Configurations of points on a line up to scaling or translation
CCONFIGURATIONS OF POINTS ON A LINE UP TO SCALINGOR TRANSLATION
ADRIAN ZAHARIUC
Abstract.
We prove that the Losev–Manin compactification of the space ofconfigurations of n points on P \{ , ∞} modulo scaling degenerates (isotriv-ially) to a compactification of the space of configurations of n points on A modulo translation. The latter resembles the compactification constructed byZiltener and Mau–Woodward, but allows the marked points to coincide, mak-ing it a G n − a -variety, which mirrors the fact that the Losev–Manin space istoric. The degeneration is compatible with the actions of G n − m and G n − a in the sense that these actions fit together globally in the total space of thedegeneration. Contents
1. Introduction 12. Preliminaries on curves and differentials 43. Stabilization 64. Marked field decorated rational trees 105. G m and G a actions on curves and their moduli 14References 181. Introduction
The goal of this note is to point out a connection between the following problems:(+) compactifying the space of n -tuples of (not necessarily distinct) points onthe affine line x , . . . , x n ∈ C modulo translation, that is,( x , . . . , x n ) ∼ ( y , . . . , y n ) if y − x = · · · = y n − x n ; and( × ) compactifying the space of n -tuples of (not necessarily distinct) points onthe punctured affine line x , . . . , x n ∈ C ∗ modulo scaling, that is,( x , . . . , x n ) ∼ ( y , . . . , y n ) if y x = · · · = y n x n . Both problems are open ended.The best answer to ( × ) may be the Losev-Manin space, which we will denote by L n , as in [LM00]. For (+), if we change the problem by instead insisting that thepoints are distinct, then a beautiful answer is given by the moduli space Q n of ‘stablescaled marked curves’ constructed by Mau and Woodward as a projective variety[MW10], after Ziltener constructed it with symplectic methods [Zi06, Zi14]. The Mathematics Subject Classification.
Key words and phrases.
Compactification, configuration space, deformation of G a to G m . a r X i v : . [ m a t h . AG ] F e b A. ZAHARIUC moduli space Q n plays a central role in the context of gauged stable maps [Wo15,GSW17, GSW18]. The first goal of this paper is to construct a compactification P n (related to Q n but simpler) which answers (+) as stated above. Definition 1.1. An n -marked G a -rational tree is a reduced, connected, complex,projective curve C of arithmetic genus 0 with at worst nodal singularities, thatis, (cid:91) O C,p (cid:39) C [[ x, y ]] / ( xy ) or C [[ x ]] at all p ∈ C ( C ), with a G a -action and n + 1nonsingular points x ∞ , x , . . . , x n ∈ C ( C ), such that x ∞ is fixed by the G a -action,but x , . . . , x n are not. The n -marked G a -rational tree is stable if any irreduciblecomponent of C which doesn’t contain any x i , 1 ≤ i ≤ n , either intersects at leasts3 other irreducible components of C , or contains x ∞ and intersects at least 2 otherirreducible components of C .We will construct a normal projective G n − a -variety (cf. [HT99, Definition 2.1]) P n such that P n ( C ) is the set of stable n -marked G a -rational trees. In fact, P n is almost certainly the fine moduli space of stable n -marked G a -rational trees, butsince P n is ultimately a combinatorial object, we will prove the weaker ‘repre-sentability’ statement which only considers families over complex varieties ratherthan families over more general schemes cf. Corollary 5.6, to avoid some lengthytechnical distractions. This weaker statement still determines P n uniquely.There is an obvious analogy between L n and P n . If ( Y n , G ) is either ( L n , G m ) or( P n , G a ), then Y n is a compactification of G n / G (where G (cid:44) → G n diagonally), withthe property that the usual G n -action on Y n = G n / G (or G n / G = G n − -action, ifwe quotient out by the trivial diagonal action) extends to all of Y n . We call this G n − -action on Y n the natural G n − -action on Y n . However, the relation between L n and P n goes beyond this analogy.Our main result is that P n deforms isotrivially to L n , in a manner compatiblewith the natural group actions. To state it, we first recall the elementary fact that G m degenerates isotrivially to G a as an algebraic group. (This seems to have beenfirst recorded in [KM78, 3.1], although the Jacobians of nodal and cuspidal cubicswere known much earlier.) Let γ : G = Spec C [ t, x ] tx +1 → Spec C [ t ] = A . Theoperation G × A G (cid:63) −→ G given on C -points by( t, x ) (cid:63) ( t, y ) = ( t, x + y + txy )with the identity section t (cid:55)→ ( t,
0) makes G an A -group scheme. Note that G t (cid:39) (cid:40) G m if t (cid:54) = 0 , G a if t = 0 , where t ∈ C = A ( C ) and G t = γ − ( t ). Theorem 1.2.
For any positive integer n , there exist a complex variety X , a flatprojective morphism ξ : X → A , and an action of G n − A = G × A · · · × A G (cid:124) (cid:123)(cid:122) (cid:125) n − copies of G on X over A such that for all t ∈ C , if X t = ξ − ( t ) , then • if t (cid:54) = 0 , then X t is isomorphic to L n , and the action of G n − t on X t isisomorphic to the natural action of G n − m on L n ; • if t = 0 , then X t is isomorphic to P n , and the action of G n − t on X t isisomorphic to the natural action of G n − a on P n . ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 3 (In fact, X \ X (cid:39) ( A \{ } ) × L n compatibly with the projections to A \{ } , andthe restriction of the G n − A -action in Theorem 1.2 to X \ X is the pullback of the G n − m -action on L n along the projection X \ X → L n .)For instance, L (cid:39) P (cid:39) P , L (cid:39) P (cid:39) P , and L is the blowup of P at 3general points, while P is the blowup of P at 3 collinear points. However, P n ismildly singular for n ≥ Q n is [MW10, Corollary 10.6and Figure 19], whereas L n is nonsingular [LM00, Theorem 2.2.a)], and they arecertainly not homeomorphic in general. Nevertheless, Theorem 1.2 still shows that L n and P n are related topologically. For instance, we have a trivial consequence: Corollary 1.3.
There exist polarizations on P n and L n for which their Hilbertpolynomials coincide, that is, χ ( L n , O L n ( m )) = χ ( P n , O P n ( m )) for all m ∈ Z . The values of χ top ( P n ) for small n can be found easily using a computer. n χ top ( P n ) 1 2 6 27 170 1390 13979Although this sequence doesn’t match any sequence in the Online Encyclopedia ofInteger Sequences, at least it illustrates the fact that P n is singular for n ≥
4, since L n is nonsingular and χ top ( L n ) = n ! [LM00]. Example 1.4.
The degeneration of L to P is illustrated in the figure below. Thegraphs in the top row represent strata of L , while the graphs in the bottom rowrepresent strata of P . The vertical arrows A → a + · · · + z indicate that the fiberover 0 of the closure relative to X of A × ( A \{ } ) is set theoretically a ∪ · · · ∪ z .0 , , , , ∞ , , , ∞ , i, j k, ∞∞ ki, j , i j, k, ∞∞ j, ki + ∞ i j k , i j k, ∞∞ ki j Although 0 and ∞ play symmetric roles in L n , we see that this is no longer true inthe context of the degeneration in Theorem 1.2.To prove Theorem 1.2, we will interpret X as a moduli space of certain objectscalled stable n -marked field-decorated rooted rational trees , cf. Definition 4.1. Theproof revolves around the observation that L n +1 and P n +1 are the universal curvesover L n and P n respectively, and relies heavily on the techniques in Knudsen’sproof that M g,n +1 is the universal curve over M g,n [Kn83], with some additionalinput from [Stacks, Tag 0E7B].A side remark on the moduli of curves point of view. Consider the operations: • glue 0 and ∞ on a curve which corresponds to a C -point in L n ; • pinch ∞ on a curve which corresponds to a C -point in P n .It seems that it is possible to carry out these two operations globally on the universalcurve over X . In this way, we may think of X as the base of a family of curves ofarithmetic genus 1. Clearly, these operations don’t compromise the group actionson the curves. A. ZAHARIUC
Acknowledgments.
This work was carried out in roughly two stages: the first one(the construction of P n ) when I was at the MSRI, and the second one (Theorem1.2) when I was at University of Windsor. Regardless, the two parts ended upcompletely interwoven. The respective grant acknowledgements are as follows.This material is based upon work supported by the National Science Foundationunder Grant No. 1440140, while the author was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Spring 2020 semester.We acknowledge the support of the Natural Sciences and EngineeringResearch Council of Canada (NSERC), RGPIN-2020-05497. Cette recherche a ´et´efinanc´ee par le Conseil de recherches en sciences naturelles et en g´enie du Canada(CRSNG), RGPIN-2020-05497.I would like to thank David Eisenbud for encouragement and interesting discus-sions during the first stage.2. Preliminaries on curves and differentials
Recall that a prestable curve is a proper flat morphism whose geometric fibersare connected curves with at worst nodal singularities, e.g. [BM96, Definition 2.1].We say that it is of genus g if all geometric fibers have arithmetic genus g . If C → S is a prestable curve and x : S → C is a section, we will often abuse notation bywriting x instead of x ( S ) for the scheme-theoretic image of x (recall that sectionsof separated morphisms are closed immersions [EGAI, 5.4.6]).Let π : C → S be a projective prestable curve. We denote the sheaf of relativedifferentials and relative dualizing sheaf by Ω C/S (or Ω π ) and ω C/S (or ω π ) respec-tively. Please see [Kn83, p. 163] for some of the fundamental properties of Ω C/S and ω C/S . Their formation commutes with base change. There is a homomorphism ψ : Ω C/S → ω C/S whose formation also commutes with base change. Please see [ACG11, Ch. 10,(2.20)] for a discussion of the kernel and cokernel of ψ . The dual ψ ∨ : ω ∨ C/S → Ω ∨ C/S is always injective, so it possible to think of ω ∨ C/S as a submodule of Ω ∨ C/S . Thisis proved on page 168 in [Kn83] for stable curves, and the same argument worksequally well for prestable curves. If S is a variety, we even have a completely explicitfiberwise description. Lemma 2.1. If S is a complex variety, then Γ( U, ω ∨ C/S ) = (cid:110) η ∈ Γ( U, Ω ∨ C/S ) : η s ∈ Γ( C s ∩ U, ω ∨ C s ) , ∀ s ∈ S ( C ) (cid:111) , where η s is the restriction of η : Ω C/S | U → O C | U to C s ∩ U , and the conditionsimply means that η s factors through ( ψ | U ) s .Proof. Both ψ and η annihilate Ω tors C/S . Using arguments similar to those in [ACG11,p. 98], we obtain the short exact sequence0 → Ω C/S / Ω tors C/S → ω C/S → ω C/S ⊗ O π sing → C/S / Ω tors C/S with ω C/S ⊗ I π sing ,C . The scheme structure on π sing isthe one implicit in loc. cit. Then the claim becomes showing that a homomorphism I π sing ,C | U → ω ∨ C/S | U which extends fiberwise to O C | U → ω ∨ C/S | U extends globally,which can be checked explicitly. The details are routine. (cid:3) ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 5
We record for future use two elementary remarks on morphisms of schemes f : X → Y such that f : O Y → f ∗ O X is an isomorphism. First, the canonical map L → f ∗ f ∗ L is an isomorphism for all invertible L . Second, if F is an O X -module,there exists a natural O Y -module homomorphism f ∗ F ∨ → ( f ∗ F ) ∨ . Indeed, anelement F| f − ( U ) → O X | f − ( U ) of Γ( U, f ∗ F ∨ ) induces mapsΓ( V, f ∗ F ) = Γ( f − ( V ) , F ) → Γ( f − ( V ) , O X ) (cid:39) Γ( V, O Y )for V ⊆ U ⊆ Y which satisfy the obvious compatibilities and are thus an elementof Γ( U, ( f ∗ F ) ∨ ). More generally, if G is an O Y -module, and φ : G → f ∗ F is an O Y -module homomorphism, then we may define a map f ∗ F ∨ → G ∨ as the composition f ∗ F ∨ → ( f ∗ F ) ∨ → G ∨ . Definition 2.2.
Let π i : C i → S be prestable curves for i = 1 ,
2, and f : C → C an S -morphism. We say that f is contractive if f : O C → f ∗ O C is an isomor-phism and R f ∗ O C = 0, and these continue to hold after any base change S (cid:48) → S .Some remarkable properties of contractive morphism are stated and proved in[Stacks, Tag 0E7B]. These properties will be essential in § Proposition 2.3.
Assume that S is a variety. If f : C → C is contractive, thenthere exists a unique O C -module homomorphism f δ : f ∗ ω ∨ C /S → ω ∨ C /S which is the identity on every open subset U ⊆ C on which f restricts to anisomorphism f − ( U ) (cid:39) U . If s ∈ S ( C ) and v is a local section of f ∗ ω ∨ C /S , then ( f δ ( v )) s = ( f s ) δ ( v s ) , where f s = f | C ,s , v s = v ⊗ κ s , etc. (We can rely on [Liu02, Definition 4.7 and Theorem 4.32] to make sense intrin-sically of the dualizing sheaf on open subsets.) Proof.
To justify uniqueness, we note that f is an isomorphism above C \ π sing2 ,and invoke the defining property on this dense open subset. To prove existence, let f ∆ : f ∗ Ω ∨ C /S → Ω ∨ C /S be the map induced by the adjoint of f ∗ Ω C /S → Ω C /S , asin the remark preceding Definition 2.2. We claim that f ∆ maps sections of f ∗ ω ∨ C to sections of ω ∨ C , and thus induces a homomorphism f δ : f ∗ ω ∨ C /S → ω ∨ C /S . If S = Spec C , then f simply contracts several P chains, and the claim can bechecked directly using the explicit description of ω ∨ C i as a sheaf of meromorphicdifferentials. In general, for each s ∈ S ( C ) and any open set U ⊆ C , we have adiagram Γ( f − ( U ) , Ω ∨ C /S )Γ( U, Ω ∨ C /S ) Γ( f − ( U ) ∩ C ,s , Ω ∨ C ,s )Γ( U ∩ C ,s , Ω ∨ C ,s )Γ( f − ( U ) , ω ∨ C /S ) Γ( f − ( U ) ∩ C ,s , ω ∨ C ,s )Γ( U ∩ C ,s , ω ∨ C ,s )whose commutativity is left to the reader to check. Then the claim follows fromthis commutativity, the special case S = Spec C , and Lemma 2.1. The construction A. ZAHARIUC of f δ is complete and the fact that it satisfies the required property is clear. Wealso note that ( f ∆ ( v )) s = ( f s ) ∆ ( v s ) amounts to commutativity of the front face inthe diagram above, and ( f δ ( v )) s = ( f s ) δ ( v s ) follows. (cid:3) Proposition 2.3 is true over more general bases too, but we don’t know a simpledirect proof and it will not be used in this greater generality.
Definition 2.4.
The homomorphism f δ in Proposition 2.3 will be called the skewlogarithmic differential of f . 3. Stabilization
The operations described in §§ §
3, we work in one of the following situations.
Situation 3.1.
Let S be a complex (irreducible) variety, π : C → S a genus 0projective prestable curve over S , x, x ∞ : S → C sections of π , and an O C -modulehomomorphism ν : ω C/S → I x ∞ ,C . Consider the situations:(0) the above, and π is smooth at x ∞ ( s ), for all s ∈ S ;(1) same as (0), but also x ( s ) (cid:54) = x ∞ ( s ) and π is smooth at x ( s ), for all s ∈ S ;(2) same as (1), but also x ∗ ν : x ∗ ω C/S → O S is an isomorphism.In all situations above, let v ∈ Γ( C, ω ∨ C/S ( − x ∞ )) corresponding to ν .Indeed, in Situation 3.1.(0), x ∞ is a relative effective Cartier divisor.3.1. Knudsen stabilization. In § § x ∞ (as opposed to n such sections in [Kn83, § v . In our situation, ‘stabilization’ may be a misnomer as the produced objects arenot stable in general. In Situation 3.1.(0), let(1) K = Coker (cid:16) O C a (cid:55)→ a ⊕ aι −−−−−→ O C ( x ∞ ) ⊕ I ∨ x,C (cid:17) , where ι : I x,C (cid:44) → O C is the inclusion, and ξ : C (cid:48) = P C K → C and π (cid:48) = πξ . (Hereand thereafter, we use the notation P X F = Proj X Sym( F ).) It can be proved byarguments analogous to those of [Kn83, §
2] that: • the morphism π (cid:48) : C (cid:48) → S is a genus 0 (projective) prestable curve; • the sections x and x ∞ lift canonically to sections x (cid:48) , x (cid:48)∞ : S → C (cid:48) ; • x (cid:48) ( s ) (cid:54) = x (cid:48)∞ ( s ) and π (cid:48) is smooth at x (cid:48) ( s ), for all s ∈ S ; • ξ is contractive, cf. Definition 2.2; • the formation of all new data commutes with base change.The curve π (cid:48) : C (cid:48) → S together with the sections x (cid:48)∞ , x (cid:48) : S → C (cid:48) will always bepart of the data obtained by Knudsen stabilization. If C ◦ = { z ∈ C : z (cid:54) = x ( π ( z )), or both π is smooth at z and z (cid:54) = x ∞ ( π ( z )) } , then C ◦ is open and K| C ◦ is invertible and hence ξ is an isomorphism above C ◦ .Let C (cid:48)◦ = ξ − ( C ◦ ) (cid:39) C ◦ . It remains to discuss v (cid:48) . Definition 3.2.
We say that v (cid:48) ∈ Γ( C (cid:48) , ω ∨ C (cid:48) /S ( − x (cid:48)∞ )) is compatible with v if v (cid:48) | C (cid:48)◦ = v | C ◦ under the obvious identification of the restrictions of the sheaves. ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 7
Proposition 3.3.
There exists an isomorphism ξ ∗ ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) (cid:39) ω ∨ C/S ( − x ∞ ) which restricts to the identity on C ◦ .Proof. Consider the skew logarithmic differential ξ δ : ξ ∗ ω ∨ C (cid:48) /S → ω ∨ C/S , cf. Defi-nition 2.4. We claim that ξ δ maps local sections of ξ ∗ ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) to local sec-tions of ω ∨ C/S ( − x ∞ ). By the last part of Proposition 2.3, it suffices to check thison fibers, when it is elementary. We thus obtain an O C -module homomorphism ξ ∗ ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) → ω ∨ C/S ( − x ∞ ) and it can be checked once more on fibers that itis actually an isomorphism. Indeed, any homomorphism from a coherent torsion-free O C -module to an invertible O C -module which is an isomorphism on fibers(of closed points) must be an isomorphism: this can be justified by first invokingNakayama’s lemma to argue that the source must be invertible because its restric-tions to all points are 1-dimensional, and then noting that all maps on stalks mustbe isomorphisms. (cid:3) Note the similarity of Proposition 3.3 with [Kn83, Lemma 1.6.a)].
Corollary 3.4.
There exists a unique v (cid:48) compatible with v . Indeed, existence follows from Proposition 3.3 while uniqueness follows from thefact that the restriction map on sections of ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) from C (cid:48) to C (cid:48)◦ is injective,which is a consequence of Proposition 3.3 once more and the density of C ◦ in C . Definition 3.5.
In Situation 3.1.(0), with notation as above, the collection of data( C (cid:48) , S, π (cid:48) , x (cid:48)∞ , x (cid:48) , v (cid:48) ) is declared to be the Knudsen stabilization of the given data. Remark . The output of this stabilization operation fits into Situation 3.1.(1).Note that the output of stabilization commutes with base changes T → S .3.2. Stabilization relative to a vector field. In § § V = Coker (cid:16) O C a (cid:55)→ av ⊕ a −−−−−−→ ω ∨ C/S ( − x ∞ ) ⊕ O C ( x ) (cid:17) , and ξ : C (cid:48) = P C V → C and π (cid:48) = πξ . Let Z ⊆ S be the scheme-theoretic vanishinglocus of x ∗ v ∈ Γ( S, x ∗ ω ∨ C/S ), C ◦ = C \ x ( Z ), and C (cid:48)◦ = ξ − ( C ◦ ). It is clear that K| C ◦ is invertible, so ξ restricts to an isomorphism on C (cid:48)◦ (cid:39) C ◦ . Since x ∞ is contained in C ◦ by the assumption x ∩ x ∞ = ∅ , it trivially lifts uniquely to a section x (cid:48)∞ : S → C (cid:48) .The canonical lift x (cid:48) : S → C (cid:48) of x corresponds to the quotient x ∗ V → x ∗ O C ( x ) → §
2, p. 173]. We also note that the formationof all data so far commutes with base change.
Proposition 3.7.
In Situation 3.1.(1), with notation as above, we have: (1)
The morphism π (cid:48) : C (cid:48) → S is a genus projective prestable curve. (2) There exists an isomorphism ω ∨ C/S ( − x − x ∞ ) (cid:39) ξ ∗ ω ∨ C (cid:48) /S ( − x (cid:48) − x (cid:48)∞ ) whichrestricts to the identity on C ◦ . (3) The sequence x (cid:48)∗ ω ∨ C (cid:48) /S → x ∗ ω ∨ C/S → x ∗ ω ∨ C/S ⊗ O Z → in which the firstmap is the pullback along x (cid:48) of the differential Ω ∨ C (cid:48) /S → ξ ∗ Ω ∨ C/S and thesecond map is the restriction, is exact.
A. ZAHARIUC
Proof.
Before proving Proposition 3.7, we make a general technical remark, statedwith independent notation. Let X be a scheme and D ⊂ X an effective Cartierdivisor. For i = 1 ,
2, let L i be an invertible O X -module and s i ∈ Γ( X, L i ). Assumethat there exists an isomorphism φ : L | D (cid:39) L | D such that φ ( s | D ) = s | D . Let V i = Coker (cid:18) O X t (cid:55)→ ( ts i ,t ) −−−−−−→ L i ⊕ O X ( D ) (cid:19) ,π i : X (cid:48) i = P X ( V i ) → X , and ι i : D (cid:44) → X (cid:48) i the ‘proper transform’ of D , constructedusing [Kn83, §
2, p. 173]. Then there exists an open cover υ : U = (cid:70) α ∈ I U α → X and a U -isomorphism τ : U × X X (cid:48) (cid:39) U × X X (cid:48) such that τ ◦ (id U × X ι ) = id U × X ι . To prove the technicality above, we may assume without loss of generality that X = Spec( R ) for some ring R , L i = O X , and D is cut out by some g ∈ R .Then s , s ∈ R , and s | D = s | D simply means that there exists h ∈ R such that s − s = hg . Then id R and (cid:20) h (cid:21) give an isomorphism between the two complexes[ R → R ], which will induce τ . The remaining details are straightforward.Let us return to the proof of Proposition 3.7. It is straightforward that allgeometric fibers of π (cid:48) have arithmetic genus 0, so to prove the first claim it remainsto show that π (cid:48) is flat. Thus all that remains to prove (for all 3 parts) is Zariskilocal on C or C (cid:48) . Let V = Coker (cid:16) O C a (cid:55)→ aπ ∗ x ∗ v ⊕ a −−−−−−−−−→ π ∗ x ∗ ω ∨ C/S ⊕ O C ( x ) (cid:17) ,ξ : C (cid:48) = P C V → C the natural projection, and C (cid:48) , ◦ = ξ − ( C (cid:48)◦ ). As above, wemay construct lifts x (cid:48) and x (cid:48) , ∞ . Since x ∗ π ∗ x ∗ ω ∨ C/S = x ∗ ω ∨ C/S and x ∗ π ∗ x ∗ v = x ∗ v ,re general technicality at the beginning of this proof shows that C (cid:48) and C (cid:48) arelocally isomorphic and the local isomorphisms are compatible with the projectionsto C and S , and with the sections x (cid:48) and x (cid:48) . Since everything left to check isZariski local, we can check it for C (cid:48) instead of C (cid:48) .The advantage C (cid:48) has over C (cid:48) is that it is easy to construct a map to thelocal model. Let h : Y = Spec C [ x, t ] → Spec C [ t ] = Z be the projection map, h : Y (cid:48) → Y the blowup at ( x, t ), r : Z → Y the zero-section, and r (cid:48) : Z → Y (cid:48) itslift. To construct the map to the local model, we proceed as follows. Let s ∈ S .Since π is smooth at x ( s ), there exists an open subset U ⊂ C such that x ( s ) ∈ U and an ´etale morphism ψ : U → A S such that π | U is the composition of ψ with theprojection A S → S . Composing with a suitable automorphism, we may assume that x maps to the 0-section. By shrinking S to x − ( U ), we may assume that x ⊂ U ,and that x is the whole preimage of the zero section of A S → S . Furthermore, wemay shrink S so that x ∗ ω ∨ C/S is trivial, and choose a trivialization. Then x ∗ v issimply a regular function on S , so it induces a morphism S → A = Z from whichwe may construct the desired map. Part 1 of the proposition follows from the factthat the composition Y (cid:48) → Y → Z is flat, part 2 follows from the existence of theisomorphism h ∗ ω ∨ Y (cid:48) /Z ( − r (cid:48) ( Z )) (cid:39) ω ∨ Y/Z ( − r ( Z ))which restricts to the identity on Y \{ } , and part 3 follows from the fact that, onthe proper transform of the zero-section, the differential Ω Y (cid:48) /Z → h ∗ Ω Y/Z vanishesonly at the point above 0. (cid:3)
ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 9
Alternatively, part (2) could have been proved by an argument similar to thatin Proposition 3.3.
Definition 3.8.
We say that v (cid:48) ∈ Γ( C (cid:48) , ω ∨ C (cid:48) /S ( − x (cid:48)∞ )) is compatible with v if x (cid:48)∗ v (cid:48) is nowhere vanishing and v (cid:48) | C (cid:48)◦ = v | C ◦ . Remark . Let ξ δ be the skew logarithmic differential of ξ (Definition 2.4). If v (cid:48) is compatible with v , then ξ δ ( v (cid:48) ) = v . Indeed, ξ δ ( v (cid:48) ) − v restricts to 0 on the denseopen C ◦ ⊂ C , so it must be 0 since ω ∨ C/S is locally free.Note that it is no longer true that such v (cid:48) is unique, even when S = Spec C .However, we’ll see that this is true up to automorphisms of the rest of the data. Lemma 3.10.
Assume that S is a variety over C . Then the group of automor-phisms φ of C (cid:48) over C such that φ ◦ x (cid:48)∞ = x (cid:48)∞ and φ ◦ x (cid:48) = x (cid:48) acts transitively onthe set of v (cid:48) compatible with v .Proof. Let v (cid:48) and v (cid:48) compatible with v . In the first case, if v (cid:48) (cid:54) = v (cid:48) and S is avariety, then x ∗ v ≡
0. Then there exists a section ρ : C → C (cid:48) of ξ which is a closedimmersion, and in fact C (cid:48) = ρ ( C ) ∪ y ( S ) P , the gluing of ρ ( C ) with P ∼ = P x ( S ) V| x ( S ) ,which is now a P -bundle over x ( S ), along a section of π (cid:48) . Then we may choose φ to be the identity on ρ ( C ) and to fix y and x (cid:48) on P . (cid:3) Proving Lemma 3.10 over arbitrary bases is the main difficulty that persuadedus to focus on the case when S is a variety. Lemma 3.11.
There is a canonical surjection ξ ∗ ( O C (cid:48) ( x (cid:48) )) → I x ( Z ) ,C ( x ) . More-over, if Z (cid:54) = S , then this map is an isomorphism.Proof. Since x ∗ is right exact as x is finite, Proposition 3.7 part 3 shows that(3) ξ ∗ x (cid:48)∗ x (cid:48)∗ ω ∨ C (cid:48) /S = x ∗ x (cid:48)∗ ω ∨ C (cid:48) /S → x ∗ x ∗ ω ∨ C/S → O C ( x ) ⊗ x ∗ O Z → x ∗ ( x ∗ ω ∨ C/S ⊗ O Z ) = x ∗ ( N x ( S ) ,C ⊗ O Z ) = x ∗ ( x ∗ O C ( x ) ⊗ O Z ) = O C ( x ) ⊗ x ∗ O Z by adjunction and the projection formula, where N stands for normal sheaf.On the other hand, there is a natural map O C (cid:48) ( x (cid:48) ) → ξ ∗ O C ( x ) obtained as thedual of the map ξ ∗ I x,C → I x (cid:48) ,C (cid:48) adjoint to I x,C → ξ ∗ I x (cid:48) ,C (cid:48) , the restriction ofthe isomorphism ξ : O C → ξ ∗ O C (cid:48) . This map pushes forward to ξ ∗ O C (cid:48) ( x (cid:48) ) → ξ ∗ ξ ∗ O C ( x ) ∼ = O C ( x ), since the fact that ξ is an isomorphism implies that ξ ∗ ξ ∗ L ∼ = L for any invertible O C -module L . We have a commutative diagram0 ξ ∗ O C (cid:48) ξ ∗ O C (cid:48) ( x (cid:48) ) ξ ∗ x (cid:48)∗ x (cid:48)∗ ω ∨ C (cid:48) /S O C O C ( x ) x ∗ x ∗ ω ∨ C/S R ξ ∗ O C (cid:48) = 0, andthe proof of commutativity is left to the reader. Then the snake lemma and (3)imply that the cokernel of ξ ∗ O C (cid:48) ( x (cid:48) ) → O C ( x ) is O C ( x ) ⊗ x ∗ O Z = O C ( x ) | x ( Z ) .The quotient map to the kernel O C ( x ) → O C ( x ) | x ( Z ) is the usual restriction, andthe existence of the desired surjection follows. This map is an isomorphism if andonly if ξ ∗ O C (cid:48) ( x (cid:48) ) → O C ( x ) is injective. The key point here is that the restrictionmaps Γ( U (cid:48) , O C (cid:48) ) → Γ( U (cid:48) \ ξ − ( x ) , O C (cid:48) ) are injective, which implies that so are therestriction maps Γ( U (cid:48) , O C (cid:48) ( x (cid:48) )) → Γ( U (cid:48) \ ξ − ( x ) , O C (cid:48) ), since O C (cid:48) ( x (cid:48) ) is invertible,from which the injectivity of ξ ∗ O C (cid:48) ( x (cid:48) ) → O C ( x ) follows easily. (cid:3) Proposition 3.12.
In Situation 3.1.(1), if Z (cid:54) = S , then there exists v (cid:48) compatiblewith v .Proof. We have ξ ∗ ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) (cid:39) I x ( Z ) ,C ⊗ ω ∨ C/S ( − x ∞ ) by Proposition 3.7 part 2,Lemma 3.11, and the projection formula. If 0 → I x ( Z ) ,C → O C → x ∗ O Z → ω ∨ C/S ( − x ∞ ) and we take global sections, we obtain an exact sequence0 → Γ( C (cid:48) , ω ∨ C (cid:48) /S ( − x (cid:48)∞ )) → Γ( C, ω ∨ C/S ( − x ∞ )) → Γ( C, ω ∨ C/S ( − x ∞ ) ⊗ x ∗ O Z ) , and since v | x ( Z ) = 0, there exists a unique v (cid:48) which maps to v . To show that x (cid:48)∗ v (cid:48) isnowhere vanishing, note that the first map in Proposition 3.7 part 3 is injective onsections in this case, as it is an isomorphism above S \ Z . Then Lemma 3.7 impliesthat x (cid:48)∗ ω ∨ C (cid:48) /S ( − x (cid:48)∞ ) (cid:39) O S given the definition of Z . Combined with the fact that x (cid:48)∗ v (cid:48) is nonzero on the dense open S \ Z ⊂ S , it implies that x (cid:48)∗ v (cid:48) is indeed nowherevanishing. (cid:3) The result of the stabilization relative to v is ( C (cid:48) , S, π (cid:48) , x (cid:48)∞ , x (cid:48) , v (cid:48) ), where all dataexcept v (cid:48) is the one constructed canonically above, and v (cid:48) is compatible with v . Wehave proved the existence of such a v (cid:48) only with some additional assumptions cf.Proposition 3.12, and we have shown that if such v (cid:48) exists then it is unique up tosuitable automorphisms of the data cf. Lemma 3.10. Existence is true in generaland will come ‘for free’ from an inductive hypothesis since we’ll see in § Marked field decorated rational trees
We will see very soon that the work done in § § S . Definition 4.1. An n -marked field-decorated rooted rational tree consists of a va-riety S , a prestable genus 0 curve π : C → S , sections x ∞ , x , . . . , x n : S → C of π ,and an O C -module homomorphism ν : ω C/S → I x ∞ ( S ) ,C , such that(1) x i ( s ) (cid:54) = x ∞ ( s ), and π is smooth at x ∞ ( s ) , x ( s ) , . . . , x n ( s ), for all s ∈ S ;(2) x ∗ i ν : x ∗ i ω C/S → O S is an isomorphism for i = 1 , . . . , n .We say that the n -marked field-decorated rooted rational tree is stable if ω C/S ( x ∞ +2 x + · · · + 2 x n ) is relatively ample.Let v be the image of ν under Hom( ω C/S , I x ∞ ( S ) ,C ) ∼ = Γ( C, ω ∨ C/S ( − x ∞ )). Thereis a natural isomorphism x ∗∞ ω C/S ( x ∞ ) ∼ = x ∗∞ Ω C/S ( x ∞ ) ∼ = O S by taking residues,or dually x ∗∞ ω ∨ C/S ( − x ∞ ) ∼ = O S . Then x ∗∞ v is simply a regular function on S , so itinduces a morphism S → A . We call this the coresidue morphism to A . Theorem 4.2.
The moduli space F n of stable n -marked field-decorated rooted ra-tional trees exists, in the sense that the the functor Var C → Set which sends S to the set of stable n -marked field-decorated rooted rational tree over S modulo S -isomorphism is representable. Moreover, F n is normal and irreducible, and thecoresidue morphism η n : F n → A is flat and projective. To prove Theorem 4.2, we will need an extension of Definition 4.1.
ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 11
Definition 4.3.
Let n be a positive integer and a = ( a , . . . , a n ) ∈ { , , } n .An a -stable field-decorated rooted rational tree (or a -FDRT) consists of a complexvariety S , a prestable genus 0 curve π : C → S , sections x , . . . , x n , x ∞ : S → C of π , and an O C -module homomorphism ν : ω C/S → I x ∞ ( S ) ,C , such that(1) x ∞ ( S ) is contained in the open subset where π is smooth;(2) if a i ≥
1, then x i ( S ) is contained in the open subset where π is smooth;(3) if a i = 2, then x ∗ i ν : x ∗ i ω C/S → O S is an isomorphism;(4) ω C/S ( x ∞ + a · x ) is π -ample, where a · x = (cid:80) nj =1 a j x j ( S ).(Note that (cid:80) nj =1 a j x j ( S ) is Cartier in (4), since x i ( S ) is Cartier if a i (cid:54) = 0.)If a is some a -stable field-decorated rooted rational tree, we will write S [ a ], C [ a ], π [ a ], x ∞ [ a ], x i [ a ], ν [ a ] for what was denoted S, C, π, x ∞ , x i , ν in Definition 4.3. Wealso write v [ a ] for the global section of ω ∨ π [ a ] ( − x ∞ [ a ]) corresponding to ν [ a ]. Forany a -FDRT a and any map φ : S (cid:48) → S [ a ], we define the pullback φ ∗ a of a along φ in the natural way: S [ φ ∗ a ] = S (cid:48) , C [ φ ∗ a ] = S (cid:48) × S C [ a ], etc. Then the class of all a -FDRT becomes a category fibered over Var C .For (cid:96) ∈ { , , } and any positive integer n , let a (cid:96),n = (2 , . . . , , (cid:96) ) ∈ { , , } n .We construct inductively a sequence of objects ( t (cid:96),n ) ≤ (cid:96) ≤ ,n ∈ Z + , such that t (cid:96),n isan a (cid:96),n -FDRT. Let t , be defined by S [ t , ] = A , C [ t , ] = A × P , π [ t , ] theprojection to A , x ∞ [ t , ]( t ) = ( t, [1 : 0]) x [ t , ]( t ) = ( t, [1 : 1]) and v [ t , ] = (1 + tx ) ∂∂x in the affine chart Y (cid:54) = 0, where x = X/Y . It is clear that t , is an a , -FDRT.Note also that the coresidue morphism for t , is the identity map on A . Then wecontinue for n ≥ • t ,n is obtained from t ,n − by pulling back along π [ t ,n − ] and taking x n to be the diagonal section C [ t ,n − ] → C [ t ,n − ] × S [ t ,n − ] C [ t ,n − ]; • t ,n is obtained from t ,n applying the stabilization of § • t ,n is obtained from t ,n applying the stabilization of § Theorem 4.4.
Let n ≥ and ≤ (cid:96) ≤ such that n + (cid:96) ≥ . Then t (cid:96),n is terminalin the category of a (cid:96),n -FDRT.Remark . Note that for all n ≥ ≤ (cid:96) ≤ n + (cid:96) ≥ a (cid:96),n -FDRT are free of automorphisms. Since we are assuming that bases arevarieties,this boils down to the elementary case when the base is Spec C . Definition 4.6. A contraction consists of the following data: (cid:96) ∈ { , } , an a (cid:96),n -FDRT a , an a (cid:96) − ,n -FDRT b , and a morphism f : C [ a ] → C [ b ] such that(1) S [ a ] = S [ b ], π [ b ] ◦ f = π [ a ], and f ◦ x α [ a ] = x α [ b ], for α = ∞ , , . . . , n ;(2) f is contractive, cf. Definition 2.2;(3) if π [ b ] sm ⊆ C [ b ] is the open subscheme where π [ b ] is smooth, then therestriction of f induces an isomorphism f − ( π [ b ] sm ) (cid:39) π [ b ] sm under which ν [ a ] | f − ( π [ b ] sm ) corresponds to ν [ b ] | π [ b ] sm in the natural sense.To prove the existence of contractions, we will rely on the powerful techniquesof [Stacks, Tag 0E7B]. Although an elementary approach is also possible, the more sophisticated techniques of [Stacks, Tag 0E7B] have some advantages; for instance,flatness comes for free, which otherwise would have taken a long calculation. Lemma 4.7.
For (cid:96) = 1 , and any n , any a (cid:96),n -FDRT admits a contraction to an a (cid:96) − ,n -FDRT, unique up to unique isomorphism.Proof. If the base S is Spec C , this is elementary. For instance, we may take C [ b ] = Proj (cid:77) j ≥ Γ (cid:0) C [ a ] , ω C [ a ] (2 x [ a ] + · · · + 2 x n − [ a ] + ( (cid:96) − x n [ a ]) ⊗ j (cid:1) , etc. Uniqueness follows from the case S = Spec C , though, to make the argu-ment completely accurate we do need to parametrize isomorphisms between thetwo contracted curves by, e.g. an open subscheme of a Hilbert scheme. To proveexistence, we note that it suffices to prove it ´etale locally on S – indeed, to glob-alize, by uniqueness, it suffices to note that the fibered category whose objects areprestable curves decorated with n arbitrary sections and a relative logarithmic vec-tor field (global section of the dual of the relative dualizing sheaf) satisfies ´etaledescent, which follows by standard arguments. To prove existence ´etale locally,let s ∈ S ( C ) be a closed point, and consider the contraction of ( s → S ) ∗ a . Weapply [Stacks, Tag 0E7C], so, after replacing S with an ´etale neighborhood of s ,we obtain an S -morphism to a ‘family of curves’ over S . After passing again toan ´etale neighborhood, we may assume that the source of this family of curves isa scheme [Stacks, Tag 0E6F]. After further shrinking S , we may assume that thefamily of curves is a prestable genus 0 curve. Although the FDRT is not fully(or even correctly) constructed yet, we already denote it by π [ b ] : C [ b ] → S . Let x α [ b ] = f ◦ x α [ a ], for α = ∞ , , . . . , n . After shrinking S , we may assume that ω π [ b ] (2 x [ b ] + · · · + 2 x n − [ b ] + ( (cid:96) − x n [ b ]) is relatively ample, since it is ample on π [ b ] − ( s ) and ampleness is an open condition in families. Let v [ b ] ∈ Γ( C [ b ] , ω ∨ π [ b ] )be the image of v [ a ] under the skew logarithmic differential f δ , cf. Definition 2.4.Given the inherently open nature of all conditions in Definition 4.3, we may confirmthat what we have obtained is indeed an a (cid:96) − ,n -FDRT, possibly after shrinking S .Shrinking S one last time, we may also confirm that condition (2) in Definition 4.6is satisfied, quoting e.g. [Stacks, Tag 0E88]. (cid:3) Contraction and stabilization are inverse operations, and now we partially provethis, ignoring the vector fields for now.
Lemma 4.8.
Let a be an a ,n -FDRT and b a contraction of a . Then b satisfiesSituation 3.1.(0) with x n [ b ] in the role of x . If C [ b ] (cid:48) , π [ b ] (cid:48) , x ∞ [ b ] (cid:48) , x [ b ] (cid:48) , v [ b ] (cid:48) is itsKnudsen stabilization ( § C [ b ] (cid:48) (cid:39) C [ a ] whichis compatible with the projections to S = S [ a ] = S [ b ] and all the sections. Note also that for i = 1 , . . . , n − x i [ b ] necessarily maps to the open subset of C [ b ] on which C [ b ] (cid:48) → C [ b ] is an isomorphism. Proof.
This is similar to the proof of [Kn83, Lemma 2.5], so we will only sketchthe argument. The claim that b satisfies Situation 3.1.(0) if x n [ b ] plays the role of x boils down immediately to the special case S = Spec C , when it’s elementary.Analogously to the proof of [Kn83, Lemma 2.5], we have an O C [ b ] -module homo-morphism f ∗ O C [ a ] ( x n [ a ] − x ∞ [ a ]) → I ∨ x n [ b ] ,C [ b ] ( − x ∞ [ b ]), and it is straightforwardto check on geometric fibers that this is an isomorphism. We claim that the adjoint ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 13 f ∗ I ∨ x n [ b ] ,C [ b ] ( − x ∞ [ b ]) α −→ O C [ a ] ( x n [ a ] − x ∞ [ a ]) of the inverse of this map is surjec-tive. Equivalently, f ∗ f ∗ O C [ a ] ( x n [ a ] − x ∞ [ a ]) → O C [ a ] ( x n [ a ] − x ∞ [ a ]) is surjective,which follows from [Kn83, Corollary 1.5]. Let J = f ∗ O C [ b ] ( x ∞ [ b ]) ⊗O C [ a ] ( − x ∞ [ a ]).Consider the (solid arrow) commutative diagram with exact rows O C [ a ] f ∗ O C [ b ] f ∗ O C [ b ] ( x ∞ [ b ]) ⊕ f ∗ I ∨ x n [ b ] ,C [ b ] f ∗ K J J ( x ∞ [ a ]) ⊕ J ( x n [ a ]) J ( x ∞ [ a ] + x n [ a ]) 0 (cid:2) id 00 α ⊗ id (cid:3) Both rows are related to the exact sequence defining the cokernel K in (1). The toprow is the f -pullback of this sequence for C [ b ]; the bottom row is this sequence for C [ a ], twisted by J . A basic diagram chase shows that there exists a unique dashedarrow which makes the diagram commute, and this map is surjective by the snakelemma. We’ve obtained a surjective map f ∗ K → O C [ a ] ( x n [ a ]) ⊗ f ∗ O C [ b ] ( x ∞ [ b ]).Of course, the target is invertible, so this induces a morphism C [ a ] → C [ b ] (cid:48) . Thecompatibilities as well as the fact that this is an isomorphism are straightforwardbecause they can be checked on fibers. (cid:3) Lemma 4.9.
Let a be an a ,n -FDRT and b a contraction of a . Then b satisfiesSituation 3.1.(1) with x n [ b ] in the role of x . If C [ b ] (cid:48) , π [ b ] (cid:48) , x ∞ [ b ] (cid:48) , x [ b ] (cid:48) is the dataconstructed canonically as part of its stabilization relative to v [ b ] (last paragraph of § C [ b ] (cid:48) (cid:39) C [ a ] which is compatible with theprojections to S = S [ a ] = S [ b ] and all the sections. A remark analogous to the one after Lemma 4.8 applies.
Proof.
The claim that b satisfies Situation 3.1.(1) if x n [ b ] plays the role of x boilsdown immediately to the special case S = Spec C , when it’s elementary. We claimthat there exists an isomorphism(4) β : f ∗ ω ∨ π [ b ] ( − x n [ b ] − x ∞ [ b ]) (cid:39) ω ∨ π [ a ] ( − x n [ a ] − x ∞ [ a ])which restricts to the identity on any open subset where f is an isomorphism. Wehave ω ∨ π [ b ] ( − x n [ b ] − x ∞ [ b ]) (cid:39) f ∗ ω ∨ π [ a ] ( − x n [ a ] − x ∞ [ a ]) by an argument ‘isomorphic’ tothat used in the proof of Proposition 3.3. Indeed, only the details of the verificationson fibers differ slightly, otherwise the argument is the same. Then (4) follows by[Kn83, Corollary 1.5].Let U = f ∗ O C [ b ] ( x n [ b ]). We have a commutative diagram with exact rows O C [ a ] f ∗ O C [ b ] f ∗ O C [ b ] ( x n [ b ]) ⊕ f ∗ ω ∨ π [ b ] ( − x ∞ [ b ]) f ∗ V U ( − x n [ a ]) U ⊕ ω ∨ π [ a ] ( − x ∞ [ a ] − x n [ a ]) ⊗ U ω ∨ π [ a ] ( − x ∞ [ a ]) ⊗ U (cid:2) id 00 β ⊗ id (cid:3) Both rows are related to the exact sequence defining the cokernel V in (2). The toprow is the pullback of this sequence for C [ b ] along f ; the bottom row is this sequencefor C [ a ], twisted by U ( − x n [ a ]). Commutativity is left to the reader. A simplediagram chase shows that there exists a unique dashed arrow which makes thediagram commute, and by the snake lemma, this map is surjective. We’ve obtaineda surjective O C [ a ] -module homomorphism f ∗ V → ω ∨ π [ a ] ( − x ∞ [ a ]) ⊗ f ∗ O C [ b ] ( x n [ b ]). The target is invertible, so it induces an S -morphism C [ a ] → ( C [ b ]) (cid:48) . The fact thatthis is an isomorphism, as well as all the required compatibilities can be checkedon fibers. (cid:3) We may now prove Theorem 4.4, which then gives Theorem 4.2.
Proof of Theorem 4.4.
We prove this by induction on 3 n + (cid:96) . The base case 3 n + (cid:96) =5 is elementary and left to the reader. For the inductive step, note that the case (cid:96) = 0 is straightforward. Assume that (cid:96) ∈ { , } and let a be an a (cid:96),n -FDRT.Note that if a map S [ a ] → S [ t (cid:96),n ] which exhibits a as a pullback of t (cid:96),n exists,then such a map must be unique. To prove that such a map exists, we proceed asfollows. Let a − be the contraction of a to an a (cid:96) − ,n -FDRT, cf. Lemma 4.7. By theinductive hypothesis, there exists a morphism φ : S [ a ] = S [ a − ] → S [ t (cid:96) − ,n ] = S [ t (cid:96),n ]which exhibits a − as a pullback of t (cid:96) − ,n , and let ψ : C [ a − ] → C [ t (cid:96),n − ] be thecorresponding map on the curves. Since the stabilization procedures of § § ψ : C [ a ] → C [ t (cid:96),n ] which exhibits C [ a ] as S [ a ] × S [ t (cid:96),n ] C [ t (cid:96),n ], and is compatiblewith the marked sections. Then, by the uniqueness part of Corollary 3.4 if (cid:96) = 1respectively Lemma 3.10 if (cid:96) = 2, we obtain that a is isomorphic to φ ∗ t (cid:96),n . (cid:3) G m and G a actions on curves and their moduli Preliminaries.
What we will need from this section is the equivariance crite-rion in Proposition 5.2 below. The proposition concerns a reflexive sheaf associatedto a
Weil divisor. We refer the reader to [Sch10, Ha94] for the theory of reflexivesheaves associated to Weil divisors, and also state a very simple functoriality resultwhich we’ll need very soon. If X is a normal variety, then we write O X ( D ) for thereflexive coherent sheaf associated to the divisor D ; however, in other sections wewill reserve this notation for the Cartier case. Remark . Let X and X (cid:48) be normal quasi-projective varieties, f : X (cid:48) → X a morphism, D = (cid:80) mk =1 a k W k a Weil divisor on X such that W i and f − ( W i ) are prime Weil divisors, and D (cid:48) = (cid:80) mk =1 a k f − ( W k ) on X (cid:48) . Then f ∗ O X ( D ) (cid:39) O X (cid:48) ( D (cid:48) ) holds provided that f − ( X sing ) has codimension at least 2in X (cid:48) . Indeed, functoriality is well known for Cartier divisors, so the statementholds in codimension 2. Then the claim follows automatically, in light of [Sch10,Theorems 1.17 and 2.10], or equivalently [Ha94, Proposition 1.11 and Theorem 1.9]. Proposition 5.2.
Let γ : G → T be a flat group scheme with reduced, connectedand rationally connected fibers with trivial Picard groups, such that both G and T are complex quasi-projective varieties, and let G t = γ − ( t ) . Let π : X → T be ageometrically integral flat projective morphism, and let X t = π − ( t ) . Assume that X is normal, that T is smooth and Pic T = 0 , and that G acts on X . Moreover,assume that for all t ∈ T such that X t is singular, all of the following hold: X t isnormal, any nonconstant regular function on G t vanishes at some point, and id G t and the constant identity-element map are ‘rationally homotopic’ in the space ofendomorphisms End ( G t ) . Finally, let D ⊂ X be a prime Weil divisor flat over T ,and with integral fibers. Then O X ( D ) admits a G -equivariant structure. Note that O X ( D ) = I ∨ D,X by [Sch10, Propositions 3.4 and 3.13.(b)]. In thestatement of Proposition 5.2, by ‘rationally homotopic’ we mean that there exist a
ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 15 rational curve B ⊂ A and a B -endomorphism of B × G t whose fibers over 0 and1 are the two endomorphisms of G t in question. Proof.
Let σ : G × T X → X be the given G -action, Y = G × T X , D = G × T D ,and D = σ − ( D ). We claim that D ∼ D . Let E = D − D , and F = O Y ( E ).For each t ∈ T , let E t = D ,t − D ,t .We will show that E t ∼
0. If X t is nonsingular, the fact that G t is rationallyconnected implies that D t ∼ g · D t for all g ∈ G t as Cartier divisors on X t , and thenthe see-saw lemma [Ha77, Exercise III.12.4] implies that O Y t ( E t ) is the pullback ofa line bundle on G t along the projection Y t (cid:39) G t × X t → G t . However, Pic G t = 0,so O Y t ( E t ) (cid:39) O Y t when X t is nonsingular. If X t is singular, the assumption thatid G t and the constant identity-element map are rationally homotopic as morphisms G t → G t implies that D ,t and D ,t are rationally equivalent, and hence linearlyequivalent as Weil divisors on X t [Fu98, § F t (cid:39) O Y t . In particular, this also implies that F is invertible because invertibility of torsion free coherent modules can be checkedfiberwise (all fibers are vector spaces of dimension 1) by a standard corollary ofNakayama’s lemma. It then follows from another application of the see-saw lemma[Ha77, Exercise III.12.4] that F is the pullback of an invertible sheaf on T alongthe projection Y → T . However, Pic T = 0, and hence F (cid:39) O Y , that is, D ∼ D .We thus obtain an isomorphism O Y ( D ) (cid:39) O Y ( D ). By Remark 5.1 once more, O Y ( D ) (cid:39) pr ∗ X O X ( D ), and O Y ( D ) (cid:39) σ ∗ O X ( D ), so the isomorphism above isan isomorphism φ : σ ∗ O X ( D ) → pr ∗ X O X ( D ). We can modify φ to a ‘unitary’isomorphism φ : σ ∗ O X ( D ) → pr ∗ X O X ( D ), that is, an isomorphism which restrictsto the identity on { e G }× T X ⊂ G × T X . This can be achieved simply by composing φ with the pullback along pr X of the restriction of φ − to { e G } × T X .Finally, we claim that φ automatically satisfies the cocycle condition. It suf-fices to check it on fibers of G × T G × T X → T . Let t ∈ T . If X t is non-singular, then O X t ( D t ) is invertible, and the cocycle condition is well-known tobe automatically satisfied, please see the proof of [MFK94, Proposition 1.5, § X t is singular, our assumption is that G t admits no nowhere vanishingregular functions. Keeping Remark 5.1 in mind, the cocycle condition amountsto the agreement of two isomorphisms from two canonically identified sheaves on G t × X t to O G t × X t ( G t × D t ). The agreement above e G t is automatic by theunitarity condition. Composing one isomorphism with the inverse of the othergives an automorphism α of O G t × X t ( G t × D t ) which restricts to the identity au-tomorphism on { e G t } × X t . By [Sch10, Proposition 3.13.(c)], α corresponds to anowhere vanishing regular function on G t × X t equal to 1 on { e G t } × X t . However,Γ( O × X t ) (cid:39) Γ( O × G t × X t ), hence α ≡ (cid:3) Proposition 5.2 will be used in conjuction with Lemma 5.3, which will also beused by itself several times.
Lemma 5.3.
Let Y be an S -scheme, and σ an action of G on Y relative to S .Let F be a G -equivariant quasi-coherent O Y -module, X = P Y ( F ) , and f : X → Y the natural projection map. Then there exists a G -action on X over S , relative towhich f is G -equivariant. Moreover, the choice is compatible with any base change S (cid:48) → S . Stabilization revisited.
In Situation 3.1.( (cid:96) ), (cid:96) ∈ { , , } , assume that thebase S is a variety, and let η : S → A be the coresidue morphism, as in §
4. Let now γ : G → A as in §
1. We say that a G -action σ : G × A C → C on C over S and A (more accurately, an action of the S -group scheme S × A G on the S -scheme C )is generated by v if its first order infinitesimal action corresponding to the vectorfield ∂∂x along the identity section of G is equal to v , and x ∞ is fixed by σ . In thissubsection, we will prove inductively that for 3 n + (cid:96) ≥ t (cid:96),n admits a G -actiongenerated by v [ t (cid:96),n ]. In the base case 3 n + (cid:96) = 5, with notation as in §
4, we maytake the action ( t, a ) · ( t, [ X : Y ]) = ( t, [ X + aY + taX : Y ]), and it is an elementarycalculation that it is generated by v [ t , ]. To prove the induction step, we need torevisit stabilization. Specifically, the induction steps will be Lemma 5.4, Lemma 5.5or the obvious fact that actions or prestable curves can be pulled back, dependingon the value of (cid:96) .5.2.1. Knudsen stabilization revisited.
In Situation 3.1.(0), assume in addition that C and S are complex normal quasi-projective varieties, that the fibers of π arenormal, and that x (cid:54) = x ∞ . It can be checked simply following trough the con-struction in § t ,n (for instance, regular incodimension 1 and lci, or more generally Cohen-Macaulay, implies normal [Stacks,Tag 0342]). Moreover, assume that an action σ on C generated by v is given. Lemma 5.4.
In the situation above, there exists a G -action σ (cid:48) on C (cid:48) such that ξ is G -equivariant relative to σ and σ (cid:48) , and σ (cid:48) generates v (cid:48) .Proof. It suffices to analyze separately two situations: x ( S ) is contained in theopen subset of C where π is smooth, respectively x ∩ x ∞ = ∅ . In the first case, x is a Cartier divisor on C and I ∨ x,C = O C ( x ). Let E = O C ( − x ) ⊕ O C ( − x ∞ ). Ourhypotheses ensure that we have a short exact sequence0 → ∧ E → E → I x ( S ) ∩ x ∞ ( S ) ,C → , so K (cid:39) I x ∞ ∩ x,C ( x ∞ + x ), where K is defined in (1), and C (cid:48) = P C I x ∞ ∩ x,C . However, I x ∞ ∩ x,C is G -equivariant, so we may apply Lemma 5.3 to construct a lift σ (cid:48) . Inthe second case, K = I ∨ x,C ( x ∞ ), and hence C (cid:48) = P C I ∨ x,C . By Proposition 5.2, I ∨ x,C is G -equivariant, so again by Lemma 5.3, there exists a lift σ (cid:48) of σ to C (cid:48) . (cid:3) Stabilization relative to a vector field revisited.
In Situation 3.1.(1), assumein addition that C and S are complex normal quasi-projective varieties, and that x ∗ v is not identically 0. These hypotheses are satisfied for t ,n and again this canbe ascertained by following through the construction in §
4. Moreover, assume thatan action σ on C generated by v is given. Lemma 5.5.
In the situation above, there exists a G -action σ (cid:48) on C (cid:48) such that ξ is G -equivariant relative to σ and σ (cid:48) , and σ (cid:48) generates v (cid:48) .Proof. It is clear that x ( Z ) is fixed (even scheme-theoretically, since it’s reduced)by σ , and in particular I x ( Z ) ,C has a natural G -equivariant structure relative to σ .Let E = O C ( − x ) ⊕ ω C/S . We have a short exact sequence0 → ∧ E → E → I x ( Z ) ,C → , hence V (cid:39) I x ( Z ) ,C ⊗ ω ∨ C/S ( x − x ∞ ), where V is defined in (2), so C (cid:48) = P C I x ( Z ) ,C .By Lemma 5.3, there exists a G -action σ (cid:48) on C (cid:48) for which ξ is G -equivariant. Inparticular, the map Γ( dξ ) sends the infinitesimal generator of σ (cid:48) to v , and it followsthat the former must be precisely v (cid:48) . In particular, x (cid:48) ( s ) is not fixed by σ (cid:48) by thesecond part of Proposition 3.12. It is easy to check that x (cid:48)∞ is fixed by σ . (cid:3) ONFIGURATIONS OF POINTS ON A LINE UP TO SCALING OR TRANSLATION 17
The G n A -action on the total space. We may now complete the proof ofTheorem 1.2. In the previous subsection, we shown that t ,n admits a G -actiongenerated by v [ t ,n ], which we will call σ . To review notation, we have S [ t ,n ] = F n .Consider the n -marked field decorated rooted rational tree y which coincides withthe pullback of t ,n along the projection (cid:36) : G n A × A F n → F n , with the soleexception of the sections x i [ y ] which are instead defined as the composition S [ y ] = G n A × A F n ( (cid:36) ,σ ) −−−−→ G n A × A F n = S [ (cid:36) ∗ t ,n ] x i [ (cid:36) ∗ t ,n ] −−−−−−→ C [ (cid:36) ∗ t ,n ] = C [ y ] , where (cid:36) : G n A × A F n → G n A is the projection to the first factor. Since S [ y ] is avariety, Theorem 4.2 shows that y induces an A -morphism ρ : G n A × A F n → F n . The fact that ρ is a group action can be checked purely formally using Theorem4.2 again. It is also easy to check that the diagonal G (cid:44) → G n A acts trivially, and wetake the quotient action simply by restricting to G n − A (cid:39) { ( g , . . . , g n ) : g (cid:63) g (cid:63) · · · (cid:63) g n = 0 } ⊂ G n A . It is straightforward to check that this restricted action restricts on F n ∼ = G n − A ⊂ F n parametrizing n -marked field decorated rooted rational trees with irreduciblesources to the tautological action.Finally, we have to argue that the fibers of the coresidue morphisms η : F n → A are L n if t (cid:54) = 0, respectively P n if t = 0. For the Losev-Manin space, this canbe checked by comparing the constructions. What the stabilization in [LM00]accomplishes in a single step, our stabilization accomplishes in two. We also notein passing that the 0-section in the universal curve over the Losev-Manin spacesextends on the universal curve over F n not to a section, but to the vanishing locusof v [ t ,n ]. Regarding P n , we simply define P n to be η − (0), but note that indeedstable n -marked field decorated rooted rational trees over Spec C with coresidue 0at x ∞ are precisely stable n -marked G a -rational trees, cf. Definition 1.1.It is straightforward to extend Definition 1.1 to prestable curve over arbitrarybases S , which we will continue to consider complex varieties. On one hand, itis straightforward to check that any stable n -marked G a -rational tree is a stable n -marked field decorated rooted rational tree with an identically 0 coresidue map,where the vector field corresponds to the infinitesimal first order automorphisminduced by ddx ∈ g a . On the other hand, the converse holds: all stable n -markedfield decorated rooted rational tree with an identically 0 coresidue map come fromstable n -marked G a -rational trees, because Theorem 4.2 shows that the pullback of t ,n to η − (0) = P n is terminal in the category of stable n -marked field decoratedrooted rational tree with an identically 0 coresidue map, and the discussion aboveshows that the curve over it is in fact a stable n -marked G a -rational tree. Corollary 5.6.
The space P n constructed above represents the functor Var C → Set which sends each S to the set of stable n -marked G a -rational trees over S up toisomorphism. We conclude the paper with the following question.
Question 5.7.
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