Effective divisors on projectivized Hodge bundles and modular forms

We construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular we construct basic modular forms for genus $2$ and $3$. We also discuss modular forms on the moduli of hyperelliptic curves. In that case the relative canonical bundle is a pull back of a line bundle on a ${\mathbb P}^1$-bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In an appendix we use our method to calculate divisor classes in the dual projectivized $k$-Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.


Introduction
Moduli spaces of curves and of abelian varieties come with a natural vector bundle, the Hodge bundle E. Starting from this vector bundle one can construct other natural vector bundles by applying Schur functors, like Sym n (E) or det(E) ⊗m .Sections of such bundles are called modular forms.For example, for the moduli space A g of principally polarized abelian varieties of dimension g these are Siegel modular forms, and for the moduli space M g of curves of genus g these are Teichmüller modular forms.If the Schur functor corresponds to an irreducible representation ρ we say that a section of E ρ is a modular form of weight ρ.The Hodge bundle extends to appropriate compactifications of such moduli spaces and in many cases the sections also extend automatically to the compactifications, e.g. for A g with g ≥ 2 by the so-called Koecher principle.
In this paper we try to construct modular forms in a geometric way.It is well-known that an effective divisor on A g or on M g with g ≥ 2 representing the cycle class mλ with λ = c 1 (det(E)) and m ∈ Z >0 yields a scalar-valued modular form of weight m, that is, a section of det(E) ⊗m .We will exploit explicit effective divisors on projectivized vector bundles to construct vector-valued modular forms.In particular, we will construct in this way certain modular forms that play a pivotal role in low genera.
For example, in the case of g = 2 there is the modular form χ 6,8 , a section of Sym 6 (E)⊗ det(E) 8 , that appeared in [4] as follows.Recall that the Torelli morphism M 2 ֒→ A 2 has dense image and we have an equality of standard compactifications M 2 = Ã2 .The moduli space M 2 has another description as a stack quotient.This derives from the fact that a smooth complete curve of genus 2 over a field k of characteristic not 2 is a double cover of P 1 ramified at six points, so can be given as y 2 = f with f a polynomial of degree 6 with non-vanishing discriminant.Writing f as a homogeneous polynomial in two variables, say f ∈ Sym 6 (W ) with W the k-vector space generated by x 1 , x 2 , and observing that we may change the basis of W , we find a presentation of M 2 as a stack quotient M 2 ∼ [W 0 6,−2 /GL(W )], where we write W a,b for the GL(W )-representation Sym a (W ) ⊗ det(W ) b .Here the space W 6,−2 can be seen as the vector space of binary sextics f with an action of GL(W ) by for a matrix ( a b c d ) ∈ GL (2).The subspace W 0 6,−2 of W 6,−2 is the space of f with nonvanishing discriminant.The twisting by det(W ) −2 is required to get the right stabilizer for the generic f , namely ±Id W .
This interpretation of M 2 was used in [4] to construct vector-valued Siegel modular forms of degree 2 by using invariant theory of binary sextics, thus extending and simplifying the description of scalar-valued Siegel modular forms by invariants by Igusa [17,18].Covariants define vector-valued modular forms and all Siegel modular forms of degree 2 on A 2 can be constructed this way.In [4] it was shown that the most basic covariant, the universal binary sextic, defines a meromorphic Siegel modular form χ 6,−2 of weight (6, −2), that is, it defines a meromorphic section of Sym 6 (E) ⊗ det(E) −2 on A 2 .After multiplying χ 6,−2 by Igusa's cusp form χ 10 one obtains the holomorphic modular form χ 6,8 , the 'first' vector-valued Siegel modular cusp form of degree 2.
In the case of g = 3, there is an analogous form χ 4,0,8 , a section of Sym 4 (E) ⊗ det(E) 8 .Here it derives from the description of the moduli space M nh 3 of non-hyperelliptic curves of genus three as a stack quotient where W is now of dimension 3 and W 0 4,0,−1 ⊂ Sym 4 (W ) ⊗ det(W ) −1 represents ternary quartics defining smooth curves.In [5] this description led to the construction of a meromorphic Teichmüller modular form χ 4,0,−1 of weight (4, 0, −1) and a (holomorphic) Siegel modular form χ 4,0,8 of degree 3 and weight (4,0,8).Also in this case all Teichmüller and Siegel modular forms of genus 3 on M 3 and A 3 can be constructed from these forms by invariant theory.
This paper arises from the desire to construct these basic forms and similar forms in a geometric way.We use cycle relations for effective divisors (or almost effective divisors) on the projectivized Hodge bundle to construct our forms.It is based on the observation that an effective divisor D on the projectivized Hodge bundle P(E) with cycle class [D] = [O(j)] + k λ − ∆ with positive integers j, k and ∆ an effective boundary class gives rise to a section of Sym j (E)⊗det(E) k vanishing on boundary divisors, that is, a modular form.This method produces the basic modular forms χ 6,8 and χ 4,0,8 of degree 2 and 3 in an efficient way.
This connection between divisors and modular forms can also be used in the other direction, obtaining cycle classes for divisors on projectivized Hodge bundles.We give some examples of this.
Another objective of this paper is to construct modular forms on moduli spaces of hyperelliptic curves of genus g.For this we work with two descriptions of the moduli, a description as a stack quotient and a description as a Hurwitz space.The latter space H g,2 has as compactification the space H g,2 of admissible degree 2 covers of genus g.In the stack description modular forms pull back to covariants for the action of GL(2) on the space of binary forms of degree 2g + 2.
In the Hurwitz space description the relative canonical bundle of the universal curve over H g,2 can be viewed as the pull back of O(g − 1) from the trivial P 1 -bundle P over H g,2 equipped with 2g + 2 non-intersecting sections.Using the theory of admissible covers, P is compactified to a space P , a fibration of rational stable curves with 2g + 2 marked points over H g,2 , and we show that the line bundle O(g − 1) on P extends to a line bundle on P with the property that its push down to H g,2 is close to the Hodge bundle.This allows us to construct modular forms on H g,2 .
When we consider projectivized bundles projectivization is meant in the Grothendieck sense, so that for a vector space V the projective space P(V ) parametrizes hyperplanes in V .
In an appendix we apply a method used in this paper to calculate the classes of certain divisors in the dual projectivized k-Hodge bundle that were determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.
Let k be a field of characteristic not 2. We consider the moduli space M 2 of curves of genus 2 over k.This is a Deligne-Mumford stack and it carries a universal curve π : C → M 2 of genus 2. The relative dualizing sheaf ω π is base point free and thus defines a morphism ϕ : C → P(E).For a curve C the map ϕ : C → P(E C ) associates to a point the space of differentials vanishing in that point.We have a commutative diagram with u the natural morphism We let M 2 be the Deligne-Mumford compactification and π : C → M 2 the corresponding universal curve.However, the extension ω π of ω π does not define an extension of the morphism ϕ to P(E) over the boundary component ∆ 1 that parametrizes reducible curves.
We consider the branch divisor D ⊂ P(E) of the morphism ϕ.The divisor D is of relative degree 6 in the P 1 -bundle P(E) over the base M 2 .We define D to be the closure of D in P(E) over M 2 .In the rational Picard group of P(E) we can write with A a class in the rational Picard group of M 2 and u : P(E) → M 2 the natural projection.
We want to determine A in terms of the generators λ, δ 0 of the Picard group of M 2 .We write λ for the first Chern class of E and δ 1 (resp.δ 0 ) for the class of ∆ 1 (resp.∆ 0 ) in the Picard group of the stack M 2 ; here ∆ 0 is the boundary component that parametrizes irreducible curves with a double point.
In order to do this we extend the morphism ϕ.It extends over a Zariski open part of ∆ 0 since ω π has no base points there.However, over ∆ i with i > 0 this system has base points.We then use a base change as described in the appendix in Section 18.After a base change we have in an open neighborhood U i of the generic point of ∆ i a semi-stable family.If we take the base to be 1-dimensional we get a semi-stable family f : C → B with as central fibre a chain C ′ + R + C ′′ with R a (−2)-curve and C ′ and C ′′ of genus 1.The extension ϕ ′ of ϕ is given by ω f (−R) with f * (ω f (−R)) = E B and the morphism ϕ ′ : C → P(E) contracts C ′ and C ′′ and is of degree 2 on R. We refer to the appendix, Section 18 for the details.The morphism ϕ ′ has the property that Proof.We write [D] = 6 [O(1)] + u * (A).We work with the above two types of 1dimensional families f : C → B. The morphism ϕ is ramified over D, and thus ϕ ′ is ramified over D and contracts C ′ and C ′′ .We denote the ramification divisor by S. We thus get (writing abusively line bundles and divisors for the corresponding divisor classes) where the first equation comes from adjunction ω f + C ′ |C ′ = O C ′ for C ′ and similarly for C ′′ , and the second one from with b 1 the special point of B. This shows that A = −2 λ + 2 b 1 .Because of the base change that we executed, we have 2 b 1 = δ 1 and we obtain A = −2 λ+δ 1 .Now we use the well-known relation 10 λ = δ 0 +2 δ 1 (see [24]) and thus get Remark 2.2.We indicate an alternative proof of this result in Remark 13.5.

An important remark is now that u
is an effective divisor on P(E).We apply u * to the corresponding section 1 of O(D).By Proposition 2.1 we see that we get a regular section χ 6,8 of the vector bundle Sym 6 (E) ⊗ det(E) 8 over M 2 .Moreover, this section vanishes on the divisors ∆ 0 and ∆ 1 .Note that the Torelli map extends to an isomorphism M 2 ∼ = Ã2 with Ã2 the standard smooth compactification of A 2 .Therefore our section defines a Siegel modular form χ 6,8 of weight (6,8) that is a cusp form.
Corollary 2.3.Let D be the closure in P(E) of the branch divisor of the canonical map for the universal curve over M 2 .The push forward u * (s), with s the natural section 1 of O(D) on P(E), defines a Siegel modular cusp form χ 6,8 of degree 2 and weight (6,8).
We now analyze the orders of vanishing along δ 1 of χ 6,8 and χ 6,−2 .When identifying M 2 with Ã2 we also write A 1,1 for δ 1 ; it is the locus of products of elliptic curves.
We analyze the orders by working locally on a family over a local base B with central fibre a general point b 1 of the boundary divisor ∆ 1 .As we mentioned before, the map ϕ : C → P(E) defined over M 2 does not extend to the whole C over M 2 due to the fact that the canonical system has base points at the nodes of the curves over the boundary divisor ∆ 1 .On the other hand, by the theory of admissible covers, the ramification divisor of the above map ϕ extends to a divisor S on C in a way that avoids the above nodal locus.Namely, over b 1 ∈ ∆ 1 the fibre is a nodal curve C which is the union of two elliptic curves C 1 and C 2 meeting at a point p.The restriction of the ramification divisor on each component is the union of the three -additional to p-ramification points of the system |O(2p)|.Therefore the extension of the map ϕ is defined on the ramification divisor S. The map ϕ maps C 1 \{p} and C 2 \{p} to two distinct points p 1 and p 2 respectively which are defined as follows.The fibre of P(E) over b 1 can be identified with The divisor D, the image of the ramification divisor under the extended map ϕ, splits then into six irreducible components denoted by D 1 , . . ., D 6 .Over our local base B we thus have the six local sections D i (i = 1, . . ., 6) of the family P(E) → B. By the above description of the extension of the map ϕ, we may conclude that D 1 , D 2 , D 3 pass through p 1 and D 4 , D 5 , D 6 through p 2 .
Lifting the sections D i locally to sections σ i of E and choosing a basis e 1 , e 2 of E over B such that e 1 and e 2 determine p 1 and p 2 in the fibre of P(E) over z = 0, we can write σ i = a i e 1 + b i e 2 for i = 1, . . ., 6. Then at z = 0 the functions b 1 , b 2 , b 3 and a 4 , a 5 , a 6 vanish, while a 1 , a 2 , a 3 , b 4 , b 5 , b 6 do not vanish.Since by blowing up once we can separate, we may assume that these sections vanish with order 1 at z = 0.By construction the section χ of Sym 6 (E) ⊗ det(E) −2 is locally given by We may write σ 1 with Λ running though the subsets of {1, . . ., 6} of cardinality i and Λ c denoting the complement.We find for these orders (3, 2, 1, 0, 1, 2, 3) for i = 0, . . ., 6, hence for the section χ given by σ 1 • • • σ 6 /z we find the orders (2, 1, 0, −1, 0, 1, 2).
These orders are in agreement with the result of [4] where χ 6,−2 was constructed by invariant theory and properties of modular forms were used to determine these orders.
A different way to construct the form χ 6,8 uses the so-called Weierstrass divisor W in the dual bundle: W := {(C, η) ∈ P(E ∨ ) : div(η) contains a Weierstrass point} over M 2 .Here C denotes a curve of genus 2 and η a differential form on C. We let W be the closure of W over M 2 .We then have an identity due to Gheorghita [12, Thm 1] where we write λ and δ i for the pullback of λ and δ i to P(E ∨ ).Now W is an effective divisor and the push forward of the section −6 .This implies that under the isomorphism of P 1 -bundles P(E) ∼ = P(E ∨ ) the isomorphism identifies [W ] with [D], and we get in the dual bundle Using push forward we find again a form of weight (6,8) vanishing on δ 1 and δ 0 .Up to a multiplicative non-zero constant this is χ 6,8 .

The class of the k-canonically embedded curve
For the calculation of the classes of effective divisors in P(E) related to the canonical image of the universal curve it is helpful to have (part of) the class of the closure of the canonical image in P(E) over M g .Without extra effort we can and will extend the calculation to the case of the k-canonically embedded curve for k ≥ 1.
We consider the universal family π : C g → M g .This comes with a natural vector bundle E k = π * (ω ⊗k π ) for k ∈ Z ≥1 and for k = 1 this is the Hodge bundle E 1 = E.We write u : P(E k ) → M g for the natural map.For k ≥ 2 the sheaf ω ⊗k π is base point free for stable curves and the surjection π * E k → ω ⊗k π defines a morphism ϕ k : C g → P(E k ).For k = 1 the sheaf ω π is base point free on M g ∪ ∆ 0 0 , with ∆ 0 0 ⊂ ∆ 0 the open locus with only disconnecting nodes, but it has base points on the nodes lying over the generic points of the boundary components ∆ i for i > 0. The appendix Section 18 describes the closure of the image over an open neighborhood of the generic point of ∆ i .
We denote by Γ k the image of ϕ k (C g ) over M g ∪ ∆ 0 0 and by Γ k the closure of Γ k in P(E k ) over M g .We can write the class of Γ k as a cycle on P(E k ) as with ) and β j a codimension j cycle on M g .Proposition 3.1.We have β 0 = 2 k(g − 1) and i=1 δ i for k = 1 and ǫ = 0 else.
i=0 δ i (by [23]) we can write β 1 as Proof.We start with the case k ≥ 2. In this case Γ k is the image of C g under ϕ k and ϕ * k (h k ) = k ω π .Since the image of the generic fibre has degree 2k(g − 1) we find ) and this settles the case k ≥ 2. The same argument works for k = 1 as long as we work on M g ∪ ∆ 0 0 .To get the coefficients of the δ i for i > 0 we work over a 1-dimensional base B in an open neighborhood U i of the generic point of ∆ i with special fibre C ′ + R + C ′′ as in the appendix Section 18 where the extension ϕ ′ of ϕ is defined by ω π ′ (−R).The contribution of δ i to 2β 1 is now π ′ * (ω(−R) 2 ), where the coefficient 2 of β 1 comes from the fact that ϕ ′ is of degree 2 on R. We get as the fibre has two singular points and R 2 = −2.Putting everything together results in the given formula.

The Case of Genus Three
Here there is no restriction on the characteristic.We consider the moduli stack M 3 of curves of genus 3 over our field k and the universal curve π : C → M 3 .The canonical map defines a morphism ϕ : C → P(E) and we thus obtain the image divisor D in P(E) over M 3 .We have a commutative diagram We consider the closure D of D in P(E) over M 3 .The canonical image of the generic curve is a quartic curve.Thus we have a relation [D] = [O(4)] + u * (A) in the rational Picard group of P(E) with A a divisor class on M 3 given in Proposition 3.1. 8that is regular outside ∆ 1 and vanishes on ∆ 0 .In view of the even powers Sym 4 and 8, this section ψ is invariant under the action of −1 on the fibres of E. As the action of −1 defines the involution of the double covering of stacks M 3 → A 3 , the section ψ descends to a section χ 4,0,8 of Sym 4 (E)⊗det(E) 8 on the image of M 3 −∆ 1 under the Torelli morphism M 3 → Ã3 , with Ã3 the standard second Voronoi compactification of A 3 .Since the image of ∆ 1 in Ã3 is of codimension 2, the section χ 4,0,8 extends to a regular section of Sym 4 (E) ⊗ det(E) 8 on all of A 3 , and then by the Koecher Principle it extends to Ã3 .Thus it defines a regular Siegel modular cusp form χ 4,0,8 of degree 3 and weight (4, 0, 8).The class H of the hyperelliptic locus H 3 in M 3 satisfies ([14, p. 140]) ( The relation (2) shows that there exists a scalar-valued Teichmüller modular form χ 9 of weight 9 on M 3 .Its square is invariant under the action of −1 on the fibres of E, hence descends to a Siegel modular form of weight 18.Up to a multiplicative scalar this is Igusa's modular form χ 18 .
If we divide χ 4,0,8 by χ 9 we obtain a meromorphic section of Sym 4 (E) ⊗ det(E) −1 on M 3 that is regular on M 3 outside the hyperelliptic locus.This form was used in [5] to construct Teichmüller modular forms and Siegel modular forms by invariant theory.

Moduli of hyperelliptic curves as a stack quotient
In this section we discuss the stack quotient description of the moduli of hyperelliptic curves.We consider hyperelliptic curves in characteristic not 2. A hyperelliptic curve of genus g is a morphism α : C → P 1 of degree 2 where C is a smooth curve of genus g.A morphism a : α → α ′ between two hyperelliptic curves is a commutative diagram A hyperelliptic curve C of genus g can be written as In fact, choosing a basis (x 1 , x 2 ) of the g 1 2 defines the morphism α.Let W = x 1 , x 2 , a vector space (over our algebraically closed base field) of dimension 2, and L = α * (O P 1 (1)).By Riemann-Roch we have dim H 0 (C, L g+1 ) = g + 3, while dim Sym g+1 (W ) = g + 2, so we have a non-zero element y ∈ H 0 (C, L g+1 ) which is anti-invariant under the involution corresponding to α.The anti-invariant subspace of H 0 (C, L g+1 ) has dimension 1.Then y 2 is invariant and lies in Sym 2g+2 (W ).Thus we find the equation y 2 = f (x 1 , x 2 ) with f homogeneous of degree 2g + 2 and with non-zero discriminant.
We have made two choices here: a generator y of H 0 (C, L g+1 ) (−1) , a space of dimension 1, and a basis of W .We can change the choice of y (by a non-zero scalar) and the choice of a basis of W by γ = (a, b; c, d) ∈ GL(W ).The action of GL(W ) is on the right via ) .If we let GL(W ) act on y by a power of the determinant, then this action preserves the type of equation.In inhomogeneous form the action by GL(W ) is by with the following effect on the equation: The last expression on the right-hand-side can be written as binary form of degree 2g +2.The stabilizer of a generic f ∈ Sym 2g+2 (W ) is µ 2g+2 , the roots of unity of order dividing 2g +2.Since we want a stabilizer of order 2 for the generic element, we consider a twisted action: define the GL(W )-representation This can be identified with Sym a (W ) as a vector space, but the action by GL(W ) is different.Inside this space W a,b we have the open subspace W 0 a,b of homogeneous polynomials of degree a with non-zero discriminant.We now distinguish two cases.
Case 1. g even.Here we consider the stack quotient This stack quotient can be identified with the moduli stack H g of hyperelliptic curves of genus g for g even.Indeed, the action of t • Id W is (on inhomogeneous equations) by hence y 2 = f maps to y 2 = t 2 f , so that the stabilizer is µ 2 , as required.Note also that the action of −1 ∈ GL(W ) is by y → −y, so y is an odd element.A basis of H 0 (C, Ω 1 C ) is given by x i dx/y, (i = 0, . . ., g − 1) .The action on dx is by (ad − bc)dx/(cx + d) 2 resulting in the action on the space of differentials by If we forget the twisted action on y, we can identify H 0 (C, Ω 1 C ) with W g−1,1 .But y 2 must be viewed as an element of W 2g+2,−g , so the action of t 1 W on y should be twisted by t −g = det −g/2 .We get Case 2. g odd.Here we take W 2g+2,−g+1 .
Remark 5.1.If we consider W 2g+2,r then r has to be even, since as above we later view y 2 as an element of W 2g+2,r and we need an action by det r/2 on y.
Here the stabilizer of a generic element is µ 4 .Now on inhomogeneous equations the action is by f → t −2g+2 f, y → y/t g+1 , hence y 2 = f maps to y 2 = t 4 f .Note that here −1 W acts by f → f and y → y.But √ −1 W acts by f → f and y → −y.To get the right stack quotient with stabilizer of the generic element of order 2, we take ] .The action on the differentials x i dx/y with i = 0, . . ., g − 1 is by For a somewhat different description see [2,Cor. 4.7,p. 654].
Recall that the moduli stack H g has as compactification the closure H g of H g inside the moduli stack M g .The Picard group of H g is known by [2] to be finite cyclic for g ≥ 2 of order 4g + 2 if g is even and 8g + 4 else.The rational Picard group of H g is known by Cornalba (see [7]) to be free abelian of rank g generated by classes δ i and ζ j for i = 0, . . ., ⌊g/2⌋ and j = 1, . . .⌊(g − 1)/2⌋.Cornalba gives also the first Chern class λ of the Hodge bundle E on H g where the generic point of the divisor ζ i has an admissible model C ′ ∪ C ′′ with two nodes C ′ ∩ C ′′ = {p, q} mapping to a union of two P 1 , with 2i + 2 marked points on C ′ , see Figure 1 in Section 7.

Modular forms on the hyperelliptic locus of genus three
Let E be the Hodge bundle on H 3 .By a modular form of weight k on H 3 we mean a section of det(E) ⊗k .The construction in the preceding section shows that a modular form of weight k on H 3 when pulled back to the stack [W 0 2,0 /(GL(W )/ ± id W )] gives rise to an invariant of degree 3k/2.Indeed, it defines a section of the equivariant bundle det(W ) 3k invariant under SL(W ), but in view of the fact that we divide by the action of GL(W )/(±id W ) this yields an invariant of degree 3k/2.
Let M k (Γ 3 ) = H 0 (A 3 , det(E) k ) be the space of Siegel modular forms of degree 3 on Γ 3 = Sp(6, Z).In [19] Igusa considered an exact sequence (2,8) with I d (2, 8) the vector space of invariants of degree d of binary octics.We can interpret Igusa's sequence in the following way.A Siegel modular form of weight k defines by restriction to the hyperelliptic locus a modular form of weight k on H 3 and it thus defines an invariant of degree 3k/2.
For each irreducible representation ρ of GL(3) we have a vector bundle E ρ made from E by a Schur functor.By a modular form of weight ρ on H 3 we mean a section of a vector bundle E ρ .We can pull back to the stack [W 0 8,−2 /(GL(W )/ ± id W )], but the situation is more involved as Sym n (Sym 2 (W )) decomposes as a representation of GL(W ).For example, we have with Here and in the rest of this section we assume that the characteristic is 0, or not 2, and high enough for the representation theory (plethysm) to work 1 .In this case we can consider the restriction of the Siegel modular form χ 4,0,8 to the hyperelliptic locus and we know that it does not vanish identically by [5,Lemma 7.7].On the other hand we have the basic covariant f 8,−2 , the diagonal section of W 8,−2 over the stack The discriminant form d of binary octics, an invariant of degree 14, does not define a modular form, but its third power d 3 does.It defines a modular form of weight 28, see [27, p. 811] and also Remark 13.1.
Proof.By restricting and projecting we obtain a covariant of bi-degree (8,12).This covariant is divisible by the discriminant and does not vanish on the locus of smooth hyperelliptic curves.Therefore, division by d gives a non-vanishing covariant of bidegree (8, −2).Taking into account the 'twisting' by det(W ) −2 , this must be a multiple of the universal binary octic.
We will discuss the other two projections later in Lemma 14.3.Note that the divisor D, the canonical image of the universal curve in P(E) that defines χ 4,0,8 , has a restriction to the locus of smooth hyperelliptic curves which is divisible by 2. Indeed, the canonical image of a hyperelliptic curve is a double conic.This suggests that we can take the 'square root' of the restriction of χ 4,0,8 to the hyperelliptic locus.However, the boundary divisors prevent this.If we take a level cover of the moduli space we can construct a modular form of weight (2, 0, 4).We will carry this out later (in Corollary 13.4), working on a Hurwitz space that we shall introduce in the next section.

The Hurwitz space of admissible covers of degree two
In this and the following sections will use the other description of the moduli of hyperelliptic curves, namely the moduli space H g,2 of admissible covers of degree 2 and genus g in the sense of [16], see [15].Thus we are looking at covers f : C → P of degree 2 with C nodal of genus g and P a stable b-pointed curve of genus 0.Here the b = 2g + 2 branch points are ordered and H g,2 → H g is a Galois cover with Galois group the symmetric group S 2g+2 .
The boundary H g,2 − H g,2 consists of finitely many divisors that we shall denote by ∆ Λ b = ∆ Λ , where we omit the index b if g is clear.Here the index Λ defines a partition {1, 2, . . ., b} = Λ ⊔ Λ c , and the generic point of ∆ Λ corresponds to an admissible cover that maps to a stable curve of genus 0 that is the union of two copies of P 1 , one containing 1 Alternatively one could use divided powers as in [1, 3.1] the points with mark in Λ, the other one those with mark in Λ c .Here we will assume that #Λ = j with 2 ≤ j ≤ g + 1.
The parity of #Λ plays an important role here.If #Λ = 2i+2 is even, then the generic admissible cover corresponding to a point of ∆ Λ is a union C i ∪ C g−i−1 that is a double cover of a union of two rational curves P 1 and P 2 with C i lying over P 1 and C g−i−1 over P 2 .Here C i (resp.C g−i−1 ) has genus i (resp.g − i − 1) with 0 ≤ i ≤ (g − 1)/2 and is ramified over the points of Λ (resp.Λ c ).
is ramified over Λ and in p, the intersection of C i and C g−i , (resp.over Λ c and in p).Note that p is a simple node.
Fig. 2: Λ odd Assuming that g and b = 2g + 2 are fixed we will write and provide the symmetric case with a factor 1/2, that is,

Divisors on the moduli of stable curves of genus zero
For later use we recall some notation and facts concerning divisors on the moduli spaces M 0,n .We refer to [20].The boundary divisors on M 0,n are denoted by S Λ n and are indexed by partitions {1, . . ., n} = Λ ⊔ Λ c into two disjoint sets with 2 ≤ #Λ ≤ n − 2 and we have S Λ n = S Λ c n .Via the natural map π n+1 : M 0,n+1 → M 0,n we may view M 0,n+1 as the universal curve and π n+1 has n sections.The generic point of S Λ n corresponds to a stable curve with two rational components, one of which contains the points marked by Λ.For pullback by π n+1 we have the relation .
The n sections of π n+1 have images S {i,n+1} n+1 with i = 1, . . ., n.We can collect these boundary divisors on M 0,n+1 via , with the convention that in view of the symmetry we add a factor 1/2 for even n and .
Later, when a fixed index k is given we will split these divisors as (and with a factor 1/2 if j = n/2).

A good model
We now will work with a 'good model' of the universal admissible cover over H g,2 .Such a model was constructed in [11,Section 4].We start with the observation that the space H g,2 is not normal, and we therefore normalize it.The result H g,2 is now a smooth stack over which we have a universal curve C → H g,2 .
We have a natural map h : H g,2 → M 0,b with b = 2g + 2 and the universal curve now fits into a commutative diagram We can construct a proper flat map that extends the relative canonical morphism C → P 1 H g,2 by taking the fibre product P of M 0,b+1 and H g,2 over M 0,b and thus obtain a commutative diagram The resulting space P is not smooth, but has rational singularities.Resolving these in a minimal way gives a model P; taking the resolution Y of the normalization Y of the fibre product of P and C over P gives us finally a commutative diagram where B is our base H g,2 or any other base mapping to it.We write π for the resulting morphism P → B, h for the natural map B → M 0,b and ν for B → H g,2 .We refer to [11,Section 4] for additional details.
In the following we will assume that we have a physical family over a base B. We will abuse the notation ∆ Λ for the pull back of the divisor ∆ Λ under ν : B → H g,2 .
In the case #Λ is odd we find a similar decomposition corresponding now to the fact that the general fibre of π has three components, one coming from the blowing up.We notice If we use the notation ∆ j = #Λ=j ∆ Λ , we find for the tautological classes λ = c 1 (E) and h * (ψ k ), simply denoted by ψ k and defined as the first Chern class of the line bundle given by the cotangent space at the kth point of our pointed curve (k = 1, . . ., b), the following formulas on our base B (see [10]) and where we use the notation (k + ) (resp.(k − )) to denote the condition k ∈ Λ (resp.k ∈ Λ) as above.The relation (4) implies the following.
Corollary 9.1.There exists a scalar-valued modular form of weight 2(2g + 1) on the moduli space H g,2 whose divisor is a union of boundary divisors.It descends to the hyperelliptic locus H g and corresponds to a power of the discriminant of the binary form of degree 2g + 2.
10. Extending the linear system The canonical system on a hyperelliptic curve is defined by the pull back of the sections of the line bundle of O(g − 1) of degree g − 1 on the projective line.We now try to extend this line bundle over our compactification.
A first attempt would be to consider the divisor (g − 1) Sk with Sk the pullback to P of the section S k of π b+1 : M 0,b+1 → M 0,b .Recall the morphism t = πf : Y → B. We can add a boundary divisor Ξ k to it such that f * O P (D k ) with D k = (g − 1) Sk + Ξ k coincides with ω t on the fibres of t, namely in view of the intersection numbers take Ξ k equal to Here Π j = #Λ=j Π Λ and Π c j = #Λ=j Π Λ c and (k + ) (resp.(k − )) indicates the condition that k ∈ Λ (resp.k ∈ Λ); moreover, we add a factor 1/2 in case j = b/2.Now f * O(D k ) and ω t agree on the fibres of t, so they differ by a pull back under t = π • f , see diagram (2).
To see the above, when e.g.#Λ = 2i+2 is even: in that case the fibre of Ỹ over t is as in Fig. 1 and ω where we indicate by i the line bundle of degree i on P 1 .One then checks that with the above choice of Ξ k the restriction of D k on the corresponding fibre of π is of type . The case where #Λ = 2i + 1 is odd, although a little more complicated, is treated similarly.
We therefore will change D k by a pull back under π.Define a divisor class on B by and define a line bundle on P by Lemma 10.1.The line bundle M does not depend on k, satisfies f * (M) = ω t and restricts to the general fibre P 1 of π as O(g − 1).For #Λ = 2i + 2 its restriction to the general fibre P 1 ∪ P 2 over ∆ Λ is of degree (i, g − i − 1), while for #Λ = 2i + 1 its restriction to the general fibre Proof.We use the section τ k of t : Y → B with f τ k = sk with sk the natural sections of the map π with image Sk .Then we have τ From this we obtain τ . We also see that the restriction of M on the fibres of π does not depend on k.Moreover, we have showing that the restrictions of O(D k + π * E k ) and O(D j + π * E j ) agree on Sj .The restrictions of the fibres of π over the general points of ∆ Λ b are easily checked.We now want to compare π * (M) with the Hodge bundle E = t * (ω t ) on B. The next proposition shows that these agree up to codimension 2. Proposition 10.2.We have an exact sequence 0 → π * (M) → E → T → 0, where T is a coherent sheaf that is a torsion sheaf supported on the boundary.Moreover, we have c 1 (π * (M)) = λ.
Proof.By Lemma 10.1 we have ω t = f * (M).But R 1 π * (M) = (0), so we have We have an exact sequence 0 → O P → f * O Ỹ → F → 0 with F a coherent sheaf of rank 1 that restricted to the smooth fibers of π has degree −(g + 1), as one sees by applying Riemann-Roch to f and O Ỹ .Tensoring the sequence with M and applying π * gives the exact sequence On the smooth fibers of π the sheaf M ⊗ F restricts to a line bundle of degree (g − 1) − (g + 1) = −2, hence π * (M ⊗ F ) is a torsion sheaf.
We now calculate c 1 (π * (M)).We apply Grothendieck-Riemann-Roch to π and We calculate and Adding π * (π * E k ) gives Substituting the formula for ψ k we find The line bundle M on P is not base point free as Proposition 10.2 shows; the restriction to the R-part has negative degree.We can make it base point free by defining Now the restriction of N to a general fibre over ∆ 2i+1 , which is a chain of three rational curves P 1 , R, P 2 , has degrees (i − 1, 1, g − i − 1) and one checks that N is base point free.
Proof.We have R 1 π * (N) = 0. Therefore the exact sequence 0 By the definition of N and the fact that R is a (−2)-curve if we take a base B of dimension 1, and thus has intersection number 0 with a fibre, we have and c 1 (N) ω π = c 1 (M) ω π since the restriction of ω π to R is trivial.

The rational normal curve
The image of a hyperelliptic curve by the canonical map is a rational normal curve, that is, P 1 embedded in P g−1 via the linear system of degree g − 1.
In our setting we can see the rational normal curve and its degenerations using the extension N of the line bundle of degree g − 1, as defined in (9), to the compactification as constructed in the preceding section.
We let u : P(E) → B be the natural projection.Now N is base point free and up to codimension 2 we have π * (N) = E, so the global-to-local map π * π * (N) → N induces a surjective map ν : π * (E) → N over P.This induces a morphism φ : P → P(E) by associating to a point of P the kernel of ν.It fits into a diagram For a point of B with smooth fibre under π the image of φ is a rational normal curve of degree g − 1.For a general point β ∈ ∆ 2i+2 with fibre P 1 ∪ P 2 the image is a union of two rational normal curves of degree i and g −i−1.For a general point β ∈ ∆ 2i+1 with fibre P 1 , R, P 2 the image is a union of three rational normal curves of degree i − 1, 1 and g − i − 1.Here we interpret the case of degree 0 as a contracted curve.
Proof.The proposition follows almost immediately from Lemma 10.1.
Remark 11.2.If i = 1 then P 1 is contracted.If also g = 2 then both P 1 and P 2 are contracted and the image of R coincides with the fibre of P(E).
Remark 11.3.The sections si : B → P for i = 1, . . ., b induce sections σ i = φ • s i : B → P(E) by sending β to the kernel of E = s * i π * (E) → s * i (N).Remark 11.4.In the case g = 2 the map φ is a birational map P → P(E) that blows down boundary components.More precisely, over ∆ 2 it blows down Π 2 and over ∆ 3 the components supported at Π 3 = Π c 3 .

Symmetrization
We have been working with the moduli space H g,2 and M 0,b and their compactifications.Here the symmetric group S b acts.We therefore make our construction symmetric.Sk .
We have the line bundle M on P defined in (6) corresponding to the divisor class D k + E k given by where ψ k is given in (6).Define the rational divisor class The divisor class of D k + E k is independent of k as observed in Lemma 10.1, but this can be seen also directly from the next lemma.
Lemma 13.2.We have the linear equivalence Proof.One checks that −ω π + Π 2 + Π 3 and D k + E k have the same restriction to fibres of π.We have Let Q be the image of φ : P → P(E), see Proposition 11.1.The map φ is the composition of a map φ ′ : P → Q with the inclusion map ι : Lemma 13.3.On P(E) we have the linear equivalence On the other hand we have By Lemma 13.2 we have Substituting this in (11) we get the desired result.
The effective divisor Q yields a modular form and Lemma 13.3 gives its weight.
Since the divisor ∆ 2 + ∆ 3 + ∆ 4 is not a pull back from the moduli space H 3 , the modular form does not descend to H 3 .Recall that the modular form χ 4,0,8 restricted to the hyperelliptic locus was associated to a divisor D that equals 2 Q.
Remark 13.5.In the same vein as above we can determine in an alternative way the result of Proposition 2.1 on class of the closure D of the ramification divisor D of the universal genus 2 curve.By the theory of admissible covers there is a natural map H 2,2 → M 2 with the property that the pull back of the Hodge bundle on M 2 is the Hodge bundle on H 2,2 associated to the corresponding family of admissible covers.Hence the pull back of the O(1) of the bundle P(E) → M 2 equals the O(1) of the bundle P(E) → H 2,2 .Let Σ = φ * ( 6 k=1 Sk ), with φ : P → P(E) the map defined in Section 11.By geometry, the pull back of D to the bundle P(E) over H 2,2 equals Σ.By Remark 11.4 we have φ * Σ = S + 2 Π 2 + 6 Π 3 .By using the formulas of Section 12, we have for g = 2: We now write [D] = O(6) + u * (a δ 0 + b δ 1 ).By pulling back to P and using the above formulas, we get (we refer to the diagram in Section 11 for notation) This implies hence a = −1/5 and b = 3/5 and the result follows by using the formula 10 λ = δ 0 + 2 δ 1 .

Comparison with the Hodge Bundle
We know by Lemma 10.3 that the line bundle N = O P (D k + E k − R) on P over H 3,2 has the property that π * (N) ∼ = E up to codimension 2. We now deal with the push forward of the tensor powers of N. Proof.We apply Grothendieck-Riemann-Roch to π and N ⊗m as in (10) in the proof of Proposition 10.2.Recall that N corresponds to the divisor(class) D k + E k − R. We use that R 1 π * N ⊗m = 0 for all m and find and using the relations ( 8) and ( 9) of the proof of Proposition 10.2 we get as required.
Proposition 14.2.On B we have the exact sequence Proof.By Lemma 13.A section of Sym j (E)⊗det(E) k over H 3 pulls back to the stack [W 0 8,−2 /(GL(W )/(±1 W ))] as a section of Sym j (Sym 2 (W )) ⊗ det(W ) k/2 for even k.We have an isotypical decomposition Sym j (Sym , where we assume here and in the rest of this section that the characteristic is 0 or not 2 and high enough for this identity to hold (or use divided powers as in [1, 3.1]).A section of Sym j (E) ⊗ det(E) k over M nh 3 pulls back to [V 4,0,−1 /GL(V )], where we now write V for the standard space of dimension 3.An identification V ∼ = Sym 2 (W ) corresponds to an embedding P 1 ֒→ P 2 with image a smooth quadric.If we view V with basis x, y, z, the kernel of the projection consists of the polynomials of degree j in x, y, z that vanish on the quadric.Thus in view of the isotypical decomposition above the exact sequence The section χ 4,0,8 of Sym 4 (E) ⊗ det(E) 8 restricted to the hyperelliptic locus allows three projections according to the decomposition Sym 4 (Sym 2 W ) ⊗ det(W ) 24 = W 8,24 ⊕ W 4,26 ⊕ W 0,28 .
Proof.The identification of E with Sym 2 (W ) corresponds to the embedding of P 1 as a conic C in P 2 .A ternary quartic Q contains C either 0, 1 or 2 times, say Q = mC + R with 0 ≤ m ≤ 2. The three projections correspond to R ∩ C and give the universal binary octic, the universal binary quartic and 1 up to twisting.The first projection was identified in Proposition 6.1.The argument for the second is similar, while the third descends to H 3 and does not vanish on H 3 .Therefore it must be a multiple of the disciminant.Taking into account the action of GL 2 /±1 W we get the indicated weights (namely 2(14 + ǫ) with ǫ = −2, −1, 0).

More Modular Forms for Genus Three
We will use more effective divisors on projectivized Hodge bundles to produce more modular forms.Note that the connection between divisors on projectivized Hodge bundles and modular forms can also be used in the other direction: obtaining results on cycle classes using modular forms.We give a few examples.To a canonical quartic plane curve C we can associate a curve Š in the dual plane of lines intersecting C equianharmonically.It corresponds to a contravariant (concomitant) σ of the ternary quartic given by Salmon in [25, p. 264] and it is defined by an equivariant GL(3) embedding W [4, 4, 0] ֒→ Sym 2 (Sym 4 (W )).It gives rise to a divisor in P(E ∨ ) and a modular form χ 0,4,16 of weight (0, 4, 16).We refer to [5, p. 54] for the relation between invariant theory of ternary quartics and modular forms.The Siegel modular form χ 0,4,16 vanishes with order 2 at infinity and order 4 along the locus A 2,1 of decomposable abelian threefolds.With ǔ : P(E ∨ ) → M 3 the projection we have ǔ * (O P(E ∨ ) (1)) = E ∨ ∼ = ∧ 2 E ⊗ det(E) −1 and we thus find an effective divisor on P(E ∨ ) over Ã3 with class [ Š] = [O P(E ∨ ) (4)] + 20 λ − 2 δ and it vanishes with multiplicity 4 along A 2,1 .We thus find on P(E ∨ ) over M 3 a relation where we identify λ and δ i with their pullbacks to P(E ∨ ).Similarly, in the dual plane we have the sextic Ť of lines intersecting the quartic curve in a quadruple of points with j-invariant 1728.The corresponding concomitant τ corresponds to W [6, 6, 0] ֒→ Sym 3 (Sym 4 (W )) and defines a modular form of weight (0, 6, 24) vanishing with multiplicity 3 at infinity and multiplicity 6 along A 2,1 .We thus get a cycle relation The concomitant σ 3 − 27 τ 2 vanishes on the locus of double conics and the corresponding modular form of weight (0, 12, 48) is divisible by χ 2  18 as can be checked using the methods of [5].Dividing by χ 2  18 gives a cusp form of weight (0, 12, 12) vanishing with multiplicity 2 at infinity and multiplicity 3 along A 2,1 .It is classically known (see e.g.[3, p. 43]) that this concomitant defines the dual curve Č to the canonical image C in P(E).We thus find an effective divisor in P(E ∨ ) containing the closure of the dual curve with class This effective divisor class can also be given by the cycle Another example of an effective divisor for genus 3 is provided by the Weierstrass divisor as given by Gheorghita in [12].Here we get a section of This gives a Teichmüller modular form of weight (0, 24, 44) vanishing with multiplicity 6 at the cusp.It descends to a Siegel modular form.
Corollary 15.1.The dual of the canonical curve defines a Siegel modular cusp form of degree 3 of weight (0, 12, 12) vanishing with multiplicity 2 at infinity.The Weierstrass divisor defines a cusp form of weight (0, 24, 44) vanishing with multiplicity 6 at infinity.

The hypertangent divisor
A generic canonically embedded curve C of genus 3 has 24 (Weierstrass) points where the tangent line intersects C with multiplicity 3. The union of these 24 lines forms a divisor in P 2 .Taking the closure of this divisor for the universal curve over M 3 defines a divisor H in P(E) over M 3 which we call the hypertangent line divisor.We calculate the class of this divisor over M 3 and also calculate the class of a corresponding divisor over H 3,2 .
The calculation over M 3 uses the divisors Š and Ť in P(E ∨ ) over M 3 as defined in the preceding section.It is a classical result that the intersection Š • Ť in the generic fibre is the 0-cycle consisting of the 24 points defining the 24 hyperflexes of the generic curve C, see [25].We consider the incidence variety Let ρ : I → P(E) and ρ : I → P(E ∨ ) be the two projections fitting in the commutative diagram We have the tautological sequence on P(E) Now note that I can be identified with the P 1 -bundle P(F ∨ ) on P(E), but also with the P 1 -bundle P( F ∨ ) on P(E ∨ ).
The tautological inclusion F → u * E induces a surjection u * E ∨ → F ∨ and this gives an inclusion P(F ∨ ) → P(u * E ∨ ) of projective bundles over P(E) which composed with natural map P(u * E ∨ ) → P(E ∨ ) gives the map ρ : I = P(F ∨ ) → P(E ∨ ).This implies this gives the identities of pullbacks of the first Chern classes Since I = P(F ∨ ) over P(E) and ρ * ȟ = c 1 (O P(F ∨ ) (1)), the Chern classes of F ∨ and the first Chern class of the tautological line bundle satisfy the relation Proof.Using relation (14) gives The other properties follow from general intersection theory.
Let now ψ be the class of the codimension 2 cocycle Š • Ť .We now claim that the codimension 2 cycle Š • Ť when restricted to the hyperelliptic locus is of the form 12 ȟ, in other words, by (2) it contains an effective codimension 2 cycle with class 12 (9 λ − δ 0 − 3 δ 1 ) ȟ + ǔ * (ξ) with ξ a codimension 2 class on M 3 .We check this using the explicit form of the two concomitants σ and τ defining Š and Ť .Here σ is a polynomial of degree 4 in a 0 , . . ., a 14 and degree 4 in the coordinates u 0 , u 1 , u 2 where a 0 , . . ., a 14 are the coefficients of the general ternary quartic.A calculation shows that σ restricted to the locus of double conics becomes a square q 2 with q of degree 2 in the u i , while τ becomes a cube q 3 .Hence the cycle S ∨ • T ∨ restricted to the hyperelliptic locus is represented by an effective cycle representing 6 q ∼ 12 ȟ.By Corollary 16.1 under ρ * ρ * this is sent to an effective cycle with class 12(9 λ − δ 0 − 3 δ 1 ).Since H is defined as the closure of the hypertangent divisor in the generic fibre, the class of H equals ρ * ρ * ψ minus 12 times the class of the hyperelliptic locus; by Lemma 16.We now work on the Hurwitz space and define and calculate the class of a hypertangent H h divisor there.It is defined by taking the eight tangent lines at the ramification points of the canonical image.More precisely, on P we have the line bundle N defined in (9).Recall that Sk for 1 ≤ k ≤ 8 is the pullback of the section S k of π 9 : M 0,9 → M 0,8 .Under restriction to the hyperelliptic locus the Weierstrass points degenerate to the ramification points.We define the corresponding hypertangent divisor H h in P(E) over H 3,2 by taking the tangents to the canonical image of the generic curve at the points of the sections Sk , k = 1, . . ., 8 over H 3,2 and then taking the closure over H 3,2 .
We now consider the bundle N(−2 Sk ) on P.This line bundle is trivial on the generic fibre of π : P → B, so π * (N(−2 Sk )) is a line bundle on B.
Lemma 16.4.We have The first statement follows by analyzing the restrictions over the boundary components.For the second we apply Grothendieck-Riemann-Roch as in the proof of Proposition 10.2.By (7) and (8) we have Pulling back to P(E) via u * and composing with the canonical surjection u * (E) → O P(E) (1) we get an induced map q : u * F k → O P(E) (1) .
The degeneracy locus of q is an effective divisor F k that is the vanishing divisor of a section of O P(E) (1) ⊗ u * F −1 k .The interpretation is as follows.The map φ defines an embedding of the generic fibre of P into the generic fibre P(E).If we identify H 0 (P 1 , O(2)) with the fibre of E and projectivize, the divisor p 1 + p 2 ∈ |O(2)| is mapped to the line through through the points φ(p 1 ), φ(p 2 ).We now sum these divisors F k and get an effective divisor H h with class where we use the formulas of Section 10 and Section 12.
We can now compare the class of the hyperelliptic hypertangent divisor H h with that of the pull back of the hypertangent divisor H to the Hurwitz space.By Proposition 16.If we view a section of Sym 2 (E) as a quadratic form on E ∨ we can take the discriminant, cf.[6].Doing this with the form χ of weight (2, 0, 0, 8) just constructed we get a scalarvalued modular form D(χ) of weight 34.This modular form vanishes on the closure of the locus of curves whose canonical model lies on a quadric cone.This locus has class 34λ − 4 δ 0 − 14 δ 1 − 18 δ 2 by Teixidor i Bigas [26,Prop. 3.1] and equals the divisor of curves with a vanishing thetanull.The modular form D(χ) is the square root of the restriction to M 4 of the product of the even theta characteristics on A 4 .

Appendix on base-point freeness
The relative dualizing sheaf ω π of the universal family π : C g → M g of genus g smooth curves is base point free and the surjection π * E → ω π gives a map ϕ : C g → P(E) over M g , which is generically an embedding.Let Γ be the image ϕ(C g ).We wish to describe the closure of the image over the generic points of the boundary components ∆ i for i = 0, . . ., [g/2].Over the general point of ∆ 0 the sheaf ω π is base point free and the map ϕ extends over this locus.But over the general point of ∆ i , i ≥ 1, which represents a nodal curve of the form C 1 ∪C 2 , with C 1 , C 2 smooth curves of genus i and g −i meeting at a nodal point x, the sheaf ω π has a base point at x.We consider a family π : Y → B of stable curves of genus g with B the spectrum of a discrete valuation ring.We assume that the central fibre C is a nodal curve C = C 1 ∪ C 2 of genera i and g − i and smooth generic fibre.After a degree 2 base change B ′ → B we get an A 1 -singularity which we resolve resulting in a semistable family π ′ : X → B ′ with a special fibre which is a chain of three curves and g − i and R a rational (−2)-curve.We have the commutative diagram The morphism v is (2 : 1) ramified at C 1 , C 2 .We have v * ω π = ω π ′ , and There is then a natural (2 : 1) map P(E B ′ ) → P(E B ). Now we will show that the system ω π ′ (−R) defines a map X → P(E B ′ ) which combined with the above (2 : 1) map gives a (2 : 1) map ϕ ′ : X → P(E B ) mapping the curves C ′ To avoid unnecessary notation we now write π : X → B for the semi-stable family denoted by π ′ : X → B ′ above.
and defines a base point free linear system on C.
Next we observe that dim H 0 (C, ω(−R)) = g 1 + g 2 + 1 with g i the genus of C i .This follows directly from ω We have the exact sequence where ) the section s is the unique section of O R (1) that vanishes at q and with s(p) = s 2 (p).We thus see dim 15) must be zero and we get an induced exact sequence Consider now a section σ of π * (ω(−R − C 1 )) with restriction (s 1 , s, s 2 ) to C. Suppose that s = 0.If we multiply σ with a local section τ of O(C 1 ) on X with divisor C 1 then ι(σ) = σ • τ |C has as restriction to R a section of O R (2) vanishing with multiplicity 2 at q and therefore it does not vanish anywhere else.Hence the subspace of the special fibre V of π * (ω(−R)) of sections vanishing on C 1 has q as only base point on R. Furthermore, the map j is surjective, and choosing a section s 1 ∈ H 0 (C 1 , ω C 1 ) with s 1 (q) = 0 we see that q is not a base point.Therefore there are no base points on R. By the surjectivity of j the restriction of V to C 1 is H 0 (C 1 , ω C 1 ) and therefore there are no base points on C 1 .By symmetry the same holds for C 2 .
Similarly to (15) we have an exact sequence and by a similar reasoning we see that we get an exact sequence This implies that given there is a unique element (s 1 , s, s 2 ) in the special fibre V of π * (ω(−R)) mapping to (s 1 , s 2 ) under j.The morphism X → P(π * (ω(−R))) is given by the surjection π * π * (ω(−R)) → ω(−R).The image of the curve C in the special fibre of P(E) consists of the canonical images of C 1 and C 2 , provided with with image of p and q and the image of R, that is, the line connecting the images of p and q.If the genus g(C i ) = 1 then the image of C i is a point.

Appendix: divisor classes of Gheorghita-Tarasca and Korotkin-Sauvaget-Zograf
Here we apply the method employed in Section 16 to determine in a relatively straightforward way the divisor classes of two divisors in P(E ∨ k ) with E k = π * (ω k π ), thus reproving a theorem of Gheorghita-Tarasca ([13, Thm 1]) and a theorem of Korotkin-Sauvaget-Zograf ([21, Thm.1.12]).The first divisor is a generalization of a divisor in P(E ∨ ) considered by Gheorghita in [12].We consider in P(E ∨ k ) over M g the divisor G k = {(C, ω) ∈ P(E ∨ k ) : div(ω) contains a Weierstrass point} and let G k be the closure of G k in P(E ∨ k ) over M g .We let ǔ : P(E ∨ k ) → M g be the natural morphism and ȟ the hyperplane class on P(E ∨ k ).Theorem 19.1.(Gheorghita-Tarasca) The class of G k is given by The second divisor is the divisor Z k in P(E ∨ k ) over M g of regular k-differentials for k ≥ 2 possessing a double zero.
For the proof of both theorems we use, as in Section 16, the incidence variety I k between P(E k ) and P(E ∨ k ) which fits in the following commutative diagram: . Similarly, I k = P(F ∨ k ) as a bundle over P(E k ), with F k the tautological rank r − 1 bundle on P(E k ).Then ρ * ( ȟ) = O P(F k ∨ ) (1).
Proof of Theorem 19.2.Here k ≥ 2, hence we have the morphism ϕ k : C g → P(E k ).Let π 1 : C g,1 → C g be the universal curve over C g and s : C g → C g,1 the tautological section the image of which we denote by S.

Corollary 4 . 2 .
Let D be the closure of the canonical curve over M 3 in P(E) and s the natural section 1 of O(D).Then χ 4,0,8 = u * (s) is a Teichmüller modular form and it descends to a Siegel modular cusp form of degree 3 and weight (4, 0, 8).
We put D = b k=1 D k and E = b k=1 E k and set M = O(D + E) , ψ = b k=1 ψ k , and S = b k=1
has a point of multiplicity 2} over M 3 and Korotkin and Zograf in [22, Thm.1] determined the class of its closure B

1 and C ′ 2
to their canonical image and R to a double line.The reduced image of the map ϕ ′ describes the closure of D over b 0 , the special point of B.