The unirationality of the moduli space of K3 surfaces of genus 22

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.


INTRODUCTION
The 19-dimensional moduli space F g of polarized K3 surfaces of genus g (or of degree 2g − 2), parametrizing pairs [S, H], where S is a K3 surface and H ∈ Pic(S) is a primitive polarization class satisfying H 2 = 2g − 2, is one of the most intriguing parameter spaces in algebraic geometry. In stark contrast to the moduli space of curves or abelian varieties, its Picard group is highly intricate, see [BLMM]. The moduli space F g is a quotient of a locally symmetric domain. Via this realization as an orthogonal modular variety one can employ automorphic methods in order to study its Kodaira dimension. In this way, Gritsenko, Hulek and Sankaran [GHS] proved that F g is a variety of general type for g > 62, as well as for g = 47, 51,53,55,58,59,61. On the other hand, using vector bundles on various rational homogeneous varieties, in a celebrated series of papers Mukai [M1], [M2], [M3], [M4], [M5] described the construction of general polarized K3 surfaces of genus g ≤ 12, as well as for g = 13, 16, 18, 20. In particular, the moduli space F g is unirational for those values of g. The case g = 14, not covered by Mukai's work, has been settled using the birational isomorphism between F 14 and the moduli space C 26 of special cubic fourfolds of discriminant 26. Nuer [Nu] first showed that F 14 is uniruled. This was then improved in [FV], where we showed that the universal K3 surface F 14,1 is rational, hence F 14 is unirational. Recently Ma [Ma] undertook a systematic study of the Kodaira dimension of the moduli space F g,n of n-pointed K3 surfaces of genus g, in the spirit of a similar analysis of the Kodaira dimension of M g,n carried out in [Log] and [F].
The aim of this paper is to study the geometry of F 22 using the connection between K3 surfaces and special cubic fourfolds of discriminant 42. We establish the following result: Theorem 1.1. The universal K3 surface F 22,1 of genus 22 is unirational.
In particular, F 22 is unirational as well. Note that 22 is the highest genus where it is known that the moduli space F g is not of general type. Our approach to F 22 relies on the relation between Noether-Lefschetz special cubic fourfolds and polarized K3 surfaces, which we explain next.
We fix a smooth cubic fourfold X ⊆ P 5 . Recall the important fact that the Fano variety of lines F (X) := ℓ ∈ G(1, 5) : ℓ ⊆ X is a hyperkähler variety of dimension 4, see [BD]. Its primitive cohomology H 4 prim (X, Z), displaying the Hodge numbers (0, 1, 20, 1, 0), looks like the Tate twist of the middle cohomology of a K3 surface, except it has signature (20, 2) rather than (19,3). When X is very general, the lattice A(X) := H 2,2 (X) ∩ H 4 (X, Z) consists only of classes of complete intersection surfaces, that is, A(X) = h 2 , where h ∈ Pic(X) is the hyperplane class, see [Voi]. Let C be the 20-dimensional coarse moduli space of smooth cubic fourfolds X ⊆ P 5 and denote by C d the locus of special cubic fourfolds X characterized by the existence of an embedding of a saturated rank 2 lattice L := h 2 , T ֒→ A(X), of discriminant disc(L) = d, where T is a codimension 2 algebraic cycle of X not homologous to a complete intersection. Hassett [Ha] showed that C d ⊆ C is nonempty and if so, an irreducible divisor, if and only if d > 6 and d ≡ 0, 2 (mod 6). A conjecture of Kuznetsov [Kuz] predicts that all cubic fourfolds [X] ∈ C 2(n 2 +n+1) are rational. This has been confirmed in the classical case d = 14, see [Fa], [BR], and more recently when d = 26 by Russo and Staglianò [RS1]. Very recently, the same authors announced a proof of the rationality of all cubic fourfolds from C 42 , see [RS2].
For d = 42, Hassett's work [Ha] implies the existence of a rational map of degree 2 where the cubic fourfold X is characterized by the existence of an isomorphism Lai's paper [Lai] represents an important first step in understanding the relation between F 22 and C 42 . We summarize its results. Starting with a polarized K3 surface [S, H] ∈ F 22 , for each point p ∈ S one considers the rational curve Under the isomorphism S [2] ∼ = F (X) described above, ∆ p corresponds to a rational curve of degree 9 inside F (X) ⊆ G(1, 5), that is, to a degree 9 scroll R p ⊆ X. The double point formula implies that, as long as it has isolated nodal singularities, R p has 8 nodes and no further singularities. This is precisely the content of [Lai,Theorem 0.3]. We denote by H scr the P GL(6)-quotient of the Hilbert scheme of 8-nodal scrolls R ⊆ P 5 of degree 9. Lai shows [Lai,Proposition 0.4] that H scr has the expected codimension 8 inside the parameter space of all scrolls of degree 9 in P 5 , in particular dim(H scr ) = 16.
One can then set up the incidence correspondence between scrolls and cubic fourfolds: For a general [R] ∈ H scr one computes that h 0 P 5 , I R/P 5 (2) = 0 and h 0 P 5 , I R/P 5 (3) = 6, It follows that X is birational to a P 5 -bundle over the variety H scr . Since π 1 is dominant, this implies that C 42 is uniruled. This is the point where Lai's paper [Lai] ends and our analysis starts. We first introduce the universal K3 surface u : F 22,1 → F 22 , then the map where R p is the degree 9 scroll contained in X corresponding to the rational curve ∆ p ⊆ F (X) under the isomorphism (1). We observe that although ϕ has degree 2, that is, for a general fourfold [X] ∈ C 42 one has two polarized K3 surfaces realizing the isomorphism (1), this ambiguity disappears once we lift to the universal K3 surface. We prove the following: Since X is a P 5 -bundle over H scr , the unirationality of F 22,1 will be implied by that of the moduli space H scr . To summarize the situation, we have the following commutative diagram: We now explain our parametrization of the moduli space of 8-nodal nonic scrolls. We start by considering the Hirzebruch surface F 1 := Bl o (P 2 ), where o ∈ P 2 , and denote by h the class of a line and by E the exceptional divisor. The smooth degree 9 scroll R ′ := S 4,5 ⊆ P 10 is the image of the linear system φ |5h−4E| : F 1 ֒→ P 10 .
We choose a 4-plane Λ ∈ G(4, 10) which is 8-secant to the secant variety Sec(R ′ ) ⊆ P 10 . We may assume Λ ∩ R ′ = ∅ and refer to [Lai,] for the proof that such a 4-plane Λ exists. Consider the restriction to R ′ of the projection π Λ with center Λ where R := π(R ′ ). Then R is an 8-nodal scroll of degree 9. If for i = 1, . . . , 8, we have that x i , y i ∩ Λ = ∅ for certain points x i , y i ∈ R ′ , then the (nodal) singularities of R appear as n i := π Λ (x i ) = π Λ (y i ). Up to the action of P GL(6) on the ambient projective space P 5 , each 8-nodal nonic scroll [R] ∈ H scr appears in this way.
It will turn out that the residual curve Γ ⊆ P 5 is a degree 14 integral curve of arithmetic genus 12 having nodes at the points n 1 , . . . , n 8 . Assuming this, let be the normalization of Γ. Then from (2) we find that C ∈ |6h − 4E|. Therefore C is a hyperelliptic curve of genus 4 which passes through the points x i , y i ∈ R ′ , for i = 1, . . . , 8. The degree 2 pencil on C is cut out by the rulings of R ′ , that is, O C (h − E) ∈ W 1 2 (C). Denoting by ι : C → C the hyperelliptic involution, we observe that R = x∈C π(x), π(ι(x)) ⊆ P 5 , that is, the degree 9 scroll R can be recovered from the curve Γ ⊆ P 5 .
We denote by P the parameter space of pairs [R, ℓ 1 + · · · + ℓ 4 ], where R ⊆ P 5 is an 8-nodal scroll of degree 9 and ℓ 1 , . . . , ℓ 4 are rulings of R, viewed as an unordered set. In the definition of P we quotient out by the P GL(6)-action on P 5 . The forgetful map P → H scr is birational to a P 1 -bundle corresponding to the choice of the four rulings, in particular dim(P) = 17. Let Hyp 4,8 be the moduli space of pairs [Γ, L], where Γ is an integral 8-nodal curve of arithmetic genus 12, whose normalization ν : C → Γ is a hyperelliptic curve of genus 4 and L ∈ W 2 8 (Γ), that is, L is a line bundle of degree 8 on Γ with h 0 (Γ, L) ≥ 3. Note that by Riemann-Roch, in this case ω Γ ⊗ L ∨ ∈ W 5 14 (Γ). We have the following result, reducing the study of F 22,1 to that of a certain moduli space of curves. Theorem 1.3. There exists a birational isomorphism χ : P ∼ = Hyp 4,8 given by Theorem 1.1 now follows once we establish the unirationality of Hyp 4,8 . We indicate how to carry this out. Start with a general element [Γ, L] ∈ Hyp 4,8 , viewed as an 8-nodal degree 14 curve Γ ⊆ P 5 embedded by the line bundle ω Γ ⊗ L ∨ . We shall show that a suitably general such curve Γ is projectively normal, thus the kernel of the multiplication map is 4-dimensional. We can write The residual curve B ⊆ P 5 is a conic such that Γ · B = 6. We denote by Π := B ⊆ P 5 the plane spanned by B. There exists a 3-dimensional subspace V ⊆ H 0 P 5 , I Γ/P 5 (2) consisting of quadrics containing the plane Π. We write where T ⊆ P 5 is a degree 7 surface lying on three quadrics that intersect along the 2-plane Π. It is not hard to see that T ∼ = Bl 9 (P 2 ) is the blow-up of P 2 at 9 general points in P 2 . Moreover, the map ϕ : Bl 9 (P 2 ) ֒→ T ⊆ P 5 implicitly defined by (3) is induced by the linear system where E 1 , . . . , E 9 are the exceptional divisors. Via the isomorphism T ∼ = Bl 9 (P 2 ), one realizes Γ as an octic plane curve with 17 nodes divided in two groups: namely the 9 points where P 2 is blown up and the remaining 8 nodes. This plane model is helpful to prove the next result: Theorem 1.4. The moduli space Hyp 4,8 is unirational.
To prove Theorem 1.4, we fix a cubic scroll Z ⊆ P 4 obtained by embedding the Hirzebruch surface F 1 := Bl o (P 2 ) by the linear system |2h − E|. We consider the parameter space Note that dim I {x 1 ,y 1 ,...,x 8 ,y 8 }/Z (6h − 4E) = 1, hence the map T → Z 8 × G(1, 4) is birationally a locally trivial P 1 -bundle over a rational variety. Therefore T is rational. The curve C is hyperelliptic of genus 4. Denoting by π ℓ : P 4 P 2 the projection with center ℓ, observe that n ′ i := π ℓ (x i ) = π ℓ (y i ) for i = 1, . . . , 8. The dominant rational map ϑ : T Hyp 4,8 needed to prove Theorem 1.4 is obtained by associating to the point t 1 , . . . , t 8 , ℓ, C ∈ T essentially the projected curve Γ := π ℓ (C). This is a nodal octic plane curve having 8 distinguished nodes at n ′ 1 , . . . , n ′ 8 , as well as 9 further nodes. The image under the map ϕ of the proper transform of Γ ′ in the blow-up of P 2 at these 9 points gives rise to an element of Hyp 4,8 . For further details on the definition of the map ϑ we refer to Theorem 3.5.
It turns out that proving directly the various transversality assumptions implicit in this sketched proof of Theorem 1.3 is not straightforward. Instead, in the rest of the paper we shall reverse the argument presented in the Introduction. First we show that Hyp 4,8 is unirational (see Theorem 3.5), then using the explicit unirational parametrization found in this way, we show that the map χ : P Hyp 4,8 is well defined, as well as birational.

Acknowledgment:
We are grateful to the referee for a very careful reading of the paper and for many good suggestions that improved the presentation.

THE MODULI SPACE
We fix a smooth cubic fourfold X ⊆ P 5 and denote by h its hyperplane class. The Hodge structure on the primitive cohomology H 4 prim (X, Z) is similar to the twist of the middle cohomology of a K3 surface. Since the signatures (with respect to the intersection form) are different, (20, 2) and (19, 3) respectively, one has to pass to sub-Hodge structures of codimension one, both having signature (19, 2), to have the possibility of realizing an isomorphism of Hodge structures between the two sides. On the cubic fourfold side one requires the existence of a class T ∈ H 2,2 (X), whereas on the K3 side one requires the existence of a polarization H ∈ Pic(S) such that the following isomorphism of Hodge structures holds Denoting by d := disc( h 2 , T ) = H 2 , it is proved in [Ha,Theorem 5.1.3] that the isomorphism (4) is realized for any d > 6 such that d ≡ 0, 2 (mod 6) that is not divisible by 4, 9 or by any prime p ≡ 2 (mod 3). When d = 2(n 2 + n + 1), the isomorphism (4) takes the geometric form (1) This opens the way to a study of the moduli spaces F n 2 +n+2 where n ≥ 2, using the concrete projective geometry of cubic fourfolds. The case n = 2 (that is, d = 14) is classical and essentially due to Fano [F]; we refer to [BD] and [BR] for a modern perspective and stronger results. The case n = 3 (that is, d = 26) has been treated in our paper [FV] as well as in [RS1], whereas this paper is devoted to the case n = 4 (that is, d = 42).
For d = 42, Hassett [Ha] constructed a degree 2 map such that the isomorphism (1) holds. Note that ϕ is defined at the level of moduli spaces of weight 2 Hodge structures and there is no direct geometric construction of the cubic fourfold one associates to a K3 surface of genus 22. Since deg(ϕ) = 2, it follows that for a general Clarifying the relation between S and S ′ is essential in order to prove Theorem 1.2.
2.1. Hilbert squares of K3 surfaces. Let (S, H) be a K3 surface with Pic(S) = Z · H and H 2 = 2g−2. We denote by S [2] the Hilbert scheme of length two zero-dimensional subschemes on S.
be the divisor consisting of zero-dimensional subschemes supported only at a single point and denote by δ : For any curve C ∈ |H|, we introduce the divisor , Z . For a point p ∈ S, we also define the curve , Z). The Beauville-Bogomolov form can be extended to a quadratic form on H 2 (S [2] , Z), by setting q(α, α) := q(w α , w α ), with w α ∈ H 2 (S [2] , Z) being the class characterized by the property α · u = q(w α , u), for every u ∈ H 2 (S [2] , Z). Here α · u denotes the intersection product.
One has the following decompositions, orthogonal with respect to q, both for the Picard group and for the group N 1 (S [2] , Z) of 1-cycles modulo numerical equivalence: We record the following more or less immediate relations: The form q takes the following values on H 2 S [2] , Z : It follows from [BM,Proposition 13.1] (see also [DM,Proposition 3.14] for this formulation) that for a polarized K3 surface [S, H] ∈ F 22 with Pic(S) = Z · H, the nef cone Nef(S [2] ) equals the movable cone Mov(S [2] ) and it is generated by the rays f and 55f − 252δ respectively. Using the terminology of [Ha], the Hilbert square S [2] is strongly ambiguous, that is, there exists another K3 surface S ′ such that there exists an isomorphism r : is not induced by an automorphism S ∼ = → S ′ . This implies r * (δ ′ ) = δ and then necessarily, the map r * : , Z interchanges the two rays of the respective nef cones, that is, Then also r * (f ) = 55f ′ − 252δ ′ and r * (δ) = 12f ′ − 55δ ′ , from which we obtain the following relations at the level of the cone of curves in S [2] and S ′[2] respectively: 2.2. Scrolls contained in special cubic fourfolds. Suppose R ⊆ X ⊆ P 5 is a rational scroll with smooth normalization having only isolated singularities and which is contained in a cubic fourfold X. The double point formula [Ful,Theorem 9.3] gives the number D(R) of singularities of R, counted appropriately: . If moreover all singularities of R are nodal, then D(R) equals the number of nodes of R.
When (7), we compute D(R) = 8. Therefore if R has only isolated nodes, then it is necessarily 8-nodal.
be an effective 1-cycle of degree 9 with respect to the Plücker embedding.
In the first case Z = ∆ p for some point p ∈ S, and in the second case r(Z) = ∆ p ′ for some point Let γ S denote the class of the Plücker line bundle O S [2] (1) with respect to the isomorphism S [2] ∼ = F (X). Since q(γ S , γ S ) = 6, one obtains Therefore 9 = Z · γ S = (af p − bf p )(2f − 9δ) = 84a − 9b, hence we can write a = 3a 1 , with a 1 ∈ Z, in which case b = 28a 1 − 1. Using [BM,Proposition 12.6], if Z is effective we also have the inequality q(Z, Z) ≥ − 5 2 , implying 7a 2 1 − 14a 1 − 1 ≤ 0. The integer solutions of this inequality are a 1 = 0, when [Z] = δ p , a 1 = 2, in which case [Z] = 6f p − 55δ p , and finally a 1 = 1. Note that in the first two cases q(Z, Z) = − 1 2 . On the other hand, a 1 = 1 implies [Z] = 3f p − 27δ p , yielding q(Z, Z) = 27 2 . But this is incompatible with the double point formula. Indeed, if R ⊆ X is the scroll associated to the curve Z under the isomorphism S [2] ∼ = F (X), then following [HT2,7.1], we have  (6) that [r * (Z)] = δ ′ p ′ . By possibly replacing S with S ′ , we may assume [Z] = δ p . We claim this implies Z is one of the smooth rational curves ∆ p , for some point p ∈ S. Indeed, from [Z] · δ = −1, it follows that Z ⊆ ∆. Moreover, Z lies in one of the fibers of the P 1 -bundle π : ∆ = P(T S ) → S. This implies that Z = ∆ p , for some p ∈ S, because otherwise π(Z) ≡ mH for some integer m > 0 and then which is a contradiction.
We are now in a position to prove Theorem 1.2. Recall the definition given in the Introduction of the parameter space X of pairs (X, R), where [X] ∈ C 42 and R ⊆ X is a degree 9 scroll. As explained, as long as R has isolated nodal singularities, R has precisely 8 nodes. We define the mapφ : F 22,1 → X given byφ ([S, H, p]) := [X, R p ], where the cubic scroll X is determined by the isomorphism F (X) ∼ = S [2] and the scroll R p ⊆ X corresponds to ∆ p , viewed as a rational curve inside F (X). Recall that it is proved in [Lai] that the projection π 1 : X → C 42 is dominant. This implies thatφ is well-defined.

MODULI OF NODAL HYPERELLIPTIC CURVES
On our way towards establishing Theorems 1.3 and 1.4 and ultimately proving Theorem 1.1, we shall reverse the construction described in the Introduction associating to a suitably general scroll [R] ∈ H scr an 8-nodal curve with hyperelliptic normalization. In order to establish the various transversality claims mentioned in the Introduction, we find it easier to start with a suitable nodal hyperelliptic curve and bring the degree 9 scroll into picture only later. We begin therefore by introducing and studying various moduli spaces of curves that will turn out to be relevant when dealing with C 42 .
Recall that for an irreducible nodal curve Y , we denote by W r d (Y ) the Brill-Noether locus consisting of line bundles L ∈ Pic d (Y ) satisfying h 0 (Y, L) ≥ r + 1.
In this Section we provide an explicit parametrization of Hyp 4,8 and conclude that this space is unirational. We begin with some preparation. We consider the Hirzebruch surface F 1 := Bl o (P 2 ) viewed as a cubic scroll via the embedding Here h denotes the pull-back of the line class under the contraction morphism F 1 → P 2 , whereas E is the exceptional divisor over the point o ∈ P 2 .
We denote by H 4 the moduli space of smooth hyperelliptic curves of genus 4 and by

Proof.
A smooth curve C ∈ |6h− 4E| is hyperelliptic of genus 4. To it we can associate the pair [C, L], where L := O C (2h − E) is a line bundle of degree 8. In other words, L = O C (1), where C ⊆ Z ⊆ P 4 is viewed as an octic curve. This construction is obviously Aut(F 1 )-invariant, hence it gives rise to a map 6h − 4E /Aut (F 1 ) Pic 8 H 4 . Conversely, we start with a general line bundle L ∈ Pic 8 (C) on a hyperelliptic curve C of genus 4. We denote by A ∈ W 1 2 (C) the hyperelliptic pencil. We may assume L does not lie in the translate ω C ⊗ A + C − C ⊆ Pic 8 (C) of the difference variety C − C ⊆ Pic 0 (C). Set O C (h) := L ⊗ A ∨ ∈ Pic 6 (C). Then h 0 (C, O C (h)) = 3 and our assumption on L guarantees an induced regular map φ |h| : C → P 2 , whose image is a sextic curve C ′ ⊆ P 2 . Set For a general L ∈ Pic 8 (C) we have h 0 (C, N ) = 1 and we write N = O C (x 1 + x 2 + x 3 + x 4 ), for points x i ∈ C. By choosing L generally in Pic 8 (C) we can arrange that the points x i are distinct. Then which is to say that the image C ′ := φ |h| (C) has a 4-fold point at o := φ |h| (x i ) for i = 1, . . . , 4.
Comparing the genera of C and C ′ we see that C ′ has no further singularities. This implies we can embed C in the blown-up surface Bl o (P 2 ) such that C ∈ |6h − 4E|.
, thus finishing the proof.
In our study of the moduli space C 42 via nodal scrolls, a special role is played by a certain degree 7 rational surface in P 5 . In what follows, we summarize its properties. If o 1 , . . . , o n ∈ P 2 are distinct points, we denote by Bl n (P 2 ) := Bl o 1 ,...,on (P 2 ) their blow-up, by E i the exceptional divisor over o i , and by h the pull-back of the line class under the contraction morphism Bl n (P 2 ) → P 2 . Proposition 3.3. Let o 1 , . . . , o 9 ∈ P 2 be points lying on a unique smooth cubic curve. Then the linear system |4h − E 1 − · · · − E 9 | is very ample on Bl 9 (P 2 ) and the image T of the embedding φ |4h−E 1 −···−E 9 | : Bl 9 (P 2 ) ֒→ P 5 is projectively normal. In particular h 0 P 5 , I T /P 5 (2) = 3. Furthermore, Bs I T /P 5 (2) = T ∪ Π, where Π is a 2-plane meeting T along a smooth elliptic curve.
Indeed, one inclusion having been already established, suppose by contradiction there is a point r ∈ Bs I T /P 5 (2) \ (T ∪ Π). We pick a general hyperplane hyperplane P 4 ∼ = H ⊆ P 5 passing through r . Then T ∩ H =: C is a smooth non-hyperelliptic curve of genus 3, where O C (1) ∈ Pic 7 (C), whereas H ∩Π =: ℓ is a line. The components C and ℓ meet along the divisor r 1 + r 2 + r 3 consisting of the points lying on the intersection J · H ⊆ Π ∩ H. Furthermore, is a 1-dimensional space, for the cubic J does not move in its linear system. It follows that the stable genus 5 curve C ∪ ℓ is not trigonal. Therefore, its canonical embedding C ∪ ℓ ֒→ P 4 is ideal-theoretically cut out by quadrics, in particular r ∈ Bs I C∪ℓ/P 4 (2) = C ∪ ℓ. In particular r ∈ T , which shows that T is scheme-theoretically cut out by quadrics.
We describe a geometric construction that will yield a parametrization of Hyp 4,8 . Recall that Z ⊆ P 4 denotes the cubic scroll defined by (8). We fix general points (t 1 , . . . , t 8 ) ∈ Z 8 and a general line ℓ ⊆ P 4 disjoint from Z. For i = 1, . . . , 8, we obtain further points x i , y i ∈ Z via the relation with the intersection being taken inside P 4 .
Clearly, T is a locally trivial P 1 -bundle over Z 8 × G(1, 4). In particular, T is a rational variety of dimension 23. Note that the 6-dimensional automorphism group Aut(F 1 ) acts on T , where the action on G(1, 4) is via the identification Aut(F 1 ) ∼ = Aut(Z) ⊆ P GL(5). Therefore the quotient T /Aut(F 1 ) has dimension 17 (and is of course unirational).
Assuming both claims (i) and (ii), we proceed with our proof. The map ϕ is an embedding and from Proposition 3.3 its image T ⊆ P 5 is a projectively normal surface. We consider the image Γ ⊆ P 5 of the strict transform of Γ ′ in Bl 9 (P 2 ) under the map ϕ. Then Γ has nodes at the points n i := ϕ(n ′ i ) ∈ P 5 for i = 1, . . . , 8 and is of degree Comparing degrees, we conclude that Γ ⊆ P 5 is a quadratic section of T . Furthermore, we have a sequence of maps C → Γ → Γ ′ , showing that the smooth hyperelliptic curve C is the normalization of Γ. Summarizing all this, the assignment We now show that each irreducible 8-nodal curve [Γ, L] ∈ Hyp 4,8 which is general in any component of Hyp 4,8 appears this way. We fix such a pair and we may assume that L ∈ W 2 8 (Γ) is base point free. Let ν : C → Γ be the normalization map and denote by Γ ′ the image of the map φ |L| : Γ → P 2 . Setting {n 1 , . . . , n 8 } = Sing(Γ), we denote by {x i , y i } := ν −1 (n i ) the inverse images of the nodes of Γ for i = 1, . . . , 8. Then [C, ν * (L)] is a general point of the universal Picard variety Pic 8 H 4 . Via Proposition 3.2 we may assume C is embedded in the cubic scroll Z ∼ = Bl o (P 2 ) ⊆ P 4 as a curve in the linear system |6h − 4E|. Furthermore where, as usual, E is the exceptional divisor at the point o. Set π := φ |L| • ν : C → P 2 .

Proof of the claims (i) and (ii).
By degeneration we exhibit a point p := (t 1 , . . . , t 8 , ℓ, C) ∈ T , where C is a reducible nodal curve, such that ϑ(p) is well-defined and both (i) and (ii) hold.
The linear system I {o 1 ,...,o 8 }/P 2 (3) is a general pencil of plane cubics. Its general member is smooth and its 12 singular members are irreducible one-nodal rational curves with singularities disjoint from the set {o 1 , . . . , o 8 }. The plane cubics through o 1 , . . . , o 8 cut out the canonical linear system on B, that is, This implies that the ninth remaining base point of the pencil I {o 1 ,...,o 8 }/P 2 (3) does not lie on B ′ , since otherwise B would have a pencil of degree one, hence B would be rational.
We now choose two general rulings ℓ 1 and ℓ 2 of Z, that is, ℓ i ≡ h−E, and set F ′ i := π ℓ (ℓ i ). Both F ′ 1 and F ′ 2 are lines in P 2 meeting in a point o 9 . Furthermore, ℓ i · B = 2 and we set (11) viewed as a nodal hyperelliptic curve of genus 4. Note that both ℓ 1 and ℓ 2 meet B in a pair of hyperelliptic conjugate points. The image curve is a reducible nodal octic, where for i = 1, 2, the intersection B ′ ·F ′ i consists of 6 nodes, namely the 2 points in π ℓ (B · ℓ i ), as well as 4 further nodes on each F ′ 1 and F ′ 2 respectively.
Since o 9 can be chosen freely in P 2 , through the points o 1 , . . . , o 9 there passes a unique smooth cubic. Therefore, the map ϕ := φ |4h−E 1 −···−E 9 | : Bl o 1 ,...,o 9 (P 2 ) ֒→ T ⊆ P 5 is an embedding. The image F i ⊆ P 5 of the strict transform in Bl o 1 ,...,o 9 (P 2 ) of F ′ i is a twisted cubic, whereas the image under ϕ of the proper transform of B ′ can be identified with the original smooth genus 2 curve B embedded by the linear system ω B ⊗ π * ℓ|B O P 2 (1) . The intersection F i · B on T is transverse and consists of the 6 points in ϕ(F ′ i · B ′ ), for i = 1, 2. Finally, F 1 and F 2 are disjoint. We consider the nodal curve (12)  Proof. Keeping the notation from the proof of Theorem 3.5, we consider the reducible nodal curve Γ = F 1 + F 2 + B defined by (12) and which appears as a quadratic section of the surface T ⊆ P 5 . We have the following commutative diagram: The bottom map in this diagram is surjective. By Proposition 3.3 the surface T is projectively normal, thus it follows that the same holds for Γ.

FROM SCROLLS OF DEGREE 9 TO NODAL HYPERELLIPTIC CURVES
Let H 9 denote the Hilbert scheme of degree 9 scrolls R ⊆ P 5 . The general point of H 9 corresponds to a smooth degree 9 scroll R ⊆ P 5 . Following e.g. [Lai,Lemma 1.5], one knows that H 9 is smooth of dimension h 0 (R, N R/P 5 ) = 59. We denote by H 8 9 the closure in H 9 of the locus of scrolls having precisely 8 (non-normal) nodes and no further singularities. Using [Lai,Proposition 0.4] it follows that H 8 9 is nonempty and has pure codimension 8 inside H 9 , that is, dim(H 8 9 ) = 51. We introduce the parameter space of unparametrized degree 9 nodal scrolls H scr := H 8 9 /P GL(6). Theorem 1.2, coupled with results from [Lai], imply that H scr is an irreducible variety of dimension 16 = dim(H 8 9 ) − dim P GL(6). Each nodal scroll [R] ∈ H scr is a projection π : P 10 P 5 of a smooth degree 9 scroll F 1 := Bl o (P 2 ) ֒→ R ′ ⊆ P 10 , embedded by the linear system φ |5h−4E| : F 1 ֒→ P 10 . Here h is the pull-back of the line class under the morphism F 1 → P 2 and E denotes the exceptional divisor corresponding to the point o ∈ P 2 . The rulings of R ′ are the fibers of the morphism φ |h−E| : R ′ → P 1 and correspond to lines in P 10 . The center of the projection π is a 4-plane Λ ⊆ P 10 which is 8-secant to the secant variety Sec(R ′ ) and the restriction π : R ′ → R of the projection map π may be regarded as the normalization of R. We denote by {n 1 , . . . , n 8 } ⊆ P 5 the set of (non-normal) nodes of R and by {x i , y i } = π −1 (n i ) ⊆ R ′ , for i = 1, . . . , 8. The projections of the rulings of R ′ ⊆ P 10 passing through x i and y i correspond to lines on R ⊆ P 5 meeting in the node n i .
Next we show that the assignment [R] → [Γ] described by (2) induces a well defined map χ : P Hyp 4,8 . For our next result, recall that we have studied in Theorem 3.5 the dominant morphism ϑ : T Hyp 4,8 .
Proof. Recall that π : R ′ → R is the normalization map and set ℓ ′ i := π −1 (ℓ i ). Since ℓ ′ i is a ruling of R ′ we have that ℓ i ≡ h − E, where we have identified F 1 and R ′ . From (2) we then obtain is the hyperelliptic linear system on C. For a general choice of the rulings, we have ℓ i ∩{n 1 , . . . , n 8 } = ∅, therefore from (2) it follows that n i ∈ Γ and hence x i , y i ∈ C.
Proof. We pick a general point r ∈ Π \ B. Then for a quadric Z ∈ H 0 P 5 , I Γ/P 5 (2) one has that Π ⊆ Z if and only if r ∈ Z. Indeed, if r ∈ Z, then the restriction of Z to Π already contains B ∪ {r }, therefore Π ⊆ Z. Since containing the fixed point r imposes one condition on |I Γ/P 5 (2)|, the conclusion follows.
We now introduce the surface T ⊆ P 5 , defined as the residual surface to Π in the complete intersection (3), that is, Bs |V | = Π + T.
Thus T is a degree 7 surface in P 5 lying on three quadrics whose intersection contains a 2-plane. Such surfaces are classified in [Io] and there are five possible families. But the geometric situation at hand helps us show that T is the surface described in Proposition 3.3. Since Γ is nondegenerate in P 5 , in particular Γ Π, hence Γ ⊆ T . It follows that Γ is the intersection of T with one of the quadrics from H 0 P 5 , I Γ/P 5 (2) \ V . Since the intersection Γ ∩ B is transverse, one has n i / ∈ B, and hence n i ∈ P 5 \ Π. We set n ′ i := p(n i ) ∈ P 2 for i = 1, . . . , 8, where (16) p = p Π : P 5 P 2 is the projection with center the 2-plane Π.
Remark 4.6. The conic B defined in (14) is the intersection of the quadric q with the 2-plane Π, whereas the cycle Γ · B of length 6 is precisely the intersection cycle Γ · J on the smooth surface T , where J ∈ |3h − E 1 − · · · − E 9 |.