Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms

Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bullet\pi_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.


INTRODUCTION
Let π: C g ,P → M g ,P be the universal curve over the moduli stack of stable marked curves, where P is a nonempty ordered set of markings. The weak Franchetta conjecture, now a theorem due to Harer [7] and Arbarello-Cornalba [1], gives an explicit description of the Picard group of the universal curve. Every divisor on C g ,P , up to a divisor pulled back from M g ,P , is rationally equivalent to (0.1) for some integers ℓ, d p and a h,S . Here K π = c 1 (ω π ) is the first Chern class of the relative dualising sheaf, σ p is the class of the p -th section, and C h,S is the class of the irreducible component, not containing the moving point, in the inverse image of the boundary divisor ∆ h,S in the universal curve (more details in Section 1).
Our main result is an explicit formula for ch(R • π * (D)), in terms of the standard generators of the tautological ring (boundary strata classes decorated with κ classes and ψ classes). In Section 2 we prove: Here the coefficients B k are the Bernoulli numbers defined by x e x −1 = k ≥0 B k k ! x k . The notation for tautological classes and for gluing maps (which we labelled ξ) is explained in Section 1. In the formula, we have adopted the conventions κ −1 · ψ s = ψ s −1 for all s ≥ 0, and ψ −1 = 0. Moreover, G(h,S ) ⊆ G g ,P is the subset consisting of stable graphs all of whose edges are separating and of type (h ,S ). This is explained in Section 1.1.
Our formula expresses ch t (R • π * (D)) as a polynomial of degree t + 1 in the unknowns ℓ, We prove Theorem 1 by applying the Grothendieck-Riemann-Roch formula to the universal curve π, as in Mumford's seminal calculation of the Chern classes of the Hodge bundle [12, Section 4].
Our main motivation is computing the pullback of (extended, cohomological) Brill-Noether classes w r d on the universal Jacobian via the Abel-Jacobi sections. Here we give a preview, full details are in Section 3.
Fix 0 ≤ d ≤ g − 1 and let J d g ,P → M g ,P be the universal Jacobian parametrising line bundles of degree d over smooth P -pointed curves of genus g . Let denote the universal (or Poincaré) line bundle on the universal curve For 0 ≤ r ≤ d /2, the universal Brill-Noether locus W r d is set-theoretically defined by g ,P , and can be endowed with the scheme structure of the (g − d + r )-th Fitting ideal of R 1 p * .
Each W r d is in general not equidimensional, and the dimension of its irreducible components is unknown. However a (cohomological) Brill-Noether cycle w r d supported on W r d and of the correct dimension can be defined, via the Thom-Porteous formula, as the (r + 1) × (r + 1) determinant The notation ∆ (p ) q c stands for the p × p determinant |c q + j −i |, for 1 ≤ i , j ≤ p and a general series c = k c k (see Section 3.2 for more details).
The discussion of the previous paragraph extends verbatim over M g ,P . One constructs a compactified universal Jacobian This problem is complicated by the fact that s is, in general, only a rational map. Theorem 1 allows one to compute s * w r d (φ) for every φ such that s is a morphism (these φ's are characterised in [11, Section 6.1]). Indeed, for every such φ, in Section 3 we will prove the equality where D(φ) is a modification of the divisor D of (0.1) obtained by replacing the coefficients a h,S with the unique coefficients a h,S (φ) such that D(φ) is φ-stable on all curves with 1 node.
Combining (0.5) with Theorem 1 and with the inversion Formula (3.9) for the Chern character, we obtain an explicit expression, for all φ such that s is a morphism, for the cohomology class s * w r d (φ) in terms of the standard generators of the tautological ring. Throughout we work over the field of complex numbers .

TAUTOLOGICAL CLASSES
Throughout we fix an integer g ≥ 1 and an ordered set of markings P = . We follow the exposition and the notation of [2, Section 17.4] to introduce the tautological ring of the moduli space M g ,P of stable P -pointed curves of genus g . We let π: C g ,P → M g ,P be the universal stable P -pointed curve. For each marking p ∈ P , we let denote the divisor class corresponding to the p -th section of π. Let ω π be the relative dualising sheaf, and set We will often simply write K instead of K π . We define the cotangent line classes, or psi classes, by . For a ≥ 0, the kappa classes have codimension a and are related by the formula [2, Formula 3.4, p. 572] The tautological ring of the moduli space of stable marked curves was originally defined by Mumford in [12,Section 4] in the unmarked case P = , and an elegant definition for all moduli spaces of stable marked curves at once was later given by C. Faber and R. Pandharipande [4]. We will give here an alternative definition to suit our purposes.
First we recall the notion of decorated boundary stratum class. For Γ = (V(Γ), E(Γ)) in the set G g ,P of (isomorphism classes of ) stable P -pointed graphs of genus g , we let and denote by ξ Γ : M Γ → M g ,P the associated clutching morphism. Here, P v is the set of edges and legs (half-edges) issuing from the vertex v , and the stability condition 2g v − 2 + |P v | > 0 is fulfilled by all v . A "decoration" θ on the graph Γ is the datum of a monomial for Γ and θ as above, are called decorated boundary strata classes. (Here and in the following, we omit writing the pullback via the projection map to the factor, and we omit writing the tensor product of classes, unless that helps identifying which factor they are pulled back from). We define R • (M g ,P ) to be the vector subspace of A • (M g ,P ) generated by these classes and then endow it with the intersection product. When θ v is trivial for all v , we simply write δ Γ . . = ξ Γ * ( )/|Aut Γ|. The collection of decorated boundary strata classes can be made into a finite set (for fixed g and P ) by only considering decorations θ that are not obviously vanishing for codimensional reasons. Even so, this collection is far from being a basis. All known relations among these generators belong to a vector space generated by the so-called Pixton's relations, but whether or not these are all the existing relations is so far unknown.
In this paper, "calculating" an element of R • (M g ,P ) will always mean expressing it as an explicit, non-unique, linear combination of decorated boundary strata classes. Example 1.1. We define the set of stable bipartitions of (g , P ) to be the collection of pairs (h ,S ) where S ⊂ P contains a distinguished "first" marking and 0 ≤ h ≤ g is such that if h = 0 then |S | ≥ 2 and if h = g then |S c | ≥ 2 (where S c = P \ S denotes the complement). With this convention, there is a bijection between the set of stable bipartitions and the set of stable graphs Γ h,S ∈ G g ,P with two vertices and one edge. The corresponding (codimension one) clutching morphism is denoted Its image is the boundary divisor ∆ h,S and its class δ Γ h,S will simply be denoted by δ h,S .

Example 1.2.
There is one more boundary divisor of M g ,P , which parametrises irreducible singular curves. That divisor is the image of the clutching morphism ξ irr attached to the stable graph Γ irr consisting of one vertex of genus g −1 with a loop, and carrying all markings P . Now for a fixed stable bipartition (h ,S ) of (g , P ), the inverse image π −1 (∆ h,S ) in C g ,P consists of two irreducible components. We will denote by C + h,S the class of the component that contains the moving point on the universal curve, and by C h,S the class of the other component. We then have the obvious relation . 1.1. Self-intersection of boundary divisors. For later use, we provide here an explicit expression for the intersections in terms of decorated boundary strata classes. An edge of a stable graph Γ ∈ G g ,P is said to be of type (h ,S ) if it disconnects the graph into a component of genus h with markings S and another component of genus g − h and markings S c . Let be the set of stable graphs all of whose edges are of type (h ,S ). A direct calculation using the general formula to compute the intersection product of two decorated boundary strata classes (see [6,Appendix]) yields The notation [α] c means taking the part of codimension c of a class α.
The self-intersection δ b h,S is easier to compute when S = P or when h < g /2. Indeed, in this case the set G (h ,S ) contains only the graph Γ h,S , so that the expansion On the other hand, when S = P and h ≥ g /2, the graphs with a nonzero contribution in (1.2) may correspond to boundary strata of codimension bigger than one. They can be characterised as follows. Define, for all t ≥ 0, the number g t = (t + 1)h − t g , and set Consider, for t > 0, the tree Γ t with a vertex v t of genus g t −1 carrying all the markings P , and with t additional vertices, all of genus g − h , each connected to v t by exactly one edge (see Figure 1) The clutching morphism attached to Γ t is The product corresponding to Γ = Γ t in right hand side of (1.2) now consists of t factors. Expanding them via (1.3) again, one finds the explicit formula Since ψ classes pull back along the clutching morphisms ξ Γ , the projection formula yields, for each a ≥ 0 and marking p ∈ P , the equality

PROOF OF MAIN THEOREM
This section provides a proof of our main result, Theorem 1, using the notation established in Section 1. We prove the theorem by following Mumford (and later Chiodo), namely by applying the Grothendieck-Riemann-Roch formula to the universal curve π. There are, in principle, different ways to approach the calculation. The main point is to compute the pushforward along π of products of divisors, and the main difficulty is to devise the computation so that all pushforwards that one actually has to deal with satisfy closed formulas with explicit coefficients in terms of decorated boundary strata classes (see (2.6) and (2.8)).
A classical argument first given by Mumford and described in [2,Chapter 17.5] shows that Td ∨ ( Σ ) −1 − 1 intersects K π trivially. Therefore where Ψ is the pushforward of Td ∨ ( Σ ) −1 − 1 by π. The latter is also explicitly computed in loc. cit. as We now need a formula for For fixed t > 0, its term of codimension t − 1 can be written as From now on our goal will be to work on each of the summands of (2.3) in order to express (2.2) in terms of decorated boundary strata classes. The first summand of (2.3) is already one of those generators.
We start out by setting e p = d p − ℓ and rewriting Using the vanishing relations σ p · C h,S = 0 for all p / ∈ S , we obtain the formula Note that when i = 0, the formula reduces to (2.4), and its pushforward via π reduces to (1.1).
Substituting i = t in the right hand side of (2.5) does not compute D t (because (2.4) does not hold for i = t ). Nevertheless, D t satisfies the same formula, where the sum over 0 ≤ b < t is replaced by a sum over 0 < b < t . However, we are interested in the pushforward of this expression via π. So, for fixed p ∈ S , consider the identity (2.6) π * σ c p · C d h,S = (−ψ p ) c −1 · δ d h,S , valid for all d ≥ 0 and c > 0. If one sets i = t and b = 0 in the right hand side of (2.5), the identity (2.6) would formally express the pushforward of σ 0 p · C t h,S as −ψ −1 p · δ t h,S . From now on we then adopt the convention ψ −1 p = 0, for every p ∈ P . This allows us to view the pushforward π * D t as a special case of the pushforward of (2.5), by substituting i = t . Using (1.1) again along with (2.6) (and the convention ψ −1 = 0), we see that for fixed t > 0 and i = 0, . . ., t the term from the right hand side of (2.2) that we need to compute is equal to The powers ψ α · δ β are expressed in terms of boundary strata classes via (1.4). The pushforwards π * K α · C β h,S , on the other hand, are taken care of by the following calculation. Lemma 2.1. Fix a stable bipartition (h ,S ) and integers α, β ≥ 0. Then Proof. The case β = 0 gives the definition of the κ class (or 0, if α = 0), so from now on we assume β > 0. We identify the universal curve C g ,P with the moduli stack M g ,P ∪x , mapping down to M g ,P by forgetting the last marking x , so the class C h,S is identified with the class of the boundary divisor δ h,S ∈ A 1 (M g ,P ∪x ). We will use the commutative diagram where η 2 is the second projection, the clutching morphisms glue the markings q and r together, and τ is shorthand for id ×τ. By [2, p. 582], one has Furthermore, by [2, Lemma 4.36, p. 583] applied to M g ,P ∪x , and taking advantage of the convention 1 ∈ S , one has (2.10) ι * C h,S = ι * δ h,S = −ψ q − ψ r , for every stable bipartition (h ,S ). Using that C h,S = ι * , we find by (2.9) and (2.10) where the last equality is obtained by expanding the binomial and by applying the Künneth decomposition. The result now follows from the string equation when α = 0 and from the dilaton equation when α > 0 (see [2, Proposition 4.9, p. 574]). To conclude the proof, we observe that if α = 0 the term corresponding to b = β −1 vanishes for dimension reasons.
We make the following convention, in order to level out the different formulas in (2.8). For all s ≥ 0, put keeping the convention ψ −1 = 0. Then Formula (2.8) becomes, uniformly for all α, β ≥ 0, (2.12) Substituting the expression for Ψ from Formula (2.1) completes the proof of Theorem 1.

Example 2.2.
Let us compute the coefficient of δ h,S for ch 1 (R • π * (D)) in the basis of decorated boundary strata classes for the rational Chow group of codimension-1 classes of M g ,P , which consists of where d S = p ∈S d p . The latter contributes The coefficient of δ h,S is therefore

PULLBACK OF BRILL-NOETHER CLASSES VIA ABEL-JACOBI SECTIONS
In this section we review the definition of compactified universal Jacobians J g ,P (φ) and then define the cohomological, universal Brill-Noether classes is an assignment, for every stable P -pointed curve (C , P ) of genus g and every irreducible component C ′ ⊆ C , of a real number φ(C , P ) C ′ such that and such that (1) if α: (C , P ) → (C ′ , P ′ ) is a homeomorphism of pointed curves, then φ(C ′ , P ′ ) = φ(α(C , P )); (2) informally, the assignment φ is compatible with degenerations of pointed curves. The notion of φ-(semi)stability was introduced in [11, Definitions 4.1 and 4.2]: Definition 3.1. Given φ ∈ V d g ,P we say that a family F of rank 1 torsion free sheaves of degree d on a family of stable curves is φ-stable if the inequality holds for every stable P -pointed curve (C , P ) of genus g of the family, and for every subcurve (i.e. a union of irreducible components) = C 0 C . Here δ C 0 (F ) denotes the number of nodes p ∈ C 0 ∩ C c 0 such that the stalk of F at p fails to be locally free. Semistability with respect to φ is defined by allowing equality in (3.1).
A stability parameter φ ∈ V d g ,P is said to be nondegenerate when φ-semistability coincides with φ-stability for all stable P -pointed curves of genus g .
For all φ ∈ V d g ,P there exists a moduli stack J g ,P (φ) of φ-semistable sheaves on stable curves, which comes with a forgetful morphism p : J g ,P (φ) → M g ,P .
When φ is nondegenerate, by [11, Corollary 4.4] the stack J g ,P (φ) is a smooth Deligne-Mumford stack, and the morphism p is representable, proper and flat.

Universal Brill-Noether classes and their pullbacks.
Let φ ∈ V d g ,P be nondegenerate. Then by [10,Corollary 4.3] and [11, Lemma 3.35] combined with our general assumption P = , there exists a tautological family (φ) of rank 1 torsion free sheaves of degree d on the total space of the universal curve π: J g ,P (φ) × M g ,P C g ,P → J g ,P (φ).
Recall the following notation from [5, Ch. 14]. Let c = k ∈ c k be a formal sum of elements in a ring R . Then the p × p determinant |c q + j −i | in R is denoted Generalising the idea of [10, Definition 3.38] (where the authors extended the universal theta divisor w 0 g −1 ), we define the (universal, cohomological) Brill-Noether class using the Thom-Porteous formula, namely by (3.2) w r d (φ) . . = ∆ (r +1) g −d +r c (−R • π * (φ)) ∈ A g −ρ (J g ,P (φ)), for ρ = g − (r + 1)(g − d + r ) the Brill-Noether number. One can interpret the class (3.2) as follows. Define the universal Brill-Noether scheme as the closed subscheme defined by the (g − d + r )-th Fitting ideal of R 1 π * (φ) (see [2,Ch. 21] for the use of Fitting ideals in Brill-Noether theory). Then the Poincaré dual of (3.2) is the class that W r d (φ) would have as its fundamental class if it were pure of the expected codimension g − ρ. The scheme (3.3) has an explicit description as a degeneracy scheme, which was already described in the proof of [8,Lemma 6] in the case r = d = 0. Fix a sufficiently π-ample divisor H , and consider the short exact sequence Pushing this forward via π yields a presentation 0 where π * u is a morphism of vector bundles whose virtual rank is which agrees with (3.3) by the general theory of Fitting ideals. Note that, by this identification, the vanishing locus (3.4) is independent of the choice of H . Note that W r d (φ) is settheoretically supported on The definition (3.2) is motivated by the following lemma.
where Σ(φ) is the closure in J g ,P (φ) of the image of the section s. The second equality of Formula (3.7) follows from the definition of pullback of an algebraic class by a rational map, and it is well-defined because J g ,P (φ) is smooth and proper.
When φ is such that the line bundle D of (0.1) is φ-stable, the map (3.6) is a well-defined morphism on M g ,P , but the converse is not true. As explained in [11, Section 6.1], for all nondegenerate φ ∈ V d g ,P there is a unique modification D(φ) of D that coincides with D on the locus parametrising smooth curves and that is φ-stable on all curves having 1 node. More explicitly, D(φ) is obtained from D by modifying the coefficients a h,S of C h,S into coefficients a h,S (φ) in the unique way that makes the resulting D(φ) a divisor that is φ-stable on all curves with 1 node. By [11, Proposition 6.4], we have that s is a well-defined morphism on the open locus We now show how Theorem 1 allows us to compute the restriction to U (φ) of the class s * w r d (φ). Chiodo's formula recovers the particular case when D(φ) equals ℓ K π + p ∈P d p σ p .
All equalities require to restrict to the locus where s is a morphism. The first follows from the fact that Chern classes commute with pullbacks. The second is cohomology and base change is tautological on U (φ) -meaning that it is the restriction of a tautological class globally defined on M g ,P . That tautological class is explicitly expressed in terms of decorated boundary strata by combining Theorem 1 with Formulas (3.8) and (3.9). We do not know whether the class Z(φ) is, in general, itself tautological on M g ,P , although we do expect that this should be the case. Except for when Z(φ) has codimension 1 or 2 (when we know that the entire cohomology of M g ,P is tautological), the only classes Z(φ) that we know to be tautological on M g ,P for general g and P are those for r = d = 0 and φ a small perturbation of 0 ∈ V 0 g ,P . This follows from the main result of [8], showing that this class coincides with the double ramification cycle. The latter is shown to be tautological in [4].

4.2.
Wall-crossing. For fixed d ∈ and for every choice of nondegenerate elements φ and φ ′ of V d g ,P one has classes w r d (φ) ∈ A • (J g ,P (φ)) and w r d (φ ′ ) ∈ A • (J g ,P (φ ′ )). A natural question is to "compute" (in terms of some natural classes) the difference where α is any birational isomorphism J g ,P (φ) J g ,P (φ ′ ) that commutes with the forgetful morphisms to M g ,P (such birational maps are esplicitly characterised in [11, Section 6.2]).
To the best of our knowledge, this question has been answered only for the case of the theta divisor, namely when r = 0 and d = g − 1, in [10].
Another natural question is to compute the difference of the pullbacks for different assignments (ℓ, d P ), (ℓ ′ , d ′ P ) and different nondegenerate φ, φ ′ ∈ V d g ,P . The case of the pullback of the theta divisor is again covered explicitly in [10]. Theorem 1 immediately allows us to generalise the result in loc. cit., in the sense that it computes explicitly, in terms of decorated boundary strata classes of M g ,P , the difference (4.1), whenever φ and φ ′ are such that the corresponding Abel-Jacobi sections s and s ′ extend to morphisms on M g ,P .
Example 2.2 shows that the results of this paper match the earlier results of [10] for the case of the pullback of the theta divisor.