LetS be a scheme and let (C, λ, ϕ) be anF-fern overS. For eachFq-subspace 06=V0 ⊂ V, let (C0, λ0, ϕ0) denote the V0-fern obtained from contracting C (Proposition 4.2). By Proposition 4.4 and Corollary 4.5, there is a natural line bundle L0 associated to C0 endowed with a fiberwise nonzeroFq-linear map λ0: V0 →L0(S). The pair (L0, λ0) thus determines a morphism of schemesS →PV0. By varying V0, we obtain a morphism
fC: S→ Y
06=V0⊂V
PV0. (7.1)
Proposition 7.6. The morphismfC depends only on the isomorphism class of(C, λ, ϕ).
Proof. Let (D, µ, ψ) be a V-fern isomorphic to (C, λ, ϕ). Let 0 6= V0 ⊂ V, and let D0 denote the contraction ofD with respect to ˆV0. The composite C→∼ D→D0 makesD0 a contraction ofCwith respect to ˆV0. We thus obtain a unique isomorphismC0 ∼→D0 of V0-ferns. Similarly, by the uniqueness of the contraction to the∞-component (Corollary 1.27), we obtain an isomorphism of ˆV0-marked curves α: (C0)∞ ∼→ (D0)∞. Recall that the pair (L0, λ0) was obtained from (C0)∞ by taking the complement of the ∞-divisor and restricting the ˆV0-marking toV0. Denoting the corresponding line bundle associated toD by (M0, µ0), the isomorphism α this induces an isomorphism of pairs
(L0, λ0)→∼ (M0, µ0).
Hence C and D induce the same morphism S→PV0.The proposition follows.
Lemma 7.7. Fix 0 6= V0 ⊂ V and let 1 6 i 6 n be the smallest integer such that V0 ⊂Vi. Let v0 ∈V0r(Vi−1∩V0). The contraction of C0 to the∞-component (C0)∞ is isomorphic to the contraction of C with respect to {0, v0,∞}.
Proof. On each fiber, the marked points of (C0)∞ are in one-to-one correspondence with V0/(Vj∩V0)∪ {∞}for some j 6i−1. The choice ofv0 thus guarantees that the 0 - and v0- and ∞-marked sections are disjoint in (C0)∞. By Proposition 1.26, we deduce that (C0)∞ is the contraction of C0 (and hence C) with respect to {0, v0,∞}.
Consider 0 6=V00 (V0 ⊂V, with corresponding pairs (L00, λ00) and (L0, λ0) as in the definition of fC above.
Lemma 7.8. There exists a morphism ψ: L00 → L0 (as schemes over S) such that λ0|V00 =ψ◦λ00.
Proof. Let 1 6 i 6 j 6 n be minimal such that V00 ⊂ Vi and V0 ⊂ Vj. Let v00 ∈ V00r(V00∩Vi−1) andv0 ∈V0r(V0∩Vj−1). Defineα :={0, v0, v00,∞}andα0 :={0, v0,∞}
andα00 :={0, v00,∞}, with corresponding contractionsCα andCα0 andCα00. LetC0 and C00denote the contractions ofC with respect to ˆV0 and ˆV00 respectively. By Lemma7.7, for the contractions to the ∞-component we have (C00)∞ ∼=Cα00 and (C0)∞ ∼=Cα0. By
construction, the line bundleL0 is the complement of the ∞-divisor inCα0 and similarly for L00 in Cα00. Moreover, the contraction morphisms C → Cα0 and C → Cα00 factor uniquely throughCα. We depict this in the following commutative diagram:
C
Cα
zz %%
L0 ⊂Cα0 Cα00 ⊃L00.
Let U00 ⊂ Cα denote the inverse image of L00. By Proposition 1.17, the contraction Cα →Cα00 induces an isomorphism U00 ∼→ L00. The composite L00 ∼→ U00 ,→ Cα → Cα0 yields a morphism ψ: L00→L0 which preserves V00-marked sections.
Figure 9: The morphism ψ: L00 → L0 on a fiber where Cα is singular. The unfilled points indicate the∞-marked points of Cα0 and Cα00, which are not included inL0 and L00. In this case, the morphism ψ maps L00 to the 0-marked point ofL0.
v0 ∞ Cα
0 v00
ψ
v0 ∞ Cα00 ⊃L00
0 v00
L0 ⊂Cα0
v0 1
2 0
Remark. For any s ∈ S for which the fiber Csα is smooth, there is a Zariski open neighborhood U of s over which Cα is smooth. The contraction morphisms Cα → Cα0 and Cα → Cα00 are isomorphisms over U and induce an isomorphism Cα0 →∼ Cα00 over U. The morphism ψ is then obtained by restriction. This implies that ψs: L00s → L0s is an isomorphism when Csα is smooth. Otherwise ψs maps L00s to the 0-marked point of L0s as in Figure 9.
The construction of the morphism ψ: L00 → L0 does not rely on the V-fern structure of C. Indeed, we can preform the same construction for an arbitrary stable α-marked curve over S, where α = {0, v, v00,∞}. Let (X, λ) be such a curve. The contractions of X with respect to α0 = {0, v0,∞} and α00 = {0, v00,∞} are both isomorphic to P1S. Denote the complement of the∞-divisor in the contractions byL0X and L00X respectively.
We endow each with a line bundle structure by taking any isomorphism to the trivial line bundle Ga,S which extends to an isomorphism Xα0 →∼ P1S or Xα00 →∼ P1S and sends the 0-marked section to 0. We then define the line bundle structures on L0X and L00X to be the unique ones induced by these isomorphisms. Note that the resulting line bundle structure does not depend on the choice of isomorphism since any two such choices differ by a linear automorphism ofGa,S. As in the proof of Lemma7.8, we obtain a morphism ψX: L00X → L0X of schemes over S via Proposition 1.17.
Lemma 7.9. With the aforementioned line bundle structures, the morphism ψX: L00X →LX is a homomorphism of line bundles.
Proof. We must show that the following diagrams, which correspond respectively to additivity and compatibility with scalar multiplication, commute:
L00X ×SL00X
We first reduce to the case whereS is integral. For this we note that the constructions of the schemes L0X and L00X and the morphism ψX are compatible with base change by Proposition 1.30. Let Cα → Mα be the universal family over the moduli space of stable α-marked curves. There exists a unique morphism f: S → Mα such that X ∼= f∗Cα. Preforming the exact same construction on Cα, we obtain line bundles (Lα)0 and (Lα)00 such that L0X ∼= (fα)∗(Lα)0 and L00X ∼= (fα)∗(Lα)00 as line bundles. Moreover, the morphism ψX is the pullback of the analogously defined morphism ψα: (Lα)00 →(Lα)0. Since Mα ∼=P1Z is integral, it thus suffices to prove the lemma whenS is integral.
Suppose thatSis integral and there exists ans∈Ssuch that the fiberXsis smooth.
ThenX is smooth over a non-empty open subset U ⊂S. The morphismψX restricts to an isomorphism over U and extends to an isomorphism Xα00|U →∼ X|U. In terms of the isomorphism of (trivial) line bundles ˜ψX: Ga,U →∼ Ga,U induced by ψX, this means that ψ˜X extends to an isomorphism of P1U and is hence linear. It follows that (7.2) and (7.3) are commutative over U. SinceL0X and L00X are reduced and separated and +0◦(ψ×ψ) and ψ◦+00 are equal over U, it follows that +0 ◦(ψX ×ψX) = ψX ◦+00 over S; hence
(7.2) commutes. A similar argument shows that (7.3) commutes. If Xs is singular for every s ∈S, then ψX must be the zero map, and it follows immediately that (7.2) and (7.3) commute.
We now return to the setting of V-ferns and the morphism ψ: L00 → L0 of Lemma 7.8.
Proposition 7.10. The morphism ψ is a homomorphism of line bundles.
Proof. This follows directly from the Lemma7.9 and the construction of the line bundle structures onL0 and L00 from the proof of Proposition 4.4, which is the same as for the line bundle structures obtained as in the discussion preceeding Lemma 7.9.
Proposition 7.11. The morphism fC factors though BV ,→Q
06=V0⊂V PV0.
Proof. By the Lemma 7.8 and Proposition 7.10, we have the following commutative diagram:
V0 λ0 //L0(S)
V? 00 λ00//
OO
L00(S).
ψ
OO
Varying V00 and V0, it then follows from Lemma 2.5 that fC factors through BV. Proposition 7.12. The morphism fC: S → BV factors through the open immersion UF ,→BV.
Proof. We just need to verify that the tuple E• associated to fC satisfies the open condition (2.2) defining UF. Fix 0 6= V00 ⊂ V0 ⊂ V such that there does not exist 16i6n withV00⊂Vi and V0 6⊂Vi. Denote the corresponding line bundles byL00 and L0. Let 1 6 j 6 n be minimal such that V00 ⊂ Vj. By assumption j is also minimal such thatV0 ⊂Vj. Choosev ∈V00r(V00∩Vj−1). It follows from Lemma7.7 that (C0)∞ and (C00)∞ are both equal to the contraction of C with respect to {0, v00,∞}. By the constructions of (L00, λV00) and (L0, λV0), we conclude that L0 = L00 and λV00 = λV0|V00. Recall that
EV0 = ker(λV0⊗ OS: V0⊗ OS → L0), where L0 denotes the sheaf of sections of L0. We wish to show that
V0 ⊗ OS =EV0 + (V00⊗ OS).
It suffices to show equality on stalks, which follows directly from the fact that λ|V00 = λV0|V00 and that if ϕ: M →L is a surjective morphism of A-modules and N ⊂ M is a submodule such that ϕ|N is still surjective, then M = ker(ϕ) +N.