MODULI OF CURVES OF GENUS ONE WITH TWISTED FIELDS YI HU AND JINGCHEN NIU Abstract.

MODULI OF CURVES OF GENUS ONE WITH TWISTED FIELDS

YI HU AND JINGCHEN NIU

Abstract. We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger’s desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in [8] and leads to a blowup-free resolution of the stable map moduli of genus two.

1. Introduction

Moduli problems are of central importance in algebraic geometry. Many moduli spaces possess arbitrary singularities [12]. Among them, the moduli MgpPⁿ, dq of degree d stable maps from genusg nodal curves into projective spacesPⁿ are particularly important. We aim to resolve the singularities of MgpPⁿ, dq, that is, to construct a new Deligne-Mumford stack that has smooth irreducible components and normal crossing boundaries and dominates M_gpPⁿ, dq properly and birationally onto the primary component (the component whose general points have smooth domain curves). The problem of resolution of singularities is arguably among the hardest ones in algebraic geometry [3, 4, 9, 10].

The stable map moduli are smooth if g 0 and singular if g ¥1 and n¥2. For g 1, a resolution was constructed by Vakil and Zinger [13], followed by an algebraic approach of Hu and Li [5]. The latter is achieved by constructing a canonical smooth blowup M€^wt₁ of the Artin stack M^wt₁ of weighted nodal curves of genus one. The method of [5] was further developed in [7] to finally establish a resolution in the case ofg2. The resolution of [7] is achieved by constructing a canonical smooth blowupPr₂ of the relative Picard stackP₂ of nodal curves of genus two.

In higher genus cases, the construction of a possible resolution of the stable map moduli may seem formidable. The constructions of the explicit resolutions in [13, 5, 7] rely on certain precise knowledge on the singularities of the moduli. For arbitrary genus, it calls for a more abstract and geometric approach. As advocated by the first author, every singular moduli space should admit a resolution which itself is also a moduli. Following this principle, we interpret the blowup stack€M^wt₁ of [5] as a smooth algebraic stack of stable weighted nodal curves of genus one with twisted fields, and consequently, the resolutionM€1pPⁿ, dq ofM1pPⁿ, dq as a Deligne-Mumford stack of genus one stable maps with twisted fields. The results in this paper are the first step to tackle the arbitrary genus case.

The main theorem of this paper is the following:

Theorem 1.1. There exits a smooth Artin stack M^tf₁ parameterizing the weighted nodal curves of genus one with twisted fields, along with a universal familyC^tfÝÑM^tf₁ and a proper and birational forgetful morphism $ : M^tf₁ ÝÑ M^wt₁ . Moreover, M^tf₁{M^wt₁ is isomorphic to the blowup stack

€M^wt₁ {M^wt₁ .

We construct the strata ofM^tf₁ and the forgetful map$in§2; see (2.13). We then glue the strata of M^tf₁ together using smooth charts in §3 and conclude thatM^tf₁ is a smooth Artin stack and is birational toM^wt₁ in Corollary 3.7. The universal familyC^tfÝÑM^tf₁ is described in Proposition 3.9.

We finally show that M^tf₁{M^wt₁ is isomorphic to M€^wt₁ {M^wt₁ in Proposition 4.3, which implies the properness of$. These results together establish Theorem 1.1.

1

We remark that a direct approach to the properness of$(i.e. without the comparison with the blowup €M^wt₁ {M^wt₁ ) is provided in the proof of [8, Theorem 2.19(p₁)], in a more general setting.

We also point out that there should exist a groupoid, represented byM^tf₁, that sends any schemeS to the set of the flat families of stable weighted nodal curves of genus 1 with twisted fields over S as in (3.33); see Remark 3.10 for some details.

According to [5], the resolution M€1pPⁿ, dq of M1pPⁿ, dq is given by M€1pPⁿ, dq M1pPⁿ, dq _M^wt

1

€M^wt₁ , where

M₁pPⁿ, dq ÝÑM^wt₁ , rC,us ÞÑ rC, c₁puO_Pⁿp1qqs and €M^wt₁ ÝÑM^wt₁ is the canonical blowup. Analogously, we take

M€₁^tfpPⁿ, dq:M₁pPⁿ, dq M^wt₁ M^tf₁,

where M^tf₁ ÝÑ M^wt₁ is the forgetful morphism aforementioned. Theorem 1.1 then leads to the following conclusion immediately.

Corollary 1.2. M€₁^tfpPⁿ, dq is a proper Deligne-Mumford stack and is isomorphic to M€1pPⁿ, dq. Via the above isomorphism and applying [5], one sees that the stack M€₁^tfpPⁿ, dq provides a resolution of M₁pPⁿ, dq. Nonetheless, without relating to M€1pPⁿ, dq, we can directly prove the resolution property of M€₁^tfpPⁿ, dq by investigating the local equations ofM₁pPⁿ, dqin [5] and their pullbacks to M€₁^tfpPⁿ, dq; see Remark 3.8.

The methods and ideas of this paper are essential to the development in [8] and forthcoming works. Based on the construction of M^tf₁, we introduce the theory of stacks with twisted fields (STF) in [8, Theorem 2.19]. To be somewhat more informative, we work on a smooth stack M that has a stratification indexed by a set Γ of graphs similar to (2.10); see [8, Definition 2.15].

The graphs in Γ need not to come from the dual graphs as in (2.10), but the stratification of M should resemble (3.2) locally. Moreover, Γ need not to consist of trees, but it should contain necessary information on the notion of the (weighted) level trees in Definition 2.1 so that we can add the twisted fields to the strata of M parallel to (2.12) and obtain a new stack M^tf; see [8, Definition 2.17]. Such M^tf enjoys desirable properties as in Corollary 3.7 and Remark 3.8. As an application of the STF theory, in [8], we construct a smooth Artin stack P^tf₂ of genus 2 nodal curves with line bundles and twisted fields, along with a proper and birational forgetful morphism P^tf₂ ÝÑP₂, such that

M€₂^tfpPⁿ, dq M₂pPⁿ, dq P2P^tf₂ ÝÑM₂pPⁿ, dq

provides a resolution. Further, we expect that they can be extended to arbitrary genus, as far as the existence of moduli of nodal curves with twisted fields is concerned. This is the main motivation of the current article.

In a related work [11], D. Ranganathan, K. Santos-Parker, and J.Wise provide a different modular perspective ofM€1pPⁿ, dq using logarithmic geometry.

Acknowledgments. We would like to thank Dawei Chen, Qile Chen, and Jack Hall for the valuable discussions.

Convention. The subscript “1” of the relevant stacks indicating the genus appears only in §1 and will be omitted starting§2, as we only deal with the genus 1 case in this paper. In particular, we will denote by

M^wt and M^tf the aforementioned stacksM^wt₁ andM^tf₁, respectively.

2. Set-theoretic descriptions

In§2.1, we discuss the combinatorics of the dual graphs of nodal curves and introduce the notion of the weighted level trees. They will be used to defineM^tf set-theoretically in §2.2.

2.1. Weighted level trees. Letγ be a rooted tree, i.e. a connected finite graph that contains no cycles, along with a special vertex o, called the root. The sets of the vertices and the edges of γ are denoted by

Verpγq and Edgpγq,

respectively. The set Verpγq is endowed with a partial order, called thetree order, so that v¡v¹ if and only ifvv¹ and vbelongs to a path betweenoand v¹. The rootois thus the unique maximal element of Verpγq with respect to the tree order.

For each ePEdgpγq, we denote byv_ePVerpγq the endpoints ofesuch thatv_e ¡v_e.Then, every vertex vPVerpγqztou corresponds to a unique

e_v PEdgpγq satisfying v_ev v.

The tree order on Verpγq induces a partial order on Edgpγq, still called thetree order, so that e¡e¹ ðñ v_e ©v_e1.

We call a pairτpγ,wqconsisting of a rooted tree γ and a function w: Verpγq ÝÑZ_¥0

a weighted tree. For such τ, we write Verpτq Verpγq and Edgpτq Edgpγq. The set of all the weighted trees is denoted byT_R^wt.

We call a map`: Verpγq ÝÑR_¤0 satisfying

`<sup>1</sup>p0qtou and `pvq¡`pv<sup>1</sup>q whenever v¡v<sup>1</sup> a level map. For eachiP`pVerpγqqzt0u, let

(2.1) i⁷min kP` Verpγq

: k¡i( ,

i.e. the level i⁷ is right “above” the leveli; see Figure 1. We remark that a rooted tree along with a level map is called a level graph with the root as the unique top levelvertex in [1,§1.5].

Definition 2.1. We call the tuple

t γ, w: VerpγqÝÑZ_¥0, `: VerpγqÝÑR_¤0

a weighted level tree if pγ,wqPT_R^wt and ` is a level map.

For every weighted level tree t as above, we write VerptqVerpγq and EdgptqEdgpγq. Set mmptq max `pvq:vPVerptq,wpvq¡0(

p ¤0q, Edgyptq ePEdgptq:`pv_eq¡m(

€ Edgptq . For any two levels i, jPR_¤0, we write

(2.2) Li, jM^t` Verptq

Xpi, jq, Ji, jM^t ` Verptq

Xri, jq. For every eP yEdgptq, let

`peq maxt`pv_eq,mu PJm,0M^t . For each leveliPJm,0Mt, we set

(2.3) E_i E_iptq eP yEdgptq:`peq¤i `pv_eq( .

In other words, E_i consists of all the edges crossing the gap between the levels iand i⁷.

We remark that all the notions in the preceding paragraph depend on the weighted level treet, although we may hereafter omit t in any of such notions when the context is clear.

Every weighted level treet determines a unique index set

Iptq I ptq \Imptq \Iptq, where I ptqJm,0Mt, ImptqteP yEdgptq:`pv_eq mu, Iptq Edgptqz yEdgptq (2.4) .

The setI ptq becomes empty ifm0, i.e. the root ois positively weighted. As mentioned before, we may simply write

IIptq, I Iptq, Im Imptq when the context is clear.

For eachI€I (possibly empty), let

I_m IXIm, IIXI. We construct a weighted level tree

(2.5) t_pIq τ_pIq, `_pIq

γ_pIq,w_pIq, `_pIq as follows:

the rooted treeγ_pIq is obtained via the edge contraction π_pIq: VerpγqVer γ_pIq such that the set of the contracted edges is

(2.6) EdgptqzEdgpt_pIqq eP yEdgptqzIm

\Im:J`peq, `pv_eqMt€I (

\I; the weight functionw_pIq is given by

w_pIq: Ver γ_pIq

ÝÑZ_¥0, w_pIqpvq ¸

v¹Pπ_pIq¹pvq

wpv¹q;

the level map`_pIq is such that for any ePEdg γ_pIq

€Edgptq ,

`_pIqpv_eq

$'

&

'%

mintiPI zI : i¥`pvq @vPπ_pI¹_qpv_equ ifeP yEdgptqzIm,

minpI zI q ifePI_m,

maxt`pvq: vPπ_pI¹_qpv_equ ifePpImzImq\I.

It is a direct check thatτ_pIqis a weighted tree and`_pIqsatisfies the criteria of a level map, hence (2.5) gives a well defined weighted level tree.

The construction oft_pIq implies I t_pIq

I zI , m t_pIq

min pI zI q\t0u , Im t_pIq

ePImzIm:`pv_e q¡m t_pIq( , I t_pIq

IzI\ ePImzIm:`pv_eq¤m t_pIq( . (2.7)

Intuitively, the weighted level tree t_pIq is obtained from t by contracting all the edges labeled by I, then lifting all the verticesv withe_vPI_m to the level m, and finally contracting all the levels inI . Sucht_pIq will be used to describe the local structure of the stack M^tf in§3.

Definition 2.2. Two weighted level trees tpγ,w, `q and t<sup>1</sup> pγ<sup>1</sup>,w<sup>1</sup>, `¹q are said to be equivalent, written as tt¹,if

(E1) pγ,wqpγ¹,w¹q as weighted trees;

(E2) for any v, wPVerpγq satisfying`pvq`pwq¥mptq, we have

`<sup>1</sup>pvq `¹pwq;

(E3) for any v, wPVerpγq satisfying`pvq¡`pwq and `pvq¥mptq, we have

`<sup>1</sup>pvq ¡`¹pwq.

0 1r1s 2r1s 3r2s 1⁷

1 2⁷ 2 3r1s 3⁷ m 3

e₃ e2

e1

ov₁v₂

v1

v2 v₃

v₃ : vertex of weight 0

: vertex of positive weight

Figure 1. A weighted level tree with chosen v₁,v₂,and v₃

It is a direct check thatis an equivalence relation on the set of weighted level trees. Intuitively, this equivalence relation records the relative positions of the verticesaboveorinthe level mptq; see Figure 1 for illustration.

We denote by T_L^wt the set of the equivalence classes of the weighted level trees. There is a natural forgetful map

(2.8) f:TL^wt ÝÑTR^wt, rγ,w, `s ÞÑ pγ,wq, which is well defined by Condition (E1) of Definition 2.2.

Iftt¹, thenmptqmpt¹q, and there exists a bijection φ_t1,t:P Iptq

ÝÑP Ipt¹q

, φ_t1,tpIq `<sup>1</sup> `¹pI q

\I_mptq\I, wherePpq denotes the power set. The next lemma follows from direct check.

Lemma 2.3. If tt¹, then t_pIqt¹_pφ

t1,tpIqq for anyI€Iptq.

2.2. Twisted fields. For every genus 1 nodal curve C, its dual graph γ_C has either a unique vertex ocorresponding to the genus 1 irreducible component of C or a unique loop. In the former case,γ_C can be considered as a rooted tree with the rooto; in the latter case, we contract the loop to a single vertex oand obtain a rooted tree with the root o. Such defined rooted tree is denoted by γC and called thereduced dual tree of C (c.f. [5, §3.4]). We call the minimal connected genus 1 subcurve ofC thecoreand denote it byC_o. Other irreducible components ofCare smooth rational curves and denoted by C_v,vPVerpγ_Cqztou. For every incident pair pv, eq, let

(2.9) qv;ePCv

be thenodal point corresponding to the edge e.

Let M^wt be the Artin stack of genus 1 stable weighted curves introduced in [5, §2.1]. Here the subscript “1” indicating the genus is omitted as per our convention. The stack M^wt consists of the pairspC,wq of genus 1 nodal curves C with non-negative weightswPH²pC,Zq, meaning that wpΣq ¥0 for all irreducible Σ€C. Here pC,wq is said to be stable if every rational irreducible component of weight 0 contains at least three nodal points. The weight of the corewpC_oqis defined as the sum of the weights of all irreducible components of the core.

EverypC,wqPM^wt uniquely determines a function

w: VerpγCq ÝÑZ_¥0, vÞÑwpCvq,

which makes the pair pγC,wq a weighted tree, called the weighted dual tree. Thus, the stackM^wt can be stratified as

(2.10) M^wt §

τPT_R^wt

M^wt_τ §

τPT_R^wt

pC,wqPM^wt:pγC,wqτ( .

If the sum of the weights of all vertices is fixed, the stability condition of M^wt then guarantees there are only finitely many τPT_R^wt so that M^wt_τ is non-empty.

Givenτpγ,wqPT_R^wt and ePEdgpτq, let

L_e ÝÑM^wt_τ

be the line bundles whose fibers over a weighted curve pC,wq are the tangent vectors of the irreducible componentsC_v

e at the nodal pointsq_v

e;e, respectively. We take (2.11) L_e L_e bL_e ÝÑM^wt_τ , L^©_e â

e¹PEdgpτq, e¹©e

L_e1 ÝÑM^wt_τ . For any direct sum of line bundlesV`mL¹_m (over any base), we write

˚PpVq: x,rvms

PPpVq:vm0 @m( . For any morphismsM1, . . . , MkÝÑS, we write

¹

1¤i¤k

pM_i{Sq :M₁SM₂S SM_k.

With notation as above, givenτPT_R^wt and rts τ, `

PT_L^wt, let

$:M^tf_rts ¹

iPI ptq

$%

˚P

à

ePEdgptq, `pveqi

L^©_e M M^wt_τ

,-

ÝÑM^wt_τ ,

E_rts ¹

iPI ptq

$% P

à

ePEi

L^©_e M M^wt_τ

,-

ÝÑM^wt_τ , (2.12)

where I ptq, L^©_e, and E_i are as in (2.4), (2.11), and (2.3), respectively. It is straightforward that both bundles in (2.12) are independent of the choice of the weighted level tree t representing rts.

Since

ePEdgptq:`pv_eqi(

€E_i @iPI ptq,

we see that M^tf_rts is a subset ofE_rts. In addition, since each stratum M^wt_τ is an algebraic stack, so areM^tf_rts and E_rts.

Using (2.12) and (2.10), we define

(2.13) M^tf: §

rtsPTL^wt

M^tf_rts ÝÑ^$ M^wt.

This is the set-theoretic definition of the proposed stackM^tf as well as the forgetful map in Theo- rem 1.1. For any xPM^wt_τ , the points of the fiber M^tf_rts

x are called the twisted fieldsoverx.

Remark 2.4. By (2.13),M€₁^tfpPⁿ, dq in Corollary 1.2 consists of the tuples C,u,rts, η

,

where pC,uq are stable maps in M₁pPⁿ, dq, rtsare the equivalence classes of weighted level trees satisfying frts γC, c1puOPⁿp1qq

, and η are twisted fields over C, c1puOPⁿp1qq .

3. The stack structure ofM^tf

In§3, we showM^tf is naturally a smooth Artin stack and describe its universal family.

3.1. Twisted charts. We first fixrts

γ,w, `

PT_L^wt and xPM^tf_rts, and write τ frts pγ,wq PT_R^wt, pC,wq $pxq PM^wt_τ . Since M^wt is smooth, we take an affine smooth chart

V V_$pxqÝÑM^wt containing pC,wq.

As in [5,§4.3] and [7, §2.5], there exists a set of regular functions

(3.1) ζePΓpOVq with ePEdgpγq,

called the modular parameters, so that for each ePEdgpγq, the locus Z_e tζ_e0u €V

is the irreducible smooth Cartier divisor on V where the node labeled by eis not smoothed. For any I€IIptq, let

V_pIq: ζ_e10 :e¹P EdgpγqzEdgpγ_pIqq (

€V, V_τpIq:V_pIqX ζ_e0 :ePEdg γ_pIq (

€V_pIq€V. Then,V_pIq is an open subset ofV. Shrinking V if necessary, we see that

V_τpIq Pπ₀ M^wt_τ

pIq XV (3.2) ,

where π0 denotes the set of the connected components. In particular, V_τpIq can be considered as a smooth chart of the stratum M^wt_τ

pIq. Rigorously, the sets I and I depend on the choice of the weighted level tree t representing rts, however, V_pIq and V_τpIq are independent of such choice; see Lemma 2.3.

Given a set of the modular parameters as in (3.1), we may extend it to a set of local parameters on V centered atpC,wq:

(3.3) tζeuePEdgpγqY tςjujPJ with pC,wq 0 : p0, . . . ,0q, whereJ is a finite set. We do not impose other conditions on ςj.

For eachePEdgpγq, we set

Bζe : pdζ_eq^_PΓpV;TM^wtq.

Lemma 3.1. For every I€I and every ePEdgpγ_pIqq, the restriction of Bζe to V_τpIq is a nowhere vanishing section of the restriction of the line bundleLe in (2.11) to V_τpIq.

Proof. Since the restriction of dζe toZ_etζe0u(V_τpIq) is identically zero, we observe that the restriction ofBζe toZ_eis a nowhere vanishing section of the normal bundle ofZ_e. It is a well-known fact of the moduli of curves that the normal bundle ofZ_e is L_e; see [2, Proposition 3.31].

For each leveliPI I ptq, we choose a special vertex

(3.4) viPVerpγq s.t. `pviqi.

We then denote by ei,e_i , and v_i respectively the edges and the vertex satisfying

(3.5) e_ie_vi, v_i v_ei, e_i e_v

i . Each iPI determines a strictly increasing sequence

(3.6) ir0s:i   ir1s:`pv<sub>i</sub> q   ir2s:`pv_ir1sq   .

We would like to remark that ir1s and i⁷ in (2.1) need not to be the same; see Figure 1 for illustration. This sequence is finite, as there is a unique steph satisfying irhs0.

By Lemma 3.1, there exist λePA with eP yEdgptq so that the fixed xPM^tf_rts over pC,wq can uniquely be written as

x

0 ; ¹

iPI

λ_e â

e©eBζe|0

:`peqi

, where

λe0 @eP yEdgptqzIm, λei1 @iPI , λe0 @ePIm. (3.7)

Let

U_x€A^I A^{Edgyptqztei:iPI u}A^I A^J be an open subset containing the point

(3.8) y_x : 0,pλ_eq_{eP yEdgptqzte}

i:iPI u,0,0 . The coordinates onU_x are denoted by

(3.9) pε_iqiPI ,pu_eq_{eP yEdgptqztei:iPI u},pz_eqePI,pw_jqjPJ

. For anyI€I, we set

U_x;pIq εi0@iPI ; ue0@ePIm ; ze0 @ePI(

€U_x, U_x;rt

pIqsU_x;pIqX εi0@iPI zI ; ue0 @ePImzIm; ze0 @ePIzI( . This gives rise to a stratification

(3.10) U_x §

I€I

U_x;rt

pIqs.

We remark that neitherU_x nor its stratification (3.10) depends on the choice of the weighted level tree t representing rts, even though the sets I and I depend on such choice. We also notice that U_x;pIq is an open subset ofU_x, but the strataU_x;rt

pIqs are not open unlessII. For eachiPI , we take

uei :1.

By (3.7), shrinking U_x if necessary, we have

(3.11) uePΓ OUx

@ eP yEdgptqzIm.

With the local parameters ζe and ςj as in (3.3), we construct a morphism θx :U_xÝÑV pÝÑM^wtq

given by

θ_xζ_e u_eu_e

`pequ_e

`peqr1s u_e

veu_e

`pvequ_e

`pveqr1s ¹

iPJ`peq,`pveqM

ε_i @eP yEdgptq; θ_xζ_e z_e @ePIEdgptqz yEdgptq; θ_xς_j w_j @jPJ.

(3.12)

The numerator and the denominator in the first line of (3.12) are both finite products, because (3.6) is always a finite sequence.

For anyI€I, it follows from (3.11) and (3.12) that

(3.13) θ_x U_x;pIq

€V_pIq, θ_x U_x;rt

pIqs

€V_τpIq, whereV_pIq and V_τpIq are described before (3.2).

FixI€I (I may be empty). With

rt_pIqs rτ_pIq, `_pIqs PTL^wt

as in (2.5), M^tf_rt

pIqs as in (2.13), and the chartV_τpIqÝÑM^wt_τ

pIq as in (3.2), let Φ_x;pIq:U_x;rt

pIqs ÝÑV_τpIqM^wt_τ

pIqM^tf_rt

pIqs ÝÑM^tf_rt

pIqs

be the morphism so that for anyyPU_x;tpIq, (3.14) Φ_x;pIqpyq

θ_xpyq; ¹

iPI zI

µ_e;i;Ipyq â

e©e, J`peq,`pv_eqM‚I

Bζe

θxpyq

:ePE_i ,

where

µe;i;I

ueu_e

`pequ_e

`peqr1s u_e

i u_e

ir1s

±

e¡ei;J`peq,`pv_eqM€I θ_xζ_e

±

e¹¡e;J`pe<sup>1</sup>q,`pv_e1qM€I θ_xζ_e1

¹

hPJ`peq,iM

εh

for alliPI zI andePE_i. Similar to (3.12), the products in the first pair of parentheses above are both finite products.

By (3.13), the description ofV_pIq above (3.2), and (2.6), we see that µ_e;i;IPΓ OU_x;pIq

@I€I, iPI zI , ePE_i. Moreover, by (3.11),

(3.15) µ_e;i;I|Ux;rtpIqs

#0 if `_pIqpv_eq  i, PΓ O_Ux;rt

pIqs

if `_pIqpv_eq i.

This, along with (3.13), Lemma 3.1, and (2.12), implies Φ_x;pIq is well-defined.

The morphisms Φ_x:pIq,I€I, together determine

Φ_x:U_x ÝÑM^tf, Φ_xpyq Φ_x;pIqpyq ifyPU_x;rt

pIqs.

We remark that Φ_x;pIq and Φx are also independent of the choice of the weighted level tree t representingrts. Moreover, we observe that

(3.16) Φ_xpy_xq Φ_x;pHqpy_xq x @xPM^tf, wherey_xPU_x is given in (3.8).

A priori Φx is just amap, for the set-theoretic definition (2.13) ofM^tf does not describe its stack structure, although each M^tf_rt1s is a stack. In §3.2, we will show such Φ_x patch together to endow M^tf with a smooth stack structure. Each Φ_x will hereafter be called a twisted chart centered at x (lying over VÝÑM^wt), although rigorously it becomes a chart of M^tf only after Corollary 3.7 is established.

Lemma 3.2. For every I€I, Φ_x;pIq:U_x;rt

pIqsÝÑ M^tf_rt

pIqs of (3.14) is an isomorphism to an open subset of M^tf_rt

pIqs.

Proof. For any iPI pt_pIqq I zI , notice that every edge in E_i of the weighted level tree t is not contracted in the construction oft_pIq (c.f. (2.6)). Thus,

E_i €Edg t_pIq

@ iPI pt_pIqq.

In particular, the edgesei,iPI pt_pIqq, can be used as the special edges oft_pIq. For conciseness, let Ert_pIqs: yEdgpt_pIqqH

e_i:iPI pt_pIqq(

\Impt_pIqq §

iPI pt_pIqq

ePEdgpt_pIqq:`_pIqpv_eqi, eei

( € yEdgpt_pIqq€ yEdgptq

; see (2.7) for notation.

LettζeuePEdgpγqYtςjujPJ be a set of the local parameters onV centered at$pxq as in (3.3). By the definition of V_τpIq above (3.2),

(3.17) tζ_euePEdgptqzEdgpt_pIqq\ tς_jujPJ

is a set of local parameters ofV_τpIq.

Recall that there existλePA,eP yEdgptq, such that

x

0 ; ¹

iPI

λ_e â

e©e

Bζe|0

:`pv_eqi

as in (3.7). Let U_x;ErtpIqs be an open subset of pAq^ErtpIqs such that pλ_eqePErt_pIqs P U_x;ErtpIqs € pAq^ErtpIqs. The coordinates ofU_x;ErtpIqs are denoted by

pµ_eq_ePErtpIqs. In addition, we set

µei 1 @ iPI pt_pIqq, µe0 @ePImpt_pIqq.

Thus, the function µ_e is defined for all eP yEdgpt_pIqq, and is nowhere vanishing onU_x;EptpIqq for all eP yEdgpt_pIqqzImpt_pIqq.

The smooth chart V_τpIqÝÑM^wt_τ

pIq in (3.2) induces a smooth chart U_x;¹ _rt

pIqs :V_τpIq U_x;ErtpIqs ÝÑM^tf_rt

pIqs

given by

z,pµ_eqePEpt_pIqq ÞÑ

z, ¹

iPI pt_pIqq

µ_e â

ePEdgpt_pIqq,e©e

Bζe|z

:`_pIqpv_eqi .

We will construct a morphism

(3.18) Ψ_x;pIq :U_x;¹ _rt

pIqs ÝÑU_x;rt

pIqs

such that Φ_x;pIqΨ_x;pIqand Ψ_x;pIqΦ_x;pIqare both the identity morphisms, which will then establish Lemma 3.2.

Given z,pµeqePEpt_pIqq PU_x;t¹

pIq,we denote its image by y :Ψ_x;pIq z,pµ_eqePEpt_pIqq

PU_x;rt

pIqs,

which is to be constructed. With the coordinates onU_x as in (3.9), we set (3.19) zepyqζepzq @ePI, wjpyqςjpzq @jPJ.

By (3.2), we see that

z_epyqζ_epzq0 ðñ ePIzI €Ipt_pIqq (3.20) .

The rest of the coordinates of y are much more complicated; we describe them by induction over the levels in I I ptq. More precisely, we will show that εipyq with iPI and uepyq with eP yEdgptqzte_i:iPI u are all rational functions in ζ_e1pzq and µ_e2, satisfying

(3.21) v

ε_ipyq0 ô iPI zI w

and v

u_epyq0 ô ePImzI_mw . In particular, (3.21) and (3.20) imply yPU_x;rt

pIqs, i.e. Ψ_x;pIq is well-defined.

The base case of the induction is for the level

i₀:maxI ptq. We take

(3.22) ε_i0pyq ζ_ei

0pzq. By (3.2), we see thatε_i0pyq satisfies (3.21). We take

u_ei

0pyq 1.

For anyeei0 with `peqi0, we set

(3.23) u_epyq

#µe ifi0RI i.e.i0PI pt_pIqq, ePEpt_pIqq

ζepzq

ζei0pzq ifi0PI i.e.ei0PEdgptqzEdgpt_pIqq .

Ifi₀PI , then by (3.17) and (3.2), we haveζ_epzq0 if and only if (mi₀and)ePImzI_m. Ifi₀RI , then µe0 if and only if (mi0 and) ePImzIm. We thus conclude that uepyq satisfies (3.21) for all eP yEdgptq with `peq i0. Moreover, such εi0pyq and uepyq are obviously rational functions in ζ_e1pzq and µ_e2. Hence, the base case is complete.

Next, for any iPI , assume that all εkpyq withk¡i and alluepyq with eP yEdgptq and `peq ¡i have been expressed as rational functions inζ_e1pzq andµ_e2, satisfying (3.21).

For the level i, we first construct ε_ipyq. The construction is subdivided into three cases.

Case 1. If iRI , then set

ε_ipyq 0.

Obviously this satisfies (3.21).

Case 2. If iPI and Ji, ir1sM‚I , theneiPEpt_pIqq, hence µei0. Let pi:min Ji, ir1sMzI

PI pt_pIqq. Intuitively,piis the level containing the image ofv_i in t_pIq. Thus,

`_pIqpeiq pi.

Letεipyq be given by

µeiεipyqu_e

ipyqu_e

ir1spyq u_e

pipyqu_e

pir1spyq

±

e¡e_pi,J`peq,`pv_eqMt€Iζepzq

±

e¹¡ei,J`pe<sup>1</sup>q,`pv_e1qM^t€Iζ_e1pzq¹

hPLi,piMt

ε_hpyq. (3.24)

The inductive assumption implies that all ε_hpyq with hPLi,piM^t, as well as all u_e

irhspyq and u_e

p irhspyq withh¥0, are non-zero and are rational functions inζ_e1pzqandµ_e2. By (3.17) and (2.6), we also see that all ζ_epzq and ζ_e1pzq in (3.24) are non-zero. Therefore, such defined ε_ipyq is a rational function inζ_e1pzq and µ_e2, satisfying (3.21).

Case 3. If Ji, ir1sM€I (hence iPI ), then we seee_iPEdgptqzEdgpt_pIqq; c.f. (2.6). Intuitively, this means e_i is contracted in the construction of t_pIq. By the description of V_τpIq above (3.2), we see that ζ_eipzq0. Letε_ipyqbe given by

ζ_eipzq ε_ipyq ¹

hPLi,piM^t

ε_hpyq. (3.25)

Mimicking the argument in Case 2, we conclude thatεipyqis a rational function in ζ_e1pzq and µ_e2, satisfying (3.21).

Next, we construct u_epyqforeP yEdgptq with `peqi. Set ueipyq 1.

Foreei, the construction is subdivided into two cases.

Case A. If Ji, `pv_eqM‚I , then

ePEpt_pIqq\Impt_pIqq yEdgpt_pIqqzte_i:iPI pt_pIqqu ,

hence µ_e exists, and µ_e0 if and only if ePImpt_pIqq. In Case A, sinceJi, `pv_eqM‚I , (2.7) further implies

(3.26) µ_e0 ðñ ePImzI_m.

Let

κκe:min Ji, `pv_eqMzI

PI pt_pIqq .

Since Ji, κM€I , we have

`_pIqpeq κ.

Letu_epyq be given by

(3.27) µ_eu_epyqu_e

ipyqu_e

ir1spyq u_e

κpyqu_e

κr1pyq s

±

e¡eκ,J`peq,`pv_eqMt€Iζepzq

±

e¹¡e,J`pe<sup>1</sup>q,`pv_e1qMt€Iζ_e1pzq ¹

hPJi,κMt

ε_hpyq. We observe that if iRI , then κiand hence ±

hPJi,κM^tεhpyq 1; if iPI , then the previous con- struction ofε_ipyq, along with the inductive assumption, guarantees ±

hPJi,κMtε_hpyq 0. Mimicking the argument of Case 2 of the construction of εipyq and taking (3.26) into account, we see that u_epyq determined by (3.27) is a rational function in ζ_e1pzq and µ_e2, and it satisfies (3.21).

Case B. If Ji, `pv_e qM€I , then (2.6) gives

(3.28) ePEdgpt_pIqq i.e.ζ_epzq0

ðñ ePImzI_m. Letu_epyq be given by

ζepzq uepyq u_e

ipyqu_e

ir1spyq ue

vepyqu_e

`pveqpyqu_e

`pveqr1spyq ¹

hPJi,`pveqM^t

εhpyq.

(3.29)

Once again, mimicking the argument of Case A of the construction of uepyq, and taking (3.28) as well as the description of V_τpIq right before (3.2) into account, we see that uepyq determined by (3.29) is a rational function inζ_e1pzqand µ_e2, and it satisfies (3.21).

The cases 1-3, A, and B together complete the inductive construction of Ψ_x;pIq. Moreover, comparing

(3.19) with the second line of (3.12),

(3.22), the second case of (3.23), (3.25), and (3.29) with the first line (3.12),

the first case of (3.23), (3.24), and (3.27) with the expressions ofµ_e;i;I right after (3.14),

we observe that Ψ_x;pIq is the inverse of Φ_x;pIq.

Corollary 3.3. Φ_x:U_xÝÑM^tf is injective.

Proof. This follows from Lemma 3.2 and the stratification (3.10) and (2.13) directly.

3.2. Stack structure. In this subsection, we will show the twisted charts Φ_x patch together to endowM^tfwith a smooth stack structure; c.f. Proposition 3.6 and Corollary 3.7. Note that a priori, Φx depends on the choices of the special vertices tviuiPI (3.4) and of the local parameters (3.3).

Nonetheless, Lemmas 3.4 and 3.5 below will guarantee that such choices do not affect the proposed stack structure of M^tf, hence will make the proof of Proposition 3.6 more concise.

Lettw_i:iPI u be another set of the special vertices satisfying (3.4), and w_i ,d_i, and d_i be the analogues ofv_i ,ei, ande_i in (3.5), respectively. As in (3.6), each level iPI similarly determines a finitesequence

ix0y i ix1y `pw<sub>i</sub> q  ix2y `pw_ix1yq   . We take an open subset

U^a_x €A^I A^{Edgyptqztdi:iPI u}A^IA^J with the coordinates

pδ_iqiPI ,pu^a_eq_{eP yEdgptqztdi:iPI u},pz^a_eqePI,pw_j^aqjPJ

. as in (3.9), and then construct

θ^a_x:U^a_x ÝÑV, µ^a_e;i;IPΓ O_U^a,

x;pIq

, Φ^a_x:U^a_xÝÑM^tf parallel to (3.12) and (3.14). LetUΦ^a_xpU^a_xqXΦ_xpU_xq.