DIVISOR CLASSES AND THE VIRTUAL CANONICAL BUNDLE FOR GENUS 0 MAPS

A. J. DE JONG AND JASON STARR

Abstract. We prove divisor class relations for families of genus 0 curves and used them to compute the divisor class of the “virtual” canonical bundle of the Kontsevich space of genus 0 maps to a smooth target. This agrees with the canonical bundle in good cases. This work generalizes Pandharipande’s results in the special case that the target is projective space, [7], [8]. Our method is completely different from Pandharipande’s.

1. Statement of results

Much geometry of a higher-dimensional complex variety X is captured by the rational curves in X. For uniruled and rationally connected varieties the parameter spaces for rational curves in X are also interesting. These parameter spaces are rarely compact, but there are natural compacitifications: the Chow variety, the Hilbert scheme and the Kontsevich moduli space. Of these, the most manageable is the Kontsevich space. Briefly \overline{M}_{0,r}(X, β) parametrizes data (C, p_{1}, …, p_{r}, f) of a proper, connected, at-worst-nodal, arithmetic genus 0 curve C, a collection p_{1}, …, p_{r} of distinct, smooth points of C, and a morphism f : C → X with f_{∗}[C] = β and satisfying a natural stability condition.

When \overline{M}_{0,r}(X, β) is irreducible and reduced and when the dimension equals the expected, or virtual, dimension, one can ask more refined questions about the geometry of \overline{M}_{0,r}(X, β). For instance, what is its Kodaira dimension? The first step in answering this and other questions is understanding the canonical bundle.

In this article we give a formula for the virtual canonical bundle of \overline{M}_{0,r}(X, β). The virtual canonical bundle is a naturally defined line bundle which equals the actual canonical bundle if \overline{M}_{0,r}(X, β) is irreducible and reduced and its dimension equals the virtual dimension.

Theorem 1.1. Assume that e := ⟨C_{1}(T_{X}), β⟩ ≠ 0. For \overline{M}_{0,0}(X, β), the virtual canonical bundle equals

\frac{1}{2e}[2eπ_{∗}f^{∗}C_{2}(T_{X}) − (e + 1)π_{∗}f^{∗}C_{1}(T_{X})^{2}+

∑_{{β′,β′′},β′+β′′=β}(⟨f^{∗}C_{1}(T_{X}), β′⟩⟨f^{∗}C_{1}(T_{X}), β′′⟩ − 4e)Δ_{β′,β′′}].

For \overline{M}_{0,1}(X, β), the virtual canonical bundle equals

\frac{1}{2e}[2eπ_{∗}f^{∗}C_{2}(T_{X}) − (e + 1)π_{∗}f^{∗}C_{1}(T_{X})^{2}+

Date: June 17, 2006.

∑_{{β′,β′′},β′+β′′=β}(⟨f^{∗}C_{1}(T_{X}), β′⟩⟨f^{∗}C_{1}(T_{X}), β′′⟩ − 4e)Δ_{β′,β′′}] + ψ_{1}.

Finally, for r ≥ 2, the virtual canonical bundle of \overline{M}_{0,r}(X, β) equals

\frac{1}{2e}[2eπ_{∗}f^{∗}C_{2}(T_{X}) − (e + 1)π_{∗}f^{∗}C_{1}(T_{X})^{2}+

∑_{{β′,β′′},β′+β′′=β}(⟨f^{∗}C_{1}(T_{X}), β′⟩⟨f^{∗}C_{1}(T_{X}), β′′⟩ − 4e)Δ_{β′,β′′}]+

\frac{1}{r − 1} ∑_{(A,B),1∈A} #B(r − #B)Δ_{(A,B)}.

In order to prove these formulas, we need to prove some divisor class relations for families of genus 0 curves. These relations are of some independent interest.

Proposition 1.2. Let π : C → M be a proper, flat family of connected, atworst-nodal, arithmetic genus 0 curves over a quasi-projective variety M or over a Deligne-Mumford stack M with quasi-projective coarse moduli space. Let D be a ℚ-Cartier divisor class on C.

(i) There is an equality of ℚ-divisor classes on M

π_{∗}(D · D) + ⟨D, β⟩π_{∗}(D · C_{1}(ω_{π})) = ∑_{{β′,β′′},β′+β′′=β}⟨D, β′⟩⟨D, β′′⟩Δ_{β′,β′′}.

(ii) Also, there is an equality of ℚ-divisor classes on C

2⟨D, β⟩D − π^{∗}π_{∗}(D · D) + ⟨D, β⟩^{2}C_{1}(ω_{π}) = ∑_{(β′,β′′)}⟨D, β′′⟩^{2}Δ̃_{β′,β′′}.

The notation Δ indicates a boundary divisor. The precise meaning of each term is given in Section 2. Roughly, Δ_{(A,B)} is the divisor in M parametrizing reducible curves where the marked points indexed by A lie in one component, and the marked points indexed by B lie in the other component. The divisor Δ_{β′,β′′} in M parametrizes reducible curves with one component of class β′ and one component of class β′′. And the divisor Δ̃_{β′,β′′} in C is the closed subset of π^{−1}(Δ_{β′,β′′}) which is the union of all components of fibers with class β′. The bundle ω_{π} on C is the relative dualizing sheaf of π. Finally the bundle ψ_{i} on M is the pullback of ω_{π} by the “i^{th} marked point” section.

The virtual canonical bundle is the determinant of the Behrend-Fantechi perfect obstruction theory for \overline{M}_{0,r}(X, β), cf. [2]. This is a complex E^{•} of O-modules on \overline{M}_{0,r}(X, β) which is perfect of amplitude [−1, 0], together with a map to the cotangent complex

ϕ : E^{•} → L_{\overline{M}_{0,r}(X,β)}^{•}

such that h^{0}(ϕ) is an isomorphism and h^{−1}(ϕ) is surjective. The determinant of a perfect complex is an invertible sheaf that, roughly, is the alternating tensor product of the determinants of the terms of the complex, cf. [5].

If the dimension of \overline{M}_{0,r}(X, β) equals the virtual dimension

⟨c_{1}(T_{X}), β⟩ + dim(X) + r − 3

and if \overline{M}_{0,r}(X, β) is reduced at all codimension 1 points, then the virtual canonical bundle equals the canonical bundle. Even when they differ, the virtual canonical bundle can be quite useful. For instance, there is a lower bound for the dimension of the space of maps from a curve C into a smooth projective variety Y mapping an effective divisor B ⊂ C to specified points of Y

dim_{[f]}Hom(C, Y; f|_{B}) ≥ ⟨−K_{Y} , f_{∗}[C]⟩ + dim(Y)(1 − p_{a}(C) − deg(B))

cf. [6, Theorem II.1.2, Theorem II.1.7]. An analogous results holds for Y = \overline{M}_{0,r}(X, β) when K_{Y} is replaced by the virtual canonical bundle and dim(Y) is replaced by the virtual dimension, cf. [4].

Proposition 1.3. [4, Lemma 2.2] Let C be a projective Cohen-Macaulay curve, let B ⊂ C be a divisor along which C is smooth, and let f : C → \overline{M}_{0,r}(X, β) be a 1-morphism. Assume that every generic point of C parametrizes a smooth, free curve in X. Then for Y = \overline{M}_{0,r}(X, β)

dim_{[f]}Hom(C, Y; f|_{B}) ≥ ⟨−K_{Y}^{virt}, f_{∗}[C]⟩ + dim^{virt}(Y)(1 − p_{a}(C) − deg(B)).

1.1. Outline of the article. There is a universal family of stable maps over \overline{M}_{0,r}(X, β)

(π : C → \overline{M}_{0,r}(X, β), σ_{1} : \overline{M}_{0,r}(X, β) → C, …, σ_{r} : \overline{M}_{0,r}(X, β), f : C → X).

The Behrend-Fantechi obstruction theory is defined in terms of total derived pushforwards under π of the relative cotangent sheaf of π and the pullback under f of the cotangent bundle of X. Thus the Grothendieck-Riemann-Roch theorem gives a formula for the virtual canonical bundle. Unfortunately it is not a very useful formula. For instance, using this formula it is difficult to determine whether the virtual canonical bundle is NEF, ample, etc. But combined with Proposition 1.2, Grothendieck-Riemann-Roch gives the formula from Theorem 1.1. The main work in this article is proving Proposition 1.2.

The proof reduces to local computations for the universal family over the Artin stack of all prestable curves of genus 0, cf. Section 4. Because of this, most results are stated for Artin stacks. This leads to one ad hoc consruction: since there is as yet no theory of cycle class groups for Artin stacks admitting Chern classes for all perfect complexes of bounded amplitude, a Riemann-Roch theorem for all perfect morphisms relatively representable by proper algebraic spaces, and arbitrary pullbacks for all cycles coming from Chern classes, a stand-in Q_{π} is used, cf. Section 3. (Also by avoiding Riemann-Roch, this allows some relations to be proved “integrally” rather than “modulo torsion”).

In the special case X = ℙ_{k}^{n}, Pandharipande proved both Theorem 1.1 and Proposition 1.2 in [8] and [7]. Pandharipande’s work was certainly our inspiration. But our proofs are completely different, yield a more general virtual canonical bundle formula, and hold modulo torsion (and sometimes “integrally”) rather than modulo numerical equivalence.

2. Notation for moduli spaces and boundary divisor classes

Denote by 𝔐_{0,0} the category whose objects are proper, flat families π : C → M of connected, at-worst-nodal, arithmetic genus 0 curves, and whose morphisms are Cartesian diagrams of such families. This category is a smooth Artin stack over Spec ℤ (with the flat topology), cf. [1]. The subcategory Δ parametrizing families with reducible fibers is a closed substack. With respect to the smooth topology, the pair (𝔐_{0,0}, Δ) is locally isomorphic to an irreducible simple normal crossings divisor in a smooth variety. Denote by Δ′ the singular locus in Δ.

Let U_{1} be 𝔐_{0,0} − Δ′ and let U_{2} be 𝔐_{0,0} − Δ. There are smooth atlases for each of these stacks as follows. For U_{2}, the family π : ℙ_{ℤ}^{1} → Spec ℤ defines a 1-morphism ζ_{2} : Spec ℤ → U_{2}. This is a smooth surjective morphism. The 2-fibered product

Spec ℤ ×_{ζ_{2},U_{2},ζ_{2}} Spec ℤ = Aut(ℙ_{ℤ}^{1})

is the group scheme PGL_{2}. Thus U_{2} is isomorphic to the quotient stack [Spec ℤ/PGL_{2}].

There is a similar atlas for U_{2}. Let V = ℤ{e_{0}, e_{1}} be a free module of rank 2. Choose dual coordinates y_{0}, y_{1} for V^{∨}. Let ℙ_{ℤ}^{1} = ℙ(V) be the projective space with homogeneous coordinates y_{0}, y_{1}. Let 𝔸_{ℤ}^{1} be the affine space with coordinate x. Denote by Z ⊂ 𝔸_{ℤ}^{1} × ℙ_{ℤ}^{1} the closed subscheme 𝕍(x, y_{1}), i.e., the image of the section (0, [1, 0]). Let ν : C → 𝔸_{ℤ}^{1} × ℙ_{ℤ}^{1} be the blowing-up along Z. Denote by E ⊂ C the exceptional divisor. Define π : C → 𝔸_{ℤ}^{1} to be pr_{Å^{1}} ∘ ν. This family defines a 1-morphism ζ_{1} : 𝔸_{ℤ}^{1} → U_{1}. This is a smooth surjective morphism. The 2-fibered product

𝔸_{ℤ}^{1} ×_{ζ_{1},U_{1},ζ_{1}} 𝔸_{ℤ}^{1} = Isom_{𝔸_{ℤ}^{2}}(pr_{1}^{∗}C, pr_{2}^{∗}C)

is equivalent to a group scheme G over 𝔸_{ℤ}^{1} whose restriction to U = 𝔸_{ℤ}^{1} − {0} is U × G_{m} × PGL_{2} and whose restriction to {0} ≅ Spec ℤ is the wreath product G_{0} := (B × B) ⋊ 𝔖_{2}. Here B ⊂ PGL_{2} is a Borel subgroup, i.e., the stabilizer of a point in ℙ^{1}, and 𝔖_{2} acts by interchanging the two components of C_{0}. Altogether U_{1} is isomorphic to the quotient stack [𝔸_{ℤ}^{1}/G].

Denote by π : ℭ → 𝔐_{0,0} the universal family. This is a proper, flat 1-morphism of Artin stacks, representable by algebraic spaces. Denote by Pic_{π} the stack parametrizing proper, flat families of connected, at-worst-nodal, arithmetic genus 0 curves together with a section of the relative Picard functor of the family. This is also an Artin stack, and the natural 1-morphism Pic_{π} → 𝔐_{0,0} is representable by (highly nonseparated) étale group schemes, cf. [9, Prop. 9.3.1].

In fact Pic_{π} has a covering by open substacks, each of which maps isomorphically to an open substack in 𝔐_{0,0}. The combinatorics of how these open pieces are glued to form Pic_{π} is straightforward. In particular, the inverse image in Pic_{π} of U_{2} is canonically isomorphic to U_{2} × ℤ.

The intersection U_{1} ∩ Δ parametrizes nodal, arithmetic genus 0 curves with two components, C = C′ ∪ C′′. For every pair of integers e′, e′′ there is a unique invertible sheaf L on C whose restriction to C′, resp. C′′, has degree e′, resp. e′′. The rule [C] ↦ [(C, L)] defines a 1-morphism U_{1} ∩ Δ → Pic_{π} representable by open immersions. Denote by Δ_{(e′,e′′)} the closure in Pic_{π} of the image of this 1-morphism. This is a Cartier divisor in Pic_{π}. The union over all (e′, e′′) of the Cartier divisors Δ_{(e′,e′′)} is precisely the inverse image in Pic_{π} of Δ.

More generally, for every integer r ≥ 0, denote by Π^{r}(Pic_{π}) the r-fold 2-fibered product of Pic_{π} with itself over 𝔐_{0,0}. Equivalently, Π^{r}(Pic_{π}) is the stack of families of genus 0 curves together with r sections of the relative Picard functor of the family.

Let (e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}) be an ordered 2r-tuple of integers. Define Δ_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})} to be the iterated 2-fiber product

Δ_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})} := Δ_{(e′_{1},e′′_{1})} ×_{Δ} ⋯ ×_{Δ} Δ_{(e′_{r},e′′_{r})}.

This is a Cartier divisor in Π^{r}(Pic_{π}).

Let π : C → M be a flat 1-morphism representable by proper algebraic spaces whose geometric fibers are connected, at-worst-nodal curves of arithmetic genus 0. This defines a 1-morphism ξ_{0} : M → 𝔐_{0,0}. Let D_{1}, …, D_{r} be Cartier divisor classes on C. This defines a 1-morphism ξ : M → Π^{r}(Pic_{π}) lifting ξ_{0}. Let f(e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}) be a function on ℤ^{2r} with values in ℤ, resp. ℚ, etc.

Notation 2.1. Denote by

∑_{(β′,β′′)} f(⟨D_{1}, β′⟩, ⟨D_{1}, β′′⟩, …, ⟨D_{r}, β′⟩, ⟨D_{r}, β′′⟩)Δ_{β′,β′′}

the Cartier divisor class, resp. ℚ-Cartier divisor class, etc., that is the pullback by ξ of the Cartier divisor class, etc.,

∑_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})} f(e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r})Δ_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})},

the summation over all sequences (e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}). If f(e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}) = f(e′′_{1}, e′_{1}, …, e′′_{r}, e′_{r}), denote by,

∑_{(β′,β′′)}′f(⟨D_{1}, β′⟩, ⟨D_{1}, β′′⟩, …, ⟨D_{r}, β′⟩, ⟨D_{r}, β′′⟩)Δ_{β′,β′′}

the pullback by ξ of,

∑_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})}′f(e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r})Δ_{(e′_{1},e′′_{1},...,e′_{r},e′′_{r})},

where the summation is over equivalence classes of sequences (e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}) such that e′_{i}+e′′_{i} = e_{i} under the equivalence relation (e′_{1}, e′′_{1}, …, e′_{r}, e′′_{r}) ∼ (e′′_{1}, e′_{1}, …, e′′_{r}, e′_{r}).

Of course these are all infinite sums, and so appear not to give well-defined Cartier divisors. However, every quasi-compact open subset of Π^{r}(Pic_{π}) intersects only finitely many of the divisors. On this subset, this is a finite sum and thus welldefined. Since Π^{r}(Pic_{π}) has a covering by quasi-compact open subsets, the Cartier divisor is defined by gluing.

Example 2.2. Let n ≥ 0 be an integer and let (A, B) be a partition of {1, …, n}. For the universal family over 𝔐_{0,n}, denote by s_{1}, …, s_{n} the universal sections. Then

∑_{β′,β′′} ∏_{i∈A}⟨Image(s_{i}), β′⟩ · ∏_{j∈B}⟨Image(s_{j}), β′′⟩Δ_{β′,β′′}

is the Cartier divisor class of the boundary divisor Δ_{(A,B)}.

3. The functor Q_{π}

Let M be an Artin stack, and let π : C → M be a flat 1-morphism, relatively representable by proper algebraic spaces whose geometric fibers are connected, atworst-nodal curves of arithmetic genus 0. There exists an invertible dualizing sheaf ω_{π}, and the relative trace map, Tr_{π} : Rπ_{∗}ω_{π}[1] → O_{M}, is a quasi-isomorphism. In particular, Ext_{O_{C}}^{1} (ω_{π}, O_{C}) is canonically isomorphic to H^{0}(M, O_{M}). Therefore 1 ∈ H^{0}(M, O_{M}) determines an extension class, i.e., a short exact sequence,

0 ⟶ ω_{π} ⟶ E_{π} ⟶ O_{C} ⟶ 0.

The morphism π is perfect, so for every complex F^{•} perfect of bounded amplitude on C, Rπ_{∗}F^{•} is a perfect complex of bounded amplitude on M. By [5], the determinant of a perfect complex of bounded amplitude is defined.

Definition 3.1. For every complex F^{•} perfect of bounded amplitude on C, define Q_{π}(F^{•}) = det(Rπ_{∗}E_{π} ⊗ F^{•}).

There is another interpretation of Q_{π}(F^{•}).

Lemma 3.2. For every complex F^{•} perfect of bounded amplitude on C,

Q_{π}(F^{•}) ≅ det(Rπ_{∗}(F^{•})) ⊗ det(Rπ_{∗}((F^{•})^{∨}))^{∨}.

Proof. By the short exact sequence for E_{π}, Q_{π}(F^{•}) ≅ det(Rπ_{∗}(F^{•}))⊗det(Rπ_{∗}(ω_{π}⊗ F^{•})). The lemma follows by duality. □

It is straightforward to compute F^{•} whenever there exist cycle class groups for C and M such that Chern classes are defined for all perfect complexes of bounded amplitude and such that Grothendieck-Riemann-Roch holds for π.

Lemma 3.3. If there exist cycle class groups for C and M such that Chern classes exist for all perfect complexes of bounded amplitude and such that Grothendieck-Riemann-Roch holds for π, then modulo 2-power torsion, the first Chern class of Q_{π}(F^{•}) is π_{∗}(C_{1}(F^{•})^{2} − 2C_{2}(F^{•})).

Proof. Denote the Todd class of π by τ = 1+τ_{1} +τ_{2} +. . .. Of course τ_{1} = −C_{1}(ω_{π}). By GRR, ch(Rπ_{∗}O_{C}) = π_{∗}(τ). The canonical map O_{M} → Rπ_{∗}O_{C} is a quasiisomorphism. Therefore π_{∗}(τ_{2}) = 0, modulo 2-power torsion. By additivity of the Chern character, ch(E_{π}) = 2 + C_{1}(ω_{π}) + \frac{1}{2}C_{1}(ω_{π})^{2} + . . .. Therefore,

ch(E_{π}) · τ = 2 + 2τ_{2} + …

So for any complex F^{•} perfect of bounded amplitude,

ch(E_{π} ⊗ F^{•}) · τ = ch(F^{•}) · ch(E_{π}) · τ =
(rk(F^{•}) + C_{1}(F^{•}) + \frac{1}{2}(C_{1}(F^{•})^{2} − 2C_{2}(F^{•})) + …)(2 + 2τ_{2} + …).

Applying π_{∗} gives,

2π_{∗}(C_{1}(F^{•})) + π_{∗}(C_{1}(F^{•})^{2} − 2C_{2}(F^{•})) + …

Therefore the first Chern class of det(Rπ_{∗}(E_{π} ⊗ F^{•})) is π_{∗}(C_{1}(F^{•})^{2} − 2C_{2}(F^{•})), modulo 2-power torsion. □

Remark 3.4. The point is this. In every reasonable case, Q_{π} is just π_{∗}(C_{1}^{2} − 2C_{2}). Moreover Q_{π} is compatible with base-change by arbitrary 1-morphisms. This allows to reduce certain computations to the Artin stack of all genus 0 curves. As far as we are aware, no one has written a definition of cycle class groups for all locally finitely presented Artin stacks that has Chern classes for all perfect complexes of bounded amplitude, has pushforward maps and Grothendieck-Riemann-Roch for perfect 1-morphisms representable by proper algebraic spaces, and has pullback maps by arbitrary 1-morphisms for cycles coming from Chern classes. Doubtless such a theory exists; whatever it is, Q_{π} = π_{∗}(C_{1}^{2} − 2C_{2}).

Let the following diagram be 2-Cartesian,

C′ ⟶^{ζ_{C}} C
π′↓  ↓^{π}
M′ ⟶^{ζ_{M}} M

together with a 2-equivalence θ : π ∘ ζ_{C} ⇒ ζ_{M} ∘ π′.

Lemma 3.5. For every complex F^{•} perfect of bounded amplitude on C, ζ_{M}^{∗}Q_{π}(F^{•}) is isomorphic to Q_{π′}(ζ_{C}^{∗}F^{•}).

Proof. Of course ζ_{C}^{∗}E_{π} = E_{π′}. And ζ_{M}^{∗}Rπ_{∗} is canonically equivalent to R(π′)_{∗}ζ_{C}^{∗} for perfect complexes of bounded amplitude. Therefore ζ_{M}^{∗}Q_{π}(F^{•}) equals det(ζ_{M}^{∗}Rπ_{∗}(E_{π}⊗ F^{•})) equals det(R(π′)_{∗}ζ_{C}^{∗}(E_{π}⊗F^{•})) equals det(R(π′)_{∗}E_{π′}⊗ζ_{C}^{∗}F^{•}) equals Q_{π′}(ζ_{C}^{∗}F^{•}).

□

Lemma 3.6. Let L be an invertible sheaf on C of relative degree e over M. For every invertible sheaf L′ on M, Q_{π}(L ⊗ π^{∗}L′) ≅ Q_{π}(L) ⊗ (L′)^{2e}. In particular, if e = 0, Q_{π}(L ⊗ π^{∗}L′) ≅ Q_{π}(L).

Proof. To compute the rank of Rπ_{∗}(E_{π} ⊗F^{•}) over any connected component of M, it suffices to base-change to the spectrum of a field mapping to that component. Then, by Grothendieck-Riemann-Roch, the rank is 2deg(C_{1}(F^{•})). In particular, Rπ_{∗}(E_{π} ⊗ L) has rank 2e.

By the projection formula, Rπ_{∗}(E_{π} ⊗ L ⊗ π^{∗}L′) ≅ Rπ_{∗}(E_{π} ⊗ L) ⊗ L′. Of course det(Rπ_{∗}(E_{π} ⊗L)⊗L′) = Q_{π}(L)⊗(L′)^{rank}. This follows from the uniqueness of det: for any invertible sheaf L′ the association F^{•} ↦ det(F^{•} ⊗ L′) ⊗ (L′)^{−rank(F^{•})} also satisfies the axioms for a determinant function and is hence canonically isomorphic to det(F^{•}). Therefore Q_{π}(L ⊗ π^{∗}L′) = Q_{π}(L) ⊗ (L′)^{2e}. □

4. Local computations

This section contains 2 computations: Q_{π}(ω_{π}) and Q_{π}(L) for every invertible sheaf on C of relative degree 0. Because of Lemma 3.5 the first computation reduces to the universal case over 𝔐_{0,0}. Because of Lemma 3.5 and Lemma 3.6, the second compuation reduces to the universal case over Pic_{π}^{0}.

4.1. Computation of Q_{π}(ω_{π}). Associated to π_{C} : C → M, there is a 1-morphism ζ_{M} : M → 𝔐_{0,0}, a 1-morphism ζ_{C} : C → C, and a 2-equivalence θ : π_{C} ∘ ζ_{C} ⇒ ζ_{M} ∘ π_{C} such that the following diagram is 2-Cartesian,

C ⟶^{ζ_{C}} C
π_{C} ↓  ↓^{π_{C}}
M ⟶^{ζ_{M}} 𝔐_{0,0}

Of course ω_{π_{C}} is isomorphic to ζ_{C}^{∗}ω_{π_{C}}. By Lemma 3.5, Q_{π_{C}} (ω_{π_{C}}) ≅ ζ_{M}^{∗}Q_{π_{C}}(ω_{π_{C}}). So the computation of Q_{π_{C}} (ω_{π_{C}}) is reduced to the universal family π : ℭ → 𝔐_{0,0}.

Proposition 4.1. (i) Over the open substack U_{1}, ω_{π}^{∨} is π-relatively ample. (ii) Over U_{1}, R^{1}π_{∗}ω_{π}^{∨}|_{U_{1}} = (0) and π_{∗}ω_{π}^{∨}|_{U_{1}} is locally free of rank 3. (iii) Over U_{2}, there is a canonical isomorphism i : det(π_{∗}ω_{π}^{∨}|_{U_{2}}) → O_{U_{2}}.

(iv) The image of det(π_{∗}ω_{π}^{∨}|_{U_{1}}) → det(π_{∗}ω_{π}^{∨}|_{U_{2}}) ⟶^{i} O_{U_{2}} is O_{U_{1}}(−Δ) ⊂ O_{U_{2}}.

(v) Over U_{1}, Q_{π}(ω_{π})|_{U_{1}} ≅ O_{U_{1}}(−Δ). Therefore on all of 𝔐_{0,0}, Q_{π}(ω_{π}) ≅ O_{𝔐_{0,0}}(−Δ).

Proof. Recall the 1-morphism ζ_{1} : 𝔸_{ℤ}^{1} → U_{1} from Section 2. Because ζ_{1} is smooth and surjective, (i) and (ii) can be checked after base-change by ζ_{1}. Also (iv) will reduce to a computation over 𝔸_{ℤ}^{1} after base-change by ζ_{1}.

(i) and (ii): Denote by ℙ_{ℤ}^{2} the projective space with coordinates u_{0}, u_{1}, u_{2}. There is a rational transformation f : 𝔸_{ℤ}^{1} × ℙ_{ℤ}^{1} −−→ 𝔸_{ℤ}^{1} × ℙ_{ℤ}^{2} by

\begin{aligned}f^{∗}x & = & x, \\ f^{∗}u_{0} & = & xy_{0}^{2}, \\ f^{∗}u_{1} & = & y_{0}y_{1}, \\ f^{∗}u_{2} & = & y_{1}^{2} \end{aligned}

By local computation, this extends to a morphism f : C → 𝔸_{ℤ}^{1} × ℙ_{ℤ}^{2} that is a closed immersion and whose image is 𝕍(u_{0}u_{2} − xu_{1}^{2}). By the adjunction formula, ω_{π} is the pullback of O_{ℙ^{2}}(−1). In particular, ω_{π}^{∨} is very ample. Moreover, because H^{1}(ℙ_{ℤ}^{2}, O_{ℙ^{2}}(−1)) = (0), also H^{1}(C, ω_{π}^{∨}) = (0). By cohomology and base-change results, R^{1}π_{∗}(ω_{π}^{∨}) = (0) and π_{∗}(ω_{π}^{∨}) is locally free of rank 3.

(iii): The curve ℙ_{ℤ}^{1} = ℙ(V) determines a morphism η : Spec (ℤ) → U_{2}. This is smooth and surjective on geometric points. Moreover it gives a realization of U_{2} as the classifying stack of the group scheme Aut(ℙ(V)) = PGL(V). Taking the exterior power of the Euler exact sequence, ω_{ℙ(V)/ℤ} = ⋀^{2}(V^{∨}) ⊗ O_{ℙ(V)}(−2). Therefore H^{0}(ℙ(V), ω_{ℙ(V)/ℤ}^{∨}) equals ⋀^{2}(V) ⊗ Sym^{2}(V^{∨}) as a representation of GL(V). The determinant of this representation is the trivial character of GL(V). Therefore it is the trivial character of PGL(V). This gives an isomorphism of det(π_{∗}ω_{π}|_{U_{2}}) with O_{U_{2}}.

(iv): This can be checked after pulling back by ζ_{1}. The pullback of U_{2} is 𝔾_{m,ℤ} ⊂ 𝔸_{ℤ}^{1}. The pullback of i comes from the determinant of H^{0}(𝔾_{m,ℤ} × ℙ_{ℤ}^{1}, ω_{π}^{∨}) = ⋀^{2}(V) ⊗ Sym^{2}(V^{∨}) ⊗ O_{𝔾_{m}}. By the adjunction formula, ω_{C/𝔸^{1}} = ν^{∗}ω_{𝔸^{1}×ℙ^{1}/𝔸^{1}}(E). Hence ν_{∗}ω_{C/𝔸^{1}}^{∨} = I_{Z}ω_{𝔸^{1}×ℙ^{1}/𝔸^{1}}. Therefore the canonical map,

H^{0}(C, ω_{C/𝔸^{1}}^{∨}) → H^{0}(𝔸_{ℤ}^{1} × ℙ_{ℤ}^{1}, ω_{𝔸^{1}×ℙ^{1}/𝔸^{1}}^{∨}),

is given by,

O_{𝔸^{1}}{f_{0}, f_{1}, f_{2}} → ⋀^{2}(V) ⊗ Sym^{2}(V^{∨}) ⊗ O_{𝔸^{1}},
f_{0} ↦ x · (e_{0} ∧ e_{1}) ⊗ y_{0}^{2},
f_{1} ↦ (e_{0} ∧ e_{1}) ⊗ y_{0}y_{1},
f_{2} ↦ (e_{0} ∧ e_{1}) ⊗ y_{1}^{2}

It follows that det(π_{∗}ω_{π}^{∨}) → O_{𝔾_{m}} has image ⟨x⟩O_{𝔸^{1}}, i.e., ζ_{1}^{∗}O_{U_{1}}(−Δ).

(v): By the short exact sequence for E_{π}, Q_{π}(ω_{π}) = det(Rπ_{∗}ω_{π}) ⊗ deg(Rπ_{∗}ω_{π}^{2}). Because the trace map is a quasi-isomorphism, det(Rπ_{∗}ω_{π}) = O_{U_{1}}. By (ii) and duality,

det(Rπ_{∗}ω_{π}^{2}) ≅ det(R^{1}π_{∗}ω_{π}^{2})^{v}ee ≅ det(π_{∗}ω_{π}^{∨}).

By (iv), this is O_{U_{1}}(−Δ). Therefore Q_{π}(ω_{π}) ≅ O_{U_{1}}(−Δ) on U_{1}. Because 𝔐_{0,0} is regular, and because the complement of U_{1} has codimension 2, this isomorphism of invertible sheaves extends to all of 𝔐_{0,0}. □

The sheaf of relative differentials Ω_{π} is a pure coherent sheaf on C of rank 1, flat over 𝔐_{0,0} and is quasi-isomorphic to a perfect complex of amplitude [−1, 0].

Lemma 4.2. The perfect complex Rπ_{∗}Ω_{π} has rank −1 and determinant ≅ O_{𝔐_{0,0}}(−Δ). The perfect complex Rπ_{∗}RHom_{O_{C}}(Ω_{π}, O_{C}) has rank 3 and determinant ≅ O_{𝔐_{0,0}}(−2Δ).

Proof. There is a canonical injective sheaf homomorphism Ω_{π} → ω_{π} and the support of the cokernel, Z ⊂ C, is a closed substack that is smooth and such that π : Z → 𝔐_{0,0} is unramified and is the normalization of Δ. Over U_{1}, the lemma immediately follows from this and the arguments in the proof of Proposition 4.1. As in that case, it suffices to establish the lemma over U_{1}. □

4.2. Computation of Q_{π}(L) for invertible sheaves of degree 0. Let M be an Artin stack, let π : C → M be a flat 1-morphism, relatively representable by proper algebraic spaces whose geometric fibers are connected, at-worst-nodal curves of arithmetic genus 0. Let L be an invertible sheaf on C of relative degree 0 over M. This determines a 1-morphism to the relative Picard of the universal family over 𝔐_{0,0},

ζ_{M} : M → Pic_{π}^{0}.

The pullback of the universal family ℭ is equivalent to C and the pullback of the universal bundle O_{ℭ}(D) differs from L by π^{∗}L′ for an invertible sheaf L′ on M. By Lemma 3.5 and Lemma 3.6, Q_{π}(L) ≅ ζ_{M}^{∗}Q_{π}(O_{ℭ}(D)).

Proposition 4.3. Over Pic_{π}^{0}, π_{∗}E_{π}(D) = (0) and R^{1}π_{∗}E_{π}(D) is a sheaf supported on the inverse image of Δ. The stalk of R^{1}π_{∗}E_{π}(D) at the generic point of Δ_{(a,−a)} is a torsion sheaf of length a^{2}. The filtration by order of vanishing at the generic point has associated graded pieces of length 2a − 1, 2a − 3, …, 3, 1.

Proof. Over the open complement of Δ, the divisor D is 0. So the first part of the proposition reduces to the statement that Rπ_{∗}E_{π} is quasi-isomorphic to 0. By definition of E_{π}, there is an exact triangle,

Rπ_{∗}E_{π} ⟶ Rπ_{∗}O_{C} ⟶^{δ} Rπ_{∗}ω_{π}[1] ⟶ Rπ_{∗}E_{π}[1].

Of course the bundle E_{π} and the canonical isomorphism Rπ_{∗}O_{C} ≅ O_{𝔐} were defined so that the composition of δ with the trace map, which is a quasi-isomorphism in this case, would be the identity. Therefore δ is a quasi-isomorphism, so Rπ_{∗}E_{π} is quasi-isomorphic to 0.

The second part can be proved (and perhaps only makes sense) after smooth basechange to a scheme. Let ℙ_{s}^{1} be a copy of ℙ^{1} with homogeneous coordinates S_{0}, S_{1}. Let ℙ_{x}^{1} be a copy of ℙ^{1} with homogeneous coordinates X_{0}, X_{1}. Let ℙ_{y}^{1} be a copy of ℙ^{1} with homogeneous coordinates Y_{0}, Y_{1}. Denote by C ⊂ ℙ_{s}^{1} × ℙ_{x}^{1} × ℙ_{y}^{1} the divisor with defining equation F = S_{0}X_{0}Y_{0} − S_{1}X_{1}Y_{1}. The projection pr_{s} : C → ℙ_{s}^{1} is a proper, flat morphism whose geometric fibers are connected, at-worst-nodal curves of arithmetic genus 0. Denote by L the invertible sheaf on C that is the restriction of pr_{x}^{∗}O_{ℙ_{x}^{1}}(a) ⊗ pr_{y}^{∗}O_{ℙ_{y}^{1}}(−a). This is an invertible sheaf of relative degree 0. Therefore there is an induced 1-morphism ζ : ℙ_{s}^{1} → 𝔐_{ℤ}(τ_{0,0}(0)).

It is straightforward that ζ is smooth, and the image intersects Δ_{b,−b} iff b = a. Moreover, ζ^{∗}Δ_{a,−a} is the reduced Cartier divisor 𝕍(S_{0}S_{1}) ⊂ ℙ_{s}^{1}. There is an obvious involution i : ℙ_{s}^{1} → ℙ_{s}^{1} by i(S_{0}, S_{1}) = (S_{1}, S_{0}), and ζ ∘ i is 2-equivalent to ζ. Therefore the length of the R^{1}pr_{s,∗}E_{pr_{s}} ⊗ L is 2 times the length of the stalk of R^{1}π_{∗}E_{π}(D) at the generic point of Δ_{a}; more precisely, the length of the stalk at each of (1, 0), (0, 1) ∈ ℙ_{s}^{1} is the length of the stalk at Δ_{a}. Similarly for the lengths of the associated graded pieces of the filtration.

Because E_{pr_{s}} is the extension class of the trace mapping, R^{1}pr_{s,∗}E_{pr_{s}} ⊗ L is the cokernel of the O_{ℙ_{s}^{1}}-homomorphisms,

γ : pr_{s,∗}(L) → Hom_{O_{ℙ^{1}_{s}}} (pr_{s,∗}(L^{∨}), O_{ℙ_{s}^{1}}),

induced via adjointness from the multiplication map,

pr_{s,∗}(L) ⊗ pr_{s,∗}(L^{∨}) → pr_{s,∗}(O_{C}) = O_{ℙ_{s}^{1}}.

On ℙ_{s}^{1} × ℙ_{x}^{1} × ℙ_{y}^{1} there is a locally free resolution of the push-forward of L, resp. L^{∨},

\begin{matrix}0 → O_{ℙ_{s}^{1}}(−1) ⊠ O_{ℙ_{x}^{1}}(a − 1) ⊠ O_{ℙ_{y}^{1}}(−a − 1) ⟶^{F} O_{ℙ_{s}^{1}}(0) ⊠ O_{ℙ_{x}^{1}}(a) ⊠ O_{ℙ_{y}^{1}}(−a) → L → 0, \\ 0 → O_{ℙ_{s}^{1}}(−1) ⊠ O_{ℙ_{x}^{1}}(−a − 1) ⊠ O_{ℙ_{y}^{1}}(a − 1) ⟶^{F} O_{ℙ_{s}^{1}}(0) ⊠ O_{ℙ_{x}^{1}}(−a) ⊠ O_{ℙ_{y}^{1}}(a) → L^{∨} → 0 \end{matrix}

Hence Rpr_{s,∗}L is the complex,

O_{ℙ_{s}^{1}}(−1)⊗_{k}H^{0}(ℙ_{x}^{1}, O_{ℙ_{x}^{1}}(a−1))⊗_{k}H^{1}(ℙ_{y}^{1}, O_{ℙ_{y}^{1}}(−a−1)) ⟶^{F} O_{ℙ_{s}^{1}}⊗_{k}H^{0}(ℙ_{x}^{1}, O_{ℙ_{x}^{1}}(a))⊗_{k}H^{1}(ℙ_{y}^{1}, O_{ℙ_{y}^{1}}(−a)).

A similar result holds for Rpr_{s,∗}L^{∨}. It is possible to write out this map explicitly in terms of bases for H^{0} and H^{1}, but for the main statement just observe the complex has rank 1 and degree −a^{2}. A similar result holds for Rpr_{s,∗}L^{∨}. Therefore R^{1}π_{∗}E_{π}(L) is a torsion sheaf of length 2a^{2}. Because it is equivariant for i, the localization at each of (0, 1) and (1, 0) has length a^{2}.

The lengths of the associated graded pieces of the filtration by order of vanishing at 𝕍(S_{0}S_{1}) can be computed from the complexes for Rpr_{s,∗}L and Rpr_{s,∗}L^{∨}. This is left to the reader. □

Corollary 4.4. In the universal case, Q_{π}(D) = − ∑_{a≥0} a^{2}Δ_{a}. Therefore in the general case of π : C → M and an invertible sheaf L of relative degree 0,

Q_{π}(L) = ∑_{β′,β′′}′⟨C_{1}(L), β′⟩⟨C_{1}(L), β′′⟩Δ_{β′,β′′}.

5. Proof of Proposition 1.2

As usual, let M be an Artin stack and let π : C → M be a flat 1-morphism, relatively representable by proper algebraic spaces whose geometric fibers are connected, atworst-nodal curves of genus 0.

Hypothesis 5.1. There are cycle class groups for C and M admitting Chern classes for locally free sheaves, and such that Grothendieck-Riemann-Roch holds for π. In particular, this holds if M is a Deligne-Mumford stack whose coarse moduli space is quasi-projective.

Proof of Proposition 1.2(i). Define D′ = 2D + ⟨D, β⟩C_{1}(ω_{π}). This is a Cartier divisor class of relative degree 0. By Corollary 4.4,

Q_{π}(D′) = ∑_{β′,β′′}′(⟨2D, β′⟩ − ⟨D, β⟩)(⟨2D, β′′⟩ − ⟨D, β⟩)Δ_{β′,β′′}.

By Lemma 3.3 this is,

\begin{matrix}4π_{∗}(D · D) + 4⟨D, β⟩π_{∗}(D · C_{1}(ω_{π}) + (⟨D, β⟩)^{2}Q_{π}(C_{1}(ω_{π})) = \\ _{∑β_{′},β_{′′}}′(4⟨D, β′⟩⟨D, β′′⟩ − (⟨D, β⟩)^{2})Δ_{β′,β′′}. \end{matrix}

By Proposition 4.1, Q_{π}(ω_{π}) = −_{∑β_{′},β_{′′}}′Δ_{β′,β′′}. Substituting this into the equation, simplifying, and dividing by 4 gives the relation. □

Lemma 5.2. For every pair of Cartier divisor classes on C, D_{1}, D_{2}, of relative degrees ⟨D_{1}, β⟩, resp. ⟨D_{2}, β⟩, modulo 2-power torsion,

\begin{matrix}2π_{∗}(D_{1} · D_{2}) + ⟨D_{1}, β⟩π_{∗}(D_{2} · C_{1}(ω_{π})) + ⟨D_{2}, β⟩π_{∗}(D_{1} · C_{1}(ω_{π})) = \\ _{∑β_{′},β_{′′}}′(⟨D_{1}, β′⟩⟨D_{2}, β′′⟩ + ⟨D_{2}, β′⟩⟨D_{1}, β′′⟩)Δ_{β′,β′′}. \end{matrix}

Proof. This follows from Proposition 1.2(i) and the polarization identity for quadratic forms. □

Lemma 5.3. For every section of π, s : M → C, whose image is contained in the smooth locus of π,

s(M) · s(M) + s(M) · C_{1}(ω_{π}).

Proof. This follows by adjunction since the relative dualizing sheaf of s(M) → M is trivial. □

Lemma 5.4. For every section of π, s : M → C, whose image is contained in the smooth locus of π and for every Cartier divisor class D on C of relative degree ⟨D, β⟩ over M, modulo 2-power torsion,

\begin{matrix}2⟨D, β⟩s^{∗}D − π_{∗}(D · D) − ⟨D, β⟩^{2}π_{∗}(s(M) · s(M)) = \\ _{∑β_{′},β_{′′}}′(⟨D, β′⟩^{2}⟨s(M), β′′⟩ + ⟨D, β′′⟩^{2}⟨s(M), β′⟩)Δ_{β′,β′′}. \end{matrix}

Proof. By Proposition 1.2(i),

\begin{matrix}2s^{∗}D + π_{∗}(D · C_{1}(ω_{π})) + ⟨D, β⟩π_{∗}(s(M) · C_{1}(ω_{π})) = \\ ∑′(⟨D, β′⟩⟨s(M), β′′⟩ + ⟨D, β′′⟩⟨s(M), β′⟩)Δ_{β′,β′′}. \end{matrix}

Multiplying both sides by ⟨D, β⟩,

\begin{matrix}2⟨D, β⟩s^{∗}D + ⟨D, β⟩π_{∗}(D · C_{1}(ω_{π})) + ⟨D, β⟩^{2}π_{∗}(s(M) · C_{1}(ω_{π})) = \\ ∑′(⟨D, β⟩⟨D, β′⟩⟨s(M), β′′⟩ + ⟨D, β⟩⟨D, β′′⟩⟨s(M), β′⟩)Δ_{β′,β′′}. \end{matrix}

First of all, by Lemma 5.4, ⟨D, β⟩^{2}π_{∗}(s(M) · C_{1}(ω_{π})) = −⟨D, β⟩^{2}π_{∗}(s(M) · s(M)). Next, by Proposition 1.2(i),

⟨D, β⟩π_{∗}(D · C_{1}(ω_{π})) = −π_{∗}(D · D) + ∑′⟨D, β′⟩⟨D, β′′Δ_{β′,β′′}.

Finally,

⟨D, β⟩⟨D, β′⟩⟨s(M), β′′⟩ + ⟨D, β⟩⟨D, β′′⟩⟨s(M), β′⟩ =

(⟨D, β′⟩ + ⟨D, β′′⟩)⟨D, β′⟩⟨s(M), β′′⟩ + (⟨D, β′⟩ + ⟨D, β′′⟩)⟨D, β′′⟩⟨s(M), β′⟩ =

⟨D, β′⟩^{2}⟨s(M), β′′⟩ + ⟨D, β′′⟩^{2}⟨s(M), β′⟩ + ⟨D, β′⟩⟨D, β′′⟩(⟨s(M), β′⟩ + ⟨s(M), β′′⟩) =

⟨D, β′⟩^{2}⟨s(M), β′′⟩ + ⟨D, β′′⟩^{2}⟨s(M), β′⟩ + ⟨D, β′⟩⟨D, β′′⟩.

Plugging in these 3 identities and simplifying gives the relation. □

Proof of Proposition 1.2(ii). Let π : ℭ → 𝔐_{0,0} denote the universal family. Let ℭ_{smooth} denote the smooth locus of π. The 2-fibered product pr_{1} : ℭ_{smooth}×_{𝔐_{0,0}}ℭ → ℭ_{smooth} together with the diagonal Δ : ℭ_{smooth} → ℭ_{smooth} ×_{𝔐_{0,0}} ℭ determine a 1-morphism ℭ_{smooth} → 𝔐_{1,0}. This extends to a 1-morphism ℭ → 𝔐_{1,0}. The pullback of the universal curve is a 1-morphism π′ : ℭ′ → ℭ that factors through pr_{1} : ℭ ×_{𝔐_{0,0}} ℭ → ℭ. Denote the pullback of the universal section by s : ℭ → ℭ′. Now ℭ is regular, and the complement of ℭ_{smooth} has codimension 2. Therefore s^{∗}O_{ℭ′}(s(ℭ)) can be computed on ℭ_{smooth}. But the restriction to ℭ_{smooth} is clearly ω_{π}^{∨}. Therefore s^{∗}O_{ℭ′}(s(ℭ)) ≅ ω_{π}^{∨} on all of ℭ.

Pulling this back by ζ_{C} : C → ℭ gives a 1-morphism π′ : C′ → C that factors through pr_{1} : C ×_{M} C → C. Let D be a Cartier divisor class on C and consider the pullback to C′ of pr_{2}^{∗}D on C ×_{M} C. This is a Cartier divisor class D′ on C′. Of course s^{∗}D′ = D. Moreover, by the projection formula the pushforward to C ×_{M} C of D′ ·D′ is pr_{2}^{∗}(D·D). Therefore (π′)_{∗}(D′ ·D′) is (pr_{1})_{∗}pr_{2}^{∗}(D·D), i.e., π^{∗}π_{∗}(D·D). Finally, denote by,

∑_{β′,β′′}⟨D, β′′⟩^{2}Δ̃_{β′,β′′},

the divisor class on C,

∑_{β′,β′′}′(⟨D, β′′⟩^{2}⟨s, β′⟩ + ⟨D, β′⟩^{2}⟨s, β′′⟩)Δ_{β′,β′′}.

The point is this: if π is smooth over every generic point of M, then the divisor class Δ̃_{β′,β′′} is the irreducible component of π^{−1}(Δ_{β′,β′′}) corresponding to the vertex v′, i.e., the irreducible component with “curve class” β′. Therefore Proposition 1.2(ii) follows from Lemma 5.4. □

Remark 5.5. If ⟨D, β⟩ ≠ 0 then, at least up to torsion, Proposition 1.2(i) follows from Proposition 1.2(ii) by intersecting both sides of the relation by D and then applying π_{∗}. This was pointed out by Pandharipande, who also proved Lemma 5.4 up to numerical equivalence in [8, Lem. 2.2.2] (by a very different method).

Lemma 5.6. Let s, s′ : M → C be sections with image in the smooth locus of π such that s(M) and s′(M) are disjoint. Then,

π_{∗}(s(M) · s(M)) + π_{∗}(s′(M) · s′(M)) = − ∑_{β′,β′′}⟨s(M), β′⟩⟨s′(M), β′′⟩Δ_{β′,β′′}.

Proof. Apply Lemma 5.2 and use s(M) · s′(M) = 0 and Lemma 5.3. □

Lemma 5.7. Let r ≥ 2 and s_{1}, …, s_{r} : M → C be sections with image in the smooth locus of π and which are pairwise disjoint. Then,

− ∑_{i=1}^{r} π_{∗}(s_{i}(M)·s_{i}(M)) = (r−2)π_{∗}(s_{1}(M)·s_{1}(M))+ ∑_{β′,β′′}⟨s_{1}(M), β′⟩⟨s_{2}(M)+⋯+s_{r}(M), β′′⟩Δ_{β′,β′′}.

Proof. This follows from Lemma 5.6 by induction. □

Lemma 5.8. Let r ≥ 2 and let s_{1}, …, s_{r} : M → C be sections with image in the smooth locus of π and which are pairwise disjoint. Then,

− ∑_{i=1}^{r} π_{∗}(s_{i}(M)·s_{i}(M)) = r(r−2)π_{∗}(s_{1}(M)·s_{1}(M))+ ∑_{β′,β′′}⟨s_{1}(M), β′⟩⟨s_{2}(M)+⋯+s_{r}(M), β′′⟩^{2}Δ_{β′,β′′}.

Combined with Lemma 5.7 this gives,

\begin{matrix}(r − 1)(r − 2)π_{∗}(s_{1}(M) · s_{1}(M)) = \\ − ∑_{β′,β′′}⟨s_{1}(M), β′⟩⟨s_{2}(M) + ⋯ + s_{r}(M), β′′⟩(⟨s_{2}(M) + ⋯ + s_{r}(M), β′′⟩ − 1)Δ_{β′,β′′}, \end{matrix}

which in turn gives,

\begin{matrix}−(r − 1) ∑_{i=1}^{r} π_{∗}(s_{i}(M) · s_{i}(M)) = \\ ∑_{β′,β′′}⟨s_{1}(M), β′⟩⟨s_{2}(M) + ⋯ + s_{r}(M), β′′⟩(r − ⟨s_{2}(M) + ⋯ + s_{r}(M), β′′⟩)Δ_{β′,β′′}. \end{matrix}

In the notation of Example 2.2, this is,

−(r − 1)(r − 2)π_{∗}(s_{1}(M) · s_{1}(M)) = ∑_{(A,B), 1∈A} #B(#B − 1)Δ_{(A,B)},

and

−(r − 1) ∑_{i=1}^{r} π_{∗}(s_{i}(M) · s_{i}(M)) = ∑_{(A,B), 1∈A} #B(r − #B)Δ_{(A,B)}.

Proof. Denote D = ∑_{i=2}^{r} s_{i}(M). Apply Lemma 5.4 to get,

\begin{matrix}2(r − 1) · 0 − ∑_{i=2}^{r} π_{∗}(s_{i}(M) · s_{i}(M)) − (r − 1)^{2}π_{∗}(s_{1}(M) · s_{1}(M)) = \\ ∑_{β′,β′′}⟨s_{1}(M), β′⟩⟨s_{2}(M) + ⋯ + s_{r}(M), β′′⟩^{2}Δ_{β′,β′′}. \end{matrix}

Simplifying,

− ∑_{j=1}^{r} π_{∗}(s_{i}(M)·s_{i}(M)) = r(r−2)π_{∗}(s_{1}(M)·s_{1}(M))+∑⟨s_{1}(M), β′⟩⟨s_{2}(M)+⋯+s_{r}(M), β′′⟩^{2}Δ_{β′,β′′}.

Subtracting from the relation in Lemma 5.7 gives the relation for (r − 1)(r − 2)π_{∗}(s_{1}(M) · s_{1}(M)). Multiplying the first relation by (r − 1), plugging in the second relation and simplifying gives the third relation. □

Lemma 5.9. Let r ≥ 2 and let s_{1}, …, s_{r} : M → C be everywhere disjoint sections with image in the smooth locus. For every 1 ≤ i < j ≤ r, using the notation from Example 2.2,

∑_{(A,B), i∈A} #B(r − #B)Δ_{(A,B)} = ∑_{(A′,B′),j∈A} #B′(r − #B′)Δ_{(A′,B′)}.

Proof. This follows from Lemma 5.8 by permuting the roles of 1 with i and j. □

Lemma 5.10. Let r ≥ 2 and let s_{1}, …, s_{r} : M → C be everywhere disjoint sections with image in the smooth locus of π. For every Cartier divisor class D on C of relative degree ⟨D, β⟩,

2(r−1)(r−2)⟨D, β⟩s_{1}^{∗}D = (r−1)(r−2)π_{∗}(D·D)+ ∑_{β′,β′′}⟨s_{1}(M), β′⟩a(D, β′′)Δ_{β′,β′′},

where,

a(D, β′′) = (r−1)(r−2)⟨D, β′′⟩^{2}−⟨D, β⟩^{2}⟨s_{2}(M)+⋯+s_{r}(M), β′′⟩(⟨s_{2}(M)+⋯+s_{r}(M), β′′⟩−1).

In particular, if r ≥ 3, then modulo torsion s_{i}^{∗}D is in the span of π_{∗}(D · D) and boundary divisors for every i = 1, …, r.

Proof. This follows from Lemma 5.4 and Lemma 5.8. □

Lemma 5.11. Let r ≥ 2 and let s_{1}, …, s_{r} : M → C be everywhere disjoint sections with image in the smooth locus of π. Consider the sheaf E = Ω_{π}(s_{1}(M) + ⋯ + s_{r}(M)). The perfect complex Rπ_{∗}RHom_{O_{C}} (E, O_{C}) has rank 3 − r and the first Chern class of the determinant is −2Δ − ∑_{i=1}^{r}(s_{i}(M) · s_{i}(M)). In particular, if r ≥ 2, up to torsion,

C_{1}(detRπ_{∗}RHom_{O_{C}} (Ω_{π}(s_{1}(M) + ⋯ + s_{r}(M)), O_{C})) =
−2Δ + \frac{1}{r−1} ∑_{(A,B), 1∈A} #B(r − #B)Δ_{(A,B)}.

Proof. There is a short exact sequence,

0 ⟶ Ω_{π} ⟶ Ω_{π}(s_{1}(M) + ⋯ + s_{r}(M)) ⟶ ⊕_{i=1}^{r}(s_{i})_{∗}O_{M} ⟶ 0.

Combining this with Lemma 4.2, Lemma 5.8, and chasing through exact sequences gives the lemma. □

6. Proof of Theorem 1.1

Let k be a field, let X be a connected, smooth algebraic space over k of dimension n, let M be an Artin stack over k, let π : C → M be a flat 1-morphism, representable by proper algebraic spaces whose geometric fibers are connected, at-worst-nodal curves of arithmetic genus 0, let s_{1}, …, s_{r} : M → C be pairwise disjoint sections with image contained in the smooth locus of π (possibly r = 0, i.e., there are no sections), and let f : C → X be a 1-morphism of k-stacks. In this setting, Behrend and Fantechi introduced a perfect complex E^{•} on M of amplitude [−1, 1] and a morphism to the cotangent complex, ϕ : E^{•} → L_{M}^{•}, [3]. If char(k) = 0 and M is the Deligne-Mumford stack of stable maps to X, Behrend and Fantechi prove E^{•} has amplitude [−1, 0], h^{0}(ϕ) is an isomorphism and h^{−1}(ϕ) is surjective. In many interesting cases, ϕ is a quasi-isomorphism. Then det(E^{•}) is an invertible dualizing sheaf for M. Because of this, det(E^{•}) is called the virtual canonical bundle. In this section the relations from Section 5 are used to give a formula for the divisor class of the virtual canonical bundle. Assume that Hypothesis 5.1 holds for π.

Denote by L_{(π,f)} the cotangent complex of the morphism (π, f) : C → M × X. This is a perfect complex of amplitude [−1, 0]. There is a distinguished triangle,

L_{π} ⟶ L_{(π,f)} ⟶ f^{∗}Ω_{X}[1] ⟶ L_{π}[1].

There is a slight variation L_{(π,f,s)} taking into account the sections which fits into a distinguished triangle,

L_{π}(s_{1}(M) + ⋯ + s_{r}(M)) ⟶ L_{(π,f,s)} ⟶ f^{∗}Ω_{X}[1] ⟶ L_{π}(s_{1}(M) + ⋯ + s_{r}(M))[1].

The complex E^{•} is defined to be (Rπ_{∗}(L_{(π,f,s)}^{∨})[1])^{∨}, where (F^{•})^{∨} is RHom(F^{•}, O). In particular, det(E^{•}) is the determinant of Rπ_{∗}(L_{(π,f,s)}^{∨}). From the distinguished triangle, det(E^{•}) is

det(Rπ_{∗}RHom_{O_{C}} (Ω_{π}(s_{1}(M) + ⋯ + s_{r}(M)), O_{C})) ⊗ det(Rπ_{∗}f^{∗}T_{X})^{∨}.

By Lemma 5.11, the first term is known. The second term follows easily from Grothendieck-Riemann-Roch.

Lemma 6.1. Assume that the relative degree of f^{∗}C_{1}(Ω_{X}) is nonzero. Then Rπ_{∗}f^{∗}T_{X}[−1] has rank ⟨−f^{∗}C_{1}(Ω_{X}), β⟩ + n, and up to torsion the first Chern class of the determinant is,

\frac{1}{2⟨−f^{∗}C_{1}(Ω_{X}),β⟩} [2⟨−f^{∗}C_{1}(Ω_{X}), β⟩π_{∗}f^{∗}C_{2}(Ω_{X})
−(⟨−f^{∗}C_{1}(Ω_{X}), β⟩ + 1)π_{∗}f^{∗}C_{1}(Ω_{X})^{2}+
∑′⟨−f^{∗}C_{1}(Ω_{X}), β′⟩⟨−f^{∗}C_{1}(Ω_{X}), β′′⟩Δ_{β′,β′′}] .

Proof. The Todd class τ_{π} of π is 1 − \frac{1}{2}C_{1}(ω_{π}) + τ_{2} + …, where π_{∗}τ_{2} = 0. The Chern character of f^{∗}T_{X} is,

n − f^{∗}C_{1}(Ω_{X}) + \frac{1}{2}(f^{∗}C_{1}(Ω_{X})^{2} − 2f^{∗}C_{2}(Ω_{X})) + …

Therefore ch(f^{∗}T_{X}) · τ_{π} equals,

n−[f^{∗}C_{1}(Ω_{X}) + \frac{n}{2}C_{1}(Ω_{π})]+\frac{1}{2} [f^{∗}C_{1}(Ω_{X})^{2} − 2f^{∗}C_{2}(Ω_{X}) + f^{∗}C_{1}(Ω_{X}) · C_{1}(ω_{π})]+nτ_{2}+…

Applying π_{∗} and using that π_{∗}τ_{2} = 0, the rank is n + ⟨−f^{∗}C_{1}(Ω_{X}), β⟩, and the determinant has first Chern class,

\frac{1}{2}π_{∗} [f^{∗}C_{1}(Ω_{X})^{2} − 2f^{∗}C_{2}(Ω_{X})] + \frac{1}{2}π_{∗}(f^{∗}C_{1}(Ω_{X}) · C_{1}(ω_{π})).

Applying Proposition 1.2 and simplifying gives the relation. □

Putting the two terms together gives the formulas in Theorem 1.1.

References

[1] K. Behrend. Gromov-Witten invariants in algebraic geometry. Invent. Math., 127(3):601–617, 1997.

[2] K. Behrend and B. Fantechi. The intrinsic normal cone. Invent. Math., 128(1):45–88, 1997.

[3] K. Behrend and Y. Manin. Stacks of stable maps and Gromov-Witten invariants. Duke Math. J., 85(1):1–60, 1996.

[4] A. J. de Jong and J. Starr. Higher fano manifolds and rational surfaces. preprint, 2006.

[5] F. F. Knudsen and D. Mumford. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand., 39(1):19–55, 1976.

[6] J. Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.

[7] R. Pandharipande. The canonical class of \overline{M}_{0,n}(P^{r}, d) and enumerative geometry. Internat. Math. Res. Notices, (4):173–186, 1997.

[8] R. Pandharipande. Intersections of Q-divisors on Kontsevich’s moduli space \overline{M}_{0,n}(P^{r}, d) and enumerative geometry. Trans. Amer. Math. Soc., 351(4):1481–1505, 1999.

[9] M. Raynaud. Spécialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math., 38:27–76, 1970.

Department of Mathematics, Columbia University, New York NY 10027

   E-mail address: dejong@math.columbia.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

   E-mail address: jstarr@math.mit.edu