The r -ELSV Formula via Localisation on the Moduli Space of Stable Maps with Divisible Ramification
TTHE r -ELSV FORMULA VIA LOCALISATION ON THE MODULISPACE OF STABLE MAPS WITH DIVISIBLE RAMIFICATION OLIVER LEIGH
Abstract.
The moduli space of stable maps with divisible ramification uses r -th roots of a canonical ramification section to parametrise stable maps whoseramification orders are divisible by a fixed integer r . In this article, a virtualfundamental class is constructed while letting domain curves have a positivegenus; hence removing the restriction of the domain curves being genus zero.We apply the techniques of virtual localisation and obtain the r -ELSV formulaas an intersection of the virtual class with a pullback via the branch morphism. Introduction
For a smooth curve X , the moduli space of stable maps with divisible ramificationwas introduced in [Le1Le1] as a natural compactification of the sub-moduli space M /r g ( X, d ) = (cid:110) (cid:2) f : C → X (cid:3) ∈ M g ( X, d ) (cid:12)(cid:12)(cid:12) R f = r · D for some D ∈ Div( C ) (cid:111) / ∼ of M g ( X, d ) where R f denotes the ramification divisor of f .The compactification of [Le1Le1] agrees with the extended concept of ramification forstable maps introduced in [FPFP] and [GV1GV1]. This extended concept is based onthe observation that, for smooth curves, the ramification divisor arises from thedifferential map df : f ∗ Ω X → Ω C . When the domains are nodal, the differentialmap can be combined with the natural morphism Ω C → ω C to obtain a morphism δ : O C −→ ω C ⊗ f ∗ ω ∨ X . (1)The extended concept of ramification is determined by δ and it is shown in [Le1Le1]that the concept of divisibility of ramification order by r is equivalent to requiring r -th roots of δ .In order to control the r -th roots, it is convenient to consider a moduli space withslightly more information. Hence, following [Le1Le1], we will refer to the followingmoduli space as the moduli space of stable maps with divisible ramification . Definition 1
For g ≥ d > M /r g ( X, d ) the moduli stack thatparameterises ( f : C → X, L, e : L ⊗ r ∼ → ω C ⊗ f ∗ ω ∨ X , σ : O C → L ) where:(i) C is a r -prestable curve of genus g (a stack such that the coarse space C is a prestable curve of genus g , where points corresponding to nodes of C are balanced r -orbifold points, and C sm ∼ = C sm ).(ii) f is a morphism such that the induced morphism f : C → X on the coarsespace is a stable map parametrised by M g ( X, d ).(iii) L is a line bundle on C and e : L ⊗ r ∼ → ω C ⊗ f ∗ ω ∨ X is an isomorphism.(iv) σ : O C → L is a section such that e ( σ r ) = δ , where δ is defined in (11). Mathematics Subject Classification.
Key words and phrases. virtual localisation; ELSV formula; moduli spaces; stable maps;twisted curves; spin structures; Hurwitz numbers. a r X i v : . [ m a t h . AG ] A p r OLIVER LEIGH
Remark 1
This article will be primarily concerned with the relative version of thespace from definition 11. We denote this by M /r g ( X, λ ) where λ is an ordered par-tition. However, we will leave the technicalities of the relative version until section 11. M /r g ( X, λ ) is a proper Deligne-Mumford stack which is non-empty only when r divides 2 g − l ( λ ) + | λ | (1 − g X ) [Le1Le1, Thm. A]. The morphism to the “normal”moduli space of stable maps (i.e. forget the r -th root and r -twisted structures), M /r g ( X, λ ) −→ M g ( X, λ ) , (2)is both flat and of relative dimension 0 onto its image.The moduli space M /r g ( X, λ ) is often not equidimensional and requires a virtualfundamental class to perform intersection theory. Also, in the case when X = P the natural C ∗ -action on P induces an action on the moduli space. A virtual classin the C ∗ -equivariant setting is a powerful computational tool. In [Le1Le1] a virtualclass is constructed for the case g = 0. The following theorem extends this resultto the equivariant setting and to include the case when g > Theorem A M /r g ( P , λ ) has a natural C ∗ -equivariant perfect obstruction theorygiving a virtual fundamental class of dimension r (2 g − l ( λ ) + | λ | ) . For the rest of the introduction we set m := m ( P , g, λ ) := r (2 g − l ( λ ) + | λ | ).It is shown in [Le1Le1] that there is a natural morphism of stacks br : M /r g ( P , λ ) −→ Sym m P which is compatible with the branch morphism br of [FPFP] via the r -th diagonalmorphism ∆ which is defined by (cid:80) i x i (cid:55)→ (cid:80) i rx i : M /r g ( P , λ ) br (cid:47) (cid:47) (cid:15) (cid:15) Sym m P (cid:15) (cid:15) M g ( P , λ ) br (cid:47) (cid:47) Sym rm P Specifically, br takes a moduli point to its branch divisor divided by r .The methods of [KoKo, GV2GV2] give an explicit description of the C ∗ -fixed loci of M g ( P , λ ). Furthermore, we will identify the C ∗ -fixed loci of M /r g ( P , λ ) as pull-backs of the C ∗ -fixed loci of M g ( P , λ ) via the forgetful morphism of (22).The C ∗ -fixed loci of M /r g ( P , λ ) can be characterised by their image under br . Fol-lowing [GV2GV2] we call the fixed loci corresponding to the point [ m · (0)] ∈ Sym m P the simple fixed locus and denote this locus by F . This contains the C ∗ -fixed mapswhere there is no degeneration of the target at ∞ . Theorem B
The virtual localisation formula of [GPGP, CKLCKL] can be applied to M /r g ( P , λ ) . The simple fixed locus F is smooth of dimension g − l ( λ ) andthere is a morphism of degree ( λ · · · λ l ( λ ) ) − to the moduli space of r -spin curves b : F −→ M r , a g,l ( λ ) where a = ( a , . . . , a l ( λ ) ) is a vector of reverse remainders with a i ∈ { , . . . , r − } defined by λ i = (cid:4) λ i r (cid:5) r + ( r − − a i ) . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 3 Moreover, the contribution to the localisation formula of (cid:2) M /r g ( P , λ ) (cid:3) vir by thevirtual normal bundle of the simple fixed locus is: e (cid:0) N vir F (cid:1) = r l +2 g − l (cid:89) i =1 λ i (cid:0) λ i r (cid:1) (cid:106) λir (cid:107) (cid:4) λ i r (cid:5) ! c rt (cid:0) − b ∗ R ρ ∗ L (cid:1)(cid:81) li =1 (cid:0) − λ i t b ∗ ψ i (cid:1) (cid:18) tr (cid:19) g − l − m where ρ and L are the universal curve and r -th root of M /r, a g,l , while l := l ( λ ) and t is the generator of the C ∗ -equivariant Chow ring of a point. We can use the branch-type morphism and the virtual fundamental class to definethe following natural Hurwitz-type formula H rg,λ := (cid:90)(cid:2) M /rg ( P ,λ ) (cid:3) vir br ∗ [ p + · · · + p m ]where m = r (2 g − l ( λ ) + | λ | ) and p i ∈ P . We can choose an equivariant lift ofthe class [ p + · · · + p m ] that corresponds to the point [ m · (0)] ∈ Sym m P . With thischoice the class br ∗ [ p + · · · + p m ] will vanish on the non-simple fixed loci. Hencewe can apply the localisation formula of theorem BB. Taking the non-equivariantlimit of the resulting intersection gives the following formula. Theorem C
There is an equality H rg,λ = m ! r m + l +2 g − l (cid:89) i =1 (cid:0) λ i r (cid:1) (cid:98) λir (cid:99) (cid:98) λ i r (cid:99) ! (cid:90) M r , a g,l c ( − R ρ ∗ L ) (cid:81) lj =1 (1 − λ i r ψ j ) where ρ and L are the universal curve and r -th root of M r , a g,l , while a = ( a , . . . , a l ) is a vector with a i ∈ { , . . . , r − } defined by λ i = (cid:4) λ i r (cid:5) r + ( r − − a i ) and l = l ( λ ) . The intersection formula in theorem CC is known as the r -ELSV formula. It appearedin [SSZSSZ] as a conjecture relating to the stationary Gromov-Witten theory of P ( r !) m (cid:90) (cid:2) M g,m ( P ,λ ) (cid:3) vir ψ r ev ∗ [pt] · · · ψ rm ev ∗ m [pt](3) = m ! r m + l +2 g − l (cid:89) i =1 (cid:0) λ i r (cid:1) (cid:98) λir (cid:99) (cid:98) λ i r (cid:99) ! (cid:90) M r , a g,l ( λ ) c ( − R ρ ∗ L ) (cid:81) lj =1 (1 − λ i r ψ j ) . This formula has since been proved using the methods of Chekhov-Eynard-Orantintopological recursion in [BKLPSBKLPS, DKPSDKPS]. In fact, the proof in [DKPSDKPS] is for ageneralised version, conjectured in [KLPSKLPS], involving a q -orbifold P . A version ofthis formula for one-part double Hurwitz numbers was recently considered in [DLDL].In the sequel to this article [Le2Le2] we will provide a geometric proof of the equalityfrom (33) using the methods of degenerated targets from [Li2Li2] and the Gromov-Witten/Hurwitz correspondence from [OPOP]. Notation
Unless otherwise stated, the following notation will be used in this article: • All stacks, schemes and varieties are over C . • X is a smooth one dimensional variety. • λ is an ordered partition with length l := l ( λ ) and | λ | > • m := m ( X, g, λ ) := r (2 g − l + | λ | (1 − g X )). • If a ∈ Z ≥ then (cid:10) ar (cid:11) is the remainder after dividing a by r (i.e. a = (cid:4) ar (cid:5) r + (cid:10) ar (cid:11) ). OLIVER LEIGH
Contents
Introduction 11Contents 44Acknowledgements 441. Background 441.1. Review of the Moduli Space of Relative Stable Maps 441.2. r -Twisted Curves 881.3. r -Twisted Curves with Roots of Line Bundles 991.4. Stable Maps with Divisible Ramification 10102. C ∗ -Action on Stable Maps with Divisible Ramification 12122.1. Natural C ∗ -Actions and Equivariant Morphisms 13132.2. Basic Properties of the Fixed Locus 13132.3. Flag Nodes and Partial Normalisation 16162.4. Forgetting r -Orbifold Structure Flag Nodes 18183. Proving Theorem BB: Localisation Formula 21213.1. Virtual Localisation Formula 21213.2. Analysis of the Contributions From M r ν r -ELSV formula 30304.1. Choice of Equivariant Lift 30304.2. Application of the Localisation Formula 31315. Proving Theorem AA - Equivariant Perfect Obstruction Theory 32325.1. Background on Equivariant Perfect Obstruction Theories 32325.2. Equivariant Perfect Obstruction Theory For M ν M /r Acknowledgements
The author wishes to thank Dan Petersen, Jim Bryan and Paul Norbury for use-ful conversations which contributed to this article and Olof Bergvall for providingfeedback on an early version of this manuscript.1.
Background
Review of the Moduli Space of Relative Stable Maps.
Let λ be an ordered partition and fix x ∈ X . Relative stable maps parameterise maps wherethe pre-image of x lies in the smooth locus of C and where the map has monodromyabove x locally given by λ . To obtain a proper space we follow [Li1Li1] and allow thetarget to degenerate in a controlled manner by allowing X to sprout a chain of P ’s. ( Degenerated Targets and Relative Stable maps ) . For a smooth curve X, wecan define the i -th degeneration X [ i ] inductively from X [0] := X by:(i) X [ i + 1] is given by the union X [ i ] ∪ P meeting at a node N i +1 .(ii) The node N is at x ∈ X . For i > N i +1 is in the i th componentof X [ i + 1], i.e. the node is not in X [ i − ⊂ X [ i + 1].Then a degenerated target is a pair ( T, t ) where T = X [ i ] for some i ≥ t is ageometric point in the smooth locus of i th component of T . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 5 A genus g stable map to X relative to ( λ, x ) is given by (cid:16) h : C −→ T, p : T −→ X, q , . . . , q l , t (cid:17) where ( C, q i ) is a l -marked genus g prestable curve, ( T, t ) is a degenerated target, h is a morphism sending q i to t and p is a morphism sending t to x such that:(i) There is an equality of divisors on C given by h − ( t ) = (cid:80) λ i q i .(ii) We have p | X is an isomorphism and p | T \ X : T \ X → { x } is constant.(iii) The pre-image of each node N of T is a union of nodes of C . At any suchnode N (cid:48) of C , the two branches of N (cid:48) map to the two branches of N , andtheir orders of branching are the same.(iv) The data has finitely many automorphisms (an automorphism is a a pairof isomorphisms a : C → C and b : T → T taking q i to q i and t to t suchthat h ◦ a = b ◦ h and p = p ◦ b ). Definition 1.1.2 ( Moduli Space of Relative Stable Maps ) . The moduli stack ofgenus g stable maps relative to ( λ, x ) is denoted M g ( X, λ ). It is the groupoid with:(i) Objects over a scheme S given by: ξ = C π (cid:15) (cid:15) S q i (cid:87) (cid:87) , T π (cid:48) (cid:15) (cid:15) S t (cid:87) (cid:87) , h : C → T, p : T → X where π and π (cid:48) are proper flat morphisms, h is a morphism over S andfor each geometric point z ∈ S we have ξ z is a genus g stable map relativeto ( λ, x ). Furthermore, we require that in a neighbourhood of a node of C z mapping to a singularity of T z we can choose ´etale-local coordinateson S , C and T with charts of the form Spec R , Spec R [ u, v ] / ( uv − a ) andSpec R [ x, y ] / ( xy − b ) respectively such that the map is of the form x (cid:55)→ αu k and y (cid:55)→ αv k with α and β units.(ii) Morphisms ξ → ξ between two appropriately labelled objects are givenby pairs of cartesian diagrams C π (cid:15) (cid:15) a (cid:47) (cid:47) C π (cid:15) (cid:15) S a (cid:48) (cid:47) (cid:47) S T π (cid:48) (cid:15) (cid:15) b (cid:47) (cid:47) T π (cid:48) (cid:15) (cid:15) S b (cid:48) (cid:47) (cid:47) S that are compatible with the other data (i.e. we have a ◦ q ,i = q ,i ◦ a (cid:48) , b ◦ t = t ◦ b (cid:48) , b ◦ h = h ◦ a and p = p ◦ b ). ( Universal Objects of M g ( X, λ )) . Use the notation M := M g ( X, λ ). There isa smooth Artin stack T which parametrises the degenerated targets and a morphismwhich forgets the map data p = ( p M , p T ) : M → M g,l × T . We have universal curves π : C → M and π T : C T → T along with universal maps h : C → C T and p : C T → X fitting into the following commuting diagram: C h (cid:47) (cid:47) π (cid:15) (cid:15) C T π T (cid:15) (cid:15) p (cid:47) (cid:47) X M p T (cid:47) (cid:47) T There also universal sections defining the marked points given by q i : M → C and t : T → C T . Moreover for convenience we define f := p ◦ h to be the universal map f : C → X . OLIVER LEIGH
Definition 1.1.4 ( Ramification Bundle ) . Using the notation from 1.1.31.1.3, there isa natural bundle defined on the universal curve C which we call the ramificationbundle and denote by R := ω log π ⊗ f ∗ ( ω log X ) ∨ , where we have also denoted ω log π = ω π (cid:0)(cid:80) q i (cid:1) and ω log X = ω X ( x ). For a family ξ ∈ M g ( X, λ ) with f := f ξ we will use the notation R f := R ξ . ( Canonical Ramification Section ) . Using the notation from 1.1.31.1.3, it is shownin [Le1Le1] that (following choices of global sections defining the divisors q i ∈ C , t ∈ C T and x ∈ X ) there is a morphism δ : O C −→ R called the canonical ramification section which has the following properties at eachgeometric point ξ = (cid:0) h : ( C, q i ) −→ ( T, t ) , p : ( T, t ) −→ ( X, x ) (cid:1) in M with f = p ◦ h and δ = δ ξ :Let B ⊂ X be the closed subset containing only x and the nodes of X . If D := f − ( B ) ⊂ C , then:(i) δ restricted to C \ D is the natural morphism O C \ D → ω C \ D ⊗ ( f | C \ D ) ∗ ω ∨ X \ B of [FPFP, Lemma 8].(ii) δ is an isomorphism locally at D .In particular, δ describes the ramification behaviour of f away from the pre-imagesof nodes of X and away from the relative divisor. Theorem 1.1.6 ( Branch Morphism [FPFP]) . There is a morphism of stacks br : M g ( X, λ ) −→ Sym rm X defined at each geometric point in ξ ∈ M g ( X, λ ) with C := C ξ and f := f ξ to be Div (cid:16) Rf ∗ (cid:2) O C δ ξ −→ R ξ (cid:3)(cid:17) where Div is Mumford’s divisor associated to the determinant of a perfect complexwhose homology is not supported on any points of depth 0 [MFKMFK, § .On the substack M g ( X, λ ) the morphism br takes a point ξ to the branch divisorof f ξ minus the sub-divisor supported at x ∈ X . Remark 1.1.7.
The construction of br in [FPFP] is given for the space of absolutestable maps M g ( X, d ). However the proofs all still work when applied to case of M g ( X, λ ) using the canonical ramification section δ of 1.1.51.1.5. ( A Natural C ∗ -action on P ) . We define (for now and the rest of this article)an action of C ∗ on P , by identifying P := P ( C ) and letting C ∗ act on C withweights 0 ,
1. Explicitly, for c ∈ C ∗ and ( x , x ) ∈ C we define the action by c · ( x , x ) = ( x , cx ) . (5)This canonically induces a C ∗ -action on P which we use throughout the rest ofthis article. The C ∗ -fixed points on P of this action are identified as0 := [0 : 1] and ∞ := [1 : 0] . The weights of the tangent spaces to P at these points are − -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 7 ( A Natural C ∗ -action on M g ( P , λ )) . We define (for now and the rest of thisarticle) an action of C ∗ on M g ( P , λ ) by acting on the image of the map c · [ f ] = [ c · f ]using the C ∗ -action on P . More precisely, the action is defined by the morphismof stacks C ∗ × M g ( P , λ ) −→ M g ( P , λ )which maps a family over a scheme S S a (cid:15) (cid:15) C ∗ , ξ = S a (cid:15) (cid:15) C ∗ , C π (cid:15) (cid:15) S q i (cid:87) (cid:87) , T π (cid:48) (cid:15) (cid:15) S t (cid:87) (cid:87) , h : C → T, p : T → P to the family of relative stable maps C π (cid:15) (cid:15) S q i (cid:87) (cid:87) , T π (cid:48) (cid:15) (cid:15) S t (cid:87) (cid:87) , h : C → T, (cid:0) ( a ◦ π (cid:48) ) · p (cid:1) : T → P . ( Fixed Loci of the C ∗ -action on M g ( P , λ )) . Following the methods of[KoKo, GPGP, GV2GV2] we identify the fixed locus as containing maps f = p ◦ h : C → P which are ´etale everywhere except possibly above 0 and ∞ . This means that if B ⊆ C is an irreducible component of C where f | B is non-constant of degree d ,then B ∼ = P and f | B is of the form [ x : x ] (cid:55)→ [ x d : x d ].Moreover, the stack-theoretic image of M C ∗ under br is reduced and equal to rm +1of points given by[( rm − n ) · (0) + n · ( ∞ )] ∈ Sym rm P , for n = 0 , . . . , rm . Simple Fixed Loci M g ( P , λ )] As observed in [FPFP, GV2GV2] there is a uniqueconnected component of M g ( P , λ ) C ∗ where the branch divisor is supported onlyat 0 ∈ P . This is also the unique connected component of M g ( P , λ ) C ∗ where thetarget is not degenerated, hence it is called the simple locus in [GV2GV2]. We denoteit by G ⊂ M g ( P , λ ) C ∗ . A geometric point in G is a morphism f with domain curve that is the union of(i) A single genus g stable curve, which we denote C v , with f | C v : C v → { } ;(ii) l copies of P denoted C i where C i ∩ C v is a single node and C i ∩ C j = ∅ for i (cid:54) = j . Also, f | C i : C i → P is a C ∗ -fixed Galois cover of degree λ i .We call the nodes arising from an intersection between C v and a C i flag nodes .It is shown in [GPGP, GV2GV2] that G is smooth and there is an isomorphism G ∼ = M g,l × P λ × · · · × P λ l (6)where M g,l is the moduli space of stable curves and P d ∼ = B µ d parametrises C ∗ -fixed Galois covers of degree d . Remark 1.1.12.
The fixed locus studied in [GPGP] was for the moduli space of ab-solute stable maps M g ( P , d ) and their version of the isomorphism (66) has differentautomorphisms. In this article we follow [GV2GV2] by using the moduli space of rel-ative stable maps M g ( P , λ ) which has labeled (i.e. ordered) fixed points in thepre-image of ∞ ∈ P . OLIVER LEIGH ( Normalisation of the Simple Locus along the Flag Nodes ) . The simple fixedlocus G of M has universal curve C G which can be partially normalised along theuniversal flag nodes to give n G : (cid:101) C G → C G . Moreover, (cid:101) C G can be decomposed into adisjoint union of closed sub-stacks: (cid:101) C G = C G ,v (cid:116) l (cid:71) i =1 C G ,i . These have geometric points ( ζ, s ) such that ζ ∈ G as described in 1.1.111.1.11 and, usingthe same notation, if ( ζ, s ) ∈ C G ,v then s ∈ C v , otherwise if ( ζ, s ) ∈ C G ,i then s ∈ C i .Moreover, using the isomorphism (66) we have the following cartesian diagrams: C G ,v (cid:15) (cid:15) (cid:47) (cid:47) C M g,l (cid:15) (cid:15) G (cid:47) (cid:47) M g,l C G ,i (cid:15) (cid:15) (cid:47) (cid:47) C P λl (cid:15) (cid:15) (cid:47) (cid:47) P (cid:15) (cid:15) G (cid:47) (cid:47) P λ l (cid:47) (cid:47) • (7)1.2. r -Twisted Curves.Definition 1.2.1 ( The Moduli Space of r -Twisted Prestable Curves ) . The modulistack of l -marked genus g r -twisted prestable curves is denoted by M rg,l . It is thegroupoid with:(i) Objects over a scheme S given by: ξ = C π (cid:15) (cid:15) S , CS s i (cid:79) (cid:79) i ∈{ ,...,l } where(a) π is a proper flat morphism from a tame stack to a scheme,(b) each s i is a section of π that maps to the smooth locus of C ,(c) the fibres of π are purely one dimensional with at worst nodal singu-larities,(d) the smooth locus C sm is an algebraic space,(e) the coarse space π : C → S with sections s i is a genus g , l -pointedprestable curve (cid:0) C, π : C → S, ( s i : S → C ) i ∈{ ,...,l } (cid:1) (f) the local picture at the nodes is given by [ U/µ r ] → T , where • T = Spec A , U = Spec A [ z, w ] / ( zw − t ) for some t ∈ A , and theaction of µ r is given by ( z, w ) (cid:55)→ ( ξ r z, ξ − r w ).(ii) Morphisms ξ → ξ between two appropriately labelled objects are givenby an equivalence class of cartesian diagrams C π (cid:15) (cid:15) a (cid:47) (cid:47) C π (cid:15) (cid:15) S a (cid:48) (cid:47) (cid:47) S that are compatible with the other data (i.e. a ◦ s ,i = s ,i ◦ a (cid:48) ) and whereequivalence is given by base-preserving natural transformations. Remark 1.2.2.
The fact that taking equivalence classes up to base-preservingnatural transformations gives a well-defined 1-category is due to [AVAV, Prop. 4.2.2]. -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 9
Theorem 1.2.3 ( Smoothness of M rg,l [AJAJ, ACVACV, OO, ChCh] ) . The stack M rg,l is asmooth proper Artin stack of dimension g − l . Remark 1.2.4.
If the twisted curves parametrised in M rg,l are also required to havestable coarse space and if 3 g − l ≥
0, the space becomes a Deligne-Mumford stack. ( Forgetting Twisted Structure ) . There is a natural morphism of stacks M rg,l −→ M g,l which forgets the r -Twisted structure. The automorphism group of a r -twistedcurve which fixes the underlying coarse curve is µ ⊕ nr where n is number of nodes[ACVACV, Prop. 7.1.1]. So for r (cid:54) = 1 the moduli space M rg,l has stacky structure that M g,l does not and the morphism M rg,l → M g,l is not an isomorphism. However, itis flat and surjective of degree 1. ( Universal Curves of r -Twisted Curves ) . Unlike M g,l , the stack M rg,l has twouniversal curves:(a) A universal r -twisted curve π M r : C M r → M rg,l which is of Deligne-Mumford type and where C M r parametrises objects ξ = C π (cid:15) (cid:15) Θ (cid:63) (cid:95) ϑ (cid:111) (cid:111) γ Θ (cid:125) (cid:125) S , CS s i (cid:79) (cid:79) i ∈{ ,...,l } where ( π, s i ) is an object in M rg,l , γ Θ is an ´etale gerbe and ϑ is a closedsub-stack.(b) A universal coarse curve π M r : C M r → M rg,l which is representable andwhere C M r parametrises objects ( π, s i ) in M rg,l as well as a section of theassociated coarse morphism π .1.3. r -Twisted Curves with Roots of Line Bundles.Definition 1.3.1 ( The Moduli Space of r -Twisted Prestable Curves with Roots ofLine Bundles ) . Let b ∈ r Z and denote by D /r,bg,l the moduli stack which has(i) Objects over a scheme S given by: ξ = (cid:16) ζ, F, L, e : L r ∼ → F (cid:17) where(a) ζ = (cid:0) π : C → S, s i : S → C (cid:1) is a family over S in M rg,l ,(b) F is a π -relative line bundle on C of degree b ,(c) L is a π -relative line bundle on C of degree br and(d) e : L r ∼ −→ F is an isomorphism.(ii) Morphisms ξ → ξ between two appropriately labelled objects are givenby triples (cid:16) a : ζ → ζ , b (cid:48) : F ∼ → a ∗ F , b : L ∼ → a ∗ L (cid:17) where a is a morphism in M rg,l , and where b (cid:48) and b are isomorphisms com-patible with the other data (i.e. if φ : a ∗ ( L r ) ∼ → ( a ∗ L ) r is the canonicalisomorphism then ( a ∗ e ) ◦ φ ◦ ( b ⊗ r ) = b (cid:48) ◦ e .) ( Partial-Normalisation at Separating Nodes [ChZChZ, § . Let ξ ∈ D /r,bg,l bea geometric point given by ξ = (cid:16) C, s i , F, L, e : L r ∼ → F (cid:17) such that C has a connecting node and C (cid:116) C is the partial normalisation at thatnode. Also, let ι i : C i (cid:44) → C be the inclusions and (cid:98) γ i : C i → (cid:98) C i be the morphismslocally forgetting the r -orbifold structure at the connecting node. Lastly, let z i ∈ (cid:98) C i correspond to the pre-images of the nodes.Then for i = 1 , (cid:98) γ i ∗ ι ∗ i L and (cid:98) γ i ∗ ι ∗ i F are locally free on (cid:98) C i and there isa unique pair b , b ∈ { , . . . , r − } such that b + b ≡ r ) with (cid:16)(cid:98) γ ∗ ι ∗ L (cid:17) r ∼ = (cid:98) γ ∗ ι ∗ F ( − b z ) and (cid:16)(cid:98) γ ∗ ι ∗ L (cid:17) r ∼ = (cid:98) γ ∗ ι ∗ F ( − b z ) . Remark 1.3.3.
The construction of 1.3.21.3.2 also works on families with a separatingnode defined by an ´etale gerbe.1.4.
Stable Maps with Divisible Ramification.Definition 1.4.1 ( Moduli Space of r -Stable Maps with Roots of Ramification ) . Recall rm = 2 g − l ( λ ) + | λ | (1 − g X ) and consider the natural morphisms(i) M g ( X, λ ) → D ,rmg,l defined by taking a family of relative stable maps ξ with π : C → S and q i : S → C to ( π, q i , R ξ ).(ii) D /r,rm g,l → D ,rm g,l defined by forgetting the r -twisted the r -th root structure.Using these morphisms, define the moduli stack M rg ( P , λ ) by the following carte-sian diagram: M rg ( P , λ ) (cid:47) (cid:47) (cid:15) (cid:15) M g ( P , λ ) (cid:15) (cid:15) D r ,rmg,l (cid:47) (cid:47) D ,rmg,l A geometric point in M rg ( P , µ ) is of the form ξ = (cid:16) ( C, q i ) , ( T, t ) , h : C → T, p : T → P , L, e : L r ∼ → R f (cid:17) where f := g ◦ h and R f := ω log C ⊗ ( p ◦ h ) ∗ ( ω log P ) ∨ and(i) (cid:16) ( C, q i ) , ( T, t ) , h : C → T, p : T → P (cid:17) ∈ M g ( P , µ ),(ii) (cid:16) ( C, q i ) , R f , L, e : L r ∼ → R f (cid:17) ∈ D /r,rmg,l . ( Simplifying Notation for Main Spaces ) . For the rest of this article we willuse the following simplifying notation for the key spaces:(i) M := M g ( X, λ ) with universal curve π : C → M .(ii) M r := M rg ( X, λ ) with universal ( r -twisted) curve π r : C r → M r .(iii) M := M g,l and M r := M rg,l .We will also denote by R and δ : O C → R the pullback to C of the correspondingbundle and morphism on C M . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 11 Definition 1.4.3 ( Stacks Parametrising Sections of Sheaves ) . Let F be a linebundle on C r . Then we have the following associated stacks which will be used inthis article:(i) Tot π r ∗ F := Spec M r (cid:16) Sym • R π r ∗ ( F ∨ ⊗ ω π r ) (cid:17) is the moduli stack with:(a) Objects (cid:16) ξ, σ : O C ξ −→ F ξ (cid:17) where ξ is an object of M r and F ξ is F pulled back to C ξ .(b) Morphisms ( ξ , σ ) → ( ξ , σ ) are morphisms a : ξ → ξ such that if φ : a ∗ F ξ ∼ → F ξ is the induced isomorphism then a ∗ σ = φ ◦ σ .(ii) Tot F := Spec C r (cid:16) Sym • F ∨ (cid:17) is the moduli stack with:(a) Objects: (cid:16) ξ, ϑ : Θ (cid:44) → C, γ Θ : Θ → S, ζ : O S −→ γ Θ ∗ ϑ ∗ F ξ (cid:17) where ξ is an object of M r over S along with the extra data foran object in C r of a closed sub-stack ϑ : Θ (cid:44) → C and ´etale gerbe γ Θ : Θ → S (see 1.2.61.2.6 for more details).(b) Morphisms ( ξ , ϑ , γ Θ , ζ ) → ( ξ , ϑ , γ Θ , ζ ) are morphisms from C r b : ( ξ , ϑ , γ Θ ) → ( ξ , ϑ , γ Θ ) such that if the induced isomorphismis ψ : b ∗ ( γ Θ ∗ ϑ ∗ F ξ ) ∼ → γ Θ ∗ ϑ ∗ F ξ then b ∗ ζ = ψ ◦ ζ . Remark 1.4.4.
Note that while we use the notation
Tot π ∗ F , it is often the casethat π ∗ F is not locally free. This space is called an abelian cone in [BFBF]. ( r -th Power Maps over M r ) . Natural examples of definition 1.4.31.4.3 arise whenthe line-bundle F is the universal r -root L and its r -th tensor power L r .Moreover, in these cases, there are natural morphisms which arise from taking the r -th power of the section. We call these morphisms the r -th power maps and denotethe related morphism by the following commutative diagrams: Tot π ∗ L τ (cid:47) (cid:47) β (cid:15) (cid:15) Tot π ∗ L r α (cid:15) (cid:15) M r M r Tot L ˇ τ (cid:47) (cid:47) ˇ β (cid:15) (cid:15) Tot L r ˇ α (cid:15) (cid:15) C r C r (8) Remark 1.4.6.
Comparing the two r -th power maps we see that that ˇ τ is a mapbetween total spaces of line bundles on C r . In fact, it is an r -fold cover ramified atthe zero section. However, in general, Tot π ∗ L is not the total space of a bundleand hence τ is more complicated than ˇ τ . Definition 1.4.7 ( Moduli Space of Stable Maps with Divisible Ramification ) . Thecanonical ramification section δ : O C → R and the universal r th root e : L r ∼ → R define a natural inclusion i (cid:48) : M r −→ Tot π ∗ L r ξ (cid:55)−→ (cid:0) ξ, e − ξ ( δ ξ ) (cid:1) . The moduli space of stable maps with divisible ramification M /r (also denoted M /r g ( P , λ )) is defined by following cartesian diagram which also defines ν and i : M r i (cid:47) (cid:47) ν (cid:15) (cid:15) Tot π ∗ L τ (cid:15) (cid:15) M r i (cid:48) (cid:47) (cid:47) Tot π ∗ L r (9) ( Universal Objects of M /r ) . The universal objects of M r pullback via themorphism ν : M /r → M r to give universal objects on M /r . For example wehave a universal ( r -twisted) curve π /r : C /r → M /r . The universal r th rootsection σ pulls back from Tot π ∗ L . Theorem 1.4.9 ( Branch-Type morphism for M /r [Le1Le1]) . There is a morphismof stacks br : M r −→ Sym m X defined at each geometric point in ξ ∈ M /r with C := C /rξ and f := f /rξ to be Div (cid:16) Rf ∗ (cid:2) O C σ ξ −→ L ξ (cid:3)(cid:17) . It commutes with the branch morphism of [FPFP] via the diagram M r br (cid:47) (cid:47) (cid:15) (cid:15) Sym m X ∆ (cid:15) (cid:15) M br (cid:47) (cid:47) Sym rm X where ∆ is defined by (cid:80) i x i (cid:55)→ (cid:80) i rx i . C ∗ -Action on Stable Maps with Divisible Ramification Notation Conventions for Main Spaces : We will use the following simplifyingnotation for the key spaces:(i) M := M g ( X, λ ) from definition 1.1.21.1.2 with π : C → M the universal curve.(ii) M r := M rg ( X, λ ) from definition 1.4.11.4.1 with π r : C r → M r the universal( r -twisted) curve.(iii) M /r := M /r g ( X, λ ) from definition 1.4.71.4.7 with π /r : C /r → M /r theuniversal ( r -twisted) curve.(iv) M := M g,l and M r := M rg,l from defintion 1.2.11.2.1.The natural forgetful morphisms are denoted by M r ν (cid:47) (cid:47) M r υ (cid:47) (cid:47) M . We will also denote by R , L and δ : O C → R the bundles and morphism on C /r pulled back from C r . Lastly, σ : O C → L is the morphism pulled back from Tot π ∗ L . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 13 Natural C ∗ -Actions and Equivariant Morphisms.2.1.1 ( C ∗ -Action on M r ) . For a ∈ C ∗ there is an isomorphism m a : P → P defining multiplication by a and for any morphism f : C → P we have a · f = m a ◦ f .This gives rise to canonical isomorphisms f ∗ ( ω log P ) ∨ ∼ ←− f ∗ m ∗ a ( ω log P ) ∨ ∼ −→ ( a · f ) ∗ ( ω log P ) ∨ which (using the notation from 1.1.41.1.4) give an isomorphism which we denote byΦ a : R f ∼ −→ R ( a · f ) . We use this canonical morphism to define a natural C ∗ -action on M r , given by themorphism of stacks C ∗ × M r −→ M r that maps a moduli point (cid:16) a, (cid:16) ( C, q i ) , ( T, t ) , h : C → T, p : T → P , L, e : L r ∼ → R f (cid:17)(cid:17) to the moduli point (cid:16) ( C, q i ) , ( T, t ) , h, a · p, L, Φ a ◦ e (cid:17) . This can be defined on families in the same way as in 1.1.91.1.9.
Remark 2.1.2.
This action has the property that the natural forgetful morphism M r → M is C ∗ -equivariant. ( C ∗ -Action on M r and Related Spaces ) . Since the C ∗ -action defined in 2.1.12.1.1does not affect the bundle L , it extends immediately to the the spaces M /r , Tot π ∗ L , Tot π ∗ L r and their universal curves by leaving the extra data unaffected.Moreover, we give the spaces M , M r and T the trivial action.The natural inclusions i (cid:48) : M r (cid:44) → Tot π ∗ L r and i : M /r (cid:44) → Tot π ∗ L appearing indefinition 1.4.71.4.7 are equivariant since the the following diagram is commutative: R f Φ a (cid:15) (cid:15) O C δ f (cid:54) (cid:54) δ ( a · f ) (cid:40) (cid:40) L re (cid:103) (cid:103) Φ a ◦ e (cid:119) (cid:119) R ( a · f ) Basic Properties of the Fixed Locus.2.2.1 ( The Simple and Non-Simple Fixed Loci ) . By the universal property of thefixed locus the commuting diagram from theorem 1.4.91.4.9 restricts to the commutingdiagram ( M r ) C ∗ (cid:101) br (cid:47) (cid:47) (cid:15) (cid:15) Sym m X ∆ (cid:15) (cid:15) M C ∗ (cid:102) br (cid:47) (cid:47) Sym rm X where (cid:102) br and (cid:102) br are the respective restrictions of br and br to the fixed loci.We know from 1.1.101.1.10 that the stack-theoretic image of M C ∗ under br is reducedand equal to a finite number of points. Since ∆ is a closed immersion we must also have that the image of ( M /r ) C ∗ under br is reduced and equal to a finite numberof points. We can identify the possible points via Im( (cid:102) br ) ∩ Im(∆) to be h n := [( m − n ) · (0) + k · ( ∞ )] ∈ Sym m P for n ∈ { , , . . . , m } , giving a decomposition( M r ) C ∗ = m (cid:71) n =0 (cid:16) ( M r ) C ∗ ∩ br − ( h n ) (cid:17) . Following 1.1.111.1.11 we split this into two cases:(i)
The Simple Fixed Locus : F := ( M r ) C ∗ ∩ br − ( h ). This is the case wherethere is no degeneration at ∞ ∈ P .(ii) The Non-Simple Fixed Loci : (cid:98) F n := ( M r ) C ∗ ∩ br − ( h n ) for n ∈ { , . . . , m } .This is the case where there is degeneration at ∞ ∈ P . ( The Simple Fixed Locus of M r ) . By the universal property of the fixed locusthe forgetful morphism υ : M r → M restricts to a morphism (cid:101) υ : ( M r ) C ∗ → M C ∗ on the fixed loci. Since this is surjective and the stack theoretic image of ( M ) C ∗ under br is reduced, we must have the following equality of stack theoretic imagesIm( br ◦ υ ) = Im( br ) ⊂ Sym rm P . Hence we have a decomposition of ( M r ) C ∗ similar to that described in 2.2.12.2.1 andwe define the simple fixed locus G r ⊂ ( M r ) C ∗ to be G r := ( M r ) C ∗ ∩ ( br ◦ υ ) − (cid:0) [ rm · (0)] (cid:1) . Lemma 2.2.3 ( The Simple Fixed Loci as Pullbacks ) . The simple fixed loci G r and F fit into the following diagram where both squares are cartesian and the verticalarrows are closed immersions: F v (cid:47) (cid:47) j (cid:15) (cid:15) G r u (cid:47) (cid:47) k r (cid:15) (cid:15) G k (cid:15) (cid:15) M r ν (cid:47) (cid:47) M r υ (cid:47) (cid:47) M (10) Proof.
Following 2.2.12.2.1 and 2.2.22.2.2, both F and G r can be constructed using pullbacksof [ rm · (0)] ∈ Sym P under the compositions involving the branch morphism andforgetful morphisms. The pullback construction allows us to construct the desireddiagram. For example we have the following commuting diagram G r (cid:47) (cid:47) (cid:44) (cid:44) (cid:15) (cid:15) G (cid:47) (cid:47) (cid:15) (cid:15) (cid:8) [ rm · (0)] (cid:9) (cid:15) (cid:15) ( M r ) C ∗ (cid:47) (cid:47) M C ∗ (cid:47) (cid:47) Sym P where the right square is cartesian. The dashed arrow arises from the properties ofthe cartesian square. We have a similar diagram for F . Hence we have a diagramof the desired form, but we must show that the squares are cartesian.If we define X := M r × M G , then we claim that G r ⊆ X is a sub-stack. To seethis we consider use the rightmost square of (1010) and combine it with the cartesiansquare defining X : G r (cid:18) (cid:114) k r (cid:36) (cid:36) u (cid:41) (cid:41) (cid:47) (cid:47) X (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) G (cid:127) (cid:95) k (cid:15) (cid:15) M r (cid:47) (cid:47) M -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 15 Hence we must have that G r → X is an immersion. In a similar way we have that F is a sub-stack of Y := M /r × M r G r .We now show that every family in X is fixed under the C ∗ action. Recall the modulispace from definition 1.3.11.3.1 and define the notation D n := D /n,rm g,l where C ∗ actswith the trivial action. Then from definition 1.4.11.4.1 we have a natural equivariantforgetful map M r −→ D r which gives the equivariant isomorphism X ∼ = D r × D G .This shows that X has trivial C ∗ action and hence X ⊆ ( M r ) C ∗ .We can similarly express Y as a cartesian product of spaces with trivial action byusing the total-space stack over D r defined by: Tot D π D ∗ L D := Spec D (cid:16) Sym • R π D ∗ ( L ∨ D ⊗ ω π D ) (cid:17) which contains objects (cid:0) ξ, σ : O ( C D ) ξ −→ ( L D ) ξ (cid:1) where ξ is an object of D r , π D : C D → D r is the universal ( r -twisted) curve and L D is the universal r -throot. (cid:3) Lemma 2.2.4.
The forgetful morphism v : F → G r from lemma 2.2.32.2.3 is ´etale ofdegree r l .Proof. We will show v is ´etale in corollary 3.4.23.4.2. We assume that 3 g − l ≥ g, l ) = (0 ,
1) and (0 ,
2) are simpler and only require minormodifications. We calculate the degree at the fibre of a geometric point ξ ∈ G r determined by the data f : C −→ P , q i , L and e : L r ∼ −→ R f . The pre-image is supported on finite collection determined by the r -th roots σ of δ . Denote by C v ⊂ C be the union of irreducible components of C mapping to0 ∈ P and denote by C i ∼ = P [ r ] the irreducible components labelled by q i where f is non-constant and the image of the r -orbifold point is 0.The map f is constant on C v so we have δ | C v = 0 and hence we must also have σ | C v = 0 as well. Thus the number of possible r -th roots σ of δ is determined bythe number of possibilities for σ | C i .We choose coordinates on C i given by C i \ C v ∼ = Spec C [ y ] and at the r -orbifoldpoint by V := [(Spec C [ z ]) /µ r ] where the action of µ r is given by z (cid:55)→ ξ r z . Then( R f ) | C i has local µ r -equivariant generator z − rλ i at the orbifold point and trivialgenerator on C i \ C v . The restriction δ | C i is then given on these open sets by z rλ i · ( z − rλ i ) and 1 respectively.The r -root bundle L is similarly given by local generators z − λ i and 1, while the r -root section σ | C i is determined by polynomials ζ ( z ) ∈ C [ z ] and ζ ( y ) ∈ C [ y ]compatible with change of coordinates such that ζ r = z rλ i and ζ r = 1. There areexactly r different choices for these, determined by the roots of unity. Hence wehave a total of r l different choices for σ .We must now examine the automorphisms and isomorphisms of objects in F . Wehave automorphisms in F and G r arising from C and f , however these are unaffectedby the morphism v . The other automorphisms of G r form a subgroup µ ⊕ ( l +1) r arisingfrom the isomorphism L ∼ → L defined by r -roots of unity and the r -orbifold nodes(discussed more in 1.2.51.2.5). The automorphisms of ξ arising from L ∼ → L and the nodes where r (cid:45) λ i do notcorrespond to automorphisms of ξ (cid:48) ∈ v − ( ξ ), but rather isomorphisms of objects.However, the automorphisms of ξ arising from nodes where r | λ i correspond to theautomorphisms of (not arising from C or f ). Hence, | v − ( ξ ) | = r l − − (cid:80) i : r | λi andfor ξ (cid:48) ∈ v − ( ξ ) we have | Aut ξ | / | Aut ξ (cid:48) | = r (cid:80) i : r | λi . Thus the degree is r l asdesired. (cid:3) Flag Nodes and Partial Normalisation.2.3.1 ( ´Etale Gerbes for the Flag Nodes in the Simple Locus ) . Consider the case3 g − l ≥
0. Let π G : C G → G and π F : C F → F be the universal ( r -twisted)curves. π G is representable and the flag nodes of G define l different sections G → C G of π G (in the case ( g, l ) = (0 ,
2) there is only one flag node). These pullback viathe map forgetful map C F → C G to define sub-stacks at the flag nodes. There arealso corresponding sections of the universal coarse curve π F : C F → F . This issummarised in the following commuting diagram where the square is cartesian and γ Θ i is an ´etale gerbe: Θ i γ Θ i (cid:15) (cid:15) (cid:31) (cid:127) ϑ i (cid:47) (cid:47) C F γ F (cid:15) (cid:15) F (cid:31) (cid:127) z i (cid:47) (cid:47) C F π F (cid:15) (cid:15) F ( Normalisation along the Flag Nodes of the Simple Locus ) . The simple fixedlocus F of M /r has universal curve C F which can be partially normalised along the´etale gerbes from 2.3.12.3.1 to give n : (cid:101) C → C F . This is the pullback via the forgetfulmap C F → C G of the partial normalisation (cid:101) C G of C G from 1.1.131.1.13. (cid:101) C can also be decomposed into a disjoint union of closed sub-stacks (cid:101) C = C v (cid:116) (cid:70) li =1 C i where C v and C i are defined from 1.1.131.1.13 via the following cartesian diagrams: C v (cid:127) (cid:95) ι v (cid:15) (cid:15) (cid:47) (cid:47) C G ,v (cid:127) (cid:95) (cid:15) (cid:15) C F (cid:47) (cid:47) C G C i (cid:127) (cid:95) ι i (cid:15) (cid:15) (cid:47) (cid:47) C G ,i (cid:127) (cid:95) (cid:15) (cid:15) C F (cid:47) (cid:47) C G (11)Moreover, combining the diagrams of (1111) with the diagrams (77) from 1.1.131.1.13 wehave the following diagrams where the squares are cartesian: C v γ v (cid:47) (cid:47) π v (cid:28) (cid:28) C v π v (cid:15) (cid:15) (cid:47) (cid:47) C G ,v π G ,v (cid:15) (cid:15) (cid:47) (cid:47) C M g,l (cid:15) (cid:15) F (cid:47) (cid:47) G (cid:47) (cid:47) M g,l C i γ i (cid:47) (cid:47) π i (cid:28) (cid:28) C i π i (cid:15) (cid:15) (cid:47) (cid:47) C G ,i π G ,i (cid:15) (cid:15) (cid:47) (cid:47) C P λi π P λi (cid:15) (cid:15) (cid:47) (cid:47) P (cid:15) (cid:15) F (cid:47) (cid:47) G (cid:47) (cid:47) P λ i (cid:47) (cid:47) • (12)Here the marked points of M g,l and 0 ∈ P correspond to the images of z i . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 17 ( Universal Bundles on the Simple Fixed Locus ) . Denote by q i : F → C F , theuniversal sections of π F : C F which correspond to the marked points from the relativestable map condition (see 1.1.31.1.3 for more details). Recall from 1.1.111.1.11 and 2.2.12.2.1 thatthe stable maps in the simple fixed locus F are not degenerated at infinity so wehave a universal relative stable map f F : C F → P . Moreover, we have isomorphisms R ∼ = ω log π F ⊗ f ∗ F (cid:0) ω log P (cid:1) ∨ ∼ = ω π F ⊗ O C F (cid:0)(cid:80) i ( λ i + 1) q i (cid:1) where the last isomorphism has used ω P ∼ = O P (cid:0) − ∞ ) (cid:1) .Letting f F : C F → P be the universal relative stable map from the universal coarsecurve such that f F = f F ◦ γ F . Then we have R ∼ = γ ∗ F (cid:16) ω π F ⊗ O C F (cid:0)(cid:80) i ( λ i + 1) q i (cid:1)(cid:17) . For 3 g − l ≥
0, restricting R to the normalisation components gives line bundles R v := ι ∗ v R ∼ = γ ∗ v ω π v (cid:0)(cid:80) i z i (cid:1) and R i := ι ∗ i R ∼ = γ ∗ i O C i (cid:0) λ i z i (cid:1) . Here we have used the right diagram of (1212) which gives that ω π i is the pullbackof ω P ∼ = O P (cid:0) − (cid:1) . Restricting L , the universal r -th root bundle on F , to thecomponents of the normalisation gives line bundles L v := ι ∗ v L and L i := ι ∗ i L . Lemma 2.3.4.
Let L be the universal r -th root bundle on F and denote the Chernpolynomial by c s ( − ) . Then we have(i) c s (cid:16) R ( γ Θ i ) ∗ ϑ ∗ i L r (cid:17) = 1 and(ii) c s (cid:16) R ( γ Θ i ) ∗ ϑ ∗ i L (cid:17) = (cid:26) if r | λ i ; otherwise.Moreover, R ( γ Θ i ) ∗ ϑ ∗ i L ∼ = 0 if r (cid:45) λ i .Proof. Using 2.3.32.3.3 and the properties that γ F is flat and γ F ∗ is exact gives R ( γ Θ i ) ∗ ϑ ∗ i L r ∼ = z ∗ i R γ F ∗ γ ∗ F (cid:16) ω π F ⊗ O C F (cid:0)(cid:80) i λ i q i (cid:1)(cid:17) ∼ = z ∗ i (cid:16) ω π F ⊗ O C F (cid:0)(cid:80) i λ i q i (cid:1)(cid:17) . The first result now follows from z ∗ i ω π F ∼ = O F and z ∗ i O C F (cid:0)(cid:80) i λ i q i (cid:1) ∼ = O F .For the second result we consider the case where r | λ i . Then locally at the flag nodethere exists a line bundle L i on F such that ϑ ∗ i L ∼ = γ ∗ Θ i L i . Then, R ( γ Θ i ) ∗ ϑ ∗ i L ∼ = L i and we also have that L ri ∼ = R ( γ Θ i ) ∗ ϑ ∗ i L r ∼ = O F . The result for this case followsfrom basic properties of the Chern polynomial.For the case where r (cid:45) λ i we show that R ( γ Θ i ) ∗ ϑ ∗ i L ∼ = 0 by showing this at everygeometric point of F . Indeed, in this case, for a geometric point ξ ∈ F we have( ϑ ∗ i L ) ξ is a trivial line bundle on (Θ i ) ξ ∼ = B µ r which has non-trivial weight for the µ r -action. Hence, there are no invariants and ( R ( γ Θ i ) ∗ ϑ ∗ i L ) ξ ∼ = 0. (cid:3) Forgetting r -Orbifold Structure Flag Nodes.2.4.1 ( Partial-Normalisation at the Flag Nodes ) . We can apply the concept from1.3.21.3.2 to the flag nodes of the simple fixed loci G r and F . For example, consider apoint in G r determined by the data f : C → P , q i , L and e : L r ∼ → ω log C ⊗ f ∗ ( ω log P ) ∨ . We let ι : B (cid:44) → C a sub-curve with a morphism (cid:98) γ : B → (cid:98) B locally forgettingthe r -orbifold structure at any points corresponding to flag nodes. In the cases( g, l ) = (0 ,
2) and 3 g − l ≥ B is an irreducible component where f | B is non-constant then (cid:98) γ ∗ ι ∗ (cid:16) ω log C ⊗ f ∗ ( ω log P ) ∨ (cid:17) ∼ = O P ( λ i ) ∼ = O P (cid:16)(cid:106) ar (cid:107) r + (cid:68) ar (cid:69)(cid:17) . and the r -th root becomes (cid:16)(cid:98) γ ∗ ι ∗ L (cid:17) r ∼ = O P (cid:16)(cid:106) ar (cid:107) r (cid:17) . (ii) If B = f − (0) and if p , . . . , p l ∈ (cid:98) B correspond to flag nodes then (cid:98) γ ∗ ι ∗ (cid:16) ω log C ⊗ f ∗ ( ω log P ) ∨ (cid:17) ∼ = ω (cid:98) B (cid:16)(cid:80) i p i (cid:17) . and the r -th root becomes (cid:16)(cid:98) γ ∗ ι ∗ L (cid:17) r ∼ = ω (cid:98) B (cid:16)(cid:80) i (cid:0) (cid:104) λ i /r (cid:105) − r (cid:1) p i + (cid:80) r | λ i rp i (cid:17) . Remark 2.4.2.
The discussion in 2.4.12.4.1 also holds for families in G r and F (as wasthe case for 1.3.21.3.2) but we have used a geometric point to simplify the exposition. ( Reverse Clutching-Type Morphism at the Flag Nodes ) . Let 3 g − l ≥ reverse remainders to be a := a ( g, λ ) := ( a , . . . , a l ) where a i ∈ { , . . . , r − } is defined by λ i = (cid:22) λ i r (cid:23) r + ( r − − a i ) . For l = l ( λ ) we denote by M /r, a g,l , the moduli space of r -spin curves twisted by a as defined in [ChCh]. That is, the moduli space of r -stable curves with r -th roots of ω C ( − (cid:80) a i q i ).Moreover, we define P /rd to be the moduli space parametrising pairs which con-sist of C ∗ -fixed Galois covers of P of degree d along with an r -th root of O P ( (cid:4) dr (cid:5) r ).Then we have a natural degree 1 morphism considered in [ChZChZ, § G r −→ M r , a g,l × P r λ × · · · × P r λ l which is defined in the following way:(i) Locally normalise the curve at the flag nodes and locally forget the stackstructure there.(ii) The map f is taken to the (local) coarse maps associate to the restrictions.(The map is forgotten on the component where f was constant.)(iii) The r -th root L on C is taken to its pullbacks on the normalised compo-nents and then locally taking the invariant sections at the pre-images ofthe nodes. As described in 2.4.12.4.1. -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 19 Note that this defines a map to M /r, b g,l × P /rλ × · · · × P /rλ l where b = ( b , . . . , b l )is defined by b i = (cid:26) a i , if r (cid:45) λ i ,a i − r = − , if r | λ i . However, we can compose with the natural isomorphism M /r, b g,l ∼ −→ M /r, a g,l whichis defined via the isomorphism L b ∼ = L a (cid:0) (cid:80) r | λ i p i (cid:1) . Corollary 2.4.4 ( Degree of Morphism to M /r, a g,l From Theorem BB ) . Consider themorphism G r → M /r, a g,l ×P /r λ ×· · ·×P /r λ l from 2.4.32.4.3. The morphism b : F → M /r, a g,l defined by the following composition has degree ( λ · · · λ l ) − : F M r , a g,l G r M r , a g,l × P r λ × · · · × P r λ l b v pr Proof.
This follows immediately from 2.4.32.4.3, lemma 2.2.42.2.4 and because the projectionmap pr has degree r − l ( λ · · · λ l ) − . (cid:3) Lemma 2.4.5.
Let a be the vector of reverse remainders from 2.4.32.4.3 and b themorphism from corollary 2.4.42.4.4. Also, let ρ be the universal ( r -twisted) curve of M /r, a g,l and L be the universal r -root.(i) There are isomorphisms (using the notation from lemma 2.2.32.2.3 and (1212)): π v ∗ ω π v ∼ = ( u ◦ v ) ∗ π G ,v ∗ ω π G ,v ∼ = b ∗ ρ ∗ ω ρ . (ii) If L v is the restriction of the r -th root bundle on C F to C v then there is adistinguished triangle b ∗ R ρ ∗ ω ρ (cid:47) (cid:47) R π v ∗ L rv (cid:47) (cid:47) (cid:76) i O F (cid:47) (cid:47) b ∗ R ρ ∗ L [1] . (iii) Using the notation from part (ii)(ii), there is a distinguished triangle b ∗ R ρ ∗ L (cid:47) (cid:47) R π v ∗ L v (cid:47) (cid:47) (cid:76) i : r | λ i O F (cid:47) (cid:47) b ∗ R ρ ∗ L [1] . Proof.
Begin by considering the following cartesian diagrams where ρ is the univer-sal coarse curve of M /r, a g,l and (cid:98) π v is defined from π v after forgetting the r -twistedstructure at the flag nodes: C v π v (cid:15) (cid:15) (cid:47) (cid:47) C M r , a g,l ρ (cid:15) (cid:15) F b (cid:47) (cid:47) M r , a g,l (cid:98) C v (cid:98) π v (cid:15) (cid:15) (cid:98) b (cid:47) (cid:47) C M r , a g,l ρ (cid:15) (cid:15) F b (cid:47) (cid:47) M r , a g,l The left diagram gives the following isomorphism π v ∗ ω π v ∼ = b ∗ ρ ∗ ω ρ . Moreover,since there is no r -twisted structure at smooth points we have the further isomor-phism b ∗ ρ ∗ ω ρ ∼ = b ∗ ρ ∗ ω ρ . The proof of part (i)(i) is then competed by observing thatthe isomorphism π v ∗ ω π v ∼ = ( u ◦ v ) ∗ π G ,v ∗ ω π G ,v arises from the middle square of theleft diagram from (1212). For part (ii)(ii) the discussion in 2.3.32.3.3 shows that γ v ∗ L rv ∼ = ω log π v where the log super-script refers to the (un-twisted) markings corresponding to the flag nodes. Since R j π v ∗ ω log π v vanishes for j (cid:54) = 0 the result for part (ii)(ii) follows from part (i)(i).For part (iii)(iii) we consider the right cartesian diagram given above. Let (cid:98) γ : C v → (cid:98) C v be the universal morphism forgetting the r -twisted structure at the flag nodes. Fromthe discussion in 2.4.12.4.1 and at the end of 2.4.32.4.3 we have that there is an isomorphism (cid:98) γ ∗ L v ∼ = (cid:98) b ∗ L ⊗ O C v (cid:0)(cid:80) i ; r | λ i r z i (cid:17) . The result for part (iii)(iii) now follows immediately. (cid:3)
Lemma 2.4.6.
The direct images R π i ∗ L i and R π i ∗ L ri are trivial bundles.Proof. Let G r −→ M /r, a g,l × P /r λ × · · · × P /r λ l be the reverse clutching morphism from 2.4.32.4.3 and define d : F → P /r λ i to be the following composition F ν (cid:47) (cid:47) G r (cid:47) (cid:47) M r , a g,l × P r λ × · · · × P r λ l pr i +1 (cid:47) (cid:47) P r λ i . Let π i be the coarse space morphism associated to π i (i.e. forget the r -twistedstructure). Also, let π P : C P → P /r λ i be the universal curve and L P be the universal r -th root bundle for P /r λ i . Then we can form the following cartesian diagram: C i π i (cid:15) (cid:15) d (cid:48) (cid:47) (cid:47) C P π P (cid:15) (cid:15) F d (cid:47) (cid:47) P r λ i If γ i : C i → C i is the morphism forgetting the twisted structure, the above cartesiandiagram gives an isomorphisms of the form d (cid:48) ∗ L P ∼ = γ i ∗ L i and d (cid:48) ∗ L r P ∼ = γ i ∗ L ri .Hence, we have isomorphisms R π i ∗ L i ∼ = d ∗ R π P ∗ L P and R π i ∗ L ri ∼ = d ∗ R π P ∗ L r P . The natural forgetful morphism P /r λ i → P λ i is the ´etale gerbe B ( µ r × µ λ i ) → B µ λ i .Combining this with the right square of the right diagram of (77) gives the followingdiagram where both squares are cartesian: C P π P (cid:15) (cid:15) (cid:47) (cid:47) d (cid:48)(cid:48) (cid:44) (cid:44) C P λi (cid:15) (cid:15) (cid:47) (cid:47) P (cid:15) (cid:15) P r λ i (cid:47) (cid:47) P λ i (cid:47) (cid:47) • This shows that C P is a quotient stack C P ∼ = (cid:2) P / ( µ r × µ λ i ) (cid:3) where ( µ r × µ λ i ) actstrivially on P .The bundle L P can now be expressed as the tensor product d (cid:48)(cid:48) ∗ O P ( (cid:98) λ i /r (cid:99) ) ⊗ π ∗P U where U is line bundle on P /r λ i ∼ = B ( µ r × µ λ i ) given by a trivial bundle where( µ r × µ λ i ) acts with weight (1 , R π P ∗ L P is also trivial where ( µ r × µ λ i )acts with weight (1 , σ : O C F → L is non-zero on fibres andhence rigidifies the µ r -action on F . Thus d ∗ U ∼ = O F and d ∗ R π P ∗ L P is trivial.The bundle L r P is given by d (cid:48)(cid:48) ∗ O P ( λ i ) with trivial ( µ r × µ λ i )-action, so R π P ∗ L r P is a trivial bundle with trivial ( µ r × µ λ i )-action. (cid:3) -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 21 Proving Theorem BB: Localisation Formula
In order to be consistent with the exposition of [GPGP], we will use the (derived) dualdescription of perfect obstruction theories for this section only. In this language,an equivariant perfect obstruction theory for a Deligne-Mumford stack X is a mor-phism T e X → F e X in D e b ( X ) satisfying properties dual to those given in [BFBF]. Areview of equivariant perfect obstruction theories is given in section 5.15.1.3.1. Virtual Localisation Formula.Theorem 3.1.1 ( Virtual Localisation Formula [GPGP, CKLCKL]) . Let X be a Deligne-Mumford stack with a C ∗ -action and let T e X → F e X be an equivariant perfect ob-struction theory. Then [ X ] vir = (cid:88) n ( ζ n ) ∗ [ X n ] vir e (N vir n ) in A C ∗ ∗ ( X ) ⊗ Q [ t, t ] where:(i) t is the generator of the C ∗ -equivariant ring of a point.(ii) The sum ranges over the connected components of the C ∗ -fixed locus X C ∗ with inclusions ζ n : X n → X .(iii) [ X n ] vir arises from the C ∗ -fixed part ( L ζ ∗ n F e X ) fix which is a perfect obstruc-tion theory for X n .(iv) N vir n is the C ∗ -moving part ( L ζ ∗ n F e X ) mov of the perfect obstruction theoryand e ( − ) is the equivariant Euler class. Remark 3.1.2.
Theorem 3.1.13.1.1 was originally due to [GPGP], however it requiredthe existence of a C ∗ -equivariant embedding of X into a smooth Deligne-Mumfordstack. The requirement of this condition was removed in [CKLCKL] and the localisationmethod was extended to include other, more general, concepts. ( Perfect Obstruction Theory ) . It will be shown in corollary 5.4.45.4.4 that thereis an equivariant perfect obstruction theory T e M /r → F e M /r for M /r which fitsinto the distinguished triangle F e ν (cid:47) (cid:47) F e M /r (cid:47) (cid:47) L ν ∗ F e M r (cid:47) (cid:47) F e ν [1]and is compatible with the distinguished triangle of equivariant tangent complexesarising from the morphism ν : M /r → M r . Here, F e M r is the perfect obstructiontheory for M r to be given in 5.2.25.2.2. F e ν is the equivariant perfect relative obstructiontheory for ν first constructed in [Le1Le1]; it is described further in 5.3.35.3.3 and 5.3.45.3.4. ( Strategy for Localisation Calculation ) . The strategy for computing the lo-calisation formula will be to use the distinguished triangle from 3.1.33.1.3 and calculatethe fixed and moving parts on each term. Indeed, pulling back via j : F (cid:44) → M /r and taking either the fixed or the moving part gives another distinguished triangle (cid:0) L j ∗ F e ν (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ F e M /r (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ L ν ∗ F e M r (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ F e ν (cid:1) f . m . [1] . This shows that we can compute ( L j ∗ E e M /r ) f . m . by computing ( L j ∗ L ν ∗ E e M r ) f . m . and ( L j ∗ E e ν ) f . m . individually. We will compute ( L j ∗ L ν ∗ E e M r ) f . m . in section 3.23.2 and( L j ∗ E e ν ) f . m . in section 3.33.3. Analysis of the Contributions From M r .Lemma 3.2.1. The fixed and moving parts of L j ∗ L ν ∗ F e M r have:(i) (cid:0) L j ∗ L ν ∗ F e M r ) fix ∼ = F for some locally free sheaf F of rank g − l .(ii) The Euler class of the moving part is the following:(a) If g = 0 and l = 1 then e (cid:16)(cid:0) L j ∗ L ν ∗ F e M r (cid:1) mov (cid:17) = t λ − λ ! λ λ − (b) If g = 0 and l = 2 then e (cid:16)(cid:0) L j ∗ L ν ∗ F e M r (cid:1) mov (cid:17) = t λ + λ r λ ! λ λ +11 λ ! λ λ +12 ( λ + λ ) (c) If g − l ≥ then e (cid:16)(cid:0) L j ∗ L ν ∗ F e M (cid:1) mov (cid:17) = r − l t − g + | λ | c t (cid:0) − b ∗ ρ ∗ ω ρ (cid:1) l (cid:89) i =1 λ i ! λ λ i +1 i (cid:18) − λ i t ψ i (cid:19) where c s is the Chern polynomial, b : F −→ M /r, a g,l ( λ ) is the morphismfrom corollary 2.4.42.4.4 and ρ is the universal curve for M /r, a g,l ( λ ) . Remark 3.2.2.
This calculation is essentially the same as in [GPGP, §
4] with changesarising from the relative condition described in [GV2GV2, § r -twisted) flag nodebeing an r -fold cover of the deformation space of the associated coarse flag node. Proof.
We prove the case of 3 g − l ≥ q : M r → M r × T that forgets alldata but the source and target families. This gives rise to the following distinguishedtriangle of perfect obstruction theories whose dual is discussed more in 5.2.25.2.2 F eq (cid:47) (cid:47) F e M r (cid:47) (cid:47) q ∗ T e M r ×T (cid:47) (cid:47) F eq [1] . Here F eq is the pullback of the perfect relative obstruction theory originally givenby Behrend in [BB] and extend to the relative stable maps case by [GV2GV2].In the simple fixed locus the target curves are not degenerated. Hence, there arenone of the complications which arise from admissibility conditions discussed in[GV2GV2, § q ∗ T e M r ×T ∼ = R π F ∗ R H om (cid:0) Ω π F ( (cid:80) i q i ) , O C F (cid:1) . For each flag node we have an ´etale gerbe γ Θ i : Θ → F which defines a sub-stackcorresponding to the r -twisted node ϑ i : Θ i (cid:44) → C F and corresponding sub-stacks ofthe normalised componentsˇ ϑ i : Θ i (cid:44) → C v and ˆ ϑ i : Θ i (cid:44) → C i . There is the following exact sequence (which is the C ∗ -equivariant and r -twistedversion of the clutching morphism from [KnKn, Thm. 3.5]):0 (cid:47) (cid:47) (cid:76) i ϑ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1) (cid:47) (cid:47) Ω e π F ( (cid:80) i q i ) (cid:47) (cid:47) n ∗ Ω e (cid:101) π ( (cid:80) i ˆ q i ) (cid:47) (cid:47) . Note that we have also tensored by O C F ( (cid:80) i q i ) and used the notation ˆ q i : F → (cid:101) C for the morphisms corresponding to q i : F → C F . We can now analyse the fixedand moving parts of q ∗ L e M r ×T by applying the functor R π F ∗ R H om ( − , O C F ) to thisexact sequence and analysing the result. -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 23 (i) Contribution from ϑ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1) :Considering this term using Serre-Grothendieck duality for twisted curves gives R π F ∗ R H om (cid:16) ϑ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1) , O C F (cid:17) ∼ = R π F ∗ R H om (cid:16) ϑ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1) ⊗ ω π F , ω π F [1] (cid:17) [ − ∼ = (cid:104) γ Θ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1)(cid:105) ∨ [ − . Denote by E := ˇ ϑ ∗ i Ω π v ⊗ ˆ ϑ ∗ i Ω π i the line bundle where the C ∗ -equivariant struc-ture is forgotten. We claim that γ Θ i ∗ E is a line bundle on F . To see this notethat γ Θ i is relative dimension 0 so we only need to show that h has constantrank 1 for all fibres. For a geometric point • (cid:44) → F the local picture of C F at theflag node is given by V := [ U/µ r ], for U = Spec C [ z, w ] /zw with action of µ r given by ( z, w ) (cid:55)→ ( ξ r z, ξ − r w ). Then the local picture of E| • has local generator dz ⊗ dw which is µ r invariant so h has constant rank 1.Since γ Θ i ∗ E is a line bundle on the C ∗ fixed locus we have that γ Θ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1) ∼ = V c ⊗ γ Θ i ∗ E where V c is a trivial bundle on F with C ∗ -weight c . We can calculate the weight c by considering the geometric point • (cid:44) → F . The equivariant bundle has localgenerator dz ⊗ dw giving that C ∗ acts with weight ( − rλ i ) − . For ( g, l ) = (0 , g, l ) = (0 ,
2) the weight is ( − rλ + rλ ) − .Hence this contribution has no fixed part and taking the Euler class gives e (cid:16)(cid:2) γ Θ i ∗ (cid:0) ˇ ϑ ∗ i Ω e π v ⊗ ˆ ϑ ∗ i Ω e π i (cid:1)(cid:3) ∨ [ − (cid:17) = e (cid:0) V c ⊗ γ Θ i ∗ E (cid:1) = e (cid:0) γ Θ i ∗ E (cid:1) − trλ i . To compute e (cid:0) γ Θ i ∗ E (cid:1) we observe that the local generator of the line bundle E ∼ = ˇ ϑ ∗ i O C v ( − ˇ ϑ i ) ⊗ ˆ ϑ ∗ i O C i ( − ˆ ϑ i ) is µ r invariant which gives E ∼ = γ Θ i ∗ γ Θ i ∗ E . Theprojection formula now gives( γ Θ i ∗ E ) ⊗ r ∼ = γ Θ i ∗ ( E ⊗ r ) ∼ = ˇ ϑ ∗ i O C v ( − r ˇ ϑ i ) ⊗ ˆ ϑ ∗ i O C i ( − r ˆ ϑ i ) ∼ = ˇ z ∗ i Ω π v ⊗ ˆ z ∗ i Ω π i where we have used the notation ˇ z i : F (cid:44) → C v and ˆ z i : F (cid:44) → C i for the morphismscorresponding to z i : F (cid:44) → C F . Since ˆ z ∗ i Ω π i is trivial as a bundle (as is the casein [GPGP, § e (cid:0) γ Θ i ∗ E (cid:1) = r ψ i . This vanishes for ( g, l ) = (0 , , (0 , Contribution from n ∗ Ω e (cid:101) π ( (cid:80) i ˆ q i ) ∼ = ι v ∗ Ω e π v ⊕ (cid:76) i ι i ∗ Ω e π i ( ˆ q i ):Considering this term using Serre-Grothendieck duality for twisted curves gives R π F ∗ R H om (cid:16) n ∗ Ω e (cid:101) π ( (cid:80) i ˆ q i ) , O C F (cid:17) [1] ∼ = R π F ∗ R H om (cid:16) n ∗ Ω e (cid:101) π ( (cid:80) i ˆ q i ) ⊗ ω π F , ω π F [1] (cid:17) ∼ = R π F ∗ R H om (cid:16) ι v ∗ (cid:0) Ω e π v ⊗ ω π v ( (cid:80) i ˇ ϑ i ) (cid:1) ⊕ (cid:77) i ι i ∗ (cid:0) Ω e π i ( ˆ q i ) ⊗ ω π i ( ˆ ϑ i ) (cid:1) , ω π F [1] (cid:17) ∼ = (cid:104) R π v ∗ (cid:0) Ω e π v ( (cid:80) i ˇ ϑ i ) ⊗ ω π v (cid:1)(cid:105) ∨ ⊕ (cid:77) i (cid:104) R π i ∗ (cid:0) Ω e π i ( ˆ ϑ i + ˆ q i ) ⊗ ω π i (cid:1)(cid:105) ∨ Considering the direct summands individually gives:(a) The term (cid:2) R π v ∗ (cid:0) Ω e π v ( (cid:80) i ˇ ϑ i ) ⊗ ω π v (cid:1)(cid:3) ∨ [ − ∼ = R π v ∗ R H om (cid:0) Ω e π v ( (cid:80) i ˇ ϑ i ) , O C F (cid:1) has C ∗ -weight 0 and was shown in [AJAJ, §
2] and [ACVACV, §
3] to be locally freeof rank 3 g − l . (b) The term (cid:2) R π i ∗ (cid:0) Ω e π i ( ˆ ϑ i + ˆ q i ) ⊗ ω π i (cid:1)(cid:3) ∨ [ −
1] is the pullback of a correspondingbundle on M . Specifically, there is an isomorphism (cid:2) R π i ∗ (cid:0) Ω e π i ( ˆ ϑ i + ˆ q i ) ⊗ ω π i (cid:1)(cid:3) ∨ [ − ∼ = (cid:2) R π i ∗ (cid:0) Ω e π i ( ˆ z i + ˆ q i ) ⊗ ω π i (cid:1)(cid:3) ∨ [ − ∼ = ( u ◦ v ) ∗ (cid:2) R π G ,i ∗ (cid:0) Ω e π G ,i ( ˆ z G ,i + ˆ q G ,i ) ⊗ ω π G ,i (cid:1)(cid:3) ∨ [ − G using the byadding G to the subscripts. This is the pullback of the bundle studied in[GPGP, §
4] and [GV2GV2, § Contribution from F eq :There is an isomorphism F eq ∼ = ( u ◦ v ) ∗ R π G ∗ f ∗ G ( ω log P ) ∨ . Hence we can use theanalysis for this term from [GPGP, §
4] with the relative changes from [GV2GV2, § u ◦ v ) ∗ π G ,v ∗ ω π G ,v ∼ = b ∗ ρ ∗ ω ρ discussed in 2.4.52.4.5 part (i)(i). (cid:3) Analysis of the Contributions Relative to ν .3.3.1 ( Description of Perfect Relative Obstruction Theory for ν ) . Consider thefollowing commutative diagram: F C F π F (cid:111) (cid:111) f (cid:47) (cid:47) Tot C F L β F (cid:47) (cid:47) τ F (cid:15) (cid:15) C F F C F π F (cid:111) (cid:111) f (cid:48) (cid:47) (cid:47) Tot C F L r α F (cid:47) (cid:47) C F (13)In this diagram α F , β F and τ F are the morphisms from 1.4.51.4.5 pulled back to F from M /r . The morphisms f and f (cid:48) closed immersions defined by the universal sections σ and σ r respectively. Using this notation we have (as described in 5.3.45.3.4) that theequivariant perfect relative obstruction theory for ν restricted to F is L j ∗ F e ν = R π F ∗ L f ∗ T e τ F . Consider the morphism n : (cid:101) C → C F (discussed in 2.3.22.3.2) normalising C F atthe distinguished nodes. This gives the distinguished triangle in D e b ( C F ) O C F (cid:47) (cid:47) n ∗ O (cid:101) C (cid:47) (cid:47) (cid:76) i ( ϑ i ) ∗ O Θ i (cid:47) (cid:47) O C F [1]where all C ∗ -linearisations are trivial. Now, taking the (derived) tensor product by L f ∗ T e τ F and pushing forward via R π F ∗ we obtain the following distinguished triangle L j ∗ F e ν (cid:47) (cid:47) F v ⊕ (cid:76) i F i (cid:47) (cid:47) (cid:76) i F Θ i (cid:47) (cid:47) L j ∗ F e ν [1]where, using the notation from 2.3.12.3.1 and 2.3.22.3.2, we have(i) F v = R π v ∗ L ι ∗ v L f ∗ T e τ F ,(ii) F i = R π i ∗ L ι ∗ v L f ∗ T e τ F ,(iii) F Θ i = R ( γ Θ i ) ∗ L ϑ ∗ i L f ∗ T e τ F . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 25 Lemma 3.3.3.
Let g − l ≥ and let log refer to the un-twisted markings on C v corresponding to the flag nodes. There is a distinguished triangle in D e b ( F ) : F v (cid:47) (cid:47) V r ⊗ R π v ∗ L v (cid:47) (cid:47) V ⊗ R π v ∗ ω log π v (cid:47) (cid:47) F v [1] where R π v ∗ L v and R π v ∗ ω log π v have trivial C ∗ -linearisations and V c is a trivial bundleon F with weight c .Proof. Consider the following diagram which is a restriction of the diagram (1313) to C v (with appropriately chosen labels): F C v π v (cid:111) (cid:111) f v (cid:47) (cid:47) Tot C v L v β v (cid:47) (cid:47) τ v (cid:15) (cid:15) C v F C v π v (cid:111) (cid:111) f (cid:48) v (cid:47) (cid:47) Tot C v L rv α v (cid:47) (cid:47) C v With this notation we have an isomorphism F v = R π v ∗ L ι ∗ v L f ∗ T e τ F ∼ = R π v ∗ L f ∗ v T e τ v and the distinguished triangle of equivariant tangent complexes arising from τ v L f ∗ v T e τ v (cid:47) (cid:47) f ∗ v T e β v (cid:47) (cid:47) f ∗ v τ ∗ v T e α v (cid:47) (cid:47) L f ∗ v T e τ v [1] . Since C v is a C ∗ -fixed locus, any equivariant line bundle will be of the form E ⊗ U c where E is a line bundle with trivial C ∗ -linearisation and U c is a trivial bundle on C v with weight c . Moreover, since F is also C ∗ -fixed we have U c ∼ = π ∗ v V c where V c is a trivial bundle on F with weight c .Forgetting the C ∗ -linearisations, f ∗ v T e β v and f ∗ v τ ∗ v T e α v correspond to the bundles L v and ω log π v respectively. Hence, we need to calculate the weight of the C ∗ -linearisation.We can calculate the weights by taking a geometric point in F given by ζ = (cid:16) C, f : C → P , L, e : L r ∼ → R f , σ : O C → L (cid:17) and considering the flag node of C where C v meets a C i . The weights must be thesame as the weighs of the corresponding sheaves at the orbifold point of C i . Thishas local picture given by V := [ U/µ r ], for U = Spec C [ z ] where the action of µ r isgiven by z (cid:55)→ ξ r z . Let Ψ be the local generator of ( L ζ ) | V with isomorphismsΨ r C [ z ] ∼ = dzz rλ i +1 C [ z ] ∼ = R f (cid:12)(cid:12) V . (14)Locally C ∗ acts on V via z (cid:55)→ c − rλi z , so the C ∗ -action on the local generator is c · Ψ = c r Ψ. Locally, the total space morphism β v is given by A → A [Ψ]Hence, we have the weight corresponding to β v is r . Similarly we can calculate theweight for α v to be 1. (cid:3) Corollary 3.3.4.
In the case g − l ≥ , the fixed part of F v is zero and themoving part has equivariant Euler class e (cid:0) F mov v (cid:1) = c rt (cid:0) b ∗ R ρ ∗ L (cid:1) (cid:0) tr (cid:1) m − g +1 − l − (cid:80) i (cid:98) λ i /r (cid:99) + (cid:80) i : r | λi c t (cid:0) b ∗ ρ ∗ ω ρ (cid:1) t g − l where c s is the Chern polynomial, b : F → M /r, a g,l is from corollary 2.4.42.4.4 and where ρ and L are the the universal curve and r -th root bundle for M /r, a g,l . Proof.
We take the equivariant Euler class of the distinguished triangle from lemma3.3.33.3.3. We begin with the term V r ⊗ R π v ∗ L v . As discussed in lemma 2.4.52.4.5 part (iii)(iii)there is a distinguished triangle b ∗ R ρ ∗ L (cid:47) (cid:47) R π v ∗ L v (cid:47) (cid:47) (cid:76) i : r | λ i O F (cid:47) (cid:47) b ∗ R ρ ∗ L [1] . This gives that e ( V r ⊗ R π v ∗ L v ) = (cid:0) tr (cid:1) (cid:80) i : r | λi e ( V r ⊗ b ∗ R ρ ∗ L ). Now, using stan-dard methods (for example [BB, Prop. 5]) one can show R ρ ∗ L is quasi-isomorphicto a two term sequence of bundles R ρ ∗ L ∼ = (cid:2) E −→ E (cid:3) in D b (cid:0) M /r, a g,l (cid:1) .Pulling back and tensoring by V r gives a quasi-isomorphim in D e b ( F ) V r ⊗ b ∗ R ρ ∗ L ∼ = (cid:2) V r ⊗ F −→ V r ⊗ F (cid:3) . where we have defined F n := b ∗ E n . Then taking the equivariant Euler class gives e (cid:0) V r ⊗ b ∗ R ρ ∗ L (cid:1) = (cid:80) rk F k =0 (cid:0) tr (cid:1) rk F − k c k ( F ) (cid:80) rk F k =0 (cid:0) tr (cid:1) rk F − k c k ( F ) = (cid:18) tr (cid:19) rk F − rk F c rt ( b ∗ R ρ ∗ L ) . The degree of L is m − l − (cid:80) i (cid:98) λ i /r (cid:99) , hence we can calculate rk F − rk F usingRiemann-Roch for twisted curves.The contribution from V r ⊗ R π v ∗ L rv is calculated in a similar manner. We have adistinguished triangle discussed in 2.4.52.4.5 part (ii)(ii) b ∗ R ρ ∗ ω ρ (cid:47) (cid:47) R π v ∗ L rv (cid:47) (cid:47) (cid:76) i O F (cid:47) (cid:47) b ∗ R ρ ∗ L [1] . However this case is simpler and it is well known that there is an isomorphism R ρ ∗ ω ρ ∼ = (cid:2) ρ ∗ ω ρ → O M /r, a g,l (cid:3) where each term in the complex is a vector bundle.The calculation now proceeds in the same way as the previous case. (cid:3) Lemma 3.3.5.
The contributions from flag nodes have no fixed part.(i) In the case g = 0 and l = 2 , there is a single flag node and e ( F Θ ) = (cid:40) r , if r | λ ; t , if r (cid:45) λ .(ii) In the case g − l ≥ , the moving part has equivariant Euler class e (cid:0)(cid:76) i F Θ i (cid:1) = t − l (cid:0) tr (cid:1) (cid:80) i : r | λi . Proof.
This proof is similar to the proof of 3.3.33.3.3. Consider the case where we have3 g − l ≥
0. There is the following diagram which is a restriction of (1313) to Θ i (with appropriately chosen labels): F Θ i γ Θ i (cid:111) (cid:111) f Θ i (cid:47) (cid:47) Tot Θ i ϑ ∗ i L β Θ i (cid:47) (cid:47) τ Θ i (cid:15) (cid:15) Θ i F Θ i γ Θ i (cid:111) (cid:111) f (cid:48) Θ i (cid:47) (cid:47) Tot Θ i ϑ ∗ i L r α Θ i (cid:47) (cid:47) Θ i With this notation we have an isomorphism F Θ i = R ( γ Θ i ) ∗ L ϑ ∗ i L f ∗ T e τ F ∼ = R ( γ Θ i ) ∗ L f ∗ Θ i T e τ Θ i -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 27 and the canonical distinguished triangle on Θ i arising from τ Θ i L f ∗ Θ i T e τ Θ i (cid:47) (cid:47) f ∗ Θ i T e β Θ i (cid:47) (cid:47) f ∗ Θ i τ ∗ Θ i T e α Θ i (cid:47) (cid:47) L f ∗ Θ i T e τ Θ i [1] . Θ i is a C ∗ -fixed locus (as in the proof of 3.3.33.3.3) and we have equivariant isomorphisms R ( γ Θ i ) ∗ f ∗ Θ i T e β Θ i ∼ = V c ⊗ R ( γ Θ i ) ∗ ϑ ∗ i L and R ( γ Θ i ) ∗ f ∗ Θ i τ ∗ Θ i T e α Θ i ∼ = V c ⊗ R ( γ Θ i ) ∗ ϑ ∗ i L r for some weights c , c ∈ Q . The weights are calculated to be r and 1 respectivelyby using the same method as the proof of 3.3.33.3.3 . The result follows from lemma2.3.42.3.4 which shows the Chern polynomials are c s (cid:16) R ( γ Θ i ) ∗ ϑ ∗ i L r (cid:17) = 1 and c s (cid:16) R ( γ Θ i ) ∗ ϑ ∗ i L (cid:17) = (cid:26) r | λ i ;0 otherwise.In the case ( g, l ) = (0 , (cid:3) Lemma 3.3.6.
The fixed part of F i is zero. In the cases g − l ≥ and ( g, l ) = (0 , the moving part has equivariant Euler class e (cid:0) F mov i (cid:1) = (cid:4) λ i r (cid:5) ! λ (cid:106) λir (cid:107) i t (cid:106) λir (cid:107) (cid:32) λ i ! λ λ i i t λ i (cid:33) − . For ( g, l ) = (0 , the moving part contribution is the above formula multiplied by t .Proof. Consider the following diagram which contains a restriction of the diagram(1313) to C i (with appropriately chosen labels): F C i π i (cid:111) (cid:111) f i (cid:47) (cid:47) Tot C i L i β i (cid:47) (cid:47) τ i (cid:15) (cid:15) C i F C i π i (cid:111) (cid:111) f (cid:48) i (cid:47) (cid:47) Tot C i L ri α i (cid:47) (cid:47) C i With this notation we have an isomorphism F i = R π i ∗ L ι ∗ i L f ∗ T e τ F ∼ = R π i ∗ L f ∗ i T e τ i and the canonical distinguished triangle on F arising from τ i : F i (cid:47) (cid:47) R π i ∗ f ∗ i T e β i (cid:47) (cid:47) R π i ∗ f (cid:48) i ∗ T e α i (cid:47) (cid:47) F i [1] . In lemma 2.4.62.4.6 it is shown that, after forgetting the C ∗ -linearisation, R π i ∗ f ∗ i T e β i and R π i ∗ f (cid:48) i ∗ T e α i are trivial bundles. Hence we may calculate their weights by con-sidering a geometric point in F .Without loss of generality we consider a geometric point • (cid:44) → F with L| • = L . Thenwe can form the following diagrams • (cid:15) (cid:15) P [ r ] p (cid:111) (cid:111) (cid:15) (cid:15) s b (cid:47) (cid:47) Tot P [ r ] L b (cid:47) (cid:47) (cid:15) (cid:15) P [ r ] (cid:15) (cid:15) F C i π i (cid:111) (cid:111) f i (cid:47) (cid:47) Tot C i L i β i (cid:47) (cid:47) C i • (cid:15) (cid:15) P [ r ] p (cid:111) (cid:111) (cid:15) (cid:15) s a (cid:47) (cid:47) Tot P [ r ] L r a (cid:47) (cid:47) (cid:15) (cid:15) P [ r ] (cid:15) (cid:15) F C i π i (cid:111) (cid:111) f (cid:48) i (cid:47) (cid:47) Tot C i L ri α i (cid:47) (cid:47) C i by taking the left and right squares of each diagram to be cartesian. (Here P [ r ] isthe standard notation for P with a r -orbifold point at 0.) These give isomorphisms (cid:0) R π i ∗ f ∗ i T e β i (cid:1)(cid:12)(cid:12) • ∼ = R p ∗ s ∗ b T e b and (cid:0) R π i ∗ f ∗ i τ ∗ i T e α i (cid:1)(cid:12)(cid:12) • ∼ = R p ∗ s ∗ a T e a . We also have a natural morphism γ ∗ γ ∗ L → L which induces a morphism on thetotal spaces h : Tot P [ r ] L → Tot P [ r ] γ ∗ γ ∗ L as well as inducing an isomorphism onglobal sections H ( P [ r ] , γ ∗ γ ∗ L ) ∼ = H ( P [ r ] , L ). Hence there exist a commutingdiagram of the following form: Tot P [ r ] L b (cid:15) (cid:15) Tot P [ r ] γ ∗ γ ∗ L b (cid:48) (cid:15) (cid:15) h (cid:111) (cid:111) P [ r ] s b (cid:84) (cid:84) P [ r ] s b (cid:48) (cid:84) (cid:84) The morphism h , of the total spaces, induces the following distinguished triangle R p ∗ s ∗ b (cid:48) T e h (cid:47) (cid:47) R p ∗ s ∗ b (cid:48) T e b (cid:48) (cid:47) (cid:47) R p ∗ s ∗ b T e b (cid:47) (cid:47) R p ∗ s ∗ b (cid:48) T e h [1] . After forgetting the C ∗ -linearisation the morphism R p ∗ s ∗ b (cid:48) T e b (cid:48) → R p ∗ s ∗ b T e b becomesthe isomorphism H ( P [ r ] , γ ∗ γ ∗ L ) ∼ = H ( P [ r ] , L ) which shows that R p ∗ s ∗ b (cid:48) T h ∼ = 0.So we must have R p ∗ s ∗ b (cid:48) T e h ∼ = 0, showing R p ∗ s ∗ b (cid:48) T e b (cid:48) → R p ∗ s ∗ b T e b is an isomorphism.In the cases 3 g − l ≥ g, l ) = (0 , γ : P [ r ] → P forgetting the stack structure. We can form the following diagrams where the topsquares are cartesian: Tot P [ r ] γ ∗ γ ∗ L (cid:47) (cid:47) b (cid:48) (cid:15) (cid:15) Tot P O P (cid:0)(cid:4) λ i r (cid:5)(cid:1) b (cid:48) (cid:15) (cid:15) P [ r ] γ (cid:47) (cid:47) p (cid:15) (cid:15) s b (cid:48) (cid:84) (cid:84) P p (cid:15) (cid:15) s b (cid:48) (cid:84) (cid:84) • • Tot P [ r ] L r (cid:47) (cid:47) a (cid:15) (cid:15) Tot P O P ( λ i ) a (cid:15) (cid:15) P [ r ] γ (cid:47) (cid:47) p (cid:15) (cid:15) s a (cid:84) (cid:84) P p (cid:15) (cid:15) s a (cid:85) (cid:85) • • Since γ is flat we have isomorphisms T e b (cid:48) ∼ = γ ∗ T e b (cid:48) and T e a ∼ = γ ∗ T e a . Furthermorethese induce isomorphisms (cid:0) R π i ∗ f ∗ i T e β i (cid:1)(cid:12)(cid:12) • ∼ = R p ∗ s ∗ b (cid:48) T e b (cid:48) and (cid:0) R π i ∗ f ∗ i τ ∗ i T e α i (cid:1)(cid:12)(cid:12) • ∼ = R p ∗ s ∗ a T e a . Here, C ∗ acts on P by c · [ x : x ] = [ x : c /λ i x ]. Hence we have:(i) R p ∗ s ∗ b (cid:48) T e b (cid:48) is the sum of trivial bundles with weights 0 , tλ i , tλ i , . . . , (cid:4) λ i r (cid:5) tλ i .(ii) R p ∗ s ∗ a T e a is the sum of trivial bundles with weights 0 , tλ i , tλ i , . . . , λ i tλ i .Taking equivariant Euler classes of the parts with non-zero weights completes theproof for the moving parts. The case ( g, l ) = (0 ,
1) is the same except the bundles O P (cid:0)(cid:4) λ i r (cid:5)(cid:1) and O P ( λ i ) and replaced with O P (cid:0)(cid:4) λ i − r (cid:5)(cid:1) and O P ( λ i − R π i ∗ f ∗ i T e β i −→ R π i ∗ f ∗ i τ ∗ i T e α i to the weightzero part is a nowhere-zero morphism between trivial bundles. Hence it is anisomorphism. The desired result now follows by taking equivariant Euler classes. (cid:3) -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 29 Results from the Analysis of the Fixed and Moving Parts.Corollary 3.4.1.
The restriction L j ∗ F M /r has fixed part (cid:0) L j ∗ F M /r (cid:1) fix ∼ = T eF ∼ = (Ω F ) ∨ which is locally free. Moreover, setting N vir F := ( L j ∗ F M /r ) mov we have(i) If g = 0 and l = 1 then e (cid:0) N vir F (cid:1) = r − (cid:0) tr (cid:1) − m λ (cid:0) λ r (cid:1) (cid:98) λ r (cid:99) (cid:4) λ r (cid:5) ! λ . (ii) If g = 0 and l = 2 then e (cid:0) N vir F (cid:1) = (cid:0) tr (cid:1) − m (cid:89) i =1 λ i (cid:0) λ i r (cid:1) (cid:106) λir (cid:107) (cid:4) λ i r (cid:5) ! λ + λ ) . (iii) If g − l ≥ then e (cid:0) N vir F (cid:1) = r l +2 g − (cid:0) tr (cid:1) g − l − m l (cid:89) i =1 λ i (cid:0) λ i r (cid:1) (cid:106) λir (cid:107) (cid:4) λ i r (cid:5) ! c rt ( − b ∗ R ρ ∗ L ) (cid:81) li =1 (1 − λ i r ψ i ) . where c s is the Chern polynomial, b : F → M /r, a g,l is from corollary 2.4.42.4.4,while ρ and L are the the universal curve and r -th root bundle for M /r, a g,l .Proof. Take the distinguished triangle of perfect obstruction theories from corollary5.4.45.4.4. Restricting to F and taking either either the fixed or the moving part gives (cid:0) L j ∗ F e ν (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ F e M /r (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ L ν ∗ F e M r (cid:1) f . m . (cid:47) (cid:47) (cid:0) L j ∗ F e ν (cid:1) f . m . [1] . We have from lemmas 3.3.33.3.3, 3.3.53.3.5 and 3.3.63.3.6 that ( L j ∗ F e ν ) fix ∼ = 0. So we have anisomorphim ( L j ∗ F e M /r ) fix ∼ = ( L j ∗ L ν ∗ F M r ) fix which is shown to be locally freein lemma 3.2.13.2.1. So, by theorem 3.1.13.1.1 the fixed part ( L j ∗ L ν ∗ F M r ) fix is a perfectobstruction theory for F . The isomorphism ( L j ∗ F M /r ) fix ∼ = (Ω F ) ∨ follows from[BFBF, Prop. 5.5].Taking the moving part of the above distinguished triangle and using the notationfrom 3.3.23.3.2 we have1 e (cid:0) N vir F (cid:1) = 1 e (cid:16)(cid:0) L j ∗ L ν ∗ F M (cid:1) mov (cid:17) e (cid:16)(cid:0) L j ∗ F ν (cid:1) mov (cid:17) = (cid:81) i e (cid:16) F mov Θ i (cid:17) e (cid:16)(cid:0) L j ∗ L ν ∗ F M (cid:1) mov (cid:17) e (cid:16) F mov v (cid:17) (cid:81) i e (cid:16) F mov i (cid:17) . These contributions are calculated in lemmas 3.2.13.2.1, 3.3.53.3.5, 3.3.63.3.6 and corollary 3.3.43.3.4.In the cases where 3 g − l < For ( g, l ) = (0 , : m = r ( λ −
1) = (cid:4) λ r (cid:5) . For ( g, l ) = (0 , : m = r ( λ + λ ) = (cid:40) (cid:4) λ r (cid:5) + (cid:4) λ r (cid:5) , if r | λ ;1 + (cid:4) λ r (cid:5) + (cid:4) λ r (cid:5) , if r (cid:45) λ .In the cases where 3 g − l ≥ c s (cid:0) ( ρ ∗ ω ρ ) ∨ (cid:1) c s (cid:0) ρ ∗ ω ρ (cid:1) = 1 , a proof of which can be found in [ACGACG, Prop. 5.16]. (cid:3) Corollary 3.4.2.
The morphism of simple fixed loci v : F → G r which forgets the r -th root section is ´etale.Proof. Consider the morphism of equivariant cotangent complexes T eF → v ∗ T eG r arising from v : F → G r . Taking the fixed part gives a morphism T eF → ( v ∗ T eG r ) fix .Applying this to the morphism of perfect obstruction theories from corollary 5.4.45.4.4gives the left square in the following commuting diagram: T eF (cid:47) (cid:47) u (cid:15) (cid:15) (cid:0) v ∗ T eG r (cid:1) fix u (cid:15) (cid:15) (cid:47) (cid:47) v ∗ T eG r u (cid:15) (cid:15) (cid:0) L j ∗ F e M /r (cid:1) fix v (cid:47) (cid:47) (cid:0) v ∗ L ( k r ) ∗ F e M r (cid:1) fix v (cid:47) (cid:47) v ∗ (cid:0) L ( k r ) ∗ F e M r (cid:1) fix The right square arises from the analysis of L j ∗ L ν ∗ F e M r ∼ = v ∗ L ( k r ) ∗ F e M r fromlemma 3.2.13.2.1 which also holds for the fixed part of L ( k r ) ∗ E e M r . Indeed, this givesan isomorphism v : (cid:0) v ∗ L ( k r ) ∗ F e M r (cid:1) fix ∼ −→ v ∗ (cid:0) L ( k r ) ∗ F e M r (cid:1) fix Now, using both corollary 3.4.13.4.1 and [BFBF, Prop 5.5] we have that F and G r aresmooth. Hence, u and u are isomorphisms. Moreover, corollary 3.4.13.4.1 shows that v is an isomorphism, showing that T eF → v ∗ T eG r is an isomorphism. (cid:3) Proving Theorem CC - The r -ELSV formula Choice of Equivariant Lift.
We use arguments based on [FPFP, § Recall the branch-type morphism from [Le1Le1] discussed earlier in 1.4.91.4.9. It isa morphism of stacks br : M r −→ Sym m P commuting with the branch morphism of [FPFP] via the diagram M r br (cid:47) (cid:47) (cid:15) (cid:15) Sym m P (cid:15) (cid:15) M br (cid:47) (cid:47) Sym rm P where ∆ is defined by (cid:80) i x i (cid:55)→ (cid:80) i rx i . We make the identification Sym m P ∼ = P m and extend the C ∗ -action so that br is equivariant with C ∗ acting on P m via c · [ y : y : · · · : y m ] = [ y : cy : · · · : c m y m ] . Let H be the hyperplane class in P m . Then, with the identification in 4.1.14.1.1the class of a point in Sym m P corresponds to the class c ( O ( H )) m . Hence, aftera choice of equivariant lift for O ( H ), we may apply the localisation formula to theintegral: (cid:90) (cid:104) M r (cid:105) vir br ∗ (cid:0) c ( O ( H )) m (cid:1) . Let y , . . . , y m be projective coordinates on P m and define the point h n ∈ P m by the ideal (cid:10) { y , . . . , y m } \ { y n } (cid:11) -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 31 Then we have h n ∈ ( P m ) C ∗ and corresponds to the point [( m − k ) · (0) + k · ( ∞ )] inSym m P . Moreover we know from the discussion in 2.2.12.2.1 that br (cid:16)(cid:0) M r (cid:1) C ∗ (cid:17) ⊆ { h , . . . , h m } . Denote by H n the unique C ∗ -linearisation of O ( H ) having weight 0 at h n . Weapply the localisation formula to (cid:90) (cid:104) M r (cid:105) vir br ∗ (cid:32) m (cid:89) n =1 c ( H n ) (cid:33) . Remark 4.1.4.
Different choices of the C ∗ -linearisations of O ( H ) will lead to dif-ferent equivariant integrands. However, they will all have the same non-equivariantlimit so we follow [FPFP, § Application of the Localisation Formula.4.2.1.
Recall from 2.2.12.2.1 that the fixed loci is decomposed as (cid:0) M r (cid:1) C ∗ = F (cid:116) (cid:70) mn =1 (cid:98) F n where F := ( M r ) C ∗ ∩ br − ( h ) and (cid:98) F n := ( M r ) C ∗ ∩ br − ( h n ). Lemma 4.2.2.
Using the linearisations in 4.1.34.1.3 we have(i) (cid:2) (cid:98) F n (cid:3) vir ∩ br ∗ (cid:16) (cid:81) mn =1 c ( H n ) (cid:17) = 0 (ii) (cid:2) F (cid:3) vir ∩ br ∗ (cid:16) (cid:81) mn =1 c ( H n ) (cid:17) = (cid:2) F (cid:3) vir · m ! t m Proof.
For both, the proof is the same as that in [FPFP, § br ∗ H n restricted to (cid:98) F n is the trivial bundle with trivial linearisation (i.e weight 0).Hence, the given intersection vanishes. For part 2 we have that br ∗ H n restrictedto F is the trivial bundle with weight n . (cid:3) Corollary 4.2.3 ( Theorem CC ) . When g − l ≥ there is an equality: (cid:90) (cid:104) M r (cid:105) vir br ∗ [ p + · · · + p m ] = m ! r m + l +2 g − l (cid:89) i =1 (cid:0) λ i r (cid:1) (cid:98) λir (cid:99) (cid:98) λ i r (cid:99) ! (cid:90) M r , a g,l c ( − R ρ ∗ L ) (cid:81) lj =1 (1 − λ i r ψ j ) where ρ and L are the universal curve and r -th root of M r , a g,l , while a = ( a , . . . , a l ) is a vector with a i ∈ { , . . . , r − } defined by λ i = (cid:4) λ i r (cid:5) r + ( r − − a i ) and l = l ( λ ) .In the special cases where g − l < we interpret this formula by defining: ( i ) (cid:90) M r , a , c ( − R ρ ∗ L )(1 − λ r ψ ) = 1 λ , when r | ( λ − and otherwise , ( ii ) (cid:90) M r , a , c ( − R ρ ∗ L ) (cid:81) j =1 (1 − λ j r ψ j ) = 1 λ + λ , when r | ( λ + λ ) and otherwise . Proof.
Considering the equivariant intersection with lifts from 4.1.34.1.3 we have (cid:90) (cid:104) M r (cid:105) vir br ∗ (cid:32) m (cid:89) n =1 c ( H n ) (cid:33) = (cid:2) F (cid:3) vir ∩ m ! t m e (cid:0) N vir F (cid:1) from lemma 4.2.24.2.2. For 3 g − l ≥ m ! r m + l +2 g − l (cid:89) i =1 λ i (cid:0) λ i r (cid:1) (cid:98) λir (cid:99) (cid:98) λ i r (cid:99) ! (cid:32)(cid:2) F (cid:3) ∩ b ∗ (cid:0) tr (cid:1) g − l c rt (cid:0) [ R ρ ∗ L ] ∨ (cid:1)(cid:81) li =1 (cid:0) − (cid:0) rt (cid:1) λ i r c ( σ ∗ i ω ρ ) (cid:1) (cid:33) where b : F → M r , a g,l ( λ ) is the degree ( λ · · · λ l ) − morphism defined in lemma2.4.42.4.4. The desired result follows from pushing forward via b and taking the non-equivariant limit. For 3 g − l < (cid:3) Proving Theorem AA - Equivariant Perfect Obstruction Theory
Notation Conventions for Main Spaces : We will use the following simplifyingnotation for the key spaces:(i) M := M g ( X, λ ) from definition 1.1.21.1.2 with π : C → M the universal curve.(ii) M r := M rg ( X, λ ) from definition 1.4.11.4.1 with π r : C r → M r the universal( r -twisted) curve.(iii) M /r := M /r g ( X, λ ) from definition 1.4.71.4.7 with π /r : C /r → M /r theuniversal ( r -twisted) curve.(iv) M := M g,l and M r := M rg,l from defintion 1.2.11.2.1.The natural forgetful morphisms are denoted by M r ν (cid:47) (cid:47) M r υ (cid:47) (cid:47) M . We will also denote by R , L and δ : O C → R the bundles and morphism on C /r pulled back from C r . Lastly, σ : O C → L is the morphism pulled back from Tot π ∗ L .5.1. Background on Equivariant Perfect Obstruction Theories.5.1.1 ( Equivariant Cotangent Complex ) . For a stack X with a C ∗ -action we denotethe derived category of equivariant coherent sheaves by D e b ( X ). For an equivariantmorphism η : X → Y there is an equivariant cotangent complex L e η ∈ D e b ( X )(originally defined by Illusie in [II, Ch. VII 2.2]) with the following properties:(i) Without the C ∗ -linearisation, L e η is the cotangent complex L η ∈ D b ( X ).(ii) If the following is a commuting diagram of equivariant morphisms X η (cid:47) (cid:47) ζ (cid:15) (cid:15) Y ζ (cid:48) (cid:15) (cid:15) W η (cid:48) (cid:47) (cid:47) Z then there is a natural morphism L η ∗ L e ζ (cid:48) → L e ζ in D e b ( X ) which is anisomorphism when the diagram is cartesian and one of ζ (cid:48) or η (cid:48) is flat.Moreover, after forgetting the C ∗ -linearisations this morphism is the nat-ural morphism L η ∗ L ζ (cid:48) → L ζ in D b ( X ). -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 33 (iii) The equivariant cotangent complex is compatible with composition. Thatis, if the following is a commuting diagram of equivariant morphisms X η (cid:47) (cid:47) ζ (cid:15) (cid:15) Y ζ (cid:48) (cid:15) (cid:15) κκκ (cid:47) (cid:47) U ζ (cid:48)(cid:48) (cid:15) (cid:15) W η (cid:48) (cid:47) (cid:47) Z κκκ (cid:48) (cid:47) (cid:47) V then there is the following commuting diagram in D e b ( X ): L ( κκκ ◦ η ) ∗ L e ζ (cid:48)(cid:48) (cid:47) (cid:47) (cid:40) (cid:40) L e ζ L η ∗ L e ζ (cid:48) (cid:59) (cid:59) (iv) If η : X → Y and ζ : Y → Z are equivariant morphisms then there is adistinguished triangle in D e b ( X ) given by L η ∗ L e ζ (cid:47) (cid:47) L e ζ ◦ η (cid:47) (cid:47) L e η (cid:47) (cid:47) Lg ∗ L e ζ [1] . Again, after forgetting the C ∗ -linearisations this is the usual distinguishedtriangle of the cotangent complex. If we have commuting diagrams of equivariant morphisms X η (cid:47) (cid:47) ζ (cid:15) (cid:15) Y ζ (cid:48) (cid:15) (cid:15) Y (cid:15) (cid:15) W η (cid:48) (cid:47) (cid:47) Z (cid:47) (cid:47) • and X ζ (cid:15) (cid:15) X (cid:15) (cid:15) η (cid:47) (cid:47) Y (cid:15) (cid:15) W (cid:47) (cid:47) • • then we can use the properties of 5.1.15.1.1 to show there is the following commutingdiagram of morphisms in D e b ( X ): L η ∗ L e Y (cid:47) (cid:47) (cid:15) (cid:15) L η ∗ L e ζ (cid:48) (cid:15) (cid:15) L e X (cid:47) (cid:47) L e ζ Definition 5.1.3 ( Equivariant Perfect Obstruction Theory, [GPGP]) . Let η : X → Y be an equivariant morphism of stacks. An equivariant perfect obstruction theoryfor η is a morphism φ e η : E e η −→ L e η in D e b ( X ) such that the associated morphism φ η : E η −→ L η in D b ( X ) (constructedby forgetting the C ∗ -linearisations) is a perfect obstruction theory for η .5.2. Equivariant Perfect Obstruction Theory For M .5.2.1 ( Equivariant Perfect Relative Obstruction Theory for q ) . Denote the mor-phisms which forget all data but the source and target families by p : M → M × T and q : M r → M r × T where p is discussed more in 1.1.31.1.3. These morphism fit in the diagram M r (cid:47) (cid:47) q (cid:15) (cid:15) (cid:102) M (cid:47) (cid:47) (cid:15) (cid:15) M p (cid:15) (cid:15) M r × T M r × T (cid:47) (cid:47) M × T (15) where (cid:102) M := M × ( M ×T ) ( M r × T ). We note that the morphism M r × T → M × T is flat and the morphism M r → (cid:102) M is ´etale (as described in [Le1Le1, Lem. 2.1.2]).We give M × T and M r × T the trivial C ∗ -action (as discussed in 2.1.32.1.3) whichmakes every morphism in the diagram of (1515) a C ∗ -equivariant morphism. Thereis an equivariant perfect relative obstruction theory φ ep : E ep → L ep for p constructedin [Li2Li2, GV2GV2]. This pulls back via the top row of (1515) to give an equivariant perfectrelative obstruction theory φ eq : E eq → L eq for q . ( Equivariant Perfect Obstruction Theory for M r ) . Using the properties ofthe equivariant cotangent complex given in 5.1.15.1.1 and the construction of 5.2.15.2.1 with[GV2GV2, § M r whichfits into the following commuting diagram with distinguished triangles for rows: q ∗ L e M r ×T (cid:47) (cid:47) E e M r (cid:47) (cid:47) φ e M r (cid:15) (cid:15) E eq φ eq (cid:15) (cid:15) (cid:47) (cid:47) q ∗ L e M r ×T [1] q ∗ L e M r ×T (cid:47) (cid:47) L e M r (cid:47) (cid:47) L eq (cid:47) (cid:47) q ∗ L e M r ×T [1](16)Moreover, this construction gives the following commutative diagram relating E e M and E e M r via the natural forgetful morphism υ : M r → M : υ ∗ E e M (cid:47) (cid:47) (cid:15) (cid:15) υ ∗ φ e M (cid:15) (cid:15) E e M r (cid:47) (cid:47) φ e M r (cid:15) (cid:15) L e υ (cid:47) (cid:47) υ ∗ E e M [1] υ ∗ φ e M [1] (cid:15) (cid:15) υ ∗ L e M (cid:47) (cid:47) L e M r (cid:47) (cid:47) L e υ (cid:47) (cid:47) υ ∗ L e M [1](17)5.3. Equivariant Perfect Relative Obstruction Theory for ν .5.3.1 ( Equivariant Perfect Relative Obstruction Theories for Total Spaces ) . Let F be a line bundle on C r and recall the spaces Tot π ∗ F and Tot F from definition1.4.31.4.3. If ψ : C Tot π ∗ F → Tot π ∗ F is the universal curve for Tot π ∗ F , then there is anatural evaluation morphism e : C Tot π ∗ F → Tot F defined by e : (cid:16) ζ, σ : O C −→ F ζ (cid:17) (cid:55)−→ (cid:16) ζ, γ Θ ∗ ϑ ∗ σ : O S −→ γ Θ ∗ ϑ ∗ F ζ (cid:17) which leads to the following commutative diagram where the left square is cartesianand all morphisms are equivariant: Tot π ∗ F ε (cid:15) (cid:15) C Tot π ∗ F ψ (cid:111) (cid:111) e (cid:47) (cid:47) (cid:98) ε (cid:15) (cid:15) Tot F ˇ ε (cid:15) (cid:15) M r C r π (cid:111) (cid:111) C r (18)Since ψ is Gorenstein and Deligne-Mumford-type, there is a natural equivariantmorphism arising from Grothendieck duality for Deligne-Mumford stacks R ψ ∗ ( ψ ∗ L e ε ⊗ ω ψ )[1] −→ L e ε . Combining this with the inverse of the isomorphism ψ ∗ L e ε ∼ → L e (cid:98) ε (arising from π being flat) and the morphism L e ∗ L e ˇ ε → L e (cid:98) ε gives the following morphism in D e b ( M r ) R ψ ∗ ( L e ∗ L e ˇ ε ⊗ ω ψ )[1] −→ L e ε . We denote this morphism by φ e ε : E e ε → L e ε . The associated morphism φ ε in D b ( M r )was shown to be a perfect relative obstruction theory for ε in [CLCL, Prop. 2.5]. -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 35 ( Equivariant Relative Obstruction Theory for τ ) . Consider the following dia-gram of equivariant morphisms by combining the r -th power maps from 1.4.51.4.5 withthe construction in 5.3.15.3.1: Tot π ∗ L r C Tot π ∗ L r Tot L r M r C r C r Tot π ∗ L C
Tot π ∗ L Tot LM r C r C r ψ e π r β (cid:98) β ˇ βϕ e (cid:48) π r α (cid:98) α ˇ ατ (cid:98) τ ˇ τ (19)Here τ and ˇ τ are the r -th power maps from 1.4.51.4.5, e and e (cid:48) are the evaluatation mapsof 5.3.15.3.1. The morphisms ϕ , ψ , (cid:98) α , (cid:98) β and (cid:98) τ are all defined by taking appropriatecartesian diagramsAs in 5.3.15.3.1 we have two natural morphisms R ψ ∗ ( ψ ∗ L e τ ⊗ ω ψ )[1] −→ L e τ and L e ∗ L e ˇ τ −→ L e (cid:98) τ ∼ = ψ ∗ L e τ . which combine to obtain a morphism in D e b ( M r ) of the form φ e τ : R ψ ∗ ( L e ∗ L e ˇ τ ⊗ ω ψ )[1] −→ L e τ . The morphism φ e τ fits into the following commutative diagram with distinguishedtriangles as rows: L τ ∗ E e α (cid:47) (cid:47) L τ ∗ φ e α (cid:15) (cid:15) E e β φ e β (cid:15) (cid:15) (cid:47) (cid:47) E e τ φ e τ (cid:15) (cid:15) (cid:47) (cid:47) L τ ∗ E e α [1] L τ ∗ φ e α [1] (cid:15) (cid:15) L τ ∗ L e α (cid:47) (cid:47) L e β (cid:47) (cid:47) L e τ (cid:47) (cid:47) L τ ∗ L e α [1](20)Moreover, it was shown in [Le1Le1, Lemma 4.1.1] that the associated morphism φ e τ inD b ( M r ) is a relative obstruction theory. ( Equivariant Perfect Relative Obstruction Theory for ν ) . Recall the followingdiagram of equivariant morphisms (99) from definition 1.4.71.4.7: M r i (cid:47) (cid:47) ν (cid:15) (cid:15) Tot π ∗ L τ (cid:15) (cid:15) M r i (cid:48) (cid:47) (cid:47) Tot π ∗ L r (21)There is a natural morphism L i ∗ L e τ → L e ν in D e b ( M /r ) which gives rise to thefollowing diagram defining the morphism φ e ν : E e ν → L e ν : L i ∗ E e τ L i ∗ φ e τ (cid:47) (cid:47) L i ∗ L e τ (cid:15) (cid:15) E e ν φ e ν (cid:47) (cid:47) L e ν (22)It was shown in [Le1Le1, Thm. 4.1.4] that the associated morphism φ ν : E ν → L ν isa perfect relative obstruction theory. ( Alternative Description for the Equivariant Perfect Relative ObstructionTheory for ν ) . The equivariant perfect relative obstruction theory for ν from 5.3.35.3.3can also be constructed by considering the following commutative diagram: M r C r π r (cid:111) (cid:111) (cid:98) f (cid:47) (cid:47) Tot C r L β r (cid:47) (cid:47) τ r (cid:15) (cid:15) C r M r C r π r (cid:111) (cid:111) (cid:98) f (cid:48) (cid:47) (cid:47) Tot C r L r α r (cid:47) (cid:47) C r (23)In this diagram α /r , β /r and τ /r are the morphisms from 1.4.51.4.5 pulled back to M /r from M r . The morphisms (cid:98) f and (cid:98) f (cid:48) closed immersions defined by the universalsections σ and σ r respectively. Using this notation E e ν = Rπ r ∗ ( L (cid:98) f ∗ T e τ r ⊗ ω π r )[1](or F e ν = R π /r ∗ L (cid:98) f ∗ T e τ /r in the (derived) dual language of 33). This was shownfor the non-equivariant case in [Le1Le1, Lem. 4.1.2] and those methods extend to theequivariant case.5.4. Perfect Obstruction Theory for M /r .5.4.1. Using the notation of (1919) and (2121) there is the following commuting diagram L i (cid:48) ∗ α ∗ L e M r (cid:47) (cid:47) ∼ = (cid:41) (cid:41) L i (cid:48) ∗ L e Tot π ∗ L r (cid:15) (cid:15) L e M r in D e b ( M ) which arises from the properties of the equivariant cotangent complexgiven in 5.1.15.1.1. The isomorphism follows from α ◦ i (cid:48) ∗ = id M r . We can extend thisdiagram to the following diagram with distinguished triangles as rows: L i (cid:48) ∗ α ∗ L e M r (cid:47) (cid:47) ∼ = (cid:15) (cid:15) L i (cid:48) ∗ L e Tot π ∗ L r (cid:15) (cid:15) (cid:47) (cid:47) L i (cid:48) ∗ L e α (cid:15) (cid:15) (cid:47) (cid:47) L i (cid:48) ∗ α ∗ L e M r [1] ∼ = (cid:15) (cid:15) L e M r L e M r (cid:47) (cid:47) (cid:47) (cid:47) L e M r [1](24)This shows that the pullback by L i (cid:48) ∗ of the natural morphism L e α → α ∗ L e M r [1] isthe zero morphism in D e b ( M r ). Lemma 5.4.2.
There exists a commuting diagram in D e b ( M /r ) of the form L i ∗ E e τ [ − (cid:47) (cid:47) L i ∗ φ e τ [ − (cid:15) (cid:15) E (cid:15) (cid:15) (cid:47) (cid:47) L ν ∗ E e M r L ν ∗ φ e M r (cid:15) (cid:15) L i ∗ L e τ [ − v (cid:47) (cid:47) L ν ∗ L i (cid:48) ∗ L e Tot π ∗ L r v (cid:47) (cid:47) L ν ∗ L e M r where E ∈ D e b ( M /r ) , v is the pullback by L i ∗ of the (shifted) connecting morphism L e τ [ − → L τ ∗ L e Tot π ∗ L r arising from the cotangent complex distinguished trianglefor τ and v is the pullback by L ν ∗ of the canonical morphism L i (cid:48) ∗ L e Tot π ∗ L r → L e M r arising from i (cid:48) . -ELSV VIA LOCALISATION USING STABLE MAPS WITH DIVISIBLE RAMIFICATION 37 Proof.
We have the following two diagrams E e τ [ − (cid:47) (cid:47) φ e τ [ − (cid:15) (cid:15) L τ ∗ E e α L τ ∗ φ e α (cid:15) (cid:15) L e τ [ − (cid:47) (cid:47) L τ ∗ L e α L i (cid:48) ∗ E e α (cid:47) (cid:47) L i (cid:48) ∗ φ e α (cid:15) (cid:15) L i (cid:48) ∗ α ∗ E e M r [1] L i (cid:48) ∗ φ e M r [1] (cid:15) (cid:15) L i (cid:48) ∗ L e α (cid:47) (cid:47) L i (cid:48) ∗ α ∗ L e M r [1]where the left comes from (2020) and the 0 in the bottom of the right diagram comesfrom the comment following (2424). We pull back the left by L i ∗ and the right by L ν ∗ to get the following diagram: L i ∗ L e τ [ − L ν ∗ L i (cid:48) ∗ L e α L ν ∗ L i (cid:48) ∗ α ∗ L e M r [1] L ν ∗ L e M r [1] L i ∗ E e τ [ − L ν ∗ L i (cid:48) ∗ E e α L ν ∗ L i (cid:48) ∗ α ∗ E e M r [1] L ν ∗ E e M r [1] ∼ =0 ∼ =0 The desired diagram (shifted by 1) is constructed by taking the cone of the verticalmorphisms. (cid:3)
Remark 5.4.3.
All the discussion and results in section 55 of this article still hold ifwe replace P with a smooth curve X and drop the word equivariant . In particularthis is true for theorem AA. We have have used P and the equivariant language tobe consistent with the other sections of the article. Corollary 5.4.4 ( Theorem AA ) . There is perfect a perfect obstruction theory φ e M r : E e M r −→ L e M r fitting into the following commutative diagram with distinguished triangles as rows: L ν ∗ E e M r (cid:47) (cid:47) L ν ∗ φ e M r (cid:15) (cid:15) E e M r φ e M /r (cid:15) (cid:15) (cid:47) (cid:47) E e ν φ e ν (cid:15) (cid:15) (cid:47) (cid:47) L ν ∗ E e M r [1] L ν ∗ φ e M r [1] (cid:15) (cid:15) L ν ∗ L e M r (cid:47) (cid:47) L e M r (cid:47) (cid:47) L e ν (cid:47) (cid:47) L ν ∗ L e M r [1] Proof.
We have the following commutative diagrams L i ∗ E τ L i ∗ φ τ (cid:47) (cid:47) L i ∗ L τ (cid:15) (cid:15) E ν φ ν (cid:47) (cid:47) L ν L i ∗ L τ [ − (cid:15) (cid:15) (cid:47) (cid:47) L i ∗ L τ ∗ L Tot π ∗ L r (cid:15) (cid:15) L ν [ − (cid:47) (cid:47) L ν ∗ L M r where the left comes from (2222). The right comes from taking the cone of thediagram in 5.1.25.1.2 when the construction is applied to square (2121). Note that the topand right morphisms of the right diagram are the bottom morphisms from lemma5.4.25.4.2. So, combining these with the diagram from lemma 5.4.25.4.2 gives the followingcommuting diagram (which defines the dashed line): L i ∗ L τ [ − L i ∗ L τ ∗ L Tot π ∗ L r L ν [ − L ν ∗ L M r L i ∗ E τ [ − E E ν [ − L ν ∗ E M r The desired morphism and diagram comes from taking the cones of the horizontalarrows in the bottom square.To see that the associated object φ M /r in D b ( M /r ) is a perfect obstruction theoryfor M /r we recall from 5.2.25.2.2 and 5.3.35.3.3 that φ M r and φ ν are perfect obstructiontheories. We have that E ν [ −
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Oliver Leigh, Matematiska institutionen, Stockholms universitet, 106 91 Stockholm,Sweden
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