SINGULARITIES OF THETA DIVISORS AND THE GEOMETRY OF A5
G. FARKAS, S. GRUSHEVSKY, R. SALVATI MANNI, AND A. VERRA
Abstract. We study the codimension two locus H in Ag consisting of principally polarized abelian varieties whose theta divisor has a sin- gularity that is not an ordinary double point. We compute the class 2 [H] ∈ CH (Ag) for every g. For g = 4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g = 5, via the Prym map we show that H ⊂A5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone ′ of A5 and show that the component N0 of the Andreotti-Mayer divisor ′ has minimal slope and the Iitaka dimension κ(A5, N0) is equal to zero.
Introduction The theta divisor Θ of a generic principally polarized abelian variety (ppav) is smooth. The ppav (A, Θ) with a singular theta divisor form the Andreotti-Mayer divisor N in the moduli space , see [AM67] and [Bea77]. 0 Ag The divisor N0 has two irreducible components, see [Mum83] and [Deb92], ′ which are denoted θnull and N0: here θnull denotes the locus of ppav for ′ which the theta divisor has a singularity at a two-torsion point, and N0 is the closure of the locus of ppav for which the theta divisor has a singularity not at a two-torsion point. The theta divisor Θ of a generic ppav (A, Θ) θnull has a unique singular point, which is a double point. Similarly, the theta∈ divisor ′ of a generic element of N0 has two distinct double singular points x and x. Using this fact, one can naturally assign multiplicities to both components− of N0 such the following equality of cycles holds, see [Mum83],[Deb92]
′ (1) N0 = θnull + 2N0.
As it could be expected, generically for both components the double point is an ordinary double point (that is, the quadratic tangent cone to the theta divisor at such a point has maximal rank g — equivalently, the Hessian matrix of the theta function at such a point is non-degenerate). Motivated by a conjecture of H. Farkas [HF06], in [GSM08] two of the authors considered the locus in θnull in genus 4 where the double point is not ordinary. In [GSM07] this study was extended to arbitrary g, considering the sublocus g−1 θnull θnull parameterizing ppav (A, Θ) with a singularity at a two-torsion point,⊂ that is not an ordinary double point of Θ. In particular it has been 1 proved that (2) θg−1 θ N ′ . null ⊂ null ∩ 0 In fact the approach yielded a more precise statement: Let φ : Xg → Ag be the universal family of ppav over the orbifold g and g be the locus of singular points of theta divisors. Note thatA canS be ⊂X viewed as a S subscheme of g given by the vanishing of the theta functions and all its partial derivatives,X see Section 1. Then decomposes into three equidimen- sional components [Deb92]: , projectingS to θ , ′, projecting to N ′ , Snull null S 0 and dec, projecting (with (g 2)-dimensional fibers) onto 1 g−1. It S − g−1 A ×A is proved in [GSM07] that set-theoretically, θnull is the image in g of the ′ A intersection null . An alternative proof of these results has been found by Smith andS Varley∩S [SV12a],[SV12b]. It is natural to investigate the non-ordinary double points on the other ′ g−1 component N0 of the Andreotti-Mayer divisor. Similarly to θnull , we define ′g−1 ′ N0 , or, to simplify notation, H, to be the closure in N0 of the locus of ppav whose theta divisor has a non-ordinary double point singularity. Note that H is the pushforward under φ of a subscheme of g given by a Hessian condition on theta functions. In particular HHcan beX viewed as a codimension 2 cycle (with multiplicities) on g. Since an explicit modular ′ A form defining N0 and the singular point is not known, we consider the cycle g−1 g−1 ′g−1 g−1 (3) N0 := θnull + 2N0 = θnull + 2H. g−1 We first note that θnull is a subset of H. Then, after recalling that the Andreotti-Mayer loci N are defined as consisting of ppav (A, Θ) with i ∈Ag dim Sing(Θ) i, we establish the set-theoretical inclusion Ni H, for i 1. From this≥ we deduce: ⊂ ≥ Proposition 0.1. For g 5 we have θg−1 H. ≥ null To further understand the situation, especially in low genus, we compute the class: Theorem 0.2. The class of the cycle H inside is equal to Ag g! [H]= (g3 + 7g2 + 18g + 24) (g + 4)2g−4(2g + 1) λ2 CH2( ). 16 − 1 ∈ Ag
As usual, λ1 := c1(E) denotes the first Chern class of the Hodge bun- dle and CHi denotes the Q-vector space parameterizing algebraic cycles of codimension i with rational coefficients modulo rational equivalence. Com- 3 paring classes and considering the cycle-theoretic inclusion 3θnull H, we get the following result, see Section 4 for details: ⊂ 3 Theorem 0.3. In genus 4 we have the set-theoretic equality θnull = H. We then turn to genus 5 with the aim of obtaining a geometric description of H 5 via the dominant Prym map P : 6 5. A key role in the ⊂ A 2 R → A study of the Prym map is played by its branch divisor, which in this case ′ equals N0 5, and its ramification divisor 6. We introduce the antiramification⊂ A divisor defined cycle-theoreticallyQ⊂R by the equality U⊂R6 P ∗(N ′ ) = 2 + . 0 Q U Using the geometry of the Prym map, we describe both and explicitly in Q U terms of Prym-Brill-Noether theory. For a Prym curve (C,η) g and an integer r 1, we recall that V (C,η) denotes the Prym-Brill-Noether∈R locus ≥− r (see Section 5 for a precise definition). It is known [Wel85] that Vr(C,η)isa Lagrangian determinantal variety of expected dimension g 1 r+1 . We − − 2 denote π : g g the forgetful map. Our result is the following: R →M Theorem 0.4. The ramification divisor of the Prym map P : 6 5 equals the Prym-Brill-Noether divisor in Q , that is, R →A R6 = (C,η) : V (C,η) = 0 . Q { ∈R6 3 } The antiramification divisor is the pull-back of the Gieseker-Petri divisor from , that is, = π∗( 1 ). The divisor is irreducible and reduced. M6 U GP6,4 Q As the referee pointed out to us, the irreducibility of (as well as that of ) also follows from Donagi’s results [Don92] on the monodromQ y of the U Prym map P : 6 5. Apart from the Brill-Noether characterization pro- vided by TheoremR →A 0.4, the divisor has yet a third (respectively a fourth!) Q geometric incarnation as the closure of the locus of points (C,η) 6 with 2 ∈R2 a linear series L W6 (C), such that the sextic model ϕL(C) P has a totally tangent conic,∈ see Theorem 8.1 (respectively as the locus⊂ of section (C,η) of Nikulin surfaces [FV11]). The rich geometry of enables ∈ R6 Q us to (i) compute the classes of the closures and inside the Deligne- Q U Mumford compactification , then (ii) determine explicit codimension two R6 cycles in 6 that dominate the irreducible components of H. In this way we find aR complete geometric characterization of 5-dimensional ppav whose 4 theta divisor has a non-ordinary double point. First we characterize θnull as the image under P of a certain component of the intersection P ∗(θ ): Q∩ null 4 Theorem 0.5. A ppav (A, Θ) 5 belongs to θnull if an only if it is lies in the closure of the locus of Prym∈A varieties P (C,η), where (C,η) is a ∈ R6 curve with two vanishing theta characteristics θ1 and θ2, such that η = θ θ∨. 1 ⊗ 2 Furthermore, θ4 is unirational and [θ4 ] = 27 44 λ2 CH2( ). null null 1 ∈ A5 ∨ Denoting by 5 6 the locus of Prym curves (C,η = θ1 θ2 ) as above, Q ⊂R 4 ⊗ we prove that 5 (and hence θnull which is the closure of P ( 5) in 5) is unirational, byQ realizing its general element as a nodal curve Q A 2 C 1 1 (5, 5) , R1 R2/P ×P ∈ I where R P1 P1 (3, 1) and R P1 P1 (1, 3) , with the vanishing 1 ∈ |O × | 2 ∈ |O × | theta-nulls θ1 and θ2 being induced by the projections on the two factors. 3 4 2 Observing that [H] =[θnull] in CH ( 5), the locus H must have extra ir- reducible components corresponding toA ppav with a non-ordinary singularity that occurs generically not at a two-torsion point. We denote by H1 5 the union of these components, so that at the level of cycles ⊂ A 4 H = θnull + H1, where [H ] = 27 49 λ2. We have the following characterization of H : 1 1 1 Theorem 0.6. The locus H1 is unirational and its general point corresponds to a Prym variety P (C,η), where (C,η) 6 is a Prym curve such that η W (C) W 1(C) and K η is very ample.∈ R ∈ 4 − 4 C ⊗
As an application of this circle of ideas, we determine the slope of 5. Let be the perfect cone (first Voronoi) compactification of — thisA is the Ag Ag toroidal compactification of g constructed using the first Voronoi (perfect) fan decomposition of the coneA of semi-positive definite quadratic forms with rational nullspace (see e.g. [Vor1908] for the origins, and [SB06] for recent progress). The Picard group of g with rational coefficients has rank 2 (for g 2), and it is generated by theA first Chern class λ of the Hodge bundle ≥ 1 and the class of the irreducible boundary divisor D := . The slope Ag −Ag of an effective divisor E Eff( ) is defined as the quantity ∈ Ag a s(E) := inf : a,b > 0, aλ1 b[D] [E]= c[D], c> 0 . b − − If E is an effective divisor on g with no component supported on the bound- ary and [E]= aλ bD, thenAs(E) := a 0. One then defines the slope (of 1 − b ≥ the effective cone) of the moduli space as s( g) := inf s(E). This A E∈Eff(Ag) important invariant governs to a large extent the birational geometry of ; Ag for instance is of general type if s( )
1. Theta divisors and their singularities In this section we recall notation, definitions, as well as some results from [GSM08]. We denote Hg the Siegel upper half-space, i.e. the set of symmetric complex g g matrices τ with positive definite imaginary part. If σ = a b × Sp(2g, Z) is a symplectic matrix in g g block form, then its action c d ∈ × −1 on τ Hg is defined by σ τ := (aτ + b)(cτ + d) , and the moduli space of complex∈ principally polarized abelian variety (ppav for short) is the quotient = H / Sp(2g, Z), parameterizing pairs (A , Θ ) with A = Cg/Zgτ +Zg, Ag g τ τ τ an abelian variety and Θτ the (symmetric) polarization bundle. We denote by A [2] the group of two-torsion points of A . Let ε,δ (Z/2Z)g, thought τ τ ∈ of as vectors of zeros and ones; then x = τε/2+ δ/2 Aτ [2], and the shifted ∗ ∈ bundle txΘ is still a symmetric line bundle. Up to a multiplicative constant the unique section of the above bundle is given by the theta function with 5 characteristic [ε,δ] defined by ε ε ε ε δ θ (τ,z) := exp πi t(m + )τ(m + ) + 2 t(m + )(z + ) . δ 2 2 2 2 m ∈Zg We shall write θ(τ,z) for the theta function with characteristic [0, 0]. The zero scheme of θ(τ,z), as a function of z A , defines the principal polar- ∈ τ ization Θτ on Aτ . Theta functions satisfy the heat equation ε ε ∂2θ (τ,z) ∂θ (τ,z) δ δ = 2πi(1 + δj,k) , ∂zj∂zk ∂τjk
(where δj,k is Kronecker’s symbol). The characteristic [ε,δ] is called even or odd corresponding to whether the scalar product tεδ Z/2Z is zero or one. Consequently, depending on the ε ∈ characteristic, θ (τ,z) is even or odd as a function of z. A theta constant δ is the evaluation at z = 0 of a theta function. All odd theta constants of course vanish identically in τ. A holomorphic function f : Hg C is called a modular form of weight k with respect to a finite index subgroup→ Γ Sp(2g, Z) if ⊂ f(σ τ) = det(cτ + d)kf(τ) τ H , σ Γ, ∀ ∈ g ∀ ∈ and if additionally f is holomorphic at all cusps of Hg/Γ. Theta constants 1 with characteristics are modular forms of weight 2 with respect to a finite index subgroup Γg(4, 8) Sp(2g, Z). We refer to [Igu72] for a detailed study of theta functions. ⊂ We denote by φ : = H Cg/(Sp(2g, Z) ⋊ Z)2g) = H / Sp(2g, Z) Xg g × →Ag g the universal family of ppav, and let Θg g be the universal theta divisor — the zero locus of θ(τ,z). Following⊂X Mumford [Mum83], we denote by := Singvert Θg the locus of singular points of theta divisors of ppav: (4)S g ∂θ = Sing Θτ = (τ,z) Hg C : θ(τ,z)= (τ,z) = 0, i = 1,...,g S ∈ × ∂zi τ ∈Ag (computationally, by an abuse of notation, we will often work locally on , thinking of it as a locus inside the cover H Cg of ). It is known S g × Xg that g is of pure codimension g +1, and has three irreducible compo- S⊂X ′ nents [CvdG00], denoted null, dec, and . Here null denotes the locus of even two-torsion points thatS lieS on the thetaS divisor,S given locally by g + 1 equations (5) := (τ,z) : θ(τ,z) = 0, z =(τε + δ)/2 for some [ε,δ] (Z/2Z)2g . Snull ∈Xg ∈ even 6 To define dec, recall that a ppav is called decomposable if it is isomorphic to a productS of lower-dimensional ppav. We denote then (6) := φ−1( ). Sdec S∩ A1 ×Ag−1 Since the theta divisor of a product (A , Θ ) (A , Θ ) is given by the 1 1 × 2 2 union (Θ1 A2) (A1 Θ2), its singular locus contains Θ1 Θ2 and is of codimension× 2 (see∪ the× work [EL97] of Ein and Lazarsfeld for× a proof of a conjecture [ADC84] of Arbarello and De Concini that Ng−2 is in fact equal to the decomposable locus). Thus the fibers of dec 1,g−1 are all of dimension g 2, and the codimension of S is→ equal A to g + 1. (We − Sdec ⊂ Xg note that any other locus of products h g−h has codimension h(g h), and contributes no irreducible componentA ×A of ). − Finally, ′ is the closure of the locus of singularS points of theta divisors of indecomposableS ppav that are not two-torsion points. Observe that , ′, Snull S and dec all come equipped with an induced structure as determinantal S g subschemes of Hg C . The Andreotti-Mayer× divisor is then defined (as a cycle) by N := φ ( )= τ : SingΘ = . 0 ∗ S { ∈Ag τ ∅} It can be shown that N0 is a divisor in g, which has at most two irreducible components, see [Deb92],[Mum83]. A The theta-null divisor θ is the zero locus of the modular form null ⊂Ag ε F (τ) := θ (τ, 0). g δ m even Geometrically, it is the locus of ppav for which an even two-torsion point lies on the theta divisor, and it can be shown that θnull = φ∗( null), viewed as an S ′ 1 ′ equality of cycles. Similarly for the other component we have N0 = 2 φ∗( ) ′ S (the one half appears because a generic ppav in N0 has two singular points x on the theta divisor). ± Remark 1.1. The two components of N0 are zero loci of modular forms (with some character χ in genus 1 and 2): θnull is the zero locus of the g−2 g ′ modular form Fg of weight 2 (2 +1), while N0 must be the zero locus of g−3 g some modular form Ig of weight g!(g + 3)/4 2 (2 + 1) (the class, and thus the weight, was computed by Mumford− [Mum83]). Unlike the explicit formula for Fg, the modular form Ig is only known explicitly for g = 4, in which case it is the so called Schottky form [Igu81a],[Igu81b]. Various approaches to constructing Ig explicitly were developed in [Yos99],[KSM02].
2. Double points on theta divisors that are not ordinary double points We shall now concentrate on studying the local structure of a theta divisor near its singular point. For this, we look at the tangent space to and the map between the tangent spaces. S 7 Proposition 2.1. Let x =(τ ,z ) be a smooth point of . Then the map 0 0 0 S (dφ)x0 : Tx0 ( ) Tτ0 (N0) is injective if and only if the Hessian matrix S → 2 2 ∂ θ (x ) ... ∂ θ (x ) ∂z1∂z1 0 ∂z1∂zg 0 . . . H(x0) := . .. . ∂2θ ∂2θ (x0) ... (x0) ∂zg∂z1 ∂zg∂zg has rank g. Proof. Since the subvariety is defined by the g + 1 equations (4), S ⊂Xg the point x is smooth if and only if the ( g(g+1) + g) (g + 1) matrix 0 2 × ∂θ ... ∂θ ∂θ ... ∂θ ∂τ11 ∂τgg ∂z1 ∂zg 2 2 2 2 ∂ θ ... ∂ θ ∂ θ ... ∂ θ ∂z1∂τ11 ∂z1∂τ ∂z1∂z1 ∂z1∂z M(τ ,z ) := gg g 0 0 ...... ...... 2 2 2 2 ∂ θ ... ∂ θ ∂ θ ... ∂ θ ∂z1∂τ11 ∂z1∂τgg ∂zg∂z1 ∂zg∂zg evaluated at x0 =(τ0,z0) has rank g + 1. We compute ∂θ (x ) ... ∂θ (x ) 0 ... 0 ∂τ11 0 ∂τgg 0 2 2 2 2 ∂ θ (x ) ... ∂ θ (x ) ∂ θ (x ) ... ∂ θ (x ) ∂z1∂τ11 0 ∂z1∂τ 0 ∂z1∂z1 0 ∂z1∂z 0 M(τ ,z )= gg g 0 0 ...... ...... ∂2θ ∂2θ ∂2θ ∂2θ (x0) ... (x0) (x0) ... (x0) ∂z1∂τ11 ∂z1∂τgg ∂zg∂z1 ∂zg∂zg g(g+1) Since the map φ is the projection on the first 2 coordinates, the propo- sition follows. Remark 2.2. From the heat equation for the theta function it follows that if the Hessian matrix H(x0) has rank g, then x0 is a smooth point of . We also note that from the product rule for differentiation and theS heat equation it follows that the second derivative ε ∂2θ (τ,z) δ z=0 ∂zj∂zk | ε restricted to the locus θ (τ, 0) = 0 is also a modular form for Γ (4, 8). δ g Since we have different, easier to handle, local defining equations (5) for , we can obtain better results in this case. Snull Proposition 2.3. A point x0 null is a smooth point of null unless ∂θ ∈ S S (x0) = 0 for all 1 i,j g. The map (dφ)x0 is injective if and only if ∂τij ≤ ≤ the Hessian matrix H(x0) has rank g. Remark 2.4. If x =(τ ,z ) is a smooth point of , while τ is singular 0 0 0 Snull 0 in θnull, this implies that at least two different theta constants vanish at τ0. 8 Using the above framework, we get a complete description of the inter- ′ section null , obtaining thus an easier proof of one of the main results of [GSM07].S ∩S Proposition 2.5. For x , the point x lies in ′ if and only if the 0 ∈ Snull 0 S rank of H(x0) is less than g. Proof. If x ′ , then it is a singular point in , hence the rank of 0 ∈ S ∩Snull S H(x0) is less than g by the above proposition. To obtain a proof in the other direction, since z0 is a two-torsion point, the matrix M(τ0,z0) appearing in the proof of the proposition above has the form ∂θ (x ) ... ∂θ (x ) 0 ... 0 ∂τ11 0 ∂τgg 0 2 2 0 ... 0 ∂ θ (x ) ... ∂ θ (x ) ∂z1∂z1 0 ∂z1∂z 0 M(x )= g 0 ...... ...... ∂2θ ∂2θ 0 ... 0 (x0) ... (x0) ∂zg∂z1 ∂zg∂zg Hence if the rank of H(x0) is less than g, x0 is a singular point of ; thus S ′ either it is a singular point of null, or it lies in the intersection null . The first case cannot happen forS dimensional reasons (the singularS locus∩S of null is codimension at least 2 within null, see also [CvdG00]), and thus we mustS have x ′. S 0 ∈Snull ∩S Corollary 2.6. Set theoretically we have φ( ′)= θg−1. Snull ∩S null Remark 2.7. From the previous proof it also follows that Sing null ′. S ⊂ Snull ∩S Our further investigation will consider the subvariety := ′g−1 := x =(τ ,z ) ′ : rk H(x ) Proposition 2.8. The Andreotti-Mayer locus N1 is contained in H. Proof. Indeed, for τ N we let z(t) Sing Θ be a curve of singular 0 ∈ 1 ⊂ τ points such that z(0) = z0 is a smooth point of the curve. Differentiating (4) with respect to t, we get g non-zero equations (the derivative of the first one will vanish): g ∂2θ(τ ,z(t)) ∂z (t) 0 j = 0. ∂z ∂z ∂t j=1 i j Denoting ∂z ∂z v := 1 ,..., g ∂t ∂t |t=0 this means that H(x ) v = 0, and since by our assumption z is a smooth 0 0 point of the curve and thus v = 0, the matrix H(x0) has a kernel, and in particular is not of maximal rank. Corollary 2.9. For g 5 the locus of Jacobians is contained in H. ≥ Jg Hence for g 5 set-theoretically θg−1 H. ≥ null Proof. Indeed, we have N H for g 5. However, since for all g the Jg ⊂ 1 ⊂ ≥ divisor θ does not contain , we must have H θg−1. null Jg Jg ⊂ \ null 3. Class computations in cohomology In this section we compute the class of the components of the expected dimension of the loci and H in Chow and cohomology rings (our compu- tation works in both,H as we only use Chern classes of vector bundles) of Xg and g, respectively. A ′ Recall that Mumford [Mum83] computed the class of N0 in the Picard group of the partial toroidal compactification of the moduli space (the Ag class of θnull is easier, and was computed previously by Freitag [Fre83]). We g−1 shall compute the classes of the codimension 2 cycles H and θnull on g. 10 A As a consequence we will obtain a complete description of H in genus 4, rederive some result of [GSM08], and reprove that for g 5 the locus H has g−1 ≥ other components besides θnull . Debarre [Deb92, Section 4] computed the ′ class of the intersection θnull N0 and used this to show that this intersection is not irreducible. In spirit∩ our computation is similar, though much more involved. For the universal family φ : we denote by Ω the relative Xg → Ag Xg/Ag cotangent bundle, by E := φ∗ΩXg/Ag we denote its pushforward — the rank g vector bundle that is called the Hodge bundle. Then the Hodge class λ1 := c1(E) is the Chern class of the line bundle of modular forms of weight one on g. The basicA tool for our computation of pushforwards is the following: Lemma 3.1. The pushforward under φ of powers of the universal theta divisor Θ can be computed as follows: ⊂Xg 0 if k [N0]= φ∗ cg ΩXg/Ag (Θ) OXg Θ . ⊗ O g−1 Recall now that is defined by the equation det H(x0)=0. On , each second derivativeS ⊂ S of the theta function is a section of Θ, and the determinantS of the Hessian matrix is known (see [GSM07],[dJ10]) to be a section of (gΘ) φ∗(det E)⊗2 . OXg ⊗ ⊗OS Using the above formula for the class of , to get H we will need to compute the pushforward S φ c Ω (Θ) (gΘ + 2φ∗λ ) ∗ g Xg/Ag |Θ 1 The computation becomes rather delicate since g−1 is not equidimen- sional. We set S ′ indec := dec = null, S S\S11 S ∪S which is then purely of codimension g +2 in , and thus we have Xg [ g−1 ]=[ ] (g[Θ]+2λ ) CHg+2( ). Sindec Sindec 1 ∈ Xg However, for dimension reasons it turns out that we often do not need to deal with the class of : Sdec Proposition 3.2. For g 4 we have the equality of codimension 2 classes on : ≥ Ag [N g−1]=[N g−1 ]:=[φ ( g−1 )] 0 0, indec ∗ Sindec Moreover this class can be computed as g! [N g−1]= (g3 + 7g2 + 18g + 24)λ2 CH2( ). 0 8 1 ∈ Ag Proof. The first statement is a consequence of the fact that the map φ has (g 2)-dimensional fiber along , and generically 0-dimensional fibers − Sdec over indec. (Note that this is the place in the argument where we are S g−1 using the assumption g 4 to ensure that indec is in fact non-empty, and that the codimension of≥ its image under φ isS lower than the codimension of .) We now compute A1 ×Ag−1 [N g−1]= φ c Ω (Θ) Θ (gΘ + 2φ∗λ ) 0 ∗ g Xg/Ag 1 = φ (Θg +Θ g −1φ∗λ +Θg−2φ ∗λ + ... ) (gΘ 2 + 2Θφ∗λ ) ∗ 1 2 1 φ∗λ2 = φ g Θg+2 +Θg+1φ∗λ +Θg 1 + (2Θg+1φ∗λ + 2Θgφ∗λ2) ∗ 1 2 1 1 g(g + 2)! g(g + 1)! g(g)! = + + +(g + 1)! + 2g! λ2 8 2 2 1 g! = (g3 + 7g2 + 18g + 24)λ2. 8 1 g−1 We now compute the class of the locus θnull : recall that a theta constant 1 is a modular form of weight 2 , and that the determinant of the Hessian ε matrix of θ (τ,z) evaluated at z = 0 is a modular form of weight g+4 δ 2 ε along the zero locus of θ (τ) (see [GSM08],[dJ10]). We thus get: δ Proposition 3.3. For g 2 we have ≥ g−1 g−3 g 2 [θnull ]=(g + 4)2 (2 + 1)λ1. Proof. Indeed, we have ε ∂2θ (τ,z) g−1 ε δ θ = τ H : [ε,δ] even ,θ (τ) = det = 0 , null ∈ g ∃ δ ∂z ∂z |z=0 j k 12 Since there are 2g−1(2g + 1) even characteristics, for each of them we get a contribution of λ1/2 (for the zero locus of the corresponding theta constant) times (g + 4)λ1/2 (for the Hessian). The proof of Theorem 0.2 comes by subtraction using the class formulas established in Propositions 3.2 and 3.3, while taking into account the relation given in formula (3). 4. The case g = 4 In this section we will work out the situation for genus 4 in detail, even- tually proving Theorem 0.3. By the above formulas for g = 4 we have [θ3 ] = 272λ2; [N 3] = 3 272λ2. null 1 0 1 g−1 ′g−1 Moreover, going back from N0 to H = N0 , we recall that for arbitrary g−1 g−1 genus by definition we have N0 = θnull +2H, and since at the intersection ′ of the two components θnull and N0 the singular points lie on both, we also have that set-theoretically θg−1 H null ⊂ As an immediate consequence we obtain: Proposition 4.1. The following identity holds at the level of codimension two cycles on 4: A 3 3 N0 = 3θnull. 3 Proof. From the formulas above we see that the cycle 2θnull appears inside ′3 3 3 2N0 , and thus that 3θnull is a subcycle of N0 . Since the Chern classes are 3 equal and θnull is equidimensional, we need to rule out the possibility of 3 N0 having an extra lower dimensional component. However, for genus 4 ′ we know geometrically that N0 is the locus of Jacobians. Using Riemann’s Singularity Theorem for genus 4 curves we then see that the period matrix ′3 of a Jacobian is in N0 if and only if its theta divisor is singular at a two- 3 torsion point, i.e. if this Jacobian lies in θnull (notice that this reproves a result of [GSM08]). The proof of Theorem 0.3 is an immediate consequence of the above facts. We can prove something more: let I4 be the Schottky modular form of weight 8 defining the Jacobian locus. Let then ∂I4 1 ∂I4 ... 1 ∂I4 ∂τ11 2 ∂τ12 2 ∂τ14 1 ∂I4 ∂I4 ... 1 ∂I4 2 ∂τ21 ∂τ22 2 ∂τ24 det (I4) := det D ...... 1 ∂I4 ∂I4 ...... 2 ∂τ41 ∂τ44 The restriction of this determinant to the zero locus of I4 is a modular form of weight 34 = 8 4 + 2. By Proposition 2.1 we know that for a point in 13 ′ N0 H the matrix (I4) is proportional to the Hessian matrix H(x0), hence \ D 3 it vanishes exactly along θnull. The class of the cycle I = det (I ) = 0 { 4 D 4 } is 8 34 λ2 = 272 λ2. Thus we obtain 1 1 Proposition 4.2. The locus θ3 is a complete intersection given by null ⊂A4 I = det (I ) = 0. 4 D 4 We observe that, by Riemann’s Theta Singularity theorem, this is the locus of Jacobians with theta divisor singular at a two-torsion point. More- over, the form √F4 (recall that Fg is the product of all even theta constants) is well-defined along the Jacobian locus and it has the same weight, hence we get a different proof of the following result recently obtained by Matone and Volpato [MV11]: Corollary 4.3. On 4 we have the equality √F4 = c det (I4) for some J 3 D constant c. The locus θnull can also be given by equations I4 = F4 = 0. Contrary to the situation in genus 4, for higher genera we know that we have other components, see Corollary 2.9. This fact can also be deduced from our class computation as follows. Proof of Proposition 0.1. Recall that the statement we are proving is that g−1 at the level of effective cycles θnull H for any g 5. We first note that the above discussion for the genus 4 case shows that the≥ cycle-theoretic inclusion holds. Secondly, since we have computed both classes, we see that for g 5 g−1 g−1 ≥ the class of N0 is not equal to 3 times the class of θnull . In fact the growth orders of the degrees of these two classes are respectively g! deg θg−1 4g−4 deg θ3 ; deg N g−1 deg N 3, null ∼ null 0 ∼ 4! 0 and one would thus expect many additional components. The rest of the paper is devoted to studying the geometry for g = 5 in detail; in this case we will be able to describe all components explicitly, and will also obtain many results describing the classical Prym geometry of the situation. 5. Prym theta divisors and their singularities While for higher g the geometry of the locus H appears quite ⊂ Ag intricate, for g = 5 one can use the Prym map P : 6 5. We begin by setting the notation and reviewing the basic facts aboutR → Pry Am varieties and their moduli, which will be used throughout the rest of the paper. Let be the moduli space of pairs (C,η) with [C] , and η a non- Rg ∈Mg zero two-torsion point of the Jacobian Pic0(C). We denote by f : C˜ C → the ´etale double cover induced by η (so the genus of C˜ is equal to 2g 1), 14 − by i : C˜ C˜ the involution exchanging the sheets of f, and by ϕ : → KC ⊗η C PH0(C, K η)∨ the Prym-canonical map. The map ϕ is an → C ⊗ KC ⊗η embedding if and only if η / C C (where we denote C := Symk(C)). ∈ 2 − 2 k We recall the definition of the Prym map P : g g−1. Consider the 2g−2 2g−2 R → A norm map Nmf : Pic (C˜) Pic (C) induced by the double cover f. The even component of the preimage→ −1 + 2g−2 ˜ 0 ˜ Nmf (KC ) := L Pic (C):Nmf (L)= KC , h (C,L) 0 mod2 ∈ ≡ 2g−2 ˜ is then an abelian variety of dimension g 1. Denoting by ΘC˜ Pic (C) the Riemann theta divisor, scheme-theoretically− we have the following⊂ equal- − ity ΘC˜ Nm 1(K )+ = 2Ξ, where Ξ is a principal polarization. The Prym | f C variety is defined to be the ppav −1 + P (C,η) := Nmf (KC ) , Ξ g−1. ∈A The polarization divisor can be described explicitly following [Mum74]: Ξ(C,η) := L Nm−1(K )+ : h0(C,L˜ ) > 0 . { ∈ f C } A key role in what follows is played by the Prym-Petri map − : 2H0(C,L˜ ) H0(C, K η), u v u i∗(v) v i∗(u), L ∧ → C ⊗ ∧ → − 0 0 ˜ − where one makes the usual identification H (C, KC η) = H (C, KC˜) with the ( 1) eigenspace under the involution i. Following⊗ [Wel85], for (C,η) −and r 1, we define the determinantal locus ∈Rg ≥− V (C,η) := L Nm−1(K ): h0(L) r + 1, h0(L) r +1 mod2 . r { ∈ f C ≥ ≡ } For a general Prym curve (C,η) , the map − is injective for every ∈ Rg L L Nm−1(K ), and dim V (C,η)= g 1 r+1 , see [Wel85]. ∈ f C r − − 2 For a point L Ξ, we recall the description of the tangent cone TCL(Ξ). 0 ∈ 0 Suppose h (C,L˜ ) = 2m 2 and we fix a basis s1 ...,s2m of H (C,L˜ ). Consider the skew-symmetric≥ matrix { } M := −(s s ) L L k ∧ j 1≤k,j≤2m m 0 and the pfaffian Pf(L) := det(ML) Sym H (C, KC η). Via the 0 ∈ ∨ ⊗ identification TL(P (C,η)) = H (C, KC η) we have the following result of [Mum74]: ⊗ Theorem 5.1. If h0(C,L˜ ) = 2 then Pf(L) = 0 is the equation of the pro- jectivized tangent space PTL(Ξ). If m 2 then L Sing(Ξ) and either Pf(L) 0, in which case mult (Ξ) ≥m + 1, or else,∈ Pf(L) = 0 is the ≡ L ≥ equation of the tangent cone PTCL(Ξ). Note that one can have L Sing(Ξ) even when m = 1 and Pf(L) is identically zero, so that the Prym∈ theta divisor Ξ can have two types of singularities, as follows: 15 Definition 5.2. For a point L Sing(Ξ), one says that ∈ (1) L is a stable singularity if h0(C,L˜ ) = 2m 4, (2) L is an exceptional singularity if L = f ∗(M≥) (B), where M Pic(C) ⊗OC˜ ∈ is a line bundle with h0(C,M) 2 and B is an effective divisor on C. ≥ st ex Let Singf (Ξ) = V3(C,η) be the locus of stable singularities and Singf (Ξ) the locus of exceptional singularities. Clearly Sing(Ξ) = Singst(Ξ) Singex(Ξ). f ∪ f Both these notions depend on the ´etale double cover f : C˜ C and are not intrinsic to Ξ. Furthermore, there can be singularities that→ are simulta- neously stable and exceptional. Every singularity of a 4-dimensional theta divisor Ξ can in fact be realized as both a stable and an exceptional singu- larity in different incarnations of (A, Ξ) as a Prym variety. ∈A5 For a decomposable vector 0 = u v 2H0(C,L˜ ), we set ∧ ∈∧ div(u) := Du + B, div(v) := Dv + B, where Du,Dv have no common components and B 0 is an effective divisor on C. The next lemma is well known, see [ACGH85,≥ Appendix C]: Lemma 5.3. For 0 = u v 2H0(C,L˜ ) the following are equivalent. ∧ ∈∧ (1) −(u v) = 0. L ∧ (2) D ,D f ∗M where M Pic(C) with h0(C,M) 2. u v ∈| | ∈ ≥ ∗ ⊗2 In such a case we write L = f (M) C˜(B), hence KC = M C (f∗(B)), 0 ⊗(−2) ⊗O ⊗O in particular h (C, KC M ) 1, and the Petri map 0(M) is not injective. In particular,⊗ Singex(Ξ) =≥ if C satisfied the Petri theorem. f ∅ 0 Suppose L V3(C,η)isa quadratic stable singularity, hence h (C,L˜ ) = 4 ∈ 5 2 0 ∨ g−2 0 ∨ and Pf(L) = 0. Setting P := P( H (L) ) and P := P(H (KC η) ), we consider the projectivized dual∧ of the Prym-Petri map ⊗ δ := P ( −)∨ : Pg−2 P5. L → The Pl¨ucker embedding of the Grassmannian G∗ := G(2,H0(L)∨) P5 −1 ∗ ⊂ is a rank 6 quadric whose preimage QL := δ (G ) is defined precisely by the Pfaffian Pf(L). Note also that rk(Q ) rk( −). On the other hand let L ≤ L G := G(2,H0(L)) P( 2H0(L)) be the dual Grassmannian. It is again a standard exercise in⊂ linear∧ algebra to show the equivalence rk(Q ) 4 G P Ker( −) = . L ≤ ⇐⇒ ∩ L ∅ For a point L Singst(Ξ) one has mult (Ξ) = 2 if and only if h0(C,M) 2 ∈ f L ≤ for any line bundle M on C such that h0(C,L˜ f ∗M ∨) 1, see [SV04]. We summarize this discussion as follows: ⊗ ≥ st Proposition 5.4. For a quadratic singularity L Singf (Ξ) the following conditions are equivalent: ∈ (1) rk(QL) 4. (2) G P ≤Ker( −) = . ∩ L ∅ (3) L Sing st(Ξ) Sing ex(Ξ). ∈ f ∩ f 16 6. Petri divisors and the Prym map in genus 6 This section is devoted to the study of singularities of Prym theta divisors of dimension 4 via the Prym map P : 6 5. We review a few facts about the Deligne-MumfordR →A compactification Rg of , and refer to [Don92] and [FL10] for details. The space is the Rg Rg coarse moduli space associated to the Deligne-Mumford stack Rg of stable Prym curves of genus g. The geometric points of correspond to triples Rg (X,η,β), where X is a quasi-stable curve with pa(X) = g, η Pic(X) is a line bundle of total degree 0 on X such that η = (1) for∈ each smooth E OE rational component E X with E X E = 2 (such a component is ⊂ ⊗2 | ∩ − | called exceptional), and β : η X is a sheaf homomorphism whose restriction to any non-exceptional→ component O is an isomorphism. Denoting π : the forgetful map, one has the formula [FL10, Example 1.4] Rg → Mg ′ ′′ (7) π∗(δ )= δ + δ + 2δram CH1( ), 0 0 0 0 ∈ Rg ′ ′ ′′ ′′ ram ram where δ0 := [∆0], δ0 := [∆0 ], and δ0 := [∆0 ] are boundary divisor classes on g whose meaning we recall. Let us fix a general point [Cxy] ∆0 correspondingR to a smooth 2-pointed curve (C,x,y) of genus g 1 and∈ the − ′ normalization map ν : C Cxy, where ν(x)= ν(y). A general point of ∆0 ′′ → (respectively of ∆0 ) corresponds to a stable Prym curve [Cxy,η], where η 0 ∗ 0 ∗ ∈ Pic (Cxy)[2] and ν (η) Pic (C) is non-trivial (respectively, ν (η) = C ). ram∈ O1 A general point of ∆0 is of the form (X,η), where X := C {x,y} P is 0 ∪ a quasi-stable curve with pa(X) = g, whereas η Pic (X) is a line bundle ⊗2 ∈ characterized by ηP1 = P1 (1) and ηC = C ( x y). For 1 i [ g ] we haveO a splitting of theO pull-back− − of the boundary ≤ ≤ 2 (8) π∗(δ )= δ + δ + δ CH1( ), i i g−i i:g−i ∈ Rg where the boundary classes δi := [∆i],δg−i := [∆g−i] and δi:g−i := [∆i:g−i] correspond to the possibilities of choosing a pair of two-torsion line bundles on a smooth curve of genus i and one of genus g i, such that the first one, the second one, or neither of the corresponding bundles− is trivial, respectively, see [FL10]. Often we content ourselves with working on the partial compactification −1 g := π ( g ∆0) of g. When there is no danger of confusion, we still R M′′ ′′∪ ramR denote by δ0 ,δ0 and δ0 the restrictions of the corresponding boundary ′ ′′ classes to . Note that CH1( )= Q λ,δ ,δ ,δram . Rg Rg 0 0 0 The extension of the (rational) Prym map P : 99K over the Rg Ag−1 general point of each of the boundary divisors of g is well-understood, ′′ R see e.g. [Don92]. The Prym map contracts ∆0 and all boundary divisors ∗ g π (∆i) for 1 i 2 . The Prym variety corresponding to a general ≤ ′′≤ ⌊ ⌋ point [Cxy,η] ∆0 as above is the Jacobian Jac(C) of the normalization. ′′ ∈ ∗ Thus P (∆0 ) = g−1. The pullback map P on divisors has recently been J 17 described in [GSM11]: one has ram δ ′ (9) P ∗(λ )= λ 0 , P ∗(D)= δ . 1 − 4 0 Remark 6.1. We sketch an alternative way of deriving the first formula in (9). For each (C,η) , there is a canonical identification of vector ∈ Rg bundles T ∨ = H0(C, K η) . The pull-back P ∗(E) of the P (C,η) C ⊗ ⊗OP (C,η) Hodge bundle can be identified with the vector bundle 1 on g with fiber 0 N R 1(C,η) = H (C,ωC η), over each point (C,η) g (we skip details showingN that this description⊗ carries over the boundary∈ R as well). Therefore P ∗(λ )= c ( )= λ 1 δram, where we refer to [FL10] for the last formula. 1 1 N1 − 4 0 ˜ f ex We have seen that for [C C] g with Singf (Ξ) = , the curve C → 1 ∈ R ∅ fails the Petri theorem. Let g,k g denote the Gieseker-Petri locus whose general element is a curveGP C⊂carrying M a globally generated pencil M W 1(C) with h0(C,M) = 2, such that the multiplication map ∈ k (M): H0(C,M) H0(C, K M ∨) H0(C, K ) 0 ⊗ C ⊗ → C is not injective. It is proved in [Far05] that for g+2 k g 1, the locus 2 ≤ ≤ − 1 has a divisorial component. As usual, we denote by r the locus GPg,k Mg,d of curves [C] such that W r(C) = . ∈Mg d ∅ In the case of 6 there are two Gieseker-Petri loci, both irreducible of pure codimensionM 1, described as follows: The locus 1 of curves [C] having a pencil M W 1(C) with • GP6,4 ∈ M6 ∈ 4 h0(C, K M ⊗(−2)) 1. We have the following formula for the class of its C ⊗ ≥ closure in , see [EH87]: M6 1 [ ] = 94λ 12δ 50δ 78δ 88δ CH1( ). GP6,4 − 0 − 1 − 2 − 3 ∈ M6 The locus 1 of curves with a vanishing theta characteristic; then • GP6,5 1 [ ] = 8 (65λ 8δ 31δ 45δ 49δ ) CH1( ). GP6,5 − 0 − 1 − 2 − 3 ∈ M6 The Prym map P : is dominant of degree 27 and its Galois R6 → A5 group equals the Weyl group of E6, see [DS81],[Don92]. The differential of the Prym map at the level of stacks ∨ (dP ) : H0(C, K⊗2)∨ Sym2 H0(C, K η) (C,η) C → C ⊗ is the dual of the multiplication map at the level of global se ctions for the Prym-canonical map ϕKC ⊗η. Thus the ramification divisor of P is a Cartier divisor on supported on the locus R6 ≇ := (C,η) : Sym2 H0(C, K η) H0(C, K⊗2) . Q ∈R6 C ⊗ −→ C The closure of P ( ) inside 5 is the branch divisor of P . At a general point (A, Θ) P ( Q) the fiberA of P has the structure of the set of lines on a one-nodal cubic∈ Q surface, that is, P −1(A, Θ) consists of 6 ramification 18 ∩Q points corresponding to the 6 lines through the node. The remaining 15 points of P −1(A, Θ) are in correspondence with the 15 lines on the one- nodal cubic surface not passing through the node. Since deg(P ) = 27 it follows that P has simple ramification and is reduced. Donagi [Don92, p. 93] established that is irreducible by showingQ that the monodromy acts Q transitively on a general fiber of P|Q. We sketch a different proof which uses the irreducibility of the moduli space of polarized Nikulin surfaces. We summarize these results as follows: Proposition 6.2. Set-theoretically, the branch divisor of the map P is equal to the closure N ′ of P ( ) in . At the level of cycles, P [ ]=6[N ′ ]. 0 Q A5 ∗ Q 0 We turn our attention to the geometry of . First we compute the class Q of its closure in , then we link it to Prym-Brill-Noether theory: R6 Theorem 6.3. The ramification divisor is irreducible. The class Q⊂R6 of its closure in equals Q R6 ′ 3 ′′ ram ′′ 1 [ ] = 7λ δ0 δ0 c δ0 CH ( 6), Q − − 2 − δ0 ∈ R where we have the estimate c ′′ 4. δ0 ≥ Proof. The irreducibility of follows from [FV11, Theorem 0.5], where it is proved that can be realizedQ as the image of a projective bundle over the irreducible moduliQ space N of polarized Nikulin K3 surfaces of genus 6. F6 To estimate the class of the closure of in , we set up two tautological Q Q R6 vector bundles and over having fibers N1 N2 R6 0 0 ⊗2 ⊗2 1(X,η) := H (X,ωX η) and 2(X,η) := H (X,ω η ) N ⊗ N X ⊗ 2 over a point (X,η) 6. There is a morphism φ : Sym ( 1) 2 between vector bundles of the∈ R same rank given by multiplicationN of Pr→Nym-canonical forms, and we denote by the degeneracy locus of φ. Using [FL10, Propo- Z sition 1.7] we have the following formulas in CH1( ) R6 1 ′ ′′ c ( )= λ δram and c ( ) = 13λ δ δ 3δram, 1 N1 − 4 0 1 N2 − 0 − 0 − 0 ′ ′′ 3 ram thus [ ]= c1( 2) 6c1( 1) = 7λ δ0 δ0 2 δ0 . By definition, = 6. Z N − N − − − ′ Q Z∩R Furthermore, φ is non-degenerate at a general point of ∆ and ∆ram, hence 0 ′′0 the difference is an effective divisor supported only on ∆0 . Z− Q ′′ Assume now that (X,η) ∆ is a generic point corresponding to a nor- ∈ 0 malization map ν : C X, where [C,x,y] 5,2 and x,y C are distinct → ∗ ∈M ∈ points such that ν(x)= ν(y). Since ν (η)= C , we obtain an identification 0 0 0 O ⊗2 ⊗2 H (X,ωX η) = H (C, KC ) whereas H (X,ωX η ) is a codimension ⊗ 0 ⊗2 ⊗ one subspace of H (C, KC (2x + 2y)) described by a residue condition at x and y. It is straightforward to check that the kernel Ker φ(X,η) = Ker Sym2 H0(C, K ) H0(C, K⊗2) C → C 19 ′′ ′′ has dimension 3. Thus [ ] [ ] 3δ is effective supported on ∆ , which Z − Q − 0 0 implies that c ′′ 4. δ0 ≥ Remark 6.4. We shall prove later that in fact c ′′ = 4. δ0 Even though the locus is defined in terms of syzygies of Prym-canonical curves, its points have aQ characterization in terms of stable singularities of Prym theta divisors. Theorem 6.5. The theta divisor of a Prym variety P (C,η) 5 has a stable singularity if and only if P ramifies at the point (C,η), that∈ A is, st = (C,η) 6 : Sing(C,η)(Ξ) = . Q ∈R ∅ Proof. Let us denote by := (C,η) 6 : V3(C,η) = the Prym-Brill- Noether locus correspondingW to{ stable∈R singularities of Pry∅}m theta divisors. Our aim is to show that = ; we begin by establishing the inclusion W Q . First note that if [C] 6 is trigonal, for any two-torsion point W⊂Q0 ∈ M ′ 1 η Pic (C)[2] C we can write KC η = A A , where A W3 (C) and∈ A′ W 1(C−). {O This} implies that (C,η)⊗ . ⊗ ∈ ∈ 7 ∈Q Fix now (C,η) and a line bundle L V (C,η). If h0(C,L˜ ) 6, then ∈R6 ∈ 3 ≥ C˜ (and hence C as well) must be hyperelliptic, so (C,η) by the previous ∈Q remark. We may thus assume that h0(C,L˜ ) = 4 and consider the associated 2 0 Pfaffian quadric QL Sym H (C, KC η). If QL = 0, then it contains the Prym-canonical model∈ ϕ (C), in particular⊗ (C,η ) . If Q 0, then KC ⊗η ∈Q L ≡ there exists M Pic(C) with h0(C,M) 3 and an effective divisor D on ˜ ∈ ∗ ≥ C, such that L = f (M) C˜(B). If deg(M) 4 then C is hyperelliptic, hence (C,η) . If deg(⊗OM) = 5, then B =≤ 0 and C is a smooth plane quintic such that∈ Q h0(C,M η) = 1. It is known, see [Don92, Section 4.3], that in this case P (C,η) is⊗ the intermediate Jacobian of a cubic threefold and the differential (dP ) has corank 2, thus once more (C,η) . (C,η) ∈Q Therefore . We claim that has at least a divisorial component, W⊂Q W which follows by exhibiting a point (C,η) 6 and a line bundle L − ∈ R ∈ V3(C,η) such that L is surjective. Assuming this for a moment, we conclude that = by invoking the irreducibility of . W Q Q To finish the proof we use a realization of Prym curves (C,η) with ∈R6 V3(C,η) = resembling [FV11, Section 2]. For a line bundle L V3(C,η) 0 ˜ ∅ + 2 0 ˜ 0 ∈ ∗ with h (C,L) = 4, if L : Sym H (C,L) H (C, KC ) denotes the i - invariant part of the Petri map, one has the following→ commutative diagram: (L,i∗L) ˜ / P3 P3 C YYYYYY × YYYYYY YYYY, f q P15 = P H0(L)∨ H0(L)∨ e e e e e e ⊗ + re µL C P/ 9 = P(Sym2 H0(L)∨) 20 In this diagram q : P3 P3 P9 is the map a b a b + b a into the projective space of× symmetric→ tensors. Reversing⊗ t →his construction,⊗ ⊗ if ι Aut(P3 P3) denotes the involution interchanging the two factors, the∈ complete intersection× of P3 P3 with 4 general ι∗-invariant hyper- 0 + × ∗ planes in H ( P3×P3 (1, 1)) and one general ι -anti-invariant hyperplane 0 I − ˜ 3 3 in H ( P3×P3 (1, 1)) is a smooth curve C P P ; the automorphism I ˜ ⊂ × − ι|C induces a double cover f : C C such that Ker( L ) has 1-dimensional → 0 − kernel corresponding to the unique element in H ( P3×P3 (1, 1)) . For more details on this type of argument, we refer to [FV11].I 7. The antiramification divisor of the Prym map In this section we describe geometrically the antiramification divisor of the map P : , defined via the equality of divisors U R6 →A5 (10) P ∗(N ′ ) = 2 + . 0 Q U For a general curve [C] 1 , if M W 1(C) denotes the pencil such ∈ GP6,4 ∈ 4 that (M) is not injective, we let x + y C be the support of the unique 0 ∈ 2 section of K M ⊗(−2). We consider the four line bundles C ⊗ L := f ∗M (x + y ) Nm−1(K ), u,v ⊗OC˜ u v ∈ f C where 1 u,v 2 and f(xu) = x, f(yv) = y. Using the parity flipping ≤ ≤ 0 lemma of [Mum74], exactly two of the quantities h (C,L˜ u,v) are equal to 2, ex the other being equal to 3, that is, Singf (Ξ) contains at least two points. Hence π∗( 1 ) . Using Theorem 6.3, equality (10), and the formula GP6,4 ⊂ U for [ 1 ] CH1( ), we compute GP6,4 ∈ M6 [ ]= P ∗([N ′ ]) 2[ ] = 108λ 14λ = π∗([ 1 ]) CH1( ). U 0 − Q − GP6,4 ∈ R6 Since the λ-coefficient of any non-trivial effective divisor class on 6 must be strictly positive, we obtain the following result: R Proposition 7.1. We have the following equality of divisors on : R6 = π∗( 1 ). U GP6,4 We now determine the pull-back of N ′ under the map P : 99K . As 0 R6 A5 usual, denotes the closure of inside . U U R6 Theorem 7.2. We have the following equality of divisors on 6: ′′ R P ∗(N ′ ) = 2 + + 20∆ . 0 Q U 0 ′ Proof. We use the formula [N0] = 108 λ1 14D, as well as Theorem 6.3 and − ∗ ′ ∗ 1 formula (9) in order to note that the effective class P (N0) 2 π ( 6,4) ′′ − Q− GP is supported only on the boundary divisor ∆0 . ′′ ∗ ′ We now prove that the multiplicity of ∆0 in P (N ) equals 20, or equiva- ′′ 0 ′′ ∗ lently mult (P (N0)) = 40, since P (∆0 )= 5 θnull. Let 5 := BlJ5 ( 5) ∆0 J A A 21 be the blowup of 5 along the Jacobian locus and denote by 5 the ex- A 2 E ⊂ A ceptional divisor. Then is a P -bundle over 5 with the fiber over a point E J ∨ (Jac(C), ΘC ) 5 being identified with the space P(I2(KC ) ) of pencils of quadrics containing∈ J the canonical curve C P4. One can lift the Prym ⊂ ′′ map to a map P˜ : 99K by setting for a general point (C ,η) ∆ R6 A5 xy ∈ 0 P˜ (C ,η ) := ((Jac(C), Θ ),q ) , xy C xy ∈ A5 where q P(I (K )∨) is the pencil of quadrics containing the union xy ∈ 2 C C x,y P4. Furthermore, P˜∗( )=∆ram, showing that ∪ ⊂ E 0 ′′ ∗ mult P (N0) = multJ5 (N0). ∆0 To estimate the latter multiplicity we consider a general one-parameter fam- ily j : U 5 from a disc U 0 such that j(0) = (Jac(C), ΘC ), with [C] →being A a general curve.∋ Let Θ := U Θ U be the rela- ∈ M5 U ×A5 → tive theta divisor over U. The image of the differential (dj)0(T0(U)) can be viewed as a hyperplane h P Sym2 H0(K ) . The variety Θ has ordinary ⊂ C U double points at those points (0,L) Θ where L Sing(Θ )= W 1(C) is ∈ U ∈ C 4 a singularity such that its tangent cone QL PI2(KC ) belongs to h. Since the assignment W 1(L) L Q PI (K ∈) is an unramified double cover 4 ∋ → L ∈ 2 C over a smooth plane quintic, we find that ΘU has 10 nodes. Using the theory of Milnor numbers for theta divisors as explained in [SV85] we obtain mult (N )= χ(θ ) χ(W (C)) + 10, J5 0 gen − 4 where χ(θgen) = 5! = 120 is the topological Euler characteristic of a gen- eral (smooth) theta divisor of genus 5. We finally determine χ(W4(C)), using the resolution C4 W4(C). From the Macdonald formula, see →g−1 2g−2 1 [ACGH85], χ(C4)=( 1) = 70, whereas χ(W (C)) = 20, − g−1 |g=5 4 − 1 1 1 because g(W4 (C)) = 11. Therefore χ(C4 ) = 2χ(W4 (C)) = 40. We find 1 1 − that χ(W4(C)) = χ(C4) χ(C4 )+ χ(W4 (C)) = 90, thus multJ5 (N0) = 120 90 + 10 = 40. − − Corollary 7.3. We have the following formula in CH1( ): R6 ′ ′′ 3 [ ] = 7λ δ 4δ δram. Q − 0 − 0 − 2 0 We exploit the geometry of the ramification and antiramification divisors of the Prym map and determine the pushforward of divisor classes on : R6 Theorem 7.4. The pushforwards of tautological divisor classes via the ra- tional Prym map P : 99K are as follows: R6 A5 ram P∗(λ) = 18 27λ1 57D, P∗(δ0 ) = 4(17 27λ1 57D), ′ ′′ − − P (δ ) = 27D, P (δ )= P (δ )= P (δ ) = 0 for 1 i g 1. ∗ 0 ∗ 0 ∗ i ∗ i:g−i ≤ ≤ − We point out that even though P is not a regular map, it can be extended in codimension 1 such that P is the morphism induced at the level of coarse moduli spaces by a proper morphism of stacks, see e.g. [Don92] p.63-64. 22 Furthermore P −1 contracts no divisors, in particular, we can pushforward divisors under P and use the push-pull formula. Perhaps the most novel ram aspect of Theorem 7.4 is the calculation of the class of the divisor P∗(∆0 ) consisting of Prym varieties corresponding to ramified double covers C C of genus 5 curves with two branch points. → Proof. We write the following formulas in CH1( ): A5 1 27λ = P P ∗λ = P (λ) P (δram), 1 ∗ 1 ∗ − 4 ∗ 0 3 ′ 6 (108λ 14D)=6[N ′ ]= P ([ ])=7P (λ) P (δram) 27P (δ ). 1 − 0 ∗ Q ∗ − 2 ∗ 0 − ∗ 0 ′ ′′ From [GSM11] it follows that P∗(δ0) = 27D, whereas obviously P∗(δ0 )=0, which suffices to solve the system of equations for coefficients. 8. The slope of A5 Using the techniques developed in previous chapters, we determine the slope of the perfect cone compactification 5 of 5 (note also that by the appendix by K. Hulek to [GSM11] this slopeA isA the same for all toroidal compactification). We begin with some preliminaries. Let D be a Q-divisor on a normal Q-factorial variety X. We say that D is rigid if mD = mD for all sufficiently large and divisible integers m. Equivalently,| the| Kodaira-{ } Iitaka dimension κ(X,D) equals zero. We denote by B(D) := Bs( mD ) the stable base locus of D. We say m | | that D is movable if codim B(D) 2. ≥ Recall that one defines the slope of g as s( g) := inf s(E). In A A E∈Eff(Ag) a similar fashion one defines the moving slope of as the slope of the cone Ag of moving divisors on , that is, Ag s′( ) := inf s(E): E Eff( ), E is movable . Ag ∈ Ag Thus s′( ) measures the minimal slope of a divisor responsible for a non- Ag trivial map from g to a projective variety. It is known that s( 4) = 8 [SM92], and as anA immediate consequence of the result about theA slope of we have that s′( )= s(θ )= 17 . In the next case, that of dimension M4 A4 null 2 g = 5, the formula [N ′ ] = 108λ 14D yields the upper bound s( ) 54 . 0 1 − A5 ≤ 7 A lower bound for the slope s( ) was recently obtained in [GSM11]. A5 We shall now prove Theorem 0.7 and establish that κ( , N ′ ) = 0, A5 0 54 in particular showing that s( 5) = 7 . To prove Theorem 0.7 we translate A ∗ ′ the problem into a question on the linear series P (N0) on 6. One can ∗ ′ | | R show that each of the components of P (N0) is an extremal divisor on 6, however their sum could well have positive Kodaira dimension. Of crucialR 23 importance is a uniruled parametrization of using sextics with a totally tangent conic. Q 2 2 2 We fix general points q1,...,q4 P , then set S := Bl 4 (P ) P {qi}i=1 4 ∈ → and denote by Eqi i=1 the corresponding exceptional divisors. We make { 15} 4 the identification P := S(6)( 2 i=1 Eqi ) , then consider the space of O − 4-nodal sextics having a totally tangent conic := (Γ,Q) P15 (2) :Γ Q = 2d, where d (Γ ) . X ∈ × |OS | ∈ reg 6 A parameter count shows that is pure of dimension 14. We define the rational map v : 99K X X R6 v(Γ,Q) := C,η := ν∗ ( (1)( d)) , OΓ − ∈R6 where ν : C Γ is the normalization map. The image v( ) is expected → X to be a divisor on 6, and we show that this is indeed the case — this construction yields anotherR geometric characterization of points in . Q Theorem 8.1. The closure of v( ) inside 6 is equal to , that is, a general Prym curve (C,η) possessesX a totallyR tangent conic.Q ∈Q Proof. We carry out a class calculation on and the result will be a con- R6 sequence of the extremality properties of the class [ ] Eff( 6). We work ′ Q ∈ R on a partial compactification 6 of 6 that is even smaller than 6. ′ 0 −1 ∗ R R R0 Let 6 := 6 π (∆0) be the open subvariety of 6, where 6 consists R R ∪ 2 R 0 R of smooth Prym curves (C,η) for which dim W6 (C)=0and h (C,L η) = 1 2 ∗ ⊗ for every L W6 (C), whereas ∆0 ∆0 is the locus of curves [Cxy], where ∈ 1 ⊂ ′ [C] 5 5,3 and x,y C. Observe that codim( 6 6, 6) = 2, ∈ M −M ∈ 1 ′ 1 R −R R in particular we can identify CH ( 6) and CH ( 6). Over the Deligne- ′ R R ′ Mumford stack R6 of Prym curves coarsely represented by the scheme 6 ′ R (observe that R6 is an open substack of R6), we consider the finite cover σ : G2 R′ , 6 → 6 2 where G6 is the Deligne-Mumford stack that classifies triples (C,η,L), with ′ 2 ′ (C,η) 6 and L W6 (C). Note that a curve [Cxy] ∆0 carries no ∈ R ∈ 6 0 ∈ non-locally free sheaves F Pic (Cxy) with h (Cxy, F ) 5, for F would 2 ∈ ≥ correspond to a g5 on the normalization C of Cxy, a contradiction. The 2 universal curve p : G6 is equipped both with a universal Prym bun- dle Pic( ) andC a → universal Poincar´eline bundle Pic( ) such that P ∈ C 2 L ∈ C |p−1(C,η,L) = L, for any (C,η,L) G6. We form the codimension 1 tauto- Llogical classes ∈ (11) a := p c ( )2 , b := p (c ( ) c (ω )) CH1(G2), ∗ 1 L ∗ 1 L 1 p ∈ 6 ⊗i and the sheaves i := p∗( ), where i = 1, 2. Both 1 and 2 are locally free. The dependenceV of aLand b on the choice of isV discussedV in [FL10]. 24 L 1 ′ 1 ′ Using the isomorphism CH (R6) = CH ( 6), one can write the following 1 ′ R formulas in CH ( 6), see [FL10, page 776]: (12) R σ (a)= 48λ+7π∗(δ ), σ (b) = 36λ 3π∗(δ ), σ (c ( )) = 22λ+3π∗(δ ). ∗ − 0 ∗ − 0 ∗ 1 V − 0 1 We also introduce the sheaf := p∗( ). Since R p∗( ) = 0(thisis 1 E P⊗L P⊗L ′ the point where we use H (C,L η)=0 for each (C,η,L) 6), applying Grauert’s theorem we obtain that⊗ is locally free and via∈R Grothendieck- Riemann-Roch we compute its ChernE classes. Taking into account that 2 ram p∗(c1( ) )= δ0 /2 and p∗(c1( ) c1( )) = 0, see [FL10, Proposition 1.6], one computesP L P δram a b (13) c ( )= λ 0 + CH1(G2). 1 E − 4 2 − 2 ∈ 6 Similarly, by GRR we find that c ( )= λ b + 2a. 1 V2 − After this preparation we return to the problem of describing the closure ′ 2 v( ) of v( ) in 6. For a point (C,η,L) G6, the two-torsion point η is X X R ∈ |L| induced by a conic totally tangent to the image of ν : C Γ P2, if and only if the map given by multiplication followed by projecti→on ⊂ χ(C,η,L): H0(C,L η) H0(C,L η) H0(C,L⊗2)/Sym2H0(C,L) ⊗ ⊗ ⊗ → is not an isomorphism. Working over the stack we obtain a morphism of 2 vector bundles over G6 χ : ⊗2 /Sym2( ), E →V2 V1 such that the class of v ( ) is (up to multiplicity) equal to X 2 ⊗2 ′ ′′ 5 ram σ∗c1 V = 35λ 5(δ + δ )+ δ = 5[ ], Sym2( ) −E − 0 0 2 0 Z V1 where we have used both (12) and (13). We recall that the cycle was Z defined in the proof of Theorem 6.3 as a subvariety of the larger space 6 ′ ′ 1 R′ with the property that 6 = 6. Thus the class [v( )] CH ( 6) Z∩R Q∩R ′′ X ∈ R is proportional (up to the divisor class δ ) to the class [ ]. It is proved 0 Q in [FV11, Proposition 3.6] that if D is an effective divisor on 6 such that ′′ R [D]= α[ ]+ β δ0 , then one has the set-theoretic equality D = . Thus we concludeQ that the closure of v( ) in is precisely . Q X R6 Q Theorem 8.2. Through a general point of the ramification divisor there Q passes a rational curve R 6 with the following numerical features: ′ ⊂ R′′ R λ = 6,R δ = 35,R δ = 0,R δram = 6,R δ = R δ = 0, 0 0 0 i i:5−i for i = 1,..., 4. In particular R < 0 and R = 0. Q U Assuming for the moment Theorem 8.2, we explain how it implies Theo- ′ rem 0.7. Assume that E Eff( 5) with s(E) s(N0). First note that one ′ ∈ A ≤ can assume that N0 supp(E), for else, we can replace E by an effective 25 divisor of the form E′ := E αN ′ with α > 0 and still s(E′) s(N ′ ). − 0 ≤ 0 After rescaling by a positive factor, we can write E N ′ ǫλ Eff( ), ≡ 0 − 1 ∈ A5 where ǫ 0. Clearly we have P ∗(E) Eff( ); observe that since N ′ is ≥ ∈ R6 0 not a component of E, the ramification divisor cannot be a component of P ∗(E) either. Thus R P ∗(E) 0, that is, Q ≥ ram ′′ δ 9ǫ 0 R P ∗(E)= R (2 + + 20δ ) ǫR λ 0 = 4 , ≤ Q U 0 − − 4 − − 2 ′ which is a contradiction. Thus s(E) = s(N0) and E must be equal to a ′ multiple of N0. Proof of Theorem 8.2. We retain the notation from Theorem 8.1 and fix a general element (C,η) corresponding to a sextic curve Γ P2 having ∈ Q ⊂ nodes at q1,...,q4. From Theorem 8.1 we may assume that there exists a 2 conic Q P such that Q Γ = 2(p1 + + p6), where p1,...,p6 Γreg. ⊂ 2 ∈ Since the points q1,...,q4 P are distinct and no three are collinear, it 1 ∈ follows that [C] / , and this holds even when C has nodal singularities. ∈ GP6,4 To construct the pencil R 6, we reverse this construction and start 2 ⊂ R with a conic Q P and six general points p1,...,p6 Q on it. On the ′ 2 ⊂ ∈ 6 blowup S of P at the 10 points q1,...,q4,p1,...,p6, we denote by Epi i=1 4 { } and by Eqj j=1 respectively, the exceptional divisors. For 1 i 6, let li { } ≤˜ ≤ ∈ Epi be the point corresponding to the tangent line Tpi (Q). If S is the blowup ′ of S at l1,...,l6, by slight abuse of notation we denote by Epi ,Eli and Eqj the exceptional divisors on S˜ (respectively the proper transforms of excep- ′ 4 6 tional divisors on S ). Then dim S˜(6) 2 j=1 Eqj i=1(Epi + Eli ) = O − − 3, and we choose a general pencil in this linear system. This pencil induces a curve R . Note that the pencil contains one distinguished element, ⊂ R6 consisting of the union of Q and two conics Q1 and Q2 passing through q1,...,q4. Considering the pushforward π∗(R) 6, after a routine calcu- lation we find: ⊂ M ′ ′′ R λ = 6,R (δ +δ +2δram)= π (R) δ = 47, π (R) δ =0 for i = 1, 2, 3. 0 0 0 ∗ 0 ∗ i ram In particular, as expected R = 0. The points in R ∆0 correspond to the case when the underlying U level 2 structure is not∩ locally free, which happens precisely when one of the points pi becomes singular. For each ram ′ 1 i 6, there is one such curve in R, hence R δ0 = 6, thus R δ0 = 35 and≤ therefore≤ using Corollary 7.3 we find: ′ 3 R = 7R λ R δ R δram = 42 35 9= 2. Q − 0 − 2 0 − − − Since the divisor N ′ is rigid, we obtain that s′( ) >s(N ′ ). Concerning 0 A5 0 the value of the moving slope s′( ), we make the following prediction: A5 ′ 70 Conjecture 8.3. s ( 5)= 9 . A 26 ′ 70 The inequality s ( 5) 9 follows from the first part of the proof of A ≥ ′ Theorem 8.2. Let E be as before, in particular N0 supp(E). After ′ rescaling by a positive factor, we can write [E]=[N0]+ ǫλ1 Eff( 5), where ǫ 0. Since R P ∗(E) 0, we find that ∈ A ≥ ≥ ram ′′ δ 9ǫ 0 R P ∗(E)= R (2 + + 20δ )+ ǫR λ 0 = 4+ . ≤ Q U 0 − 4 − 2 Hence ǫ 8 and this gives s′( ) 70 . Thus Conjecture 8.3 boils down ≥ 9 A5 ≥ 9 to constructing a movable effective divisor on having slope 70 . A5 9 ′ Remark 8.4. Theorem 0.7 implies that the divisor N0 can be contracted via a birational map having as its source. Especially from the point of A5 view of the Minimal Model Program for 5, it would be interesting to find a new compactification of the moduli spaceA of ppav ∗, and a birational map A5 f : 99K ∗ such that f contracts N ′ . A5 A5 0 9. The Prym realization of the components of H ′ For each irreducible component of H = N 4 in , we describe an explicit 0 A5 codimension 2 subvariety of 6 which dominates it via the Prym map. As a consequence, we prove thatR H consists of two irreducible components, both unirational and of dimension 13. We define two subvarieties of R6 corresponding to Prym curves (C,η) such that ϕKC ⊗η lies on a quadric of rank at most 4, cutting a (Petri special) pencil on C. Depending on the degree of this pencil, we denote these loci by and respectively. Q4 Q5 Definition 9.1. We denote by the closure in of the locus of curves Q5 R6 (C,η) 6 such that C carries two vanishing theta characteristics θ1,θ2 W 1(C)∈R with η = θ θ∨. ∈ 5 1 ⊗ 2 Equivalently, KC η = θ1 θ2, which implies that the Prym-canonical model ⊗ ⊗ 4 of C lies on a quadric Q P of rank 4, whose rulings induce θ1 and θ2 respectively. ⊂ Definition 9.2. We denote by 4 the closure in 6 of the locus of curves (C,η) such that η W 1(CQ) W (C) and KR η is very ample. ∈R6 ∈ 4 − 4 C ⊗ ′ 1 ′ 1 Equivalently, KC η = A A , where A W4 (C) and A W6 (C), and ⊗ ⊗ ∈ 4 ∈ then the image ϕKC ⊗η(C) lies on a quadric Q P of rank at most 4, whose rulings cut out A and A′ respectively. ⊂ Remark 9.3. Along the same lines, one can consider the locus 3 of curves ′ 1 Q′ 1 (C,η) 6 such that KC η = A A , where A W3 (C) and A W7 (C). Observe∈R that = π−1( ⊗1 ), where⊗ 1 is the∈ trigonal locus inside∈ . Q3 M6,3 M6,3 M6 In particular, codim( 3, 6) = 2. However from the trigonal construction [Don92, Section 2.4],Q it followsR that P ( )= , that is, P blows-down Q3 J5 Q3 and thus 3 plays no further role in describing the components of H in 5. Q 27 A First we show that 4 lies both in the ramification and the antiramifica- tion divisor of the PrymQ map: Proposition 9.4. . Q4 ⊆Q∩U Proof. We choose a point (C,η) 4 general in a component of 4 and ∈ Q 1 Q write η = M C ( D), where M W4 (C) and D C4 is an effective divisor. Then⊗O we compute− ∈ ∈ h0(C, K η( D)) = h0(K M ∨) = 3, C ⊗ − C ⊗ that is, ℓ := D is a 4-secant line to the Prym canonical model ϕKC ⊗η(C). Moreover ℓ is contained in the rank 4 quadric Q whose rulings cut out on C ∨ the pencils M and KC η M respectively. The line ℓ is not contained in a plane of Q belonging⊗ to the⊗ ruling Λ that cuts out on C the pencil M, for else it would follow that η = 0. Then ℓ is unisecant to the planes in Λ and if d M is a general element, then D + d is a hyperplane in P4. Thus M ∈| | M K η = (d + D + x + y), C ⊗ OC M where x,y C, that is, H0(C, K M ⊗(−2)) =0, and [C] 1 . ∈ C ⊗ ∈ GP6,4 Proposition 9.5. The locus is unirational and of dimension 13. Q4 1 Proof. Since 4 , we use the fact that every curve [C] 6,4 is a quadratic sectionQ ⊂Q∩U of a nodal quintic del Pezzo surface. In the∈ course GP of proving Theorem 8.2 we observed that a general Prym curve (C,η) is characterized by the existence of a totally tangent conic. We show∈ that U a similar description carries over to the case of 1-nodal del Pezzo surfaces. 2 2 We fix collinear points q1,q2,q3 P , a general point q4 P , then denote 2 ′ ∈ 2 2 ∈ by ℓ := q1,q2,q3 P , by S := Bl 4 (P ) P the surface whose ⊂ {qi}i=1 → ′ 4 image by the linear system S (2) i=1 Eqi is a 1-nodal del Pezzo O − 3 3 ′ ′ ′ ∗ quintic. Set PS := S (3) i=1 Eqi 2Eq4 . Note that Aut(S )= C . O − − We consider the 10-dimensional rational variety := (Q,p ,...,p ): Q P2 (2) , p ,...,p Q V { 1 5 ∈ |O | 1 5 ∈ } and the rational map p : 99K P2 given by p((Q,p ,...,p )) := p , where p V 1 5 6 6 is the residual point of intersection of Q with the unique cubic E P2 (3) ∈ |O | passing through q1,...,q4,p1,...,p5. We consider the linear system 4 ′ P(Q,p1,...,p5) := Γ S (6) 2 Eqi :Γ Q = 2(p1 + + p6) . ∈ O − i=1 Claim: For a general (Q,p ,...,p ) , we have dim P = 4, that 1 5 ∈V (Q,p1,...,p5) is, the points q1,...,q4,p1,...,p6 fail to impose one independent condition on 4-nodal sextic curves. 3 ′ Since ℓ + Q + PS P(Q,p1,...,p5), to conclude that dim P(Q,p1,...,p5) 4, it suffices to find one curve⊂ Γ P that does not have ℓ as a compo-≥ ∈ (Q,p1,...,p5) nent. We choose (Q,p1,...,p5) general enough that the corresponding ∈V 28 cubic E is smooth. Then 2E P and obviously ℓ 2E. To fin- ∈ (Q,p1,...,p5) ish the proof of the claim, we exhibit a point (Q,p1,...,p5) such that dim(P ) = 4. We specialize to the case p ℓ and∈ let V Q be the (Q,p1,...,p5) 1 ∈ 2 conic determined by p2,...,p5 and q4. The cubic E must equal ℓ + Q2 and p ℓ Q, so that E Q = p + + p . Then P = ℓ + P′, where 6 ∈ ∩ 1 6 (Q,p1,...,p5) 3 ′ P ′ := Y S (5) Eqi 2Eq4 : Y Q2 = p1+p6+2(p2+p3+p4+p5) . ∈ O − − i=1 2 ′ Because p2,...,p 5 P are general, dim( P ) = 20 3 3 2 8 = 4, which completes the proof∈ of the claim. − − − − We now consider the P4-bundle := (Q,p ,...,p , Γ):Γ P , P 1 5 ∈ (Q,p1,...,p5) together with the map u : 99K 6, given by P R u((Q,p ,...,p , Γ)) := C,η := (1)( p p ) , 1 5 OC − 1 − − 6 ′ 4 where C S is the normalization of Γ. Then M := C (2)( i=1 Eqi ) ⊂ 4 O − 6 ∈ W 1(C) is Petri special and M η = ′ (3)( E p ) = , 4 | ⊗ | ∼ |OS − i=1 qi − j=1 j | ∅ hence u( ) . Therefore there is an induced map u ¯ : // Aut( S′) 99K P ⊂Q4 P Q4 between 13-dimensional varieties. Since every curve (C,η) 4 has a totally ′ ∈Q 1 tangent conic and can be embedded in S , it follows that any M W4 (C) with h0(C,M η) 1 appears in the way described above, which∈ finishes the proof. ⊗ ≥ Another distinguished codimension 2 cycle in is the locus R6 ′ := (C,η) : η W (C) W (C) Q4 { ∈R6 ∈ 2 − 2 } of Prym curves (C,η) for which ϕKC ⊗η fails to be very ample. Writing 1 η = C (a + b p q), with a, b, p, q C, then M := C (2a + 2b) W4 (C) O − − ∈ O ∈ 4 and the 2-nodal image curve ϕKC ⊗η(C) lies on a pencil of quadrics in P , thus also on a singular quadric of type (4, 6). We show however, that this quadric is not the projectivized tangent cone of a quadratic singularity L Singst (Ξ), hence points in ′ do not constitute a component of P −1(H∈). (C,η) Q4 ′ Proposition 9.6. We have 4 . In particular, all singularities of Q U ′ the Prym theta divisor corresponding to a general point of Q4 are ordinary double points, that is, P ( ′ ) H. Q4 Proof. Note that ′ is not contained in , then use Proposition 5.4. Q4 U Proposition 9.7. The locus dominates via the Prym map the locus H , Q4 1 that is, P ( ) H . Q4 ⊃ 1 ′ Proof. We start with a point x0 =(τ0,z0) , corresponding to a singular point z Θ such that rk H(x ) 4∈S and x is a general point of a 0 ∈ τ0 0 ≤ 0 component of H θnull. In particular (Aτ0 , Θτ0 ) can be chosen outside any − ′ subvariety of 5 having codimension at least 3. Since each component of A ′ S maps generically finite onto N0, we find a deformation xt =(τt,zt) t∈T ′, parameterized by an integral curve T 0, such that{ for all t T } 0⊂, S 29 ∋ ∈ − { } the corresponding theta divisor Θt has only a pair of singular points, that is, Sing(Θ ) = z . Since P ( ) is dense in N ′ , after possibly shrinking t {± t} Q 0 T , we can find a family of triples (Ct,ηt,Lt) t∈T , such that (Ct,ηt) for all t T , while for t = 0 the{ line bundle}L V (C ,η ) corresponds∈ Q ∈ t ∈ 3 t t to the singularity zt Sing(Ξt). If we set (C,L,η) := (C0,L0,η0), by ∈ 0 semicontinuity we obtain that h (C,L) 4. Since rk H(L) = rk QL 4, it follows from Proposition 5.4 that L ≥Singst (Ξ) Singex (Ξ), which≤ ∈ (C,η) ∩ (C,η) implies that the Prym-canonical line bundle can be expressed as a sum of two pencils. Since L is not a theta characteristic and P (C,η) / 5, we obtain ∈ J ′ that the Prym-canonical bundle can be expressed as KC η = A A , where 1 ⊗ ⊗ A W4 (C). From Proposition 9.6 it follows that KC η can be assumed to∈ be very ample, that is, (C,η) . ⊗ ∈Q4 Corollary 9.8. P ( ) is a unirational component of H, different from θ4 . Q4 null 4 9.1. A parametrization of θnull. Our aim is to find an explicit unirational 4 parametrization of θnull. Proposition 9.9. P ( )= θ4 , where the the closure is taken inside . Q5 null A5 Proof. This proof resembles that of Proposition 9.7. If φ : 5 5 denotes X →A ′ the universal abelian variety, recall that we have showed that φ∗( null )= θ4 . Thus a point (τ,z) ′ corresponding to a generalS ∩S point null ∈ Snull ∩ S (A , Θ ) of a component of θ4 is a Prym variety P (C,η), where (C,η) τ τ null ∈ is a Prym curve such that z Sing(Θτ ) corresponds to a singularity LQ∩USingst (Ξ) Singex (Ξ). Then∈ L = f ∗(θ ), where θ Pic5(C) is ∈ (C,η) ∩ (C,η) 1 1 ∈ a vanishing theta-null. Since h0(C,L˜ ) = h0(C,θ )+ h0(C,θ η) 4, we 1 1 ⊗ ≥ find that θ2 := θ1 η is another theta characteristic, that is, (C,η) 5. 4 ⊗ ∈ Q Therefore θnull P ( 5). The reverse inclusion being obvious, we finish the proof. ⊆ Q We can now complete the proof of Theorem 0.5. We consider the smooth quadric Q := P1 P1 and the linear systems of rational curves × 7 7 P := P1 P1 (3, 1) and P := P1 P1 (1, 3) . 1 O × 2 O × Over P7 P7 we define the P5-bundle 1 × 2 P7 2 := (R1,R2, Γ) : Ri i for i = 1, 2, Γ R1 R2/Q(5, 5) . U ∈ ∈ |I | There is an induced rational map ψ : 99K given by U Q5 ψ(R ,R , Γ) := (C,p∗ (1) p∗ ( 1)) , 1 2 1O ⊗ 2O − ∈R6 1 where ν : C Γ is the normalization map and p1,p2 : C P are the composition of→ν with the two projections. → A general pair (R ,R ) P7 P7 corresponds to smooth rational curves 1 2 ∈ 1 × 2 such that the intersection cycle R1 R2 = o1 + + o10 consists of distinct 2 points. For any curve Γ R1 R2 (5, 5) we have R1 Γ = R2 Γ = 2(o1 + ∈ |I 30 | ∗ + o10). Since ν : R1 R2 (3, 3) KC is an isomorphism, it follows that both p∗ (1) and p∗ |I(1) are vanishing|→| theta-nulls,| hence ψ( ) . 1O 2O U ⊂Q5 Theorem 9.10. The rational map ψ : 99K is generically finite and U Q5 dominant. In particular (and thus θ4 = P ( )) is unirational. Q5 null Q5 Proof. We start with a point (C,θ1,θ2) 5 moving in a 13-dimensional family. In particular, the image Γ of the induced∈ Q map ϕ : C P1 P1 (θ1,θ2) → × is nodal and we set Sing(Γ) = o1,...,o10 . ′ { } ′ We choose divisors D,D θ1 , corresponding to lines ℓ,ℓ Q(1, 0) such that ν∗(Γ ℓ)= D and ν∗∈(Γ | ℓ|′)= D′ respectively. Then D+∈D |O′ K |, ∈| C | and since the linear system o1+ +o10 (3, 3) cuts out the canonical system on C, it follows that there exists|I a cubic curve| E (3, 3) such that ∈ |OQ | 10 E Γ= D + D′ + 2 o . i i=1 By B´ezout’s Theorem, both ℓ and ℓ′ must be components of E, that is, we ′ can write E = ℓ + ℓ + R1, where R1 Q(1, 3) is such that R1 Γ = 10 ∈ |O | 2 i=1 oi. Switching the roles of θ1 and θ2, there exists R2 Q(3, 1) such 10 −∈1 |O | ∨ that R2 Γ = 2 i=1 oi. It follows that (R1,R2, Γ) ψ ((C,θ1 θ2 )). 5 7 7 ∈ ⊗ The variety being a P -bundle over P1 P2 is unirational, hence 5 is unirational asU well, thus finishing the proof.× Q Acknowledgments: Research of the first author is supported by the Sonder- forschungsbereich 647 “Raum-Zeit-Materie” of the DFG. The first author is grateful to INdAM and Universit`aRoma Tre for providing financial support and a stimu- lating environment during a stay in Rome in which parts of this paper have been completed. Research of the second author was supported in part by National Sci- ence Foundation under the grant DMS-10-53313. Research of the third author is supported in part by Universit`a“La Sapienza” under the grant C26A10NABK. Re- search of the fourth author is supported in part by the research project “Geometry of Algebraic Varieties” of the Italian frame program PRIN-2008. References [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris. Geometry of algebraic curves. 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Humboldt Universitat¨ zu Berlin, Institut fur¨ Mathematik, Unter den Lin- den 6, 10099 Berlin, Germany E-mail address: [email protected] Stony Brook University, Department of Mathematics, Stony Brook, NY 11790-3651, USA E-mail address: [email protected] Universita` “La Sapienza”, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185, Roma, Italy E-mail address: [email protected] Universita` “Roma 3”, Dipartimento di Matematica, Largo San Leonardo Murialdo I-00146 Roma, Italy E-mail address: [email protected] 33