Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

We prove that infinitely many irreducible components of the moduli space of polarized Enriques surfaces are unirational (resp. uniruled), characterizing them in terms of decompositions of the polarization as an effective sum of isotropic classes. In particular, this applies to components of arbitrarily large genus g and ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-invariant of the polarization.


Introduction
Let E denote the smooth 10-dimensional moduli space parametrizing smooth Enriques surfaces over C. A polarized (resp.numerically polarized) Enriques surface is a pair made of an Enriques surface together with an ample linear (resp.numerical) equivalence class on it.For integers g > 1 and φ > 0, let E g,φ (resp., E g,φ ) denote the moduli space of polarized (resp.numerically polarized) Enriques surfaces (S, H) (resp.(S, [H])) such that H 2 = 2g − 2 and φ(H) = φ, where (1) φ(H) Thus g is the arithmetic genus of all curves in the linear system |H|.There is an étale double cover ρ : E g,φ → E g,φ mapping the two pairs (S, H) and (S, H + K S ) to (S, [H]).We refer to §2 for more details.
The space E is irreducible and rational, as has been shown by Kondō [13], and the forgetful maps E g,φ → E are étale.Nevertheless the spaces E g,φ and E g,φ are in general reducible, and it is an open problem to determine their Kodaira dimensions (cf.[8]).
It is known that E 3,2 is irreducible and rational (cf.[3]), that E 4,2 is irreducible and rational (this is the classical case of Enriques sextics, cf.[8, §3]) and that E 6,3 is irreducible and unirational (cf.[16]), and it has been conjectured that the moduli spaces of polarized Enriques surfaces are all unirational (or at least, of negative Kodaira dimension), see [8, §4].This was disproved by Gritsenko and Hulek in the recent paper [9], where the existence of infinitely many irreducible components of general type of the moduli space of numerically polarized Enriques surfaces is established.On the other hand, they show that all components of E g,φ have negative Kodaira dimension for g 17.
In the present paper we improve these results.Our interest lies in the moduli spaces of polarized Enriques surfaces E g,φ : We give a description of their irreducible components in terms of decompositions of the polarization as an effective sum of isotropic classes and prove their unirationality (resp.uniruledness) in infinitely many cases (for arbitrarily large g and φ).
To explain our results, we introduce some notions.Any effective line bundle H with H 2 0 on an Enriques surface may be written as (cf.Corollary 4.6 below) (2) (where '≡' denotes numerical equivalence), such that all E i are effective, non-zero, isotropic (i.e., E 2 i = 0) and primitive (i.e., indivisible in Num(S)), all a i are positive integers, n 10 and (3) for all other indices i = j, up to reordering indices.We call this a simple isotropic decomposition, cf.Definition 4.1.
We say that two polarized (respectively, numerically polarized) Enriques surfaces (S, H) and (S ′ , H ′ ) in E g,φ (resp., (S, [H]) and (S, [H ′ ]) in E g,φ ) admit the same simple decomposition type (cf.Definition 4.13) if one has simple isotropic decompositions (4) We call n the length of the decomposition (type).If, possibly after reordering indices, there exists r n such that a 1 = • • • = a r and E i • E j = 1 for all 1 i r and 1 j n, i = j, then we say that (S, H) and (S ′ , H ′ ) admit the same simple r-symmetric decomposition type.
We note that in (4) the case ε = 1 is only needed when all a i 's are even, otherwise one may substitute any E i having odd coefficient with E i + K S .Also note that a given line bundle may admit decompositions of different types, cf.Remark 4.14, but nevertheless the property of admitting the same decomposition type is an equivalence relation on E g,φ (and E g,φ ), cf.Proposition 4. 15.
The main result of this paper is the identification of many unirational (resp.uniruled) irreducible components of E g,φ .These are components of E g,φ parametrizing precisely those pairs (S, H) with H admitting the same simple decomposition type.
Theorem 1.1.The locus of pairs (S, H) ∈ E g,φ admitting the same simple decomposition type of length n 4 is an irreducible, unirational component of E g,φ .
The locus of pairs (S, H) ∈ E g,φ admitting the same simple decomposition type of length 5 is an irreducible component of E g,φ , which is unirational if all E i • E j = 1 for all i = j, and uniruled otherwise.
We stress that there are line bundles satisfying the assumptions of these statements with arbitrarily large g and φ.Moreover, there are decomposition types of all possible lengths 1 n 10 to which these results apply.For small values of g or φ they actually provide all irreducible components of E g,φ , as stated in the following corollaries: Corollary 1.3.When φ 4 the different irreducible components of E g,φ are precisely the loci parametrizing pairs (S, H) admitting the same simple decomposition type and they are all unirational.
Corollary 1.4.When g 20 the different irreducible components of E g,φ are precisely the loci parametrizing pairs (S, H) admitting the same simple decomposition type.Moreover, they are all unirational, except possibly E 16,5 and E 17,5 , which are in any event irreducible and uniruled.
As a further example, our results can also be used to describe the irreducible components of E g,φ for the highest values of φ with respect to g, cf.Corollary 5. 8.
We note that the proofs of our results do not rely on the construction of E g,φ in [9].By the above results the (equivalence class of) simple decomposition type seems to be the correct invariant to distinguish all the irreducible components of the moduli space of polarized Enriques surfaces.This is indeed true for numerical polarizations: we prove in Proposition 4.16 that the various irreducible components of E g,φ are precisely the loci of pairs admitting the same simple decomposition type.We do not know if the same holds for linear polarizations in full generality, cf.Question 4.17 1 .
As another application we answer [9, Question 4.2] about the irreducibility of the preimage by ρ : E g,φ → E g,φ of a component of E g,φ under the assumptions of Theorems 1.1 and 1.2: Corollary 1.5.Let C ⊂ E g,φ be an irreducible component parametrizing classes admitting the same simple decomposition type of length 5 or being 6-symmetric.Then ρ −1 (ρ(C)) is reducible if and only if C parametrizes pairs (S, H) such that H is 2-divisible in Num(S).
Note that a class is 2-divisible in Num(S) if and only if all coefficients in any simple isotropic decomposition are even, cf.Lemma 4.8.
It is an interesting question whether this last corollary holds in general, that is, without any assumption on the decomposition types 2 .
An interesting feature of our approach via simple isotropic decompositions is that it enables one to write down efficiently the complete list of all possible decompositions within a given numerical range.(Note that the datum of such a decomposition prescribes of course the genus, but also the φ-invariant, cf.Remark 4.12).As an illustration of our methods we catalogue all the irreducible components of all the moduli spaces E g,φ with g 30 in an appendix; for almost all of them we are able to determine the number of corresponding irreducible components of E g,φ , as well as unirationality or uniruledness.We do not make use of this list in the present paper, but this information is needed in our paper [4] on moduli of curves on Enriques surfaces.Moreover, the approach to moduli spaces of polarized Enriques surfaces via simple isotropic decompositions also plays a central role in our subsequent work [5] on Severi varieties on Enriques surfaces.
Our proofs of Theorems 1.1 and 1.2 are based on the fact that a general Enriques surface has a model in P 3 as an Enriques sextic, i.e., a sextic surface singular along the six edges of a tetrahedron; such a model corresponds to the datum of an isotropic without any assumptions on the decomposition types.The proof of [10,Thm. 4.2] is different from our proof of Corollary 1.5, but still relies on our results on simple isotropic decompositions in §4.
sequence (E 1 , E 2 , E 3 ) with E i • E j = 1 for i = j, the E i 's corresponding to three edges of some face of the tetrahedron.The idea is then to exhibit various irreducible and rational (resp.uniruled) families F of elliptic curves in P 3 with prescribed intersection numbers with the edges of some fixed tetrahedron, such that a general Enriques sextic singular along this particular tetrahedron contains a member of F. One thus gets incidence varieties that are irreducible and rational (resp.uniruled) and dominate the corresponding components of the moduli space of polarized Enriques surfaces.This whole construction, which is very geometric in nature, is done in §5, where the proofs of our theorems and corollaries stated in this introduction are given; in particular, Theorems 1.1 and 1.2 are consequences of Propositions 5.5 and 5.6.Before this, in §4, we prove the existence of simple isotropic decompositions together with related technical results needed in §5.

Background results on moduli spaces
Let E, E g,φ and E g,φ be as in the introduction.The moduli space E is an open subset of a 10-dimensional orthogonal modular variety, cf.[2, VIII §19-21].The moduli spaces E g,φ of polarized Enriques surfaces exist as quasi-projective varieties by [15,Thm. 1.13].
We have the forgetful map whose differential at a point (S, H) is the linear map 6) is an isomorphism, hence E g,φ is smooth and the map ( 5) is an étale cover.The moduli spaces E g,φ exist by [9].More precisely, fixing an orbit h of the action of the orthogonal group in the Enriques lattice U ⊕ E 8 (−1), in [9] the authors construct (irreducible) moduli spaces M a En,h parametrizing isomorphism classes of numerically polarized Enriques surfaces (S, [H]) with [H] in the orbit h ⊂ U ⊕ E 8 (−1) ≃ Num(S) (see [2,Lemma VIII.15.1]).The spaces M a En,h are open subsets of suitable orthogonal modular varieties.Then our space E g,φ is the union of all M a En,h where h varies over all orbits with h 2 = 2g − 2 and φ(h) = φ, cf.(1).It follows by [9,Prop. 4.1] that there is an étale double cover ρ : E g,φ → E g,φ mapping (S, H) and (S, H + K S ) to (S, [H]).

Background results on line bundles on Enriques surfaces
Any irreducible curve C on an Enriques surface S satisfies C 2 −2, with equality if and only if C is smooth and rational.An Enriques surface containing such a curve is called nodal, otherwise it is called unnodal.On an unnodal Enriques surface, all divisors are nef and all divisors with positive self-intersection are ample.It is well-known that the general Enriques surface is unnodal, cf.references in [6, p. 577].
Recall that a divisor E is said to be isotropic if E 2 = 0 and E ≡ 0. By Riemann-Roch, either E or −E is effective.It is said to be primitive if it is non-divisible in Num(S).On an unnodal surface, any effective primitive isotropic divisor E is represented by an irreducible curve of arithmetic genus one.
Let H be an effective line bundle with H 2 > 0 and φ(H) as in (1).One has We recall the following from [7, p. 122]: Definition 3.2.An isotropic r-sequence on an Enriques surface S is a sequence of isotropic effective divisors {E 1 , . . ., E r } such that It is well-known that any Enriques surface contains such sequences for every r 10 and that there are no such sequences with r > 10 (cf.[7, p. 175]); moreover, by [7, Cor.2.5.6],we have Proposition 3.3.Any isotropic r-sequence with r = 9 can be extended to a 10-sequence.
We will also make use of the following result: , for an isotropic 10-sequence {E 1 , . . ., E 10 } consisting precisely of all isotropic divisors computing φ(D) up to numerical equivalence.Moreover, if F is a divisor satisfying F 2 = 0 and F • D = 4, then F ≡ E i,j for some i = j, where E i,j is defined by (9).

Simple, isotropic decompositions
One of the aims of this section is to prove the existence of simple isotropic decompositions stated in the introduction (see Corollary 4.6) and prove that the isotropic divisors occurring in such decompositions can always be extended to an isotropic 10-sequence plus one of the divisors E i,j occurring in Lemma 3.4 (see Corollary 4.7).The latter will be needed in the proof of our main results, see the comment right after Proposition 5.6.We will also deduce several results on simple isotropic decompositions, like for instance the fact that 2-divisibility can be read off any isotropic decomposition (see Lemma 4.8) and the fact that the property of admitting the same decomposition type as defined in the introduction is an equivalence relation on E g,φ and E g,φ (see Proposition 4.15).
We start by recalling the following from the introduction: Definition 4.1.Let H be an effective line bundle H with H 2 0 on an Enriques surface S.
• An expression , where all a i are positive integers, n 10 and all E i are primitive, effective, isotropic divisors is called a simple isotropic decomposition if (3) is satisfied, up to reordering indices.
• An expression , satisfying the same conditions will also be called a simple isotropic decomposition.
• The number n is the length of the decomposition.
• The decomposition is r-symmetric if, possibly after reordering indices, there exists r n such that a 1 = • • • = a r and E i • E j = 1 for all 1 i r and 1 j n, i = j (equivalently, there is a set of r isotropic divisors occurring in the decomposition with the same coefficient and each having intersection 1 with the remaining isotropic divisors in the decomposition).
Example 4.2.Consider, in the notation of Lemma 3.4, the simple isotropic decomposition H ≡ E 1,2 + E 1 + 2E 2 + E 3 + E 4 .This is 2-symmetric but not 3-symmetric.Indeed, the set {E 3 , E 4 } has the property that each member occurs in the decomposition with coefficient 1 and intersects the remaining isotropic divisors in the decomposition in one point.There is no larger such set, since E 1 • E 1,2 = 2 and E 2 occurs with coefficient 2.
We recall [12, This lemma guarantees the existence of an effective decomposition satisfying almost all the conditions of a simple isotropic decomposition; indeed, what is missing, cf.(3), is the additional requirement that n = 9 in case (i) and that n = 10 in case (ii).(9) for some i = j.
Note that since any simple isotropic set of n elements contains members of an isotropic (n − 1)-sequence, any simple isotropic set contains at most 11 elements (cf.[7, p. 179]).If it contains 11 elements, then they necessarily satisfy (iii) in Lemma 4.3, possibly after permuting indices.It will follow from Proposition 4.5 right below (cf. the footnote) that simple isotropic sets of 11 elements are precisely the maximal simple isotropic sets.Also note that by [6, Rem.p. 584] any maximal simple isotropic set generates Num(S).
The following is a key result, which generalizes Proposition 3.3.
Proposition 4.5.Any simple isotropic set J can be extended to a maximal simple isotropic set. 3 Furthermore, if J = {F 1 , . . ., F n } with n 9, F 1 • F 2 = 2 and F i • F j = 1 for {i, j} = {1, 2}, then J can be extended to maximal simple isotropic sets such that either of F 1 or F 2 equals E i,j .
We postpone the proof until the very end of the section to discuss some consequences.The first one yields the existence of simple, isotropic decompositions: Corollary 4.6.Any effective line bundle H with H 2 0 on an Enriques surface has a simple isotropic decomposition.
Proof.By Lemma 4.3, we are done unless possibly if we end up in case (i) with n = 9 or in case (ii) with n = 10.We treat these two cases separately and prove that in both cases we will find a different isotropic decomposition of H satisfying condition (iii) of Lemma 4.3, thus being a simple isotropic decomposition as desired.
Assume first that where all a i −m 0 for 1 i 8, at least one being zero.Thus, the latter decomposition satisfies condition (iii) of Lemma 4.3.
In case (b), then, by Proposition 4.5, the set {E 1 , . . ., E 9 } can be extended to a maximal simple isotropic set {E 1 , . . ., E 9 , E 10 , E 11 }.This set contains an isotropic 10sequence by definition, which cannot contain {E 1 , . . ., E 9 } by assumption.Possibly after reordering indices, we may thus assume that {E 2 , . . ., E 9 , E 10 , E 11 } is an isotropic 10sequence, where all a i −m 0 for 2 i 9, at least one being zero.Thus, the latter decomposition satisfies condition (iii) of Lemma 4. 3 where all a i − m 0 for i 3, at least one being zero.Thus, the latter decomposition satisfies condition (iii) of Lemma 4.3.
The next result yields a "canonical" way of writing any simple isotropic decomposition in Pic(S), which will be central in our proofs.
More precisely, given any simple isotropic decomposition we may find an expression (10) such that each F i occurs in it (up to numerical equivalence) with coefficient b i (and the remaining coefficients in (10) are zero).Moreover, if F i •F j = 2 for only one pair of indices i, j, then we may find isotropic 10-sequences satisfying either of the conditions 10.By Proposition 4.5 there is a maximal simple isotropic set {E 1 , . . ., E 10 , E 1,2 } containing the set {F 1 , . . ., F n }.Moreover, if F i • F j = 2 for only one pair of indices i, j, then n 9 by definition of a simple isotropic decomposition, so that we can make sure, still by Proposition 4.5, that either of F i or F j equals E 1,2 .Thus, we may write , where each F i occurs (up to numerical equivalence) with coefficient b i , the other coefficients are zero and where This gives an expression of H as in (10) with ε ∈ {0, 1}.Note that a i = 0 for at least one i.We may furthermore by symmetry assume that We claim that either a 0 = 0 or a 10 = 0. Indeed, if a 0 > 0 and a 10 > 0, then we must have a 2 = 0.If a 1 = 0, then the length of the decomposition is 9 and this contradicts the first line in (3).If a 1 > 0, then the length of the decomposition is 10 and this contradicts the second line in (3).This proves our claim.Moreover, if a 0 = 0, the first line in (3) implies that #{i | i ∈ {1, . . ., 10}, a i > 0} = 9.Thus, ( 11) is satisfied.
If not all a i are even, we can, possibly after replacing one E i having odd coefficient a i by E i + K S , assume ε = 0. We can thus make sure that ( 12) is satisfied.
The condition in (12) concerning the parity of the coefficients a i is related to divisibility properties of H, by the following: Lemma 4.8.A line bundle H on an Enriques surface S is numerically 2-divisible (that is, its class in Num(S) is 2-divisible) if and only if all coefficients in any simple isotropic decomposition of H in Num(S) are even.Furthermore, in this case, (S, H) and (S, H + K S ) belong to different irreducible components of the moduli space of polarized Enriques surfaces.
Proof.The if part of the first assertion is clear.To prove the converse, assume that H is numerically 2-divisible and let be a simple isotropic decomposition in the form of Corollary 4.7 (modulo numerical equivalence); in particular, all a i 0 and a 0 = 0 or a 10 = 0. We consider these two cases separately and let E i,j be defined as in (8).
Assume a 0 = 0. Since (E i,j − E i ) • H = 2a i + a j , for i = j, and H is numerically 2-divisible, we must have all a j even, as desired.
Assume a 10 = 0.For i = 1, 2 we have (E i,10 − E 10 ) • H = a i , hence a 1 and a 2 are even.For i 3 we have and since a 1 , . . ., a 9 are all even, also a 0 is even.
To prove the last assertion, assume, to get a contradiction, that (S, H) and (S, H +K S ) belong to the same irreducible component of the moduli space of polarized Enriques surfaces.Then H and H + K S are either both 2-divisible in Pic(S) or not.However, we know that in the present case only one of them is 2-divisible, a contradiction.Notation 4.9.When writing a simple isotropic decomposition (2) verifying (3) (up to permutation of indices), we will usually adopt the convention that E i , E j , E i,j are primitive isotropic satisfying for k = i, j.This notation has already been used in Lemma 3.4 and Example 4.2.(By Corollary 4.7, there is no ambiguity in this notation.)Remark 4.10.The requirement that a simple isotropic decomposition satisfies n = 9 in case (i) and n = 10 in case (ii) of Lemma 4.3, which is equivalent to condition (11), is crucial in the last proof.Indeed, take where a 1 is an even nonnegative integer, and a 0 , a 3 , . . ., a 10 are odd positive integers.If a 1 = 0, then this decomposition is as in case (i) of Lemma 4.3 with n = 9, and if a 1 > 0, then it is as in case (ii) with n = 10.Hence, this is not a simple isotropic decoposition according to our definition.On the other hand H is numerically 2-divisible.Indeed, the claim is equivalent to using Lemma 3.4(a), the claim follows.Note that by Lemma 4.8 we have that (S, H) and (S, H + K S ) belong to different irreducible components of the moduli space of polarized Enriques surfaces.Remark 4.11.By Lemma 4.8 we get that ( 12) is equivalent to This means that the 'ε' in expression (12) only depends on H and not on the simple isotropic decomposition.
Remark 4.12.Writing a simple isotropic decomposition of H as in (10) has the advantage that φ(H) is calculated by one among E 1,2 , E 1 , . . ., E 10 .More precisely, setting a := 10 i=0 a i , one has ( 14) Indeed, for any nontrivial isotropic effective By symmetry, arguing as in the proof of Corollary 4.7, one can furthermore make sure that ( 15) in which case ( 16) We next recall the following from the introduction:  Proof.The property is clearly reflexive and symmetric, so we have left to prove transitivity.Assume therefore that (S, H) and (S ′ , H ′ ) admit the same simple decomposition type and (S ′ , H ′ ) and (S ′′ , H ′′ ) admit the same simple decomposition type.We will prove that so do (S, H) and (S ′′ , H ′′ ).The same proof will work for numerical polarizations.
By assumption, using the notation of Corollary 4.7, we have Here all a i and b i are nonnegative integers, . By (19) we also have  Conversely, it is proved in [9] that the irreducible components of E g,φ correspond precisely to the different orbits of the action of the orthogonal group on U ⊕ E 8 (−1).Since this group acts transitively on the set of isotropic 10-sequences by [7, Lemma 2.5.2], and

and
we see that any two numerical polarizations admitting the same simple decomposition type lie in the same irreducible component of E g,φ , as claimed.Question 4.17. 4 Theorems 1.1 and 1.2 give a positive answer in the case of simple decomposition types that are of length 5 or 6-symmetric.
The following lemma classifies all possible equivalence classes of simple decomposition types with φ 5. Note that all decomposition types do exist on any Enriques surface, by Lemma 3.4(a) and the existence of isotropic 10-sequences.

Lemma 4.18. Assume H is an effective line bundle on an Enriques surface S such that
, the line bundle H has one and only one of the following simple isotropic decompositions: Proof.The proof is tedious but straightforward and similar to [12, pf. of Prop.1.4 in §2.2], and we therefore will leave most of it to the reader.The idea is to pick an effective, isotropic E such that E • H = φ(H), find a suitable integer k so that φ(H − kE) < φ(H) (in which case we use the classification for lower φ), or so that φ(H − kE) = φ(H) and H − kE is as in Proposition 3.1(i) or (ii).As a sample, we show how this works in the case φ(H) = 5 and g ≡ 3 mod 5.
Because of the different values of φ(H − kE), it is again not possible that H can be written both as in (25) and as in (23) or (24).
In case (e) we have Assume φ(H − kE) = 2.By the classification in the case φ = 2, we have the two possibilities: (f) Assume finally φ(H − kE) = 1.By the classification in the case φ = 1, we have Remark 4.19.We will later use the observation immediately deduced from parts (i)-(ii) of Lemma 4.18 that for φ(H) 2 there are at most three distinct numerical, effective, isotropic classes E such that E • H 2.

Proof. The divisor
To prove this, assume by contradiction that n Then, for k = 1 or 2, there is an isotropic 10-sequence Proof.Assume first that r 7. By Proposition 3.3, the set A of A ∈ Pic(S) such that

Assume, to get a contradiction, that n
6 and B has a simple isotropic decomposition containing at least two summands.None of these may be F 2 , since B − F 2 = A − kF 1 has negative square, unless k = 0, in which case B = F 2 + A is not a simple isotropic decomposition.
Since F 2 • B = n − 2k, the intersection of F 2 with each of the summands in the simple isotropic decomposition of B is smaller than n.Since F 1 • B = 3, there is at least one of these summands, say E ′ , such that F 1 • E ′ = 1.If r = 0, since F 2 • E ′ < n, the curve E ′ contradicts the minimality of A and finishes the proof in this case.
Case k = 0. Then n = 3, B ∼ A + F 2 , B 2 = 6 and φ(B) = E i • B = 2. Thus, by Lemma 4.18(ii), B can be written as a sum of three isotropic divisors, containing all E i for i ∈ {1, . . ., r}.This implies r 3. Since F i • B = 3, for i = 1, 2, each summand has intersection one with F i , for i = 1, 2. This implies r = 3.Indeed, if r < 3, then at least one of the summands of B, say E ′ , is different from the E i s, and has for all i ∈ {1, . . ., 7}, we find a contradiction to the minimality of A. Subcase (n, B 2 ) = (5,4).As E i • B = 1, for i ∈ {2, . . ., r}, by Lemma 4.18(i) we have r = 1 and But then we get the contradiction Therefore, we have proved the claim that A • F 2 2.
Assume now that A•F 2 = 2.By Lemma 4.20, the isotropic sequence {F 1 , A, E 1 , . . ., E r } can be extended to an isotropic 10-sequence such that F 2 • F 1 = F 2 • A = 2 and F 2 has intersection one with the remaining divisors in the sequence.Hence, we are done.
For the rest of the proof we therefore let r = 8.Then we can by Proposition 3. Proof of Proposition 4.5.We first prove the first statement.Consider the simple isotropic set {E 1 , . . ., E r } satisfying (3).If E i •E j = 1 for all i = j, and if r = 9, we apply Proposition 3.3.If instead r = 9, we apply Lemmas 4.21 and 3.4(b).If E 1 •E 2 = 2 and otherwise E i • E j = 1 for i = j, we apply Lemmas 4.22 and 3.4(b).Finally, if and otherwise E i • E j = 1 for i = j, we apply Lemmas 4.20 and 3.4(b).

Irreducibility, unirationality and uniruledness of moduli spaces
To prove our results, we extend a construction from [16].First we recall some basic facts about classical Enriques sextic surfaces in P 3 (see [7]).
Consider the linear system S of surfaces of degree 6 that are singular along the edges of T .They are called Enriques sextic surfaces and have equations of the form where Q(x 0 , ..., x 3 ) = i j q ij x i x j and c 0 , . . ., c 3 , q ij are constants.This shows that dim(S) = 13 and we may identify S with the P 13 with homogeneous coordinates q = (c 0 : c 1 : c 2 : c 3 : q 00 : q 01 : q 02 : q 03 : q 11 : q 12 : q 13 : q 22 : q 23 : q 33 ).
If Σ ∈ S is a general surface, its normalization ϕ : S → Σ is an Enriques surface and ) is an ample divisor class with H 2 = 6 and φ(H) = 2.More precisely, H ∼ E 1 + E 2 + E 3 , with the usual Notation 4.9, and the edges ℓ i and ℓ ′ i of T are the images by ϕ of the curves E i and E ′ i ∼ E i + K S , with i = 1, 2, 3. (Recall that for a primitive, isotropic E, the complete linear system |E + K S | has a unique element.) We thus have a natural rational map assigning to a general surface Σ ∈ S the pair (S, H), where ϕ : S → Σ is the normalization and H = ϕ * (O Σ (1)).Composing with the forgetful map E 4,2 → E, we have a rational map S E, which is dominant.Indeed, given a general, whence unnodal, Enriques surface S, we can find a 3-isotropic sequence and the linear system |H| determines a morphism ϕ H : S → P 3 , cf., e.g., [7, Thm.4.6.3 and 4.7.2],and, up to a change of coordinates, Σ = ϕ H (S) is an Enriques sextic surface.Accordingly, the map p is dominant.If (S, H) is a point of E 4,2 , the fibre p −1 (S, H) consists of the orbit of Σ = ϕ H (S) via the 3-dimensional group of projective transformations fixing T .
Denote by v the vertex of T not contained in the face spanned by ℓ 1 , ℓ 2 , ℓ 3 .We define F i , i = 0, 1, 2, to be the family of irreducible cubic (resp., quartic, quintic) curves F ⊂ P 3 of arithmetic genus 1 such that v ∈ F and F meets • all edges of T exactly once, if i = 0; • the edges ℓ 1 and ℓ ′ 1 of T exactly twice, and the remaining edges exactly once, if i = 1; • the edges ℓ 3 and ℓ ′ 3 of T exactly once, and the remaining edges exactly twice, if i = 2.Note that if S is an Enriques surface that is the normalization of a sextic Σ ∈ S containing an elliptic curve F as above, then {E 1 , E 2 , E 3 , F } is a simple isotropic set on S, where we still denote by F the strict transform of F ⊂ Σ in S. In particular, since such simple isotropic sets exist with F irreducible on an unnodal Enriques surface, the families F i are non-empty.

Lemma 5.1. (a)
The family F 2 is irreducible, 10-dimensional and rational, and each F ∈ F 2 is contained in a 3-dimensional linear system of Enriques sextics.
(b) The family F 1 is irreducible, 8-dimensional and rational, and each F ∈ F 1 is contained in a 5-dimensional linear system of Enriques sextics.
(c) The family F 0 is irreducible, 6-dimensional and rational, and each F ∈ F 0 is contained in a 7-dimensional linear system of Enriques sextics.
Proof.We first prove (b) (resp.(c)).Let F ∈ F 1 (resp.F ∈ F 0 ).The linear system S cuts out on F a linear system of divisors with base locus (containing) T ∩F and a moving part g of degree (at most) 8 (resp., 6).Note that S contains the 9-dimensional linear system formed by surfaces of the form T +Q, where Q is a general quadric in P 3 : looking at equation ( 28), these are the surfaces obtained by setting c i = 0, for i = 1, . . ., 4. Since quadrics cut out on F a complete linear system, we see that g is complete, of dimension 7 (resp.5).This proves that the linear system of Enriques sextics containing F has dimension 5 (resp., 7).
We now prove the rest of (b).Given F ∈ F 1 , the intersection of F with the edges of T is a subscheme Z of length 8 of the union of these edges off the vertices of T .Let Z be the Hilbert scheme of such subschemes.We have a restriction morphism γ : F 1 → Z.
Claim.The morphism γ is injective and dominant.
Indeed, let F be in F 1 and let Z = γ(F ).To prove the injectivity, it suffices to prove that the linear system of quadrics passing through Z has dimension 1. Suppose this is false.Then there would be a net Q of quadrics through these 8 points.Fix the attention on a face Π of T containing four of these points (on three edges).Then the quadrics in Q containing two fixed general points of Π contain Π, because there is no conic containing the four points of Z on Π and two general points of Π.Consequently, the remaining four of the eight points should be coplanar, a contradiction, proving the injectivity.The dominance is then clear because, consequently, the quadrics containing a general Z in Z form a pencil whose base locus is in , the claim yields that F 1 is irreducible, rational of dimension 8.This proves (b).
We next prove the rest of (c).If F ∈ F 0 , then F spans a plane Π F ⊂ P 3 , which intersects the set of edges of T in six distinct points.These six points are the vertices of the quadrilateral cut out on Π F by the faces of T .Hence the cubic F is smooth at these points, otherwise it would contain one of the sides of the above quadrilateral.Now we claim that the set of plane cubics through these six points is a linear system of dimension 3. Indeed, otherwise, the cubics in this linear system, of dimension r 4, would cut out on F , off the six points, a g r−1 3 with r − 1 3, which is impossible, since F has arithmetic genus 1.Thus, F 0 is a P 3 -bundle over an open subset of |O P 3 (1)| ≃ P 3 , and is therefore irreducible, rational and 6-dimensional.This proves (c).
As for item (a), the fact that F 2 is irreducible, 10-dimensional and rational is proved in [16,Prop. 1.1 and §2].The rest of the assertion is proved exactly in the same way we did it for cases (b) and (c) above.
We next define F 00 to be the family of ordered pairs (F, F ′ ) of irreducible cubic curves F, F ′ ⊂ P 3 of arithmetic genus 1 such that F, F ′ ∈ F 0 and F and F ′ intersect exactly in one point not on T , with distinct tangent lines.
Note that if S is an Enriques surface that is the normalization of a sextic Σ ∈ S containing a pair (F, F ′ ) of elliptic curves as above, then {E 1 , E 2 , E 3 , F, F ′ } is a simple isotropic set on S, where we still denote by F and F ′ the respective strict transforms of F and F ′ in S. As above, since such isotropic sets of irreducible curves exist on an unnodal Enriques surface, we have that F 00 is non-empty.
Lemma 5.2.The family F 00 is irreducible, 11-dimensional and rational and each pair (F, F ′ ) ∈ F 00 is contained in a 2-dimensional linear system of Enriques sextics.
Proof.The family F 00 can be constructed in the following way: fix a pair of general planes Π and Π ′ in P 3 intersecting along a line ℓ, and fix a point p ∈ ℓ.Consider in both Π and Π ′ the family of cubic curves passing through p and the six intersection points of Π and Π ′ , respectively, with the edges of T ; each of these is a two-dimensional linear system.Varying Π, Π ′ and p and taking the two families of cubic curves, we obtain all elements of F 00 .This description shows the rationality and the dimension.Now fix (F, F ′ ) ∈ F 00 and let S F +F ′ be the linear system of Enriques sextics containing F ∪ F ′ .First we prove that dim(S F +F ′ ) 2. Indeed, the linear system S F of Enriques sextics containing F is 7-dimensional by Lemma 5.1(c).It cuts on F ′ a linear system of divisors with base locus (containing) T ∩ F and p = F ∩ F ′ and a moving part of degree (at most) 5, hence of dimension at most 4. Therefore, containing F ′ imposes at most 5 conditions on S F .
Next we prove that dim(S F +F ′ ) 2, which will finish our proof.Consider the pair F ⊂ Π and F ′ ⊂ Π ′ in F 00 , with the planes they span.Set ℓ = Π∩Π ′ and F ∩ℓ = {a, b, p} and F ′ ∩ ℓ = {a ′ , b ′ , p}.Let Σ ∈ S F +F ′ be general.Then ℓ intersects Σ in six points, among these are {a, b, a ′ , b ′ , p}; call p ′ the sixth point.The surface Σ intersects Π (resp., Π ′ ) in a cubic G off F (resp., G ′ off F ′ ), passing through a ′ , b ′ and p ′ (resp., a, b and p ′ ), in addition to the six intersection points of Π (resp., Π ′ ) with the edges of T .Then G (resp.G ′ ) is uniquely determined by the condition of passing through the six points Ξ (resp.Ξ ′ ) of intersection of Π (resp.Π ′ ) with the edges of T and through a ′ , b ′ , p ′ (resp., a, b, p ′ ).Let us prove this for G (the proof for G ′ is identical).Suppose there is a pencil of cubics containing Ξ and a ′ , b ′ , p ′ .Since a ′ , b ′ , p ′ lie on ℓ, there is a cubic in the pencil containing ℓ.The remaining conic component of this cubic should pass through Ξ, and this is clearly impossibile.Consequently, as Σ varies in S F +F ′ , the intersection Σ ∩ (Π∪ Π ′ ) may at most vary with the point p ′ ∈ ℓ.Thus the restriction S Π∪Π ′ of S F +F ′ to Π ∪ Π ′ is at most one-dimensional.Consider the restriction map S F +F ′ S Π∪Π ′ , which is linear, rational and surjective by assumption.Its indeterminacy locus is the unique surface T ∪ Π ∪ Π ′ .Since dim(S Π∪Π ′ ) 1, we deduce that dim(S F +F ′ ) 2, as desired.
We next define F 0i , for i = 1, 2, to be the family of ordered pairs of irreducible curves (F, F ′ ) in P 3 of arithmetic genus 1 such that F ∈ F 0 , F ′ ∈ F 1 and F and F ′ intersect exactly in i points not on T , with distinct tangent lines.
Note that, as before, if S is an Enriques surface that is the normalization of a sextic Σ ∈ S containing a pair (F, F ′ ) of elliptic curves as above, then {E 1 , E 2 , E 3 , F, F ′ } is a simple isotropic set on S, where we still denote by F and F ′ the respective strict transforms of F and F ′ in S. Since such isotropic sets of irreducible curves exist on an unnodal Enriques surface, we have that each F 0i is non-empty.
Lemma 5.3.The family F 0i is irreducible, uniruled and (14 − i)-dimensional and each pair (F, F ′ ) ∈ F 0i is contained in a linear system S F +F ′ of Enriques sextics of dimension at least i − 1.If (F, F ′ ) ∈ F 0i is contained in an Enriques sextic Σ whose normalization S is an Enriques surface, then S F +F ′ has dimension exactly i − 1, unless F + F ′ is contained in only nodal Enriques sextics (that is, Enriques sextics whose normalizations are nodal).
Proof.We have a natural dominant map q : F 0i → F 1 × (P 3 ) ∨ sending the pair (F, F ′ ) to F ′ ∈ F 1 and the plane Π F spanned by F in (P 3 ) ∨ .
For i = 1, the fiber of q over (F ′ , Π) consists of the union of four 2-dimensional linear systems of cubics in Π through the six intersection points of Π with the edges of T and one of the four intersection points of Π with F ′ .This proves the irreduciblity because the monodromy action of the four intersection points is the symmetric group (see [1, Lemma on p. 111]), and shows also the uniruledness.The dimension also follows easily.
For i = 2, the fiber of q over (F ′ , Π) consists of the union of six 1-dimensional linear systems of cubics in Π through the six intersection points of Π with the edges of T and two of the four intersection points of Π with F ′ .As above, this proves irreduciblity, uniruledness and the dimension.
The dimension of the linear system of Enriques sextics S F ′ containing a fixed F ′ ∈ F 1 is 5 by Lemma 5.1(b).Containing an additional cubic F ∈ F 0 intersecting F ′ in i points, imposes at most 6 − i conditions, arguing as in the proof of Lemma 5.2.Therefore, the linear system of Enriques sextics S F +F ′ containing a pair (F, Let Σ be an Enriques sextic containing F + F ′ such that its normalization ϕ : S → Σ is an unnodal Enriques surface.The linear system S cuts on Σ a linear system whose pull-back on S via ϕ is the sublinear system of |6(E 1 + E 2 + E 3 )| with base locus twice the sum of the pullback of the edges of the tetrahedron, which is Hence, the free part is |2(E 1 + E 2 + E 3 )|.So we have a linear, rational restriction map whose indeterminacy locus is just the surface Σ.

Consider now the incidence varieties
for i = 0, 1, 2, and which are irreducible, rational and 13-dimensional, by Lemmas 5.1 and 5.2.Similarly, for i = 1, 2, let which are irreducible, uniruled and 13-dimensional, by Lemma 5.3.Proposition 5.4.If G is any of the incidence varieties G i , for i = 0, 1, 2, G 00 , G 0i , for i = 1, 2, the obvious projection π : G → S is dominant, hence generically finite.Accordingly, if ξ ∈ G is a general point, then Σ = π(ξ) is a general element of S and its normalization S is a general Enriques surface.
Proof.We prove the assertion for G = G 00 , the proof in the other cases being similar.
Let S be a general Enriques surface.There is an isotropic 5-sequence {E 1 , . . ., E 5 } on S. Set H = E 1 + E 2 + E 3 .Then ϕ H : S → Σ ⊂ P 3 maps S, up to a projective transformation, to a general surface in S. Moreover E 4 , E 5 are mapped to two elliptic cubic curves F, F ′ meeting at a point.This proves the assertion.
We now define various maps from these incidence varieties to some E g,φ s, for various g and φ, which we eventually prove to be dominant, establishing irreducibility and unirationality or uniruledness.
Consider a general element (F, Σ) of G i , for i = 0, 1, 2. Then the normalization S of Σ is an Enriques surface and on S we have the three curves E 1 , E 2 , E 3 , plus the strict transform of F which, by abuse of notation, we still denote by F .Similar convention we introduce for G 0i , for i = 0, 1, 2.
We use the following definition in the proofs of Propositions 5.5 and 5.6.
Definition 5.7.For an isotropic 3-sequence I = {E 1 , E 2 , E 3 } on the Enriques surface S, we let F i (I), i = 0, 1, 2, be the set of all primitive, isotropic divisors F on S satisfying and F 0i (I), i = 0, 1, 2, the set of all pairs (F, F ′ ) of primitive, isotropic divisors F, F ′ on S such that F ∈ F 0 (I) and Proof of Proposition 5.5.Let (S, H) be as in either of the statements of the proposition.
In particular, H admits a simple decomposition type of length n, with 2 n 5.By Corollary 4.7, if n 4, we may write an isotropic 3-sequence and F ∈ F i (I), possibly allowing some of the α i s to be 0. If n = 5, we may write H ∼ α 1 E 1 + α 2 E 2 + α 3 E 3 + α 4 F + α 5 F ′ + εK S with (F, F ′ ) ∈ F 0i (I).We may assume (S, H) to be general, in particular, S is unnodal.Then by [7,Thm. 4.6.3 and 4.7.2] the complete linear system |E 1 + E 2 + E 3 | maps S birationally onto an Enriques sextic in P 3 , with double lines along the edges of the tetrahedron T defined by the images of all E i and E ′ i := E i + K S .Under this map, F (respectively, (F, F ′ )) is mapped to an element of F i (resp., F 0i ), finishing the proof.
We now give the proofs of the three corollaries in the introduction.7), with equality φ = 6 possible only for H 2 = 36 by Proposition 3.1, in which case the simple decomposition type has length 2. Thus the result follows from Theorem 1.1 in this case.
We have left to treat the cases where φ 5.By Lemma 4.18, all cases with φ 5 and g 20 have decomposition types of length n 5, except for the type E 1 + E 2 + E 3 + E 4 + E 5 + E 6 for (g, φ) = (16,5), which is the only type occurring for these values of g and φ.Hence E 16,5 is irreducible and uniruled by Theorem 1.2.Again by Lemma 4.18, all remaining cases with φ 5 and g 20 admit simple decomposition types of length n 4 or of length 5 with all nonzero intersections occurring equal to one, except for the type 2E 1 + E 2 + E 3 + E 4 + E 1,5 for (g, φ) = (17, 5), which is the only type occurring for these values of g and φ.Hence E 17,5 is irreducible and uniruled and all irreducible components of the remaining E g,φ are unirational by Theorem 1.1.
Proof of Corollary 1.5.If [H] ∈ Num(S) is not 2-divisible, then by Lemma 4.8 some simple decomposition types of H and H + K S have not all even coefficients in front of the isotropic, primitive summands.Hence, by substituting one E i with odd coefficient with E i + K S , we see that H and H + K S admit the same simple decomposition type, and thus belong to the same irreducible component of E g,φ , by Theorems 1.1 or 1.2 and the assumption on the decomposition types.Hence ρ −1 (ρ(C)) is irreducible.
Conversely, assume [H] ∈ Num(S) is 2-divisible.Then H and H + K S do not lie in the same irreducible component of E g,φ by the last assertion in Lemma 4.8, whence ρ −1 (ρ(C)) consists of two disjoint components.
The cases of the latter corollary are of particular interest from a Brill-Noether theoretical point of view, since they are precisely the cases where the gonality of a general curve in the complete linear system |H| is less than both 2φ and ⌊ g+3 2 ⌋, the first being the lowest degree of the restriction of an elliptic pencil on the surface, the latter being the gonality of a general curve of genus g, cf.[12,Cor. 1.5].
Appendix: Irreducible components of E g,φ and E g,φ for g 30 Using Proposition 4.16 (and Notation 4.9) we list all irreducible components of the moduli spaces E g,φ for g 30, and describe the properties of ρ −1 of these components obtained by Theorems 1.1 and 1.2 and Corollary 1.5.We thus obtain information about all irreducible components of the moduli spaces E g,φ , with few exceptions 5 .The various decomposition types can be obtained from Lemma 4.18 and Proposition 3.1, and an ad hoc treatment as in the proof of Lemma 4.18 for the cases φ = 6 and 7.The fact that all decomposition types below are in different equivalence classes can be checked by computing suitable intersections as in the proof of Lemma 4.18, and the fact that they all do exist on any Enriques surface follows from Lemma 3.4(a).

Lemma 2.12]: Lemma 4.3. Any
effective line bundle H with H 2 0 on an Enriques surface can be written asH ≡ a 1 E 1 + • • • + a n E n ,where all a i are positive integers, 1

Definition 4.4. A
set {E 1 , . . ., E n } of primitive isotropic divisors on an Enriques surface is called a simple isotropic set if it satisfies one of the conditions (i)-(iii) in Lemma 4.3, possibly after permuting indices.It is called a maximal simple isotropic set if it is of the form {E 1 , . . ., E 10 , E i,j }, where {E 1 , . . ., E 10 } is an isotropic 10-sequence and E i,j is defined up to numerical equivalence as in . Assume next that H ≡ a 1 E 1 +• • •+a 10 E 10 with E 1 •E 2 = 2 and E i •E j = 1for all other indices i = j.By Proposition 4.5, the set {E 1 , . . ., E 10 } can be extended to a maximal simple isotropic set {E 1 , . . ., E 10 , E 11 }.Possibly after interchanging E 1 and E 2 , we may assume that E 1

.
Remark 4.14.A decomposition type is not necessarily unique within the same linear or numerical equivalence class, even imposing the conditions (15) on the coefficients.Moreover, also properties such as the length or being r-symmetric may vary with the different ways of writing the decompositions.Consider for instance the decomposition type H ≡ 2E 1 +E 2 +E 3 +E 4 +E 5 +E (8)E 1,7 (with g = 30 and φ(H) = 7), written in the form of Corollary 4.7, that is, {E 1 , ..., E 6 } may be extended to an isotropic 10-sequence {E 1 , ..., E 10 } so that E 1,7 is defined as in (8).This has length 7 and is 5-symmetric, but not 6-symmetric.It therefore does not satisfy the conditions of Theorems 1.1 and 1.2.Let also E 7,8 be as defined by(8).It follows that E 1 + E 1,7 ∼ E 8 + E 7,8 .Thus, we may also writeH ≡ E 1 + E 2 + E 3 + E 4 + E 5 + E 6 + E 8 + E 7,8, which has length 8 and is 6-symmetric.This decomposition satisfies the conditions of Theorem 1.2.Proposition 4.15.The property of admitting the same simple decomposition type defines equivalence relations on E g,φ and E g,φ , respectively.
Proposition 4.16.Two numerically polarized Enriques surfaces (S, [H]) and (S ′ , [H ′ ]) lie in the same irreducible component of E g,φ if and only if they admit the same simple decomposition type.Proof.Since Num(S) ≃ U ⊕ E 8 (−1) is constant among all S ∈ E, the only if part is immediate.
Does Proposition 4.16 also hold for polarized Enriques surfaces?In other words, is it true that (S, H) and (S, H ′ ) lie in the same irreducible component of E g,φ if and only if H and H ′ admit the same simple decomposition type?(The "only if" part follows as in the first lines of the proof of 4.16, as Pic 8}, contradicting Remark 4.19.This proves (26), whence the lemma.Let F 1 and F 2 be isotropic divisors such that F 1 •F 2 = 2 and {E 1 , . . ., E r } be an isotropic r-sequence, with 0 r 8, such that 3 extend {E 1 , . . ., E 8 } to an isotropic 10-sequence {E 1 , . . ., E 10 }.We claim that {F 1 , E 1 , . . ., E 8 , E 9 } is the desired isotropic 10-sequence.