Smoothability of relative stable maps to stacky curves

Using log geometry, we study smoothability of genus zero twisted stable maps to stacky curves relative to a collection of marked points. One application is to smoothing semi-log canonical fibered surfaces with marked singular fibers.


Introduction
Consider the moduli space Maps • n of degree n maps f : C → P 1 from a smooth rational curve C such that f is unramified over infinity.Marking the preimage f −1 (∞) = {p 1 , . . ., p n } of such a map induces a locally closed embedding Maps • n → M 0,n (P 1 , n) of the S n -torsor Maps • n → Maps • n parametrizing such f : C → P 1 as well as a labeling of f −1 (∞) into the space of n-pointed degree n genus 0 stable maps to P 1 .The image of this embedding is the locus of stable maps (f : C → P 1 , p 1 , . . ., p n ) such that C is smooth and f (p i ) = ∞ for all i.Let Maps n denote the closure of this locus.
Question 1.Is there a combinatorial description of the boundary Maps n \ Maps • n ?There are several natural combinatorial conditions (see Proposition 2.4 and the discussion preceding it) that are necessary for a stable map (f : C n:1 − −− → P 1 , p 1 , . . ., p n ) to lie inside Maps n : (1) The evaluations satisfy f (p i ) = ∞ for all i.
(2) Every point of f −1 (∞) ⊂ C either is a marked point or is on a contracted component.
(3) For each maximal connected closed subvariety T ⊂ C contracted to ∞ by f , we have # marked points on T = ramification index of f at the nodes T ∩ (C \ T ).
(3) For each maximal connected closed subvariety T ⊂ f −1 (x), we have where the first sum is over marked points contained in T and e f (q) denotes the ramification of f | C\T at the point q (see Figure 1 ).
This question in much greater generality was studied by Gathmann [Gat02, Proposition 1.14], building off previous work of Vakil [Vak00,Theorem 6.1].In particular, Gathmann showed that Conditions ( * ) relative to ∞ are both necessary and sufficient for a stable map to lie in M (d 1 ,...,d n ) .As a consequence, the set of points of M 0,n (P 1 , d) satisfying Conditions ( * ) relative to ∞ are the points of an irreducible closed substack.See also the balancing condition of Gross-Siebert [GS13, Definition 1.12 and Lemma 1.15].
In [AV02], Abramovich and Vistoli introduced moduli spaces of twisted stable maps, which allows the target to instead be a Deligne-Mumford stack.To form a compact moduli space, the source curves obtain a stacky structure.Let K 0,n (X, d) denote the moduli space of n-pointed genus 0 degree d twisted stable maps to a Deligne-Mumford stack X.The goal of this paper is to study the analogue of Question 1 for genus 0 twisted stable maps to a weighted projective line P(a, b), or more generally a genus 0 Deligne-Mumford curve.
When X is P(a, b) and ∞ ∈ P(a, b) is a fixed point away from [0 : 1] and [1 : 0], we have the following (see Theorem 1.5 for a more general result).For this example, the reader can keep in mind M 1,1 = P(4, 6) and the point j = ∞ parametrizing a nodal elliptic curve.
Theorem 1.2.Let (f : C → P(a, b), p 1 , . . ., p n ) be an n-pointed genus 0 twisted stable map such that the coarse map (g : C → P 1 , q 1 , . . ., q n ) satisfies Conditions ( * ) with respect to ∞.Then f is smoothable in a family with generic fiber satisfying: f (p i ) = ∞ and f is ramified to order d i at p i for all i = 1, . . ., n.
In what follows, we set up notation needed to state our more general results.Let (X, x 1 , . . ., x r ) be a smooth and proper 1-dimensional genus 0 Deligne-Mumford curve, and suppose the x i ∈ X are points where the coarse moduli space map is étale (equivalently, the x i have the same stabilizer as the generic point of X ).For each j = 1, . . ., r, let Γ j = (d j1 , . . ., d jn j ) be a tuple of positive integers, and fix n 0 ≥ 0. Set n = r j=0 n j and and let Γ = (n 0 , {Γ 1 , x 1 }, . . ., {Γ r , x r }) be the tuple of combinatorial data.
Definition-Notation 1.3.For any tuple of combinatorial data Γ , let M Γ (X) be the locally closed substack of K 0,n (X, d) parametrizing n-pointed genus 0 degree d twisted stable maps (f : C → X, {{p jk } n j k=1 } r j=0 ) such that (1) the p jk are marked points with stabilizer of order a jk lying over smooth points q jk of the coarse space C, (2) for each j > 0, the image of p jk is f (p jk ) = x j , (3) for each j > 0, the coarse map h : C → X is ramified to order d jk at q jk , and (4) C is smooth.
Similarly, we let N Γ (X) be the locally closed substack of K 0,n (X, d) of maps satisfying conditions (1), (2) and (3) above as well as the condition (4 ) C is smooth in a neighborhood of f −1 (x j ) for all j.
Theorem 1.5.Let X and Γ be as above.Then we have an equality N Γ (X) = K Γ (X).That is, every twisted stable map whose coarse moduli map satisfies the relative condition for {Γ j , x j } for each j = 1, . . ., r is smoothable to a twisted stable map parametrized by N Γ (X).In particular, K Γ (X) is the set of points of a closed substack of K 0,n (X, d).
Remark 1.6.Note that Theorem 1.5 is local on the target.Indeed, the definitions of both K Γ (X) and N Γ (X) are in terms of local conditions around the points x i ∈ X.Therefore, the theorem has a natural generalization to higher-genus maps to a higher-genus target X provided we only consider those maps for which the 1-dimensional components of the preimages f −1 (x i ) are rational curves.In this case, the theorem reads that any such map which is also contained in K Γ (X) is smoothable in a family with generic fiber contained in N Γ (X).
When the target is a weighted projective line, we obtain the following stronger statement.
Theorem 1.7.Let X = P(a, b) be a weighted projective line, and let Γ be a tuple of combinatorial data as above.Then we have an equality M Γ (X) = K Γ (X).That is, every twisted stable map which satisfies the relative condition for {Γ j , x j } for each j = 1, . . ., r is smoothable to a family of stable maps from a smooth rational curve satisfying f (p jk ) = x j and with ramification d jk at p jk .

Applications to moduli of fibered surfaces
Our original motivation for writing this paper came from studying compactifications of the moduli space of fibered surfaces.Twisted stable maps are used in [AV00,AB19b] to construct a compactification F v g,n (γ, ν) of the moduli space of genus γ fibrations over a genus g curve with ν marked sections and n marked fibers.The objects of the boundary are certain semi-log canonical unions of birationally fibered surfaces called twisted surfaces.This compactification is closely related to the compactification via stable log varieties from the minimal model program.In [AB19b, Section 1.4], we proposed the problem of using log geometry to give the main component a moduli-theoretic interpretation and classify the boundary components.The present paper solves this problem for elliptic fibrations with marked singular fibers (we refer the reader to [AB21] and [AB19a, Section 4]).A key observation is that Conditions ( * ) relative to ∞ ∈ M 1,1 as well as the choice of stabilizers on the marked points translate to conditions on the configuration of singular fibers on the components of the twisted elliptic surface; see [AB19a, Propositions 4.1 and 4.4].
Theorem 1.8.Theorem 1.7 gives necessary and sufficient combinatorial conditions for a twisted elliptic surface over a genus 0 curve with marked singular fibers to be smoothable to an elliptic surface over P 1 with marked singular fibers.
For convenience, we work over an algebraically closed field of characteristic 0.
along which to attach C 2 , and the configuration of the n + 1 special points on C 2 (which is parametrized by M 0,n+1 ), yielding the dimension count dim Maps n + n − 2.

Genus 0 relative stable maps to (P 1 , ∞)
In this section, we prove the special case of Theorem 1.2 where the target is P 1 ; i.e., a = b = 1.This special case was originally proved by Gathmann [Gat02].Our approach differs from that of [Gat02] in that we give a direct construction of smoothings of comb-type maps (see Propositions 2.7 and 2.11) rather than appealing to [Vak00, Theorem 6.1].This will be a key step in the proof of the general case of Theorem 1.2.For the remainder of this section, fix positive integers Γ = (d 1 , . . ., d n ) with d = d i .
Example 2.1.We begin with some examples motivating Conditions ( * ) when Γ = (1, . . ., 1).First, it is clear that Condition ( * )(1) is required as the evaluation condition is closed.However, consider a degree n map f 1 : (C 1 , q 1 ) → P 1 from a smooth rational curve C 1 with f (q 1 ) = ∞.Let C = C 1 ∪ q 1 ,q 2 C 2 be a nodal union of two rational curves, and let p 1 , . . ., p n ∈ C 2 \ q 2 be n marked points.Then there is an n-marked degree n stable map (f : C → P 1 , p 1 , . . ., p n ) given by taking the map f 1 on C 1 and the constant map with image ∞ on C 2 .While this map satisfies Condition ( * )(1), a simple dimension count shows that the dimension of this locus inside M 0,n (P 1 , n) is equal to dim Maps n + n − 2, and so this condition alone is not enough to cut out the locus Maps n at least for n ≥ 3 (see Figure 2).
The above example motivates Condition ( * )(2), which in this case requires that the map f 1 be totally ramified at q 1 so that there are no other points of C 1 in f −1 (∞).Requiring f 1 to be totally ramified at q 1 means that we impose the vanishing of n derivatives, which is a codimension n condition on Maps n .Thus the locus of maps of this combinatorial type satisfying both Conditions ( * )(1) and ( * )(2) has dimension dim Maps n − 2. Note that such maps automatically satisfy Condition ( * )(3) as well.However, the following example illustrates that Conditions ( * )(1) and ( * )(2) do not imply Condition ( * )(3) in general.
Example 2.2.Consider a degree 3 map f 1 : (C 1 , q 1 , q 2 ) → P 1 from a smooth rational curve such that f (q i ) = ∞, and suppose that f has ramification index i at q i .Let (C 2 , q 1 , p 1 , p 2 ) be a smooth pointed rational curve, and let C = C 1 ∪ q 1 ,q 1 C 2 .Then we have a 3-pointed degree 3 stable map (f : C → P 1 , q 2 , p 1 , p 2 ) where This map satisfies Conditions ( * )(1) and ( * )(2) but not ( * )(3).Note furthermore that this stable map cannot lie in Maps n as f 1 is ramified to order 2 at the marked point q 2 , but the only way this can happen in the limit of a family in Maps • n is if two marked points collided at q 2 .Remark 2.3.Note that a degenerate special case of Condition ( * )(3) is when T = q is itself a point.Since T must be a maximal closed subvariety contracted by f , this means that q cannot lie on a contracted component.Therefore, by Condition ( * )(2), q must be a marked point.Then Condition ( * )(3) reads that f must be unramified at q. Indeed, we saw this was necessary in the above Example 2.2 (see Figure 3).
Proposition 2.4.Conditions ( * ) (1), (2), and (3) are necessary for a stable map to lie in Proof.It is clear that (1) is necessary as evaluation is continuous so the condition f (p i ) = ∞ is closed, so we proceed to (2) and (3).Consider a 1-parameter family of stable maps over the spectrum S of a DVR with generic fiber lying in Maps • n , and denote by (f 0 : C 0 → P 1 , p 1 , . . ., p n ) the central fiber.
First we show (2).Let U ⊂ C denote the open complement of the locus of components contracted by the map f , and consider D := f −1 (∞).As the total space C is normal and f is non-constant, D is a Cartier divisor.Moreover, note that any π-vertical component of D is contracted by f and therefore is not in U .Thus, restricting to U , we see that D| U is a Cartier divisor which is horizontal over S, and therefore any point of D| U lying over 0 ∈ S must be in the closure of the marked points of the generic fiber.
Finally, we show (3).Ramification corresponds to a polynomial having a multiple root at a point, and this multiplicity takes into account precisely how many points collided, i.e., the number of marked points on the relevant contracted component.More formally, we can consider the intersection product T • D. By the projection formula, T • D = 0. On the other hand, D = σ i + E, where E is the π-vertical component.Computing T • ( σ i + E) = 0 in terms of local multiplicities gives exactly the equality in Condition ( * )(3).
Our task now is to show that Conditions ( * )(1)-(3) are sufficient for a stable map (f : C → P 1 , p 1 , . . ., p n ) to lie in M (d 1 ,...,d n ) .We will do this by constructing a smoothing of the marked curve and a linear series on the total space which restricts to f on the central fiber but whose generic fiber lies in We begin with a preliminary lemma that shows that nodal curves can be smoothed to surfaces admitting A m singularities for any m at the nodes.Lemma 2.5.Let C 0 be a genus 0 nodal curve with k nodes q 1 , . . ., q k , and let m 1 , . . ., m k ≥ 1.Then there exists a smooth C → S, where S = Spec(R) is the spectrum of a DVR, such that the total space C has an A m i −1 singularity at q i for each i = 1, . . ., k.
Proof.The miniversal deformation space of a pointed prestable curve of genus 0 is k t 1 , . . ., t k , where t i is the smoothing parameter of the i th node; i.e., formally locally around the i th node, the miniversal family of curves looks like xy = t i inside A 2 k t 1 ,...,t k .Let R = z be the DVR in the statement.To prove the lemma, it suffices to note we can construct a map of rings k t 1 . . .t k → k z such that t i → z m i .Then formally locally around the i th node, the family of curves over Spec(R) will be isomorphic to xy = z m i , as required.
We now show that it suffices to consider the case of a stable map where every maximal connected subtree contracted by f (as in Condition ( * )(3)) is irreducible.
Lemma 2.6.Let (f 0 : C 0 → P 1 , p 1 , . . ., p n ) be a stable map satisfying Conditions ( * )(1), (2), and (3).Then there exists a deformation to a family of stable maps (f : 1), (2), and (3), and , and write C 1 , . . ., C r for the connected components of the closure of the complement . Each E i is pointed by (q ij , p il ), where the p il are the marked points that lie on E i and the q ij are the points of E i along which E i is glued to the non-contracted components of C. Then each E i is a pointed genus 0 prestable curve, and there exists a smoothing E i → Spec(R) with sections τ ij smoothing the q ij and σ il smoothing the p il .Now consider the constant family l C l R with constant sections corresponding to the marked points and the points q ij along which the q ij are glued.
We can glue l C l R with t E t by identifying the sections q ij × Spec(R) with τ ij for all (i, j) and call the result C → Spec(R), which is now a family of pointed curves.By construction, this is a partial smoothing of the pointed curve (C 0 , p 1 , . . ., p n ).Moreover, there is a stable map f : for each l and taking the constant map ∞ on each E t .This descends to a map f as desired since f (q ij ) = ∞.Conditions ( * )(1), (2) are satisfied by f η by construction, and ( * )(3) is satisfied since the maximal irreducible components Ẽi η of f −1 η (∞) have the same number of marked points as E i and f η has the same ramification at (q ij ) η as f 0 at q ij .We now restrict ourselves to the case where the connected components of f −1 (∞) are irreducible.Let (f : C 0 → P 1 , p 1 , . . ., p n ) be such a map, suppose that E 1 , . . ., E k are the 1-dimensional connected components of f −1 (∞), and write E = i E i .By the above reduction, each E i is a smooth rational curve.
We write the closure of the complement of E in C 0 as a union of connected components C 1 , . . ., C r .Thus each C i is a tree of rational curves such that (f 0 )| C i is non-constant and such that the preimage of ∞ is 0-dimensional.For each i, let I i ⊂ {1, . . ., n} be the subset of indices k such that p k lies on E i .If E i and C j intersect, we let q ij ∈ E i and t ij ∈ C j be the points of E i and C j , respectively, at which they are glued, and we let e ij be the ramification index of (f 0 )| C j at t ij .Note that Condition ( * )(3) then reads that for each i, where the left side is a definition and the right sum is over the j such that E i meets C j .
Consider a smoothing (C → S, σ 1 , . . ., σ n ) of (C 0 , p 1 , . . ., p n ) over the spectrum of a DVR, and let D = d i σ i denote the divisor of Γ -weighted marked sections on this surface.We wish to construct a rank 1 linear series on such a smoothing so that the central fiber agrees with f 0 and the generic fiber satisfies that D| η is the preimage of ∞.Our strategy is to consider the map π : C → C which contracts E to a point and instead construct the appropriate linear series on C .Proposition 2.7.In the setting above, there exist a smoothing (C → S, σ 1 , . . ., σ n ) and a Cartier divisor D on C , where π : C → C is the contraction of the E i such that (1) π * D = D + a i E i for some a i , where D is the divisor of marked sections, and (2) a i E i • C j = e ij t ij for all i and j such that E i meets C j .
Proof.For each i, we let a i = r j=1 e ij and let m ij = k j e ik .By Lemma 2.6, there exists a smoothing (C → S, σ 1 , . . ., σ n ) of (C 0 , p 1 , . . ., p n ) such that C has an A m ij −1 singularity at the node t ij .Computing intersection products in the surface C, we have since locally around the node t i , the curves E i and C j are distinct lines through an A m ij −1 singularity.Therefore, where the sum is over those j for which C i and E j meet.Here, we go back and forth between viewing these intersection products as numbers or as divisors on the curves depending on whether it is convenient to emphasize the particular intersection points.Now consider the divisor D + a i E i .We can compute that D • E i = k∈I i d k = n i since D is the divisor of weighted marked sections and E i contains p k for k ∈ I i .On the other hand, Finally, Therefore, there exists a section s ∈ H 0 (L) which pulls back to the section cutting out D + a i E i on C, and so D = div(s) is the required Cartier divisor.
Remark 2.8.Fix a list of positive integers r ij for i and j such that E i meets C j .Then in the construction of of the smoothing in Proposition 2.7, we can replace e ij with r ij e ij in the definitions of a i and m ij .In this way, we get a smoothing satisfying the properties of Proposition 2.7, where the total space has A m ij −1 singularities with m ij divisible by any fixed choice of integers.
Let 0 ∈ P 1 be a general point of the target, and consider the divisor B 0 ⊂ C 0 given by f −1 0 (0).Then B 0 consists of a union of points on C 0 which are disjoint from E and contained in the locus where f 0 is étale.
Since C → S is a family of genus 0 curves birational to P 1 S , we can extend the collection of points B 0 into a Cartier divisor B on C which is horizontal and satisfies B| C 0 = B 0 .Proposition 2.9.With notation as above, D + a i E i ∼ B are linearly equivalent Cartier divisors on C.
Proof.Let ψ : C → S denote the morphism to the base, and consider ψ * N , where First we claim that H 1 (C 0 , N | C 0 ) = 0. Indeed, since C 0 is a nodal genus 0 curve, it suffices to show that the degree of N restricted to each component of C 0 is at least 0. Now N | E = O E by construction since (D + a i E i ) • E = 0 and B avoids E. On the other hand, for each i, we have , so these divisors are linearly equivalent.Therefore, we in fact have that N is trivial on each component of C 0 .Hence, by cohomology and base change, ψ * N is a vector bundle whose formation commutes with base change.Moreover, over the generic fiber, C η P 1 η , and B and D + a i E i are divisors of the same degree n by construction.Therefore, H 0 (C η , N | C η ) = 1.Since any line bundle on the spectrum of a DVR is trivial, we conclude that ψ * N O S .Consequently, N has a non-vanishing section which exhibits the required linear equivalence Remark 2.10.Note that the vanishing claimed in the above proof does not hold if we replace C 0 with a genus 0 Deligne-Mumford stack.See Remark 5.7 for an example.Now let B be the image of B under π, which is also a Cartier divisor since B is contained in the locus where π is an isomorphism.By the exact sequence (2.1), the linear equivalence in Proposition 2.9 is equivalent to a linear equivalence D ∼ B .In particular, D and B form a basepoint-free rank 1 linear subseries of Proposition 2.11.With notation as above, let g : C → P 1 be the morphism induced by the basepoint-free linear series B , D .Then the composition Proof.The map (g • π) contracts E to ∞ by construction.Moreover, on each C j , the restrictions of B and D satisfy Putting these together, we conclude the following.
Theorem 2.12.Let (f 0 : C 0 → P 1 , p 1 , . . ., p n ) be a stable map satisfying Conditions ( * )(1), (2), and (3).Then there exist a smoothing (C → S, σ 1 , . . ., σ n ) over S = Spec(R), the spectrum of a DVR, and a stable map In particular, Conditions ( * )(1), (2), and (3) are both necessary and sufficient for a stable map to be contained in Proof.By Propositions 2.7 and 2.11, there exist a smoothing (C → S, σ i ) and a map f : where D is the divisor of weighted marked sections and the

Log twisted curves and maps
We refer the reader to [ACG + 13] for the standard definitions in log geometry and to [CJR + 18, Section 2] for log twisted stable maps.We always work with fine and saturated (fs) log structures.We will use M X to denote the log structure of X, which will be implicit if no confusion arises.The characteristic of the log structure will be denoted by M X .

Log geometry
We will need the following results regarding log smooth morphisms.
• If a morphism of log schemes f : X → pt is log smooth (where the point has the trivial log structure), then U ⊂ X is a toroidal embedding, where U is the locus with trivial log structure and M X is the divisorial log structure.
where the sum is over marked points and N i is the divisorial log structure associated to the i th marked point; we call M C the canonical log structure associated to a pointed curve if there is no confusion.
There exists a log cotangent complex L log X/Y (see [Ols05] or [ACG + 13, Section 7]) for morphisms of log schemes f : X → Y , and deformation theory of log schemes is controlled by the log cotangent complex (see [Ols05, Theorem 5.2]).
Remark 3.2.If f is a log smooth morphism, then the log cotangent complex L log X/Y is represented by the sheaf of log differentials (see [Ols05, Section 1.1(iii)]).There does not exist a distinguished triangle in general; however, Olsson constructs a distinguished triangle for log flat or integral morphisms (see [Ols05, 1.3]).

(log) Twisted curves
Our main reference is [Ols07].To compactify moduli spaces of maps f : C → M, where M is a Deligne-Mumford stack, one needs to allow C to be a stack as well, known as a twisted curve (see e.g.[AV02,AB19b]).Definition 3.3.A twisted curve is a purely 1-dimensional Deligne-Mumford stack C, with at most nodes as singularities, satisfying the following conditions: where ξ ∈ µ m acts by (z, w) → (ξz, ξ −1 w).
Definition 3.4.An n-pointed twisted curve C/S marked by disjoint closed substacks {Σ i } n i=1 in C is assumed to satisfy the following: (1) The Σ i are contained in C sm .
(2) Each Σ i is a tame étale gerbe over S.
(3) The map Let X be a Deligne-Mumford stack.We say that a fine log structure M X is locally free if for every geometric point x → X, the characteristic sheaf satisfies M X N r for some r.Definition 3.5.In the above situation, we say that a morphism of sheaves of monoids M → M is simple if for every geometric point x → X, we have where φ is given by (m 1 , . . ., m r ).
Definition 3.6.An n-pointed twisted log curve over S is the data where • M S is the minimal log structure for the family C → S, • the a i : S → Z >0 are locally constant, and • l is a simple morphism.

Theorem 3.7 (cf. [Ols07, Theorem 1.8]). The fibered category of n-pointed twisted curves is naturally equivalent to the stack of n-pointed log twisted curves.
There is a natural map from the stack of twisted curves to the stack of (pre)stable curves induced by taking the coarse space π : C → C. The induced map on local deformation spaces Def(C) → Def(C) can be described as a root stack of order m i along the boundary divisor {t i = 0}, where t i is the deformation parameter of the i th -node of C and m i is the stabilizer order of C at the i th node.The stabilizer orders m i correspond via Theorem 3.7 to the data of the simple extension l as in Definition 3.5.For more details, see the discussion following [Ols07, Theorem 1.9 and Remark 1.10].
Example 3.8.Consider a smoothing of a nodal curve as in Lemma 2.5, and let (C, M C ) → (S, M S ) be the minimal log structure.The appearance of A m i −1 singularities on the total space of the smoothing at the nodes of C 0 is equivalent to the existence of a simple extension M S → M S with φ = (m 1 , . . ., m k ).Thus, by Olsson's Theorem 3.7, such a smoothing is the coarse space of a smoothing of the twisted curve with stabilizer µ m i at the i th node of C 0 .Note that we can see this directly by taking the canonical stack of the total space C which introduces a stabilizer µ m i at the A m i −1 singularity.
Let us briefly explicate the construction of Theorem 3.7 as we will use it further on.If π : C → C is the coarse map of a twisted curve over S, then the simple extension l : M S → M S is the map between the minimal log structures of h : C → S and f : C → S, respectively.The pushforward log structure π * MC sits in a pushout square Each a i keeps track of the stabilizer of C/C along σ i , and we have simple extensions N C,i → π * N C,i corresponding to multiplication by a i on characteristic monoids.
Proposition 4.1.Let g : C → X be a prestable map with (X, M X ) as above.Suppose that the 1-dimensional connected components of the support of f * (M X ) are smooth genus 0 curves.Then g lifts to a prestable logarithmic map.That is, there exist log structures M C on C and M on S = Spec(k) and a map g : Proof.The question is étale local on the target, so we may pull back to a chart for the log structure (X, M X ).
Hence without loss of generality, we may assume that the target is (A 1 , 0) = (X, D) with its toric log structure and that g : C → A 1 is a proper map satisfying the assumptions of the proposition.Moreover, we write g −1 (0) = {p 1 , . . ., p k , E 1 , . . ., E r } as a disjoint union of points and smooth rational curves.Let us denote by q ij the nodes of C along E i and by e ij the ramification of g at these points.First we claim that g factors through an expansion of the target To construct such a factorization, it suffices to construct non-constant maps h i : E i → E with h i (q ij ) = 0 and such that the order of tangency of h i at q ij is equal to e ij .Since the E i are smooth rational curves, such maps exist.Therefore, we have a factorization h : C → X, and it suffices to show that h can be lifted to a log map where since ϕ underlies a log map ( X, M X ) → (X, M X ) and a composition of log maps is a log map.Now we have reduced to the case where the support of h * (M X /N) is {p 1 , . . ., p k , p k+1 , . . ., p n }, a union of isolated points.Here the copy of N in the quotient is the pullback of the minimal log structure on Spec(k) along X → Spec(k).In a neighborhood of the smooth p i , there is a unique divisorial log structure such that the map lifts, encoding the order of tangency.In a neighborhood of the nodal p i , the map lifts to a log map by the tangency condition above, which is precisely the predeformability condition for a map to an expansion (see for example [ACG + 13, Section 12]).
Remark 4.2.Proposition 4.1 may be generalized to the case where the 1-dimensional connected components of the support of f * M X are trees of rational curves at the expense of allowing C to be blown up at the nodes.
To do this, one needs to consider the map g trop of tropical curves and subdivide the source and target to make g trop tropically transverse (see [Ran22, Sections 2.5 and 2.6] for more details).Such a subdivision corresponds to choosing a possibly larger expansion X → X and a sequence of log blowups C → C after which we may lift g to a map h : C → X which underlies a log map as in the proposition.Since this is not strictly necessary for our stated goal of constructing smoothings, we leave the details to the reader.
Next we address the question of lifting the log structure to a twisted map from its coarse space.More precisely, consider a proper morphism f : C/S → X from a twisted curve to a smooth Deligne-Mumford curve X.Let M X be a log structure making (X, M X ) a log twisted curve with coarse space (X, M X → M X ).Here (X, M X ) is log smooth, and M X → M X is the simple extension encoding the stabilizers of X. Suppose furthermore that there exist log structures M C and M S on the coarse space C of C and on S, respectively, such that is a prestable log map.Proposition 4.3.Let f : C/S → X and f : (C, M C )/(S, M S ) → (X, M X ) be as above.Then there exist a root stack π : C → C and log structures MC on C and M S on S such that ( C, MC) → (S, M S ) is a log smooth curve, and there exists a map f : (f • π) * M X → MC making f into a prestable log map.
Proof.Let g : C → X be the coarse map.By Theorem 3.7, we have simple extensions of log structures M C → M C , M S → M S , and M X → M X on C, S, and X, respectively, such that (C → S, {a i , σ i }, M S → M S ) and (X → Spec(k), {b i , x i }, k * → k * ) are log twisted curves.Here the σ i : S → C correspond to the marked gerbes of C → S with order of stabilizer given by a i .Similarly, the x i ∈ X correspond to the marked gerbes of X with stabilizer order b i .
The question is local on X, so we can restrict to a neighborhood of one of the marked points x i with stabilizer µ b i on X and suppose without loss of generality that there is a single marked point x ∈ X with stabilizer µ b .This point is exactly the support of M X = N and M X = N, and the simple extension is simply the map 1 → b.
By assumption, there exists a map g : g * M X → M C making (g, g ) into a log map.By Theorem 3.7, it suffices to show that there exist a further simple extension M C → M C which is an isomorphism away from finitely many smooth points and a map of log structures f : g * M X → M C such that the square , where e is the element corresponding to a local parameter of x ∈ X and e is the generator of the copy of N. By the universal property of this pushout, to construct f , it suffices to show that s(g (e)) is divisible by b.However, we can guarantee this by taking a large enough simple extension s.

Deformations of log twisted maps and their coarse spaces
be a stable log map from a log twisted curve, and suppose that (X, M X ) is a log smooth curve.We define the critical locus of f to be the support of the cokernel of the natural map Note that this map, viewed as a two-term complex, is a presentation of the log cotangent complex L log f relative to S, so we may equivalently define the critical locus to be the support of H 0 (L log f ).The critical locus is a union of nodes of C, branch points of f , and contracted components of f .By contrast, the kernel of the above map, or equivalently H −1 (L log f ), is supported solely on the contracted components of f .
Let A 1 , . . ., A k be the connected components of the critical locus and U 1 , . . ., U k be affine étale neighborhoods of the A i with log structure pulled back from C. Suppose each U i avoids A j for j i.Let Def (f ,f ) be the versal deformation space of the log map (f , f ).Since the moduli space is a Deligne-Mumford stack, Def (f ,f ) is pro-representable.Let Def i be the miniversal deformation space of the restriction We have the following proposition (see also [Vak00,Proposition 4.3]).Proposition 4.4.In the setting above, the natural map Since the g i : Since both L log f and L log f i are two-term complexes, we have distinguished triangles given by restricting f to U i .This map corresponds to the map on deformation spaces Def (f ,f ) → Def i .Taking direct sums yields the diagram of distinguished triangles where the vertical maps are equalities via the identifications K = ⊕K i and Q = ⊕Q i .The middle map corresponds to the product of the restriction map on deformation spaces Since the vertical maps on the ends are equalities, the middle map is an isomorphism by the properties of distinguished triangles, or equivalently the five lemma, from which we conclude.
Remark 4.5.Via Diagram (4.1), we see that in fact the deformation spaces Def i of an étale neighborhood U i of the component A i of the critical locus is independent of the choice of U i since the Q i -and K i -terms are independent of U i .Now we address the question of lifting log smooth deformations along the coarse space map of a log twisted curve.Let be a prestable log map, and suppose that there exists a simple extension of log structures M S → M S such that (C 0 /S 0 , (σ i , a i ), l 0 : M S 0 → M S 0 ) is a log twisted curve.Let (C 0 , Σ i ) → S 0 be the corresponding twisted curve with log map (C 0 , M C 0 ) → (S 0 , M S 0 ), and let g 0 : (C 0 , M C 0 ) → (X, M X ) be the composition of h 0 with the coarse space map π 0 : C 0 → C. Let (S 0 , l 0 : M S 0 → M S 0 ) → (S, l : M S → M S ) be a strict closed immersion of log schemes with simple extensions defined by a square 0 ideal I ⊂ O S .Proposition 4.6.With notation as above, suppose we have a log smooth deformation of h 0 over (S, M S ).Then h lifts uniquely to a log smooth deformation of g 0 factoring through h.
Proof.We can summarize the situation with the commutative diagram below: where the middle square is a pullback in the category of fs log schemes by [Ols07, Lemma 4.6, Diagram 4.6.2, and Lemma 4.7].Given a strict square 0 thickening (S, l : M S → M S ) of (S 0 , l 0 : M S 0 → M S 0 ) with ideal I and a log smooth deformation h as in Diagram (4.2), we can form the pullback in the category of fs log schemes to obtain a log smooth deformation of h 0 • (id, s 0 ).Then M C → M C is a simple extension, and by Theorem 3.7, this is equivalent to a log smooth deformation g.

Smoothing twisted stable maps
Our goal now is to lift the smoothings constructed in Section 2 to the case of twisted maps.More precisely, suppose we have a genus 0 twisted stable map (f 0 : C 0 → X, p 1 , . . ., p n ) such that • X is a smooth stacky curve with coarse map τ : X → X = P 1 , • the coarse map (h 0 : C 0 → X, q 1 , . . ., q n ) satisfies Conditions ( * ) relative to ∞ ∈ X, and • τ is étale over ∞.The situation can be summarized in the following diagram, where S 0 = Spec(k): By Theorem 2.12, there is a smoothing h : (C, σ i )/S → X of h 0 over S = Spec(R), the spectrum of a DVR, with generic fiber contained in M (d 1 ,...,d n ) , and we wish to lift this to a smoothing f : (C, Σ i )/S → X of f 0 with generic fiber contained in M Γ (X).That is, we wish to fill in the dotted arrows in the diagram The difficulty is that there are global obstructions to lifting from a map to X to a map to X.For example, the j-invariant of a map to M 1,1 must satisfy that the discriminant is a sum of a square and a cube which cuts out a high-codimension locus inside the space of maps to P 1 .The key observation is that τ is étale.Thus we can lift such a smoothing locally around ∞ (see Theorem 5.1) to obtain a partial smoothing of f 0 .Then we can construct a further global smoothing when X has some positivity (see Theorem 5.6).
Theorem 5.1.In the situation above, there exists a partial smoothing f : (C, Σ i )/S → X of f 0 such that , and • the coarse map on the generic fiber h η is ramified to order d i at (σ i ) η .
Proof.We wish to use the log deformation theory results of Section 4.2.To do this, we begin by showing that the smoothing of the coarse map h : (C, σ i )/S → X can be endowed with the structure of a stable log map.Lemma 5.2.Let (h 0 : C 0 → P 1 , q 1 , . . ., q n ) be as above, and let be a smoothing constructed in Theorem 2.12.Denote by M the divisorial log structure on P 1 corresponding to the point ∞.Then there exist log structures M C and M S on C and S and a map h : h * M → M C such that h is a stable log map over S.
Proof.First note that C → S is log smooth, where we equip S = Spec(R) with the standard log structure and C with the divisorial log structure induced by the union of the central fiber and marked sections.Indeed, this follows since C → S is a toroidal morphism of the corresponding toroidal embeddings.Endow P 1 with the divisorial log structure for ∞ ⊂ P 1 ; the pullback of the monomial x ∞ cutting out ∞ is locally a sum of monomials for the toroidal structure on C since h * [∞] is supported on the toroidal boundary.Thus, there is a natural map h of divisorial log structures recording the exponents of the monomial h * x ∞ .
Suppose (R, m) is complete, and let h n : (C n , σ i,n )/S n → X be the compatible system of deformations over S n = Spec(R/m n ) obtained by truncating the smoothing over S. We will lift these to a compatible system of deformations of h 0 in two steps.
First let (g 0 , g 0 ) : (C 0 , M C 0 ) → (X, M) be the composition h 0 • π 0 , where we note that h 0 is a log map by Lemma 5.2 and π 0 is a log map by the definition of log twisted curves.By Lemma 4.6 and induction on n, the map h n extends to a log smooth deformation g n : (C n , M C n ) → (X, M X ) over (S n , M S n ), where M S → M S a simple extension and (S n , M S n ) → (S, M S ) is a strict closed embedding.Moreover, the g i are compatible by construction.In this way, we get an R-point of the formal deformation space SpfR → Def (g 0 ,g 0 ) .
Next we will use Proposition 4.4 to lift this R-point of Def (g 0 ,g 0 ) , at least in a neighborhood of ∞, to a smoothing of f 0 .Note that f 0 lifts to a log map (f 0 , f 0 ) by Proposition 4.3, where we endow X with the pullback log structure τ * M. Equivalently, this is the divisorial log structure for ∞ but is different from the canonical log structure on a log twisted curve.Let U ⊂ P 1 be an étale neighborhood of ∞ such that τ| V : V → U is étale, where V = τ −1 (U ), and such that U avoids all the components of the critical loci of both f 0 and g 0 away from ∞.Let Def f 0 ,∞ be the miniversal deformation space of the restriction of (f 0 , f 0 ) to an étale neighborhood of f −1 0 (∞) and similarly for Def g 0 ,∞ .Note that by the independence of the chosen étale neighborhood (see Remark 4.5), we have that these are the miniversal deformation spaces of (f 0 )| U : f −1 0 (U ) → U ⊂ P 1 and (g 0 )| V : g −1 0 (V ) → V ⊂ X.We set U = f −1 0 (U ) = g −1 0 (V ) and denote by U a neighborhood of the critical loci of both f 0 and g 0 away from ∞.
By Proposition 4.4, we have where Def f 0 , ∞ (resp.Def g 0 , ∞ ) is the miniversal deformation space of a neighborhood of the critical loci of (f 0 , f 0 ) (resp.(g 0 , g 0 )) which are away from ∞.
Proof.We proceed as in Proposition 4.4 and see that there is a natural isomorphism and similarly We now compute . Since τ| V is étale, we have that g * 0 Ω V g * 0 τ * Ω U = f * 0 Ω U by the functorality of pullbacks.Therefore, the natural map where the first map is the projection from Proposition 4.4 and the second map is the inverse of the isomorphism in Lemma 5.3.There is a canonical splitting of the projection Def (f 0 ,f 0 ) → Def f 0 ,∞ via the isomorphism of Proposition 4.4 given by picking the constant deformation in Def f 0 , ∞ .Further composing with this splitting gives us a formal deformation SpfR → Def (f 0 ,f 0 ) .Since Def (f 0 ,f 0 ) is the formal neighborhood of a point of a locally of finite type algebraic stack (see [Che14,GS13]), this formal deformation algebraizes to a family (f , f ) : (C, M C )/(S, M S ) → (X, τ * M) of log twisted stable maps (see for example [Bha16, Theorem 1.1] and [BHL17, Corollary 1.5]).Let us denote by f : (C, Σ i )/S → X the underlying twisted stable map.By construction, the coarse map of f agrees with h over an étale neighborhood of ∞, so we conclude that f satisfies the required conditions.Remark 5.4.By construction, the smoothing of Theorem 5.1 induces the constant infinitesimal deformation of the restriction of f 0 away from ∞.More precisely, suppose A is a component of the critical locus of f 0 which is disjoint from f −1 0 (∞), and let U → C be an étale neighborhood of A which is disjoint from all other critical loci.Then f | U mod m n is the constant infinitesimal deformation of f 0 | U × C C 0 .
Remark 5.5.In Theorem 5.1 and the situation above, we can drop the assumption that τ is étale over ∞ at the cost of replacing C 0 with a root stack.Indeed, if τ is not étale, then τ factors through a non-trivial root stack along ∞.In this case, one obtains a corresponding simple extension of log structures M → M on P 1 .Then by Proposition 4.3, the map f 0 can be extended to a log map after taking a further root stack C 1 → C 0 .The rest of the argument goes through as written to produce a partial smoothing of the composition C 1 → C 0 → X with the required properties as in the theorem.We leave the details to the reader since our main interest is in the case of (M 1,1 , ∞), where τ is étale over ∞.

Global smoothings for target weighted projective lines
Next we consider the problem of smoothing twisted stable maps to a weighted projective line P = P(a, b) without any tangency conditions.Theorem 5.6.Let (f 0 : C 0 → P, Σ i,0 ) be a genus 0 twisted stable map to a weighted projective line P = P(a, b).Then there exists a smoothing Proof.Suppose (R, m) is complete.We will construct such a smoothing by building a compatible system of deformations over R/m n .Since the stack of not necessarily representable twisted stable maps to P is a locally of finite type algebraic stack, such a compatible system automatically algebraizes as in the proof of Theorem 5.1.First note that if a = ka and b = kb with a , b coprime, then the canonical map ρ : P(a, b) → P(a , b ) is étale, and so infinitesimal deformations lift uniquely along ρ.Thus, without loss of generality, we may assume that a and b are coprime and P is a twisted curve.First we reduce to the case where every connected component in the fibers of f 0 : C 0 → P is irreducible.We can argue as in Lemma 2.6.If E is a connected component of the preimage f −1 0 (p) for some p, then we can view f 0 | E as a map to the residual gerbe G p where E is pointed by the nodes.Then since G p is étale over Spec(k), it suffices to smooth the composition E → Spec(k) as a marked twisted curve and then glue together the smoothing of E with the constant family of maps f | C 0 \E along the markings corresponding to nodes exactly as in the proof of Lemma 2.6 to obtain a partial smoothing where each connected component of the contracted locus is irreducible.Therefore, without loss of generality, we may suppose that every connected component of the contracted locus of f 0 is irreducible.
By Propositions 4.1 and 4.3, there exists a twisted curve C0 with a partial coarse space map π 0 : C0 → C 0 such that g 0 := f 0 • π 0 : C0 → P can be endowed with a log structure where P has the toric log structure and ( C0 , MC 0 ) → (Spec(k), M) is log smooth.Since (P, M P ) is toric, the log cotangent complex L log P O P is trivial.Moreover, since C0 is genus 0, (5.1) Ext 1 (g * 0 L log P , OC 0 ) = H 1 ( C0 , OC 0 ) = 0. Therefore, the log deformation space of (g 0 , g 0 ) is log smooth over the log deformation space of the curve ( C0 , MC 0 ).By Theorem 3.7, Def( C0 , MC 0 ) is a root stack over Def(C 0 , p 1 , . . ., p n ), where the p i are the smooth marked points lying under Σ i,0 .In particular, we can build a smoothing (C/S, σ i ) of (C 0 , p i ) whose truncations (C n /Spec(R/m n ), σ i,n ) lift to a formal smoothing of ( C0 , MC 0 ), after possibly extending the log structure on S. By the vanishing obstruction group (5.1), the map (g 0 , g 0 ) can then be lifted to a compatible family (g n : Cn → P, g n ) of log smooth deformations over R/m n .Thus, after algebraizing, we have a smoothing (g : C/S → P, Σ i ) of g 0 .Taking the relative coarse space of g gives a flat family of maps f : C/Spec(R) → P whose formation commutes with base change (see [AOV11,Proposition 3.4] and [AOV08, Corollary 3.3]).Therefore, we conclude that the central fiber of f agrees with f 0 and that the generic fiber is smooth since it is a partial coarse space of the smooth generic fiber of C/S.Remark 5.7.For the proof of Theorem 5.6, it is necessary to use log deformation theory even though the statement does not make any reference to log geometry.The reason is that the stack P is not convex in the classical sense.The tangent bundle is not necessarily effective even though it has positive degree, and there are examples of genus 0 twisted stable maps where the vanishing (5.1) does not hold for the usual tangent bundle.For example, consider C = [E/τ], where E is an elliptic curve with hyperelliptic involution τ : E → E, π : C → P 1 is the coarse space, and p i ∈ P 1 are the points lying under the fixed points of τ.Then the line bundle L = O C 1 2 p 1 − 1 2 p 2 + 1 2 p 3 − 1 2 p 4 with sections 0 ∈ L and 1 ∈ L ⊗2 induces a map f : C → P(1, 2) with f * T P = L and h 1 (C, L) = h 1 (P 1 , π * L) = h 1 (P 1 , O(−2)) 0.

Proofs of the main theorems
We use the notation from the introduction.Let (X, x 1 , . . ., x r ) be a smooth and proper pointed stacky curve, and suppose that the coarse map τ : X → X is étale at the points x i (see also Remark 5.5).Fix a tuple of discrete data Γ = (n 0 , {Γ 1 , x 1 }, . . ., {Γ r , x r }).Proposition 6.1.The loci M Γ (X) and N Γ (X) (see Definition 1.3 ) are locally closed substacks of K 0,n (X, d).
Proof.We run through the conditions of Definition 1.3.Condition (1) is both open and closed by the definition of a twisted curve, and condition (2) is closed.Conditions (4) and ( 4) are open by the openness of the smooth locus.For condition (3), we consider the cohomology sheaves of the cotangent complex L log f .The locus where f is constant is given by the support of H −1 (L log f ).Since the universal family of curves over the moduli space is proper, the locus where the marked points avoid SuppH −1 (L log f ) is open, and therefore so is the condition of f being non-constant along the marked points.Finally, under this condition, the order of tangency of f at x i is measured by the length of coker(f * Ω 1 X → Ω 1 C ) x , and by semicontinuity, there is a locally closed subset where this length is constant d i .
Proof of Theorem 1.5.The marked points p 0k for k = 1, . . ., n 0 with no tangency conditions do not change the outcome of the problem.Indeed, if we can construct such a smoothing with n 0 > 0, then by forgetting and stabilizing, we obtain a smoothing with n 0 = 0, and, conversely, if we have a partial 1-parameter smoothing over a base S with n 0 = 0, then up to a finite base change S → S, we can pick generic sections passing through p 0k to obtain a smoothing with n 0 > 0. So without loss of generality, suppose n 0 = 0.
We proceed by induction on r.The base case is precisely Theorem 5.1.In general, let (f : C → X, {{p jk } n j k=1 } r j=1 ) be a map in K Γ (X).Applying Theorem 5.1 to (f : C → X, p r1 , . . ., p rn r ) produces a 1-parameter deformation whose generic fiber is a relative map to (X, x r ) with tangency Γ r which also satisfies that the curve is smooth along the preimage of x r .Moreover, by Remark 5.4, the generic fiber is contained in K Γ (X).Therefore, without loss of generality, we may assume that C is smooth in a neighborhood of f −1 (x r ).By the inductive hypothesis, there exists a 1-parameter deformation of f with generic fiber contained in N Γ (X), where Γ = ({Γ 1 , x 1 }, . . ., {Γ r−1 , x r−1 }).Moreover, by Remark 5.4, this partial smoothing induces the constant deformation in an étale neighborhood of f −1 (x r ), and so the generic fiber is contained in N Γ (X).

Figure 1 .
Figure 1.In this example, the contracted component is attached to a double branched, unbranched, and triply branched point, respectively, and thus must have 2 + 1 + 3 = 6 marked points.

Figure 2 .
Figure 2. Maps of this type are given by choosing the mapf 1 = f | C 1 (which is parametrized by Maps n ), a point of the finite set f −1 1 (∞)along which to attach C 2 , and the configuration of the n + 1 special points on C 2 (which is parametrized by M 0,n+1 ), yielding the dimension count dim Maps n + n − 2.

Figure 3 .
Figure 3.The map in Example 2.2 which satisfies Conditions (1) and (2) but not (3) since the map is unramified where the contracted component C 2 is attached and ramified over ∞ elsewhere.
k j e ik = n i − j e ij = 0 by Condition ( * )(3).Here we have used that E k • E i = 0 if i k.Therefore, we need to show that D + a i E i descends to a Cartier divisor D along the contraction π : C → C .Note that π exists, C is a normal quasiprojective surface, and π * O C = O C by [KM98, Theorem 3.7].Moreover, we have an exact sequence (2.1) 0 / / Pic(C ) π * / / Pic(C) / / Z, where the map Pic(C) → Z is the restriction L → L| E ; see [KM98, Corollary 3.17].In particular, since O C (D + a i E i ) is in the kernel of this map, there exists a line bundle L on C such that π * L = O C (D + aE).On the other hand, π * O C = O C , and so by the projection formula,

( 1 )
If π : C → C denotes the coarse space morphism, then C sm = π −1 C sm , and π is an isomorphism over a dense open subset of C. (2) If x → C is a node such that the strictly henselian local ring O C, x is the strict henselization of k[x, y]/(xy), then • If C is a curve, then log smooth is equivalent to nodal; see [ACG + 13, Theorem 5].• Given a family f : C → S of nodal curves, there is a minimal log structure M S on S such that any other log structure that makes f log smooth is pulled back from it; see [ACG + 13, Section 7.3].We call the corresponding log structure and structure morphism f : f * M S → MC the canonical log structure of f : C → S. Note here that MC does not include marked points.If f is equipped with marked points, then we denote the natural log structure by