariant of line bundles on Enriques surfaces that generalizes theφ- invariant introduced by Cossec [8]. In particular, we get a one-to-one correspondence between the irreducible components ofEgand 11-tuples of integers satisfying particular conditions. This makes it possible, in principle, to list all irreducible components of Eg for eachg≥2.

1. Introduction

For any integer g≥2, letE_g (resp.,Eb_g) denote the moduli space of complex polarized (resp. numerically polarized) Enriques surfaces (S, L) (resp. (S,[L])) of(sectional) genus g, that is, such thatL² = 2g−2. (Thus, g is the arithmetic genus of all curves in the linear system |L|.) The moduli spacesEg exist as quasi-projective varieties by Viehweg’s theory, cf. [28, Thm. 1.13]. The moduli spacesEbgexist by results of Gritsenko and Hulek [13]; more precisely, for each orbithof the action of the orthogonal group in theEnriques lattice N:=U⊕E₈(−1) (see [1, Lemma VIII.15.1]), there is an irreducible moduli space M^a_En,h parametrizing isomorphism classes of numerically polarized Enriques surfaces (S,[L]) with [L] in the orbit h ⊂ N ≃ NumS. The space Ebg in our notation is thus the union of all M^a_En,h where h varies over all orbits with h² = 2g −2. It follows from [13, Prop. 4.1] that there is an ´etale double cover ρ_g :Eg → Ebg identifying (S, L) and (S, L+K_S). Note that in general the spaces Eg and Ebg have many irreducible components.

In this paper we answer the following fundamental questions:

(1) Given an irreducible component ofEbg, is its inverse image byρ_g is irreducible or not (cf. [13, Question 4.2])?

(2) How can one determineall the irreducible components of E_g?

Regarding the first question, for each irreducible component Eb^′ of Ebg either ρ⁻¹_g Eb^′ is irreducible or it consists of two disjoint components, according to whether (S, L) and (S, L+K_S) lie in the same component of Eg or not for (S,[L])∈Eb^′. We will prove:

Theorem 1.1. Let Eb^′ be an irreducible component of Ebg. Then ρ⁻¹_g (Eb^′) is reducible if and only if Eb^′ parametrizes pairs (S,[L]) such that[L]is 2-divisible in NumS.

Date: March 18, 2020.

1

We remark that a much weaker version of this theorem was obtained in [4, Cor. 1.5], with a completely different approach.

Regarding question (2) above, one can start by fixing another fundamental invariant in addition to the genus, namely the φ-invariant

(1) φ(L) := min

E·L|E²= 0, E >0 ∈Z₊,

introduced by Cossec [8], which has interesting geometrical interpretations, cf., e.g., [9, 18, 15, 26]. Then one may, as in [4], consider the moduli spaces Eg,φ and Ebg,φ

parametrizing pairs with L² = 2g−2 and φ(L) = φ, which in general still have many different irreducible components. Also recall that not all possible pairs (g, φ) occur; for instance it is known that φ² 6 2g−2 by [9, Cor. 2.7.1], and that there are no cases satisfying φ² <2g−2 < φ²+φ−2 by [17, Prop. 1.4], but a complete classification of all possible pairs (g, φ) is still missing.

Some irreducibility results have been known in low genus for a while; for instanceE3,2, E_4,2 and E_6,3 are irreducible, see [2], [10, §3] and [27]. In [4] all irreducible components of Eg,φwere determined forφ≤4 org≤20 and described in terms of decompositions of the line bundles they parametrize into effective, primitive isotropic decompositions (that is, into effective classes of square zero that are indivisible in NumS), cf. §2 below. As a sample, which was classically known, E5,2 has three irreducible components, denoted by E_5,2^(I), E_5,2^(II)+ and E_5,2^(II)− in [5], corresponding to the following decompositions of L into effective, primitive isotropic classes:

E_5,2^(I) L∼2E₁+E_1,2, E₁·E_1,2 = 1;

E_5,2^(II)+ L∼2E₁+ 2E₂, E₁·E₂ = 1;

E_5,2^(II)− L∼2E₁+ 2E₂+K_S, E₁·E₂ = 1

(where ’∼’ denotes linear equivalence). The components can also be distinguished by studying the projective models of its general members, which is classical, cf. [9, Prop.

4.1.2, Prop. 4.5.1, Thm. 4.6.3, Prop. 4.7.1, Thm. 4.7.1]. These cases also furnish a nice sample of Theorem 1.1: under ρ₅ :E5 →Eb5, the two componentsE_5,2^(II)+ and E_5,2^(II)− are identified, whereas E_5,2^(I) is mapped two-to-one onto one irreducible component of Eb5.

To explain our results and our answer to question (2), letLbe an effective line bundle on an Enriques surface satisfying L² >0. Set

(2) ε_L=

(0, ifL+K_S is not 2-divisible in PicS, 1, ifL+K_S is 2-divisible in PicS.

We will prove (cf. Theorem 5.7) that there exist unique nonnegative integers a_i, de- pending on L, satisfying

a₁≥ · · · ≥a₇ and a₉+a₁₀≥a₀ ≥a₉≥a₁₀ such that L can be written as¹

(3) L∼a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S,

1The reason for choosing to write (3) without the term “a8E8” is because then one automatically has E1·L≤ · · · ≤E10·L.

for an isotropic 10-sequence {E1, . . . , E₁₀} of effective divisors (cf. Definition 2.1) and an effective isotropic divisor E_9,10 ∼ ¹₃(E₁+· · ·+E₁₀)−E₉ −E₁₀ (cf. Lemma 2.2).

We call (3) a fundamental presentation of L and the coefficients a_i = a_i(L) and ε_L the fundamental coefficients of L (cf. Definitions 5.1 and 5.8). We will prove that the irreducible components of Eg are precisely the loci parametrizing pairs of genus g with the same fundamental coefficients (cf. Theorem 5.9).

As an alternative description of the irreducible components of Eg we will introduce a new function on the Enriques lattice N that generalizes the φ-function defined in (1). On the set of ordered 10-tuples of integers one has an order relation by setting (a₁, . . . , a₁₀)<(b₁, . . . , b₁₀) if either P10

i=1a_i <P10

i=1b_i or P10

i=1a_i =P10

i=1b_i and there is an n∈ {1, . . . ,9} such thata_i =b_i for alli∈ {1, . . . , n−1} and a_n< b_n.

Definition 1.2. Let L be an effective line bundle on an Enriques surface S such that L² >0. The φ-vector associated to L, denoted by φ(L) = (φ₁(L), . . . , φ₁₀(L))∈Z¹⁰

+, is the minimal value of all (E1·L, . . . , E₁₀·L) under the above mentioned order relation, where (E₁, . . . , E₁₀) runs over all isotropic 10-sequences satisfying 0 < E₁ ·L ≤ · · · ≤ E₁₀·L.

We say that an isotropic 10-sequence {E1, . . . , E₁₀} computes φ(L) if E_i·L =φ_i(L) for all i∈ {1, . . . ,10}.

Thus, theφ-vector function measures the “lowest” intersection numbers of line bundles with respect to entire isotropic 10-sequences, generalizing Cossec’s φ-function, since, as proved in the next result, φ₁(L) =φ(L). We will prove the following properties:

Theorem 1.3. Let φ(L) = (φ₁, . . . , φ₁₀) be the φ-vector associated to an effective line bundle L withL²>0 on an Enriques surface S. Then

(a) 0< φ₁≤ · · · ≤φ₁₀; (b) P10

i=1φ_i is divisible by 3;

(c) φ₁+· · ·+φ₇≥2 (φ₈+φ₉+φ₁₀);

(d) L² = ¹₉P10 i=1φ_i2

−P10 i=1φ²_i;

(e) φ₁, . . . , φ₈ are the eight lowest intersection numbers with Lachieved by numeri- cally distinct effective isotropic divisors on S; in particular φ₁ =φ(L);

(f ) L is numerically 2-divisible if and only ifφ_i is even for all i∈ {1, . . . ,10};

(g) the isotropic10-sequences computingφ(L)are, up to numerical equivalence, pre- cisely the ones appearing in fundamental presentations of L.

Conversely, for any Enriques surface S and for any 10-tuple of integers(φ1, . . . , φ₁₀) satisfying (a)-(c), there is an [L]∈NumS such that L² >0 and φ(L) = (φ₁, . . . , φ₁₀).

In particular, we remark that the set of values (g(L), φ(L)) occurring as genus and φ-vector of polarized Enriques surfaces (S, L) are completely determined, and this a posteriori determines all possible values of pairs (g(L), φ(L)) by property (e).

Our anwer to question (2) can now be summarized as:

Theorem 1.4. The irreducible components ofE_g are in one-to-one correspondence with the set of 11-tuples of integers (φ₁, . . . , φ₁₀, ε) satisfying

(i) 0< φ₁≤ · · · ≤φ₁₀, (ii) P10

i=1φ_i is divisible by 3,

(iii) φ₁+· · ·+φ₇≥2 (φ₈+φ₉+φ₁₀),

(iv) ε∈ {0,1}, with ε= 0 occurring if at least one φ_i is odd, (v) 2g−2 = ¹₉P10

i=1φ_i2

−P10 i=1φ²_i.

Precisely, the irreducible component of E_g corresponding to a specific (φ₁, . . . , φ₁₀, ε) parametrizes all pairs (S, L) with φ(L) = (φ₁, . . . , φ₁₀) and ε_L =ε, which are precisely the pairs with the following fundamental presentation, setting s:= ¹₃P10

j=1φ_j: L∼

X7 i=1

(φ₈−φ_i)E_i+ (s−2φ₈−φ₉)E₉+ (s−2φ₈−φ₁₀)E₁₀+ (s−3φ₈)E_9,10+εK_S, for an isotropic 10-sequence{E₁, . . . , E₁₀}, withE_9,10∼ ¹₃(E₁+· · ·+E₁₀)−E₉−E₁₀.

We propose the following notation for the irreducible components of Eg:

• Eg;φ1,...,φ10 corresponds to (φ₁, . . . , φ₁₀) and ǫ= 0, if at least one φ_i is odd.

• E_g;φ⁺ _1,...,φ10 corresponds to (φ1, . . . , φ₁₀) and ǫ= 0, if allφ_i are even.

• E_g;φ⁻ _1,...,φ10 corresponds to (φ₁, . . . , φ₁₀) and ǫ= 1, if allφ_i are even.

Thus, by property (f) in Theorem 1.3, the components E_g;φ1,...,φ10 parametrize pairs (S, L) with L not numerically 2-divisible, E_g;φ⁺ _1,...,φ10 parametrize pairs (S, L) with L 2-divisible in PicS, and E_g;φ⁻ _1,...,φ10 parametrize pairs (S, L) with L+K_S 2-divisible in PicS. In particular, by Theorem 1.1 we obtain that the map ρ_g : Eg → Ebg identifies E_g;φ⁺ _1,...,φ10 withE_g;φ⁻ _1,...,φ10 and is two-to-one on Eg;φ1,...,φ10. We propose to use the nota- tion Ebg;φ1,...,φ10 for the images by ρ_g of E_g;φ^± _1,...,φ10 and Eg;φ1,...,φ10. As an application of our results we obtain a specific component ofEbg dominating all others:

Proposition 1.5. The irreducible componentEb477;26,27,28,29,30,31,32,33,33,34 dominates ev- ery irreducible component of Ebg for allg≥2.

It would be interesting to know whether this component is the same as the dominant- ing component found by Gritsenko and Hulek [13], cf. Question 5.10.

The paper is organized as follows. In §2 we recall and improve some results from [4]

on effective decompositions of line bundles on Enriques surfaces into isotropic divisors.

In§3 we study reducible surfaces that are a transversal union of a rational surface and a surface birational to the second symmetric product of an elliptic curve, considered first in [6]. Our main result is Theorem 3.7, which says that projective models in P^g−1 of those reducible surfaces by line bundles of degree 2g−2 (as described in Proposition 3.4) are smoothable to Enriques surfaces of degree 2g−2; more precisely, they represent smooth points in the Hilbert scheme of such surfaces. This result is the Enriques version of [7, Thm. 1] forK3 surfaces and we believe that it is of independent interest and that it will have further applications; indeed, although degenerations of Enriques surfaces have been widely studied (cf., e.g., [19, 22, 20, 25]), a concrete result such as Theorem 3.7 has not been available yet, cf. Remark 5.11. A second key result is Proposition 3.9 stating that under certain conditions (which will turn out to be equivalent to numerical non-2-divisibility) the projective models by both a line bundle and its adjoint lie in the same irreducible component of the Hilbert scheme.

In §4 we prove Theorem 1.1 by degeneration, using the results from §3. Finally, in

§5 we introduce the notions of fundamental presentation andφ-vector mentioned above and prove Theorems 1.3 and 1.4, as well as Proposition 1.5.

Acknowledgements. I thank Klaus Hulek for interesting correspondence about [13] and useful comments on a preliminary draft of this paper, Christian Liedtke, Frank Gounelas and Marian Aprodu for asking inspiring questions during a talk on mine on [4] at TU M¨unchen, and Ciro Ciliberto, Thomas Dedieu, Concettina Galati, Michael Hoff, Yeon- grak Kim, Frank-Olaf Schreyer and Alessandro Verra for useful conversations on the topic. I acknowledge support from the Trond Mohn Foundation (project “Pure Mathe- matics in Norway”) and grant 261756 of the Research Council of Norway.

2. Isotropic 10-sequences and simple isotropic decompositions

Let us explain some notions from [4]. Any effective line bundleL withL²>0 on an Enriques surface may be written as (cf. [4, Cor. 4.6])

(4) L∼a₁E₁+· · ·+a_nE_n+εK_S,

such that all E_i are effective, non–zero, isotropic (i.e., E_i² = 0) and primitive (i.e., indivisible in NumS), alla_i are positive integers, ε∈ {0,1},n610 and

(5)







either n6= 9, E_i·E_j = 1 for alli6=j,

or n6= 10,E₁·E₂ = 2 and E_i·E_j = 1 for all other indicesi6=j, or E₁·E₂ =E₁·E₃ = 2 and E_i·E_j = 1 for all other indicesi6=j,

up to reordering indices. We call this asimple isotropic decomposition, cf. [4, Def. 4.1].

We say that two polarized Enriques surfaces (S, L) and (S^′, L^′) in Eg admit the same simple decomposition type (cf. [4, Def. 4.13]) if one has simple isotropic decompositions (6) L∼a₁E₁+· · ·+a_nE_n+εK_S and L^′ ∼a₁E₁^′+· · ·+a_nE_n^′ +εK_S′, with ε∈ {0,1}

such that E_i·E_j =E_i^′·E^′_j for alli6=j. Similarly, we say that two numerically polarized Enriques surfaces (S,[L]) and (S,[L^′]) inEb_g admit the same simple decomposition type if (6) holds modulo K_S and K_S′.

We note that ε = 1 is only needed in (6) when all a_is are even, otherwise one may substitute anyE_i having odd coefficient withE_i+K_S. Also note that a given line bundle may admit decompositions of different types, cf. [4, Rmk. 4.14], but nevertheless the property of admitting the same decomposition type is an equivalence relation onEg and Ebg, cf. [4, Prop. 4.15].

We recall the following from [9, p. 122]:

Definition 2.1. An isotropic 10-sequence on an Enriques surface S is a sequence of isotropic effective divisors {E1, . . . , E₁₀} such thatE_i·E_j = 1 for i6=j.

It is well-known that any Enriques surface contains such sequences. Note that we, contrary to [9], require the divisors to be effective, which can always be arranged by changing signs. We will also make use of the following result, cf. [4, Lemma 3.4(a)], [8, Lemma 1.6.2(i)] or [9, Cor. 2.5.5]:

Lemma 2.2. Let {E1, . . . , E₁₀} be an isotropic10-sequence. Then there exists a divisor D on S such that D² = 10, φ(D) = 3 and 3D∼E₁+· · ·+E₁₀. Furthermore, for any i6=j, we have

(7) D∼E_i+E_j +E_i,j, withE_i,j effective isotropic, E_i·E_i,j =E_j·E_i,j = 2,

and E_k·E_i,j = 1 for k6=i, j. Moreover, E_i,j·E_k,l =

(1, if{i, j} ∩ {k, l} 6=∅, 2, if{i, j} ∩ {k, l}=∅.

In particular, for i, j, k distinct, we have E_i+E_j +E_i,j ∼E_i+E_k+E_i,k, so that

(8) E_j+E_i,j ∼E_k+E_i,k.

The next result yields a “canonical” way of writing simple isotropic decompositions:

Proposition 2.3. Let Lbe any effective line bundle on an Enriques surfaceS such that L² > 0. Then there is an isotropic 10-sequence {E1, . . . , E₁₀} (depending on L) such that there is a simple isotropic decomposition

(9) L∼a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S,

where E_9,10∼ ¹₃(E₁+· · ·+E₁₀)−E₉−E₁₀ anda₀, a₁, . . . , a₁₀ are nonnegative integers satisfying

a₁≥ · · · ≥a₇, and (10)

a₉+a₁₀≥a₀ ≥a₉≥a₁₀. (11)

Proof. By [4, Cor. 4.7] combined with [4, Rem. 4.11], after renaming indices, there is an isotropic 10-sequence {E1, . . . , E₁₀}and nonnegative integers a₀, a₁, . . . , a₁₀such that

L∼a₁E₁+· · ·+a₁₀E₁₀+a₀E_9,10+ε_LK_S

with a₀ = 0 or a₈ = 0. We have left to prove that we can make sure the coefficients satisfy (10) and (11).

Assume a₀ = 0. By renaming indices we may assume a₁ ≥ · · · ≥a₁₀, so that (10) is satisfied. If a₈= 0, then a₉=a₁₀= 0 and (11) is satisfied. If a₈ >0, then, using (7):

L ∼ a₁E₁+· · ·+a₁₀E₁₀+ε_LK_S

∼ a₈(E₁+· · ·+E₁₀) + (a₁−a₈)E₁+· · ·+ (a₁₀−a₈)E₁₀+ε_LK_S

∼ 3a8(E9+E₁₀+E_9,10) + (a1−a₈)E1+· · ·+ (a10−a₈)E10+ε_LK_S

∼ X7

i=1

(ai−a₈)Ei+ (2a8+a₉)E9+ (2a8+a₁₀)E10+ 3a8E_9,10+ε_LK_S. Setting a^′_i :=a_i−a₈ fori∈ {1, . . . ,7}, a^′_i := 2a8+a_i fori∈ {9,10} and a^′₀ := 3a8, we see that a^′₁ ≥ · · · ≥a^′₇ and

a^′₉+a^′₁₀= 4a₈+a₉+a₁₀≥3a₈ =a^′₀≥2a₈+a₉ =a^′₉≥2a₈+a₁₀=a^′₁₀, so the coefficients a^′_i satisfy (10) and (11).

Assume henceforth thata₀ >0, so thata₈= 0. By renaming indices we may assume a₁ ≥ · · · ≥a₇, so that (10) is satisfied, anda₉ ≥a₁₀. We see that (11) is satisfied unless a₀ < a₉ or a₉+a₁₀< a₀. We treat these two cases separately.

Case a₀ < a₉. We set b:= min{a9−a₀, a₇}. Recalling (7) and (8), we have L ∼ a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S

∼ b(E₁+· · ·+E₁₀) + X7 i=1

(a_i−b)E_i−bE₈+a₀(E₉+E_9,10) +(a₉−a₀−b)E₉+ (a₁₀−b)E₁₀+ε_LK_S

∼ 3b(E₈+E₁₀+E_8,10) + X7 i=1

(a_i−b)E_i−bE₈+a₀(E₈+E_8,10) +(a₉−a₀−b)E₉+ (a₁₀−b)E₁₀+ε_LK_S

∼ X7 i=1

(a_i−b)E_i+ (a₉−a₀−b)E₉

+(a₀+ 2b)E₈+ (a₁₀+ 2b)E₁₀+ (a₀+ 3b)E_8,10+ε_LK_S. By definition ofb, we see that at least one amongE₇ andE₉ appears with coefficients 0. Hence, in the expression P7

i=1(a_i −b)E_i + (a₉ −a₀ −b)E₉ there are at most 7 nonzero terms, and we may rearrange them so that the coefficients appear in decreasing order, that is, so that (10) is satisfied. Moreover, we see that (a₀+ 2b) + (a₁₀+ 2b) = a₀ +a₁₀+ 4b ≥ a₀+ 3b. If a₀ ≥ a₁₀, we see that also a₀ + 3b ≥ a₀+ 2b ≥ a₁₀+ 2b, whence (11) is satisfied. If insteada₀< a₁₀, we setE₉^′ :=E₁₀,a^′₉ :=a₁₀+ 2b,E₁₀^′ :=E₈, a^′₁₀:=a₀+ 2b,E_9,10^′ :=E_8,10and a^′₀ :=a₀+ 3b; then we rewrite

(a₀+ 2b)E₈+ (a₁₀+ 2b)E₁₀+ (a₀+ 3b)E_8,10=a^′₉E₉^′ +a^′₁₀E^′₁₀+a^′₀E_9,10^′ ,

with a^′₉ +a^′₁₀ ≥ a^′₀, a^′₉ > a^′₁₀ and a^′₀ ≥ a^′₁₀. If a^′₀ ≥ a^′₉ (which happens if and only if a₀+b≥a₁₀), we are done. If a^′₀ < a^′₉, we repeat the process from the start, which this time will give the desired decomposition, sincea^′₀ ≥a^′₁₀.

Case a₉+a₁₀< a₀. Recalling (8), we have

L ∼ a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S

∼ X7

i=1

a_iE_i+a₉(E9+E_9,10) +a₁₀(E10+E_9,10) + (a0−a₉−a₁₀)E9,10+ε_LK_S

∼ X7

i=1

a_iE_i+a₉(E₈+E_8,10) +a₁₀(E₈+E_8,9) + (a₀−a₉−a₁₀)E_9,10+ε_LK_S

∼ X7

i=1

a_iE_i+ (a9+a₁₀)E8+a₉E_8,10+a₁₀E_8,9+ (a0−a₉−a₁₀)E9,10+ε_LK_S.

We note that {E1, . . . , E₇, E_8,10, E_8,9, E_9,10} is an isotropic 10-sequence, and E₈ ∼

1

3(E₁+· · ·+E₇+E_8,10+E_8,9+E_9,10)−E_8,10−E_8,9 (cf. Lemma 2.2). Thus, setting E₈^′ :=E_9,10,E₉^′ :=E_8,10,E₁₀^′ :=E_8,9 and E_9,10^′ :=E₈, and b:= min{a7, a₀−a₉−a₁₀},

we may rewrite as L ∼

X7 i=1

a_iE_i+ (a₀−a₉−a₁₀)E₈^′ +a₉E₉^′ +a₁₀E₁₀^′ + (a₉+a₁₀)E_9,10^′ +ε_LK_S

∼ b(E₁+· · ·+E₇+E₈^′ +E₉^′ +E₁₀^′ ) + X7

i=1

(a_i−b)E_i+ (a₀−a₉−a₁₀−b)E₈^′ +(a₉−b)E₉^′ + (a₁₀−b)E₁₀^′ + (a₉+a₁₀)E_9,10^′ +ε_LK_S

∼ 3b(E₉^′ +E₁₀^′ +E_9,10^′ ) + X7 i=1

(a_i−b)E_i+ (a₀−a₉−a₁₀−b)E₈^′ +(a₉−b)E₉^′ + (a₁₀−b)E₁₀^′ + (a₉+a₁₀)E_9,10^′ +ε_LK_S

∼ X7

i=1

(a_i−b)E_i+ (a₀−a₉−a₁₀−b)E₈^′

+(a₉+ 2b)E₉^′ + (a₁₀+ 2b)E^′₁₀+ (3b+a₉+a₁₀)E_9,10^′ +ε_LK_S. By definition of b, we see that at least one of E₇ and E₈^′ appears with coefficient 0.

Hence, in the expression P7

i=1(a_i −b)E_i+ (a₀ −a₉−a₁₀−b)E₈^′ there are at most 7 nonzero terms, and we may rearrange them so that the coefficients appear in decreasing order, that is, so that (10) is satisfied. We also see that the coefficients in front of E₉^′, E₁₀^′ andE_9,10^′ satisfy the conditions (11).

Finally, the fact that (9) is a simple isotropic decomposition is easy to check.

Remark 2.4. Recalling (2), we have by [4, Lemma 4.8] that ε_L=

(0, if some a_i is odd 0 or 1, if all a_i are even, with the a_is as in Proposition 2.3.

3. Flat limits of Enriques surfaces

Let E be a smooth elliptic curve. Denote by ⊕ (and ⊖) the group operation on E and by e₀ the neutral element. Let R := Sym²(E) and π :R → E be the (Albanese) projection map sending x+y to x⊕y. We denote the fiber of π over a pointe∈E by

f_e:=π⁻¹(e) ={x+y∈Sym²(E)|x⊕y=e(equivalently,x+y∼e+e₀)}, which is theP¹defined by the linear system|e+e0|. We denote the algebraic equivalence class of the fibers by f.

For each e∈E, we define the curve s_e (called D_e in [3]) as the image of the section E → R mapping x to e+ (x⊖e). We let s denote the algebraic equivalence class of these sections, which are the ones with minimal self-intersection, namely 1, cf. [3]. We note for later use that for x6=y we have

(12) s_x∩s_y ={x+y}.

We also note that we have

(13) K_R∼ −2s_e0 +f_e0.

For any of the three nonzero 2-torsion points η of E the map E → R defined by mapping eto e+ (e⊕η) realizesE as an unramified double cover of its image curve

T_η :={e+ (e⊕η)|e∈E}.

We have

(14) T_η ∼ −KR+fη−f_e0 ∼2se0−2fe0 +f_η, by [3, (2.10)]. In particular,

(15) T_η 6∼ −KR and 2T_η ∼ −2KR.

We henceforth fixηand setT :=T_η. For later use we gather a couple of lemmas here:

Lemma 3.1. We have hⁱ(TR(−T)) = 0for all i.

Proof. We first note that (14), Serre duality and Riemann-Roch imply that (16) hⁱ(OR(−KR−T)) =hⁱ(OR(−T)) = 0 for all i.

Then the lemma follows from the sequence

0−→ OR(−KR−T)−→ TR(−T)−→ OR(−T)−→0,

which is the dual of the sequence of relative differentials of π tensored byO_R(−T).

Lemma 3.2. We have

(17) sx+fy ∼sy+fx for all x, y∈E.

In particular,

(18) s_x+f_η ∼s_x⊕η+f_e0 for all x∈E.

Proof. Restricting tos_e0 and using the isomorphismπ|se0 :s_e0 →E, we have (sx−sy)|se0 ∼π|^∗_se

0(x−y)∼(fx−fy)|se0,

and (17) follows from the special case of [3, Prop. (2.11)] stating that two line bundles on R with the same restriction to a section are isomorphic. Setting y :=x⊕η in (17) and using the group law (x⊕y∼x+y−e₀) on E, we obtain (18).

Embed T as a cubic in P². Consider distinct pointsx₁, . . . , x₉∈T such that (19) x₁+· · ·+x₉ ∈ |N_{T /R}⊗ N_{T /}P²|.

Let σ_R : Re → R be the blow up at x₁, x₉ and σ_P : Pe → P := P² be the blow up at x₂, . . . , x₈. We will always assume the pointsx_i to be sufficiently general, so that (20) Re is Del Pezzo (whence contains no (−2)-curves), and

(21) x₁ and x₉ lie on distinct fibers of π:R→E.

We denote by ℓ on Pe the pullback of a general line on P² and by e_i the exceptional divisor over x_i,i∈ {2, . . . ,8}. By abuse of notation we denote bysand f the pullbacks of sections and fibers on Reand bye_i the exceptional divisor over x_i,i∈ {1,9}. We still denote byπ the composed mapRe→R→E. By abuse of notation we denote byT the strict transform of T in both Re and Pe. We have (cf. (14)-(15))

T ∼2s_e0−2f_e0 +f_η−e₁−e₉ 6∼ −K_Re, 2T ∼ −2K_Re on R,e (22)

T ∼3ℓ−e2− · · · −e8 ∼ −K_Pe on P .e (23)

Define X := Re ∪T Pe as the surface obtained by gluing Re and Pe along T. The first cotangent sheaf T_X¹ :=ext¹_O

X(Ω_X,OX) ofX (cf. [24, Cor. 1.1.11] or [11,§2]), satisfies (24) T_X¹ ≃ N_{T /Re}⊗ N_{T /Pe} ≃ OT.

by [11, Prop. 2.3], because of (19). Thus, X issemi-stable, cf. [11, Def. (1.13)] and [12, (0.4)]. We will denote by D the family of surfacesX obtained in this way. It is easy to see that D is irreducible of dimension 9.

We recall that a Cartier divisor, or a line bundle, L ∈PicX, is a pair (L_Re, L_Pe) such that L_Re ∈PicR,e L_Pe ∈PicPe and L_Re|T ≃L_Pe|T. We remark that since T is numerically equivalent to the anticanonical divisor on both Re andPe, we have

(25) L²=L²_e

R+L²_e

P = 2pa(L_Re)−2 + 2pa(L_Pe)−2 + 2d, d:=L_Re·T =L_Pe·T, so is even. As OT(K_Re+T)≃ω_T ≃ OT, the canonical divisor K_X is represented by (26) K_X = (K_Re+T,0) = (f_η−f_e0,0) in PicRe×PicP .e

In particular, by (22)-(23) we have

(27) K_X 6= 0 and 2KX = 0.

By [12, (3.3)] the surface X also carries a Cartier divisor ξ represented by the pair (28) ξ= (T,−T)∼(2se0 −2fe0+fη−e1−e9,−3ℓ+e2+· · ·+e8) in PicRe×PicP .e

We now find special effective primitive isotropic divisors onX that will be used later.

For j∈ {1,9}, the linear system|ℓ⊗ Jxj|onPe is a pencil inducing ag¹₂ onT, which has, by Riemann-Hurwitz, two members that also belong to a fiber of π_|T :T →E. In other words, there are two fibers f_αj and f_α′

j of π:Re→E such that (fαj∪ej)∩T ∈ |ℓ||T and (f_α^′

j∪ej)∈ |ℓ||T, j ∈ {1,9}.

One easily verifies thatα^′_j =α_j⊕η. In particular, there are two uniquely defined points α_j and α_j⊕η on E such that the pairs

(f_αj+e_j, ℓ) and (f_αj⊕η+e_j, ℓ), j ∈ {1,9},

define Cartier divisors on X. It is easy to check, using the group law on E and (26), that one is obtained from the other by tensoring with K_X. We define

E₉ := (f_α9 +e₉, ℓ) and E₉+K_X = (f_α9⊕η+e₉, ℓ).

Similarly, for each i∈ {2, . . . ,8} there are two uniquely defined pointsα_i and α_i⊕η on E such that the pairs

E_i:= (f_αi, ℓ−e_i) and E_i+K_X = (f_αi⊕η, ℓ−e_i), for i∈ {2, . . . ,8}, define Cartier divisors on X.

Considering each point x_i ∈ T, for i ∈ {1,9} as a point in R = Sym²(E) we may write x_i =p_i+ (pi⊕η). There are two sections inRpassing through x_i, namelyspi and s_pi⊕η, cf. (12). Thus, on Re the pairs

(29) (s_pi−e_i,0) and (s_pi⊕η−e_i,0), i∈ {1,9}

define Cartier divisors on X. Using (18) and (26) one checks that one is obtained from the other by tensoring with K_X. We define

(30) E₁ := (s_p1 −e₁,0) and E₁+K_X = (s_p1⊕η−e₁,0).

We haveπ(x_i) =p_i⊕pi⊕ηandf_π(xi)is the unique fibre ofπ:Re→E passing through x_i. Its second intersection point with T is x^′_i := (pi⊕η₁) + (pi⊕η₂), where η₁ and η₂ are the two nonzero 2-torsion points of E in addition to η. The g¹₂ cut out on T by the pencil of lines through x^′_i has, again by Riemann-Hurwitz as above, two elements that are fibers of π_|T :T →E, say the fibers over β_i ∈E and β_i⊕η∈E. It follows that for i∈ {1,9} there are two uniquely defined pointsβ_i and β_i⊕η onE such that the pairs (31) (f_π(xi)+f_βi−ei, ℓ) and (f_π(xi)+f_βi⊕η−ei, ℓ), i∈ {1,9},

define Cartier divisors on X. It is again easy to check that one is obtained from the other by tensoring with K_X. We define

(32) E₁₀:= (f_π(x1)+f_β1 −e₁, ℓ) and E₁₀+K_X = (f_π(x1)+f_β1⊕η −e₁, ℓ).

Note that we have E_i² = 0 for all iand E_i·E_j = 1 for alli6=j.

Lemma 3.3. We have E₁+· · ·+E₁₀+ξ∼3(E₉+E₁₀+E_9,10), with (cf. (29)) (33) E_9,10= (sp9 −e9,0) and E_9,10+K_X = (sp9⊕η−e9,0).

Proof. By (17) we haves_e0 ∼s_p9+f_e0 −f_p9 and s_p1 ∼s_p9 +f_p1−f_p9. Hence one finds (E₁+· · ·+E₁₀+ξ)−3(E₉+E₁₀)∼(3(s_p9 −e₉) +A,0)

with

(34) A:=f_p1+f_η+f_α2 +· · ·+f_α8−3f_p9−2f_α9 −2f_π(x1)−2f_β1 ≡0.

Now E_9,10 and E_9,10+K_X are defined up to making a choice between p₉ and p₉⊕η.

Interchanging them has the effect of adding K_X, which has the effect of twisting A by K_Re+T, cf. (26). Hence, up to interchangingp₉ and p₉⊕η, the lemma follows ifA∼0 or A+ (K_Re+T)∼0. We claim that the latter follows in turn from

(35) A|T ∼0.

Indeed, since h⁰(A−T) =χ(A−T) =χ(A+K_Re) = 0, we see from 0 ^//O_Re(A−T) ^//O_Re(A) ^//OT(A) ^//0

that if (35) holds butA6∼0, thenh¹(A−T) =h²(A−T)>0. Buth²(A−T) =h⁰(K_Re+ T−A), whence it follows thatA∼K_Re+T. Thus, one hasA+(K_Re+T)∼2(K_Re+T)∼0 by (22). It thus suffices to prove (35).

To prove (35), we remark that by construction we have (36) OT(f_αi)∼ OT(ℓ)(−xi) for i∈ {1, . . . ,9}

and OT(fαi+f_π(xi)+f_βi)∼ OT(2ℓ), whence using (36):

(37) OT(f_π(x1)+f_β1)∼ OT(ℓ)(x1).

From (17) and the fact thatOT(spi)∼ OT(xi) fori∈ {1,9}by construction, we deduce (38) O_T(f_pi)∼ O_T(f_η−s_η)(x_i), i∈ {1,9}.

Inserting (36)-(38) into (34) we find A|T ∼ OT(2s_η −f_η)(3ℓ)(−x1− · · · −x₉). Since OT(3ℓ−x₂− · · · −x₈) ⁽²³⁾∼ N_{T /Pe} ⁽²⁴⁾∼ N_{T /}^∨_Re⁽²²⁾∼ OT(−2se0+ 2f_e0 −f_η)(x₁+x₉)

(17)∼ OT(−2sη +f_η)(x₁+x₉),

we get A|T ∼0, as desired.

Thus, we may similarly to (7) define (39) E_i,j := 1

3(E₁+· · ·+E₁₀+ξ)−E_i−E_j for each i6=j.

Hence (8) holds on X. In particular, we remark for later that

(40) E_1,9 ∼(f_π(x9)+f_β9 −e₉, ℓ) and E_1,9+K_X ∼(f_π(x9)+f_β9⊕η−e₉, ℓ) (cf. (31), possibly after interchanging β₉ and β₉⊕η).

Proposition 3.4. Let X =Re∪_T Pe be a member of D and

(41) L∼a₀E_9,10+a₁E₁+a₂E₂+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀, with all a_i ≥0 satisfying

a₉+a₁₀≥a₀ ≥max{a₉, a₁₀}, (42)

a₀+ min{a1, a₂}>0, (43)

min{a1, a₂} ≥a₃ ≥ · · · ≥a₇, (44)

a₀+ min{a1, a₂}+a₃+· · ·+a₇+a₉+a₁₀≥3.

(45)

Then the complete linear system |L| defines a morphism ϕ_L : X → P^g−1 that is an isomorphism onto its image except for the contraction of (−1)-curves on Re and Pe and it contracts at least one such curve, namely e₈ on Pe. Its image is X :=R∪_T P, where R and P are the images of Re and P, respectively, and intersect transversally and onlye along T :=ϕ_L(T)≃T.

Furthermore,

(i) H^j(X,O_X) = 0 for j = 1,2;

(ii) K_X is Cartier and represented by(K_R+T ,0), whence K_X 6= 0 and2K_X = 0.

Proof. SetL_Re:=L|_Re and L_Pe:=L|_Pe. Denoting numerical equivalence by ’≡’ , we have E_9,10≡(s−e₉,0), E₉≡(f+e₉, ℓ), E₁₀≡(2f−e₁, ℓ),

E₁ ≡(s−e₁,0), E_i≡(f, ℓ−e_i), i∈ {2, . . . ,7}.

Claim 3.4.1. L_Re is nef, L²_e

R ≥ 5 and L_Re ·T ≥ 5. In particular, L_Re +T is nef with (L_Re+T)² ≥13.

Proof of claim. We have an effective decomposition

(46) L_Re≡a₀(s−e₉) +a₁(s−e₁) +a₁₀(f−e₁) + (a₂+· · ·+a₇+a₉+a₁₀)f+a₉e₉. The only negative components are e₉ and f−e₁. Since e₉ ·L_Re = a₀ −a₉ ≥ 0 and (f−e₁)·L_Re =a₀−a₁₀≥0 (using (42)), we see thatL_Re is nef. From (46) we find (47) L²_e

R= 2(a₀+a₁)(a₂+· · ·+a₇+a₉+a₁₀)+a₀(2a₁+a₉+a₁₀)+a₉(a₀−a₉)+a₁₀(a₀−a₁₀).

One now readily checks that conditions (42) and (45) implyL²_e

R≥5, as desired.

Finally, recalling (22), we have T²=−2 and T·L_Re = 2(a₂+· · ·+a₇) + 3(a₉+a₁₀).

Again (42) and (45) yield that T ·L_Re ≥ 5. Since T is irreducible with T² = −2, it follows that L_Re+T is nef with (L_Re+T)² ≥13.

Claim 3.4.2. L_Re and L_Pe are globally generated and each defines a morphism that is an isomorphism except for the contraction of (−1)-curves; moreover, L_Pe·e₈ = 0.

Proof of claim. We first considerL_Re. By Claim 3.4.1 we have thatL_Re−K_Re≡L_Re+T is big and nef. Therefore, if |L_Re| fails to separate a scheme Z of length ≤ 2, then by Reider’s theorem [23, Thm. 1] there exists an effective divisorF containingZ such that (48) F·(L_Re+T), F²

∈ {(0,−1),(1,0),(0,−2),(1,−1),(2,0)},

with the latter three occuring only if degZ = 2. We will show that the only possibility is the fourth one, with F·T = 1 andF ·L_Re = 0.

To prove this, note that by (21) the only negative curves in fibers f of Re are the (−1)-curves e₁,e₉,f−e₁,f−e₉, which have intersections

e₁·(L_Re+T) =a₁+a₁₀+ 1≥1, e₉·(L_Re+T) =a₀−a₉+ 1≥1, (f−e₁)·(L_Re+T) =a₀−a₁₀+ 1≥1, (f−e₉)·(L_Re+T) =a₁+a₉+ 1≥1 (using (42)). Moreover, we have T ·(L_Re+T)≥3 by the above, and

f·(L_Re+T) = a₀+a₁+ 2≥3,

(s−e₉)·(L_Re+T) = a₁+· · ·+a₇+ 2a₉+ 2a₁₀≥3, (s−e1)·(L_Re+T) = a₀+a₂+· · ·+a₇+a₉+a₁₀≥3,

using (42), (43) and (45). All other curves D 6≡ e₁,e₉,f−e₁,f−e₉,f,s−e₁,s−e₉, T intersect f, (s−e₉) and (s−e₁) positively andT nonnegatively, whence

D·(L_Re+T)≥D·L_Re ≥a₀+a₁+· · ·+a₇+a₉+a₁₀≥3,

using again (45). This proves that the only possibility in (48) is F² = −1, F ·T =

−F·K_Re = 1 andF ·L_Re= 0. In particular, it shows that |L_Re|defines a morphism that is an embedding except for the contraction of (−1)-curves, as desired.

We then consider L_Pe. We have

(49) L_Pe ∼a₂(ℓ−e₂) +· · ·+a₇(ℓ−e₇) +a₉ℓ+a₁₀ℓ.

In particular L_Pe·e₈= 0 and|L_Pe|defines a birational morphism that is an isomorphism outside finitely many contracted (−1)-curves. This follows e.g. from (20) and [14, Prop.

3.10], as −L_Pe·K_Pe =L_Pe·T =L_Re·T ≥5 by Claim 3.4.1.

We note thatT is nef onPe. AsL_Pe∼(L_Pe+T)+K_Pe, we haveh^j(L_Pe) = 0, j= 1,2. As L_Re(−T)≡L_Re+K_Re we have h^j(L_Re(−T)) = 0, j= 1,2.From the short exact sequence (50) 0 ^//L_Re(−T) ^//L ^//L_Pe ^//0