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BY 4.0 license Open Access Published by De Gruyter April 1, 2022

Degeneration of curves on some polarized toric surfaces

Karl Christ, Xiang He and Ilya Tyomkin ORCID logo

Abstract

We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show the irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

1 Introduction

In algebraic geometry, degenerations are one of the main tools in the study of algebraic curves. They play a central role in enumerative questions [4], Brill–Noether theory [17, 12], irreducibility problems [13], etc. In the current paper, we investigate degenerations of curves on toric surfaces. Our main motivation comes from the Severi problem and Harris’ approach to it, which we briefly recall. Roughly speaking, Severi varieties parametrize integral curves of a fixed geometric genus in a given linear system. The problem of the irreducibility of Severi varieties in the planar case, the classical Severi problem, had been a long-standing open problem until settled in characteristic 0 by Harris in 1986 [13]. In our preceding paper [7], we extended his result to arbitrary characteristic. The present paper is concerned with the analogous problem for toric surfaces.

Harris’ approach goes back to the original ideas of Severi, and consists of two parts: first, one proves that the closure of any irreducible component of the Severi variety contains the Severi variety V0,dirr parametrizing integral rational curves; this amounts to a statement about the possible degenerations of curves in the given linear system. Second, one describes the branches of the original Severi variety along V0,dirr, and uses monodromy arguments to show that all branches belong to the same irreducible component. A similar approach was used by the third author in [24] to prove the irreducibility of Severi varieties on Hirzebruch surfaces in characteristic 0.

In this paper, we prove the degeneration part of the argument. We restrict our attention to the case of polarized toric surfaces associated to so-called h-transverse polygons. This is a rich class of surfaces that includes the projective plane, Hirzebruch surfaces, and many other examples. In particular, this class includes examples of toric surfaces admitting reducible Severi varieties [27, 20]; see Example 7.2. Our main result asserts that any irreducible component of a Severi variety on such a surface contains in its closure the Severi variety parametrizing integral rational curves, provided that the characteristic of the ground field is either 0 or large enough. Moreover, any integral curve of geometric genus g in the linear system associated to an h-transverse polygon can be degenerated to an integral curve of any genus 0gg.

In the planar case, the monodromy part of the argument is relatively easy, and the main difficulty lies in the degeneration step. We shall mention, however, that in general the monodromy argument is subtle, and there are examples of toric surfaces for which it fails. Moreover, some toric surfaces admit reducible Severi varieties, and the structure of the monodromy groups seems to be the main reason for their reducibility. An interested reader can find more about such examples in the recent paper of Lang and the third author [20], and about the computation of the monodromy groups in the case of toric surfaces in Lang’s [19].

1.1 The main results

We work over an algebraically closed field K. To formulate the main result, we need the notion of the width of an h-transverse polygon. Recall that an h-transverse polygon in 2 is a convex lattice polygon Δ, whose horizontal slices at integral heights y=k are either empty or intervals with integral boundary points. We define the width of an h-transverse polygon Δ to be the maximal length of its horizontal slices and denote it by 𝚠(Δ).

Main Theorem.

Let ΔR2 be an h-transverse lattice polygon, and (SΔ,LΔ) the associated polarized toric surface. If char(K)=0 or char(K)>12w(Δ), then any irreducible component of the Severi variety parametrizing integral curves of geometric genus g0 in the linear system |LΔ| contains in its closure the Severi variety parametrizing integral rational curves in |LΔ|.

We refer the reader to Theorem 5.6 for a stronger version of this statement, in which we allow for some fixed tangency conditions with the toric boundary of SΔ. The bound

char(K)>𝚠(Δ)2

is sharp for char(K)3: In Proposition 5.14 we provide for every prime p an explicit example of a polarized toric surface with h-transverse polygon of width max{6,2p} and a component of the corresponding Severi variety parametrizing genus 1 curves, that violates the assertion of the theorem if char(K)=p.

We give two applications of the Main Theorem. First, we generalize Zariski’s Theorem in Theorem 6.1; that is, we show under the assumptions of the Main Theorem, except requiring char(K)>𝚠(Δ) instead of char(K)>12𝚠(Δ), that a general element in any irreducible component of a Severi variety is a nodal curve. This was previously known only in characteristic 0 (see [18]), and there are examples of polarized toric surfaces, for which the assertion fails in positive characteristic (see [26]). In fact, the bound char(K)>𝚠(Δ) in Zariski’s Theorem is sharp as Example 6.4 shows. Our argument goes as follows: we prove by explicit calculations that a general integral rational curve is nodal. By the Main Theorem, any irreducible component of a Severi variety contains in its closure the general integral rational curves, and the claim follows since being nodal is an open condition.

The second application returns to our original motivation, and we deduce irreducibility of Severi varieties whenever the assumptions of the Main Theorem are satisfied and the results of Lang [19] ensure that the monodromy acts as a full symmetric group on the set of nodes of a general integral rational curve; that is when also the monodromy part in Harris’ approach is known. This is the case when char(K)=0, the primitive outer normals to the sides of an h-transverse polygon Δ generate the whole lattice 2, and the polarization is ample enough; see Theorem 7.1 for a precise statement. In [2, Corollary 1.11], Bourqui obtains irreducibility for some smooth polarized toric surfaces. Note however the crucial difference that, unlike in [2, Corollary 1.11], the ampleness condition in our Theorem 7.1 is independent of the fixed geometric genus g.

The tangency conditions we allow for in Theorem 5.6 recover for SΔ=2 some of the generalized Severi varieties of Caporaso and Harris [4]; namely, the ones where the position of the point with tangency to the line is not fixed; see Remark 5.3 for details. Recently, the irreducibility of generalized Severi varieties of irreducible curves has been proven in characteristic 0 by Zahariuc [28]. We expect that our methods can be used to obtain a more detailed understanding of degenerations of curves with fixed tangency profiles on toric surfaces; this however was not the focus of the current paper and the tangencies we allow for in Theorem 5.6 are the ones we get for free in an argument designed for curves that intersect the toric boundary transversally. Note also, that we ensure in the current paper that the curves of smaller geometric genus constructed in each degeneration are irreducible, which we did not require in [7]; this is a second aspect, in which we get a more detailed picture even in the planar case.

The degeneration argument of [7] works for a larger class of toric surfaces, and not just 2. While this class does not contain all surfaces with h-transverse polygons considered in the present paper, it does contain for example Hirzebruch surfaces and toric del Pezzo surfaces. In an upcoming third part [8] of the series, we give a characteristic-free tropical proof that involves no monodromy arguments in these cases, thus establishing the irreducibility of Severi varieties for such surfaces in any characteristic. As a consequence, we obtain the irreducibility of Hurwitz schemes in any characteristic.

1.2 The approach

Our approach is based on tropical geometry and develops further the techniques introduced in [7]. The main idea is to control the degenerations by studying the tropicalizations of one-parameter families of algebraic curves and the induced maps to the moduli spaces of tropical curves. The polygon Δ being h-transverse allows us to work with floor decomposed tropical curves – a particularly convenient class of tropical curves introduced by Brugallé and Mikhalkin [3]. Let us explain our approach in some detail.

For the closure V of a given irreducible component of a Severi variety, we consider the locus VFD of algebraic curves, whose tropicalizations are floor decomposed, weightless, 3-valent, and immersed except at contracted legs; see Section 2.3 for the definition of weight function and weightless tropical curves. We analyze the tropicalizations of one-parameter families of curves in V and show that in the closure of VFD there are curves, whose tropicalization contains either a contracted loop or a contracted edge that belongs to a cycle of the tropicalization. Proceeding similarly to [7], we show that the length of the contracted edge/loop can be increased indefinitely, and conclude that the closure of VFD contains general irreducible curves of smaller genera. From this, the Main Theorem follows by induction and a dimension count.

Although the strategy of the proof is similar to the proof in [7], it is more complicated. In the planar case discussed in [7], the one-parameter family connecting a general tropical curve with one that contains a contracted edge can be constructed such that all its fibers are weightless, 3-valent tropical curves, except for possibly one 4-valent vertex. In the more general setting of the current paper, this is no longer the case, and we have to allow for additional types of tropical curves. For each of them, we have to prove a local liftability result to ensure that the curves we construct in the tropical family are tropicalizations of curves in the closure of VFD. Among the new types considered in this paper is the case of tropical curves, which are weightless and 3-valent except for a unique 2-valent vertex of weight 1. To show local liftability in this case, we introduce a local version of Speyer’s well-spacedness condition [23] in Proposition 3.8. Then we use an argument that uses properties of modular curves parametrizing elliptic curves together with a torsion point of a fixed order, that depends on the characteristic; see Lemma 4.6 for details. This is the reason why unlike the result in [7], the Main Theorem is not stated for arbitrary characteristic. As mentioned above, the theorem may fail in low characteristic: a related construction (also involving modular curves) gives explicit examples when this happens; see Proposition 5.14 for details.

1.3 The structure of the paper

In Section 2 we recall basic definitions and fix some terminology. In Section 3 we discuss h-transverse polygons, floor decomposed curves, and the tropicalization of parametrized curves. We establish in Proposition 3.8 a local version of Speyer’s well-spacedness condition for realizable tropical curves. In Section 4 we recall the local liftability results established in [7] and prove in Lemmas 4.5 and 4.6 the additional ones needed in the degeneration argument for h-transverse Δ. In Section 5 we prove Theorem 5.6, which immediately implies the Main Theorem. In Proposition 5.14 we show that in low characteristic the claim of the theorem may fail. In Section 6 we prove Zariski’s Theorem, and in Section 7 we combine the results with those of [19] to establish irreducibility of Severi varieties in some cases.

2 Notation and terminology

We follow the conventions of [7]. For the convenience of the reader, we recall them here and refer to [7] for further details.

2.1 Families of curves

By a family of curves we mean a flat, projective morphism of finite presentation and relative dimension one. By a collection of marked points on a family of curves, we mean a collection of disjoint sections contained in the smooth locus of the family. A family of curves with marked points is prestable if its fibers have at-worst-nodal singularities. It is called (semi-)stable if so are its geometric fibers. A prestable curve with marked points defined over a field is called split if the irreducible components of its normalization are geometrically irreducible and smooth and the preimages of the nodes in the normalization are defined over the ground field. A family of prestable curves with marked points is called split if all of its fibers are so. If UZ is open, and (𝒳,σ) is a family of curves with marked points over U, then by a model of (𝒳,σ) over Z we mean a family of curves with marked points over Z, whose restriction to U is (𝒳,σ).

2.2 Toric varieties and parametrized curves

For a toric variety S, we denote the lattice of characters by M, of cocharacters by N, and the monomial functions by xm for mM. The polarized toric variety associated to a lattice polytope ΔMM is denoted by (SΔ,Δ), and the facets of Δ are denoted by Δi. In the current paper, we mostly work with toric surfaces. For simplicity, we will always fix an identification M=N=2 such that the duality between the lattices is given by the usual dot product pairing.

A parametrized curve in a toric variety S is a smooth projective curve with marked points (X,σ) together with a map f:XS such that f(X) does not intersect orbits of codimension greater than one, and the image of X(iσi) under f is contained in the dense torus TS. A family of parametrized curvesf:𝒳SΔ over K consists of the following data:

  1. a smooth, projective base curve (B,τ) with marked points,

  2. a family of stable marked curves (𝒳B,σ), smooth over BB(iτi), and

  3. a rational map f:𝒳SΔ, defined over B, such that for any closed point bB the restriction 𝒳bSΔ is a parametrized curve.

2.3 Tropical curves

The graphs we consider have vertices, edges, and half-edges, called legs. We denote the set of vertices of a graph 𝔾 by V(𝔾), of edges by E(𝔾), and of legs by L(𝔾). We set E¯(𝔾):=E(𝔾)L(𝔾). By a tropical curve Γ we mean a finite graph 𝔾 with ordered legs equipped with a length function:E(𝔾)>0 and a weight (or genus) functiong:V(𝔾)0. We say that a curve is weightless if the weight function is identically zero. For any leg lL(𝔾) we set (l):=. We usually view tropical curves as polyhedral complexes by identifying the edges of 𝔾 with bounded closed intervals of corresponding lengths in and identifying the legs of 𝔾 with semi-bounded closed intervals in .

For eE¯(𝔾) we denote by e the interior of e, and use e to indicate a choice of orientation on e. The legs are always oriented away from the adjacent vertex. For vV(𝔾), we denote by Star(v) the star of v, i.e., the collection of oriented edges and legs having v as their tail. In particular, Star(v) contains two oriented edges for every loop based at v. The number of edges and legs in Star(v) is called the valence of v. The genus of Γ is defined to be g(Γ)=g(𝔾):=1-χ(𝔾)+vV(𝔾)g(v), where χ(𝔾):=b0(𝔾)-b1(𝔾) denotes the Euler characteristic of 𝔾. Finally, a tropical curve Γ is called stable if every vertex of weight zero has valence at least three, and every vertex of weight one has valence at least one.

A parametrized tropical curve is a balanced piecewise integral affine map h:ΓN from a tropical curve Γ to N, i.e., a continuous map, whose restrictions to the edges and legs are integral affine, and for any vertex vV(Γ) the following balancing condition holds: eStar(v)h/e=0. Note that the slope h/eN is not necessarily primitive, and its integral length is the stretching factor of the affine map h|e. We call it the multiplicity of h along e, and denote it by 𝔴(h,e), or 𝔴(e) if the parametrization h is clear from the context. In the literature, the multiplicity 𝔴(e) is often called the weight of e, but we shall not use this terminology in the current paper.

A parametrized tropical curve (Γ,h) is called stable if so is Γ. Its combinatorial type Θ is defined to be the weighted underlying graph 𝔾 with ordered legs equipped with the collection of slopes h/e for eE¯(𝔾). We denote the group of automorphisms of a combinatorial type Θ by Aut(Θ), and the isomorphism class of Θ by [Θ].

The degree of a parametrized tropical curve is the sequence (h/li), where li are the legs not contracted by h. If rank(N)=2, then we say that a degree is dual to a polygon Δ, if each slope h/lj is a multiple of an outer normal of Δ and the multiplicities of slopes that correspond to a given side Δi in this way sum to the integral length of Δi.

2.4 Moduli and families of parametrized tropical curves

We denote by Mg,n,trop the moduli space of parametrized tropical curves of genus g, degree , and with n contracted legs. We always assume that the first n legs l1,,ln are the contracted ones. The space Mg,n,trop is a generalized polyhedral complex. It is stratified by subsets M[Θ], that are indexed by combinatorial types Θ with the fixed invariants g,, and n. If 𝔾 is the underlying graph of a combinatorial type Θ, then M[Θ]=MΘ/Aut(Θ), where MΘ is a polyhedron in |E(𝔾)|×N|V(𝔾)| parametrizing tropical curves h:ΓN of type Θ. The e-coordinate for an edge eE(𝔾) is the length (e) of e; the v-coordinates for a vertex vV(𝔾) are the coordinates of h(v)N.

Let Λ be a tropical curve without loops. By a family of parametrized tropical curves over Λ we mean a continuous family of curves hq:ΓqN for qΛ such that the following holds:

  1. The degree and the number of contracted legs are the same for all fibers.

  2. Along the interior of an edge/leg e of Λ, the combinatorial type of the fibers is constant.

  3. If a vertex vV(Λ) is adjacent to an edge/leg e of Λ, there is a fixed contraction of weighted graphs with marked legs φe:𝔾e𝔾v of the underlying graphs of the fibers.

  4. The length of an edge γE(𝔾e) in Γq along an edge/leg e of Λ is given by an integral affine function. Here we identify edges over an adjacent vertex of e with those over e via φe, setting their length to zero in case they get contracted.

  5. Similarly, the coordinates of the images h(u) for a vertex u of 𝔾e in h(Γq) along an edge/leg e of Λ is given by an integral affine function, where we again identify vertices over the interior of e with those over vertices adjacent to e via φe.

Any family of parametrized tropical curves h:ΓΛN induces a piecewise integral affine map α:ΛMg,n,trop by sending qΛ to the point parametrizing the isomorphism class of the fiber hq:ΓqN. Furthermore, α lifts to an integral affine map from the interior of each edge/leg e of Λ to the corresponding stratum MΘ. We refer the reader to [7, Sections 3.1.3 and 3.1.4] for formal definitions.

2.5 Valued fields

For a valued field K, we denote by K0 the ring of integers, by K00K0 the maximal ideal, and by K~ the residue field. As a general convention, we will similarly indicate when working over K0 and K~ by adding a superscript 0, respectively an overlining tilde; for example, X0 for a model over K0 of a curve XSpec(K) and X~ for its reduction.

3 Realizable tropical curves and h-transverse polygons

In this section and the next, we will work over the algebraic closure K of a complete, discretely valued field, with valuation ν:K{}.

3.1 h-transverse polygons

The toric surfaces SΔ that we consider in this paper are the ones with h-transverse polygon Δ. We restrict ourselves to h-transverse polygons since the theory of floor decomposed tropical curves has been developed only for this case in [3].

Definition 3.1.

A polygon ΔM=2 is called h-transverse, if its normal vectors all have integral or infinite slopes with respect to the x-coordinate.

Clearly, whether Δ is h-transverse depends only on its normal fan; thus the notion encodes a property of the surface SΔ and does not depend on the polarization Δ. On the other hand, it does depend on a choice of coordinates, i.e., on the identification M2. For the results we will discuss, it, of course, suffices that there is one choice of coordinates, for which Δ is h-transverse.

More explicitly, h-transversality of a polygon Δ is characterized by the following. Denote by y0:=minqΔyq the minimal y-coordinate of lattice points in Δ. Set yi:=y0+i with i up to yk:=maxqΔyq, and denote by Yi the horizontal line given by y=yi. Then Y0 and Yk intersect Δ in a bounded interval, possibly of length 0. Now, Δ is h-transverse if and only if the lines Yi for 0ik intersect the boundary of Δ at two lattice points; or in other words, the intervals YiΔ are lattice intervals for all i. See Figure 1 for an illustration. We denote by 𝚠(Δ) the width of Δ, that is, the maximum length of the intersections YiΔ. If they are not of length 0, we call the lattice intervals Y0Δ and YkΔ the horizontal sides of Δ.

Example 3.2.

Many smooth toric surfaces are induced by h-transverse polygons, for example, 2, Hirzebruch surfaces, toric del Pezzo surfaces, and similarly blow-ups of Hirzebruch surfaces at any subset of the four torus invariant points. An interesting class of examples of singular polarized toric surfaces induced by h-transverse polygons are the surfaces associated to the so-called kites introduced in [19]; see also [20, Section 1.1]. These are lattice polygons with vertices (0,0),(k,±1) and (k+k,0) for non-negative integers k,k with kk and k>0 (the choice of coordinates differs from [19]), see Figure 1. Some of these surfaces admit reducible Severi varieties, and we will discuss them in more detail in Proposition 5.14 and Example 7.2 below.

Example 3.3.

Let SΔ be the weighted projective plane with Δ given by vertices (0,1), (1,-1) and (6,-1); see Figure 1. Then Δ is not h-transverse for any choice of coordinates. Indeed, if there was such a choice of coordinates, then the x-coordinate would give a non-zero linear function on N, that would assign each of the three primitive vectors along the rays of the dual fan of Δ values ±1 or 0. The above conditions on linear functions on N correspond to three triples of parallel lines in M, and one checks that for Δ as above no three have a common point of intersection other than the origin; hence Δ is not h-transverse for any choice of coordinates. Note also that the same holds for any Δ whose dual fan is a refinement of the dual fan of Δ. Thus, an appropriate toric blow-up SΔ of SΔ gives a smooth toric surface that is not h-transverse for any choice of coordinates.

Figure 1

The kite (k,k)=(1,2) and two triangles. The right triangle is not h-transverse in any coordinate system, cf. Example 3.3, while the middle triangle is h-transverse with respect to the x-axis parallel to the blue lines, but not with respect to the usual x-axis.

3.2 Floor decomposed curves

We recall that a floor decomposed tropical curve is defined to be a parametrized tropical curve h:ΓN=2, in which the x-coordinate of the slope of each edge/leg is either 0 or ±1. In the former case, a non-contracted edge/leg is called an elevator. If a connected component of the graph obtained from Γ by removing the interiors of all elevators contains a non-contracted edge/leg, it is called a floor.

Lemma 3.4.

Let h:ΓNR be a floor decomposed tropical curve of degree dual to an h-transverse polygon Δ. Then the multiplicity of any elevator eE¯(Γ) is bounded above by the width w(Δ).

Proof.

Recall that the multiplicity 𝔴(h,e) is defined to be the integral length of h/e. Therefore, for embedded tropical curves, this bound follows from Legendre duality. In general, we argue as follows. For any mM, consider the parametrized tropical curve hm:Γ given by the composition of h with the linear functional m:N. Let dm be the sum of multiplicities of the legs of Γ having positive hm-slope. If p is a point that is not an image of a vertex of Γ, then by balancing, the number of preimages hm-1(p) counted with edge multiplicities equals dm. Thus, to show the claim, it suffices to find an mM for which dm=𝚠(Δ) and hm does not contract elevators of Γ.

In order to do so, note first that we can inscribe Δ in an h-transverse integral parallelogram M with two horizontal sides and width 𝚠()=𝚠(Δ). We now claim that the primitive integral vector m along a non-horizontal side of and having positive y-component satisfies the assertion. Indeed, hm does not contract elevators, since the y-coordinate of m cannot be 0. To calculate the degree of hm, pick a point p<minvV(Γ){hm(v)} in . All preimages hm-1(p) are contained in the set of legs li of Γ, whose slopes in N have negative inner product with m, i.e.,

(hli,m)<0.

Let l- (resp., r-) be the connected component of Δ in the bottom left (respectively, bottom right) of the parallelogram ; both are (possibly empty, non-convex) polygons. Note that outer normals along l- and r- sum to zero, and m has y-coordinate 1 since Δ is h-transverse. Thus, the sum of multiplicities 𝔴(hm,li) over the legs li, whose slopes have negative inner product with m, equals the sum of the integral lengths of the bottoms of Δ, l- and r-. The latter is nothing but the length of the bottom of , i.e., 𝚠()=𝚠(Δ). Thus, for the tropical curve hm:Γ, we have dm=𝚠(Δ), as claimed. ∎

In our setup, a vertical segment connecting two floors in a floor decomposed tropical curve can contain several elevators. It will be convenient to give a name to such segments, which are special cases of elevators in the sense of [3]:

Definition 3.5.

Let h:ΓN be a floor decomposed tropical curve and EΓ a subgraph such that h(E) is a bounded interval with constant x-coordinate in N. We will call E a basic floor-to-floor elevator if it satisfies the following properties:

  1. for any edge/leg e of ΓE adjacent to E, h(e)h(E) is one of the endpoints of h(E),

  2. for each of the two endpoints of h(E) there is a unique vertex ui of E that gets mapped to it, and

  3. the ui have exactly two adjacent edges/legs not contained in E and their slopes have x-coordinates 1 and -1.

3.3 Tropicalization and realizability

In this subsection, we will give a local version of Speyer’s well-spacedness condition for the realizability of elliptic curves [23]; see also [15, Theorem 1.1], [1, Theorem 6.9] and [22, Theorem D] for other generalizations.

If a parametrized curve f:XSΔ is defined over the valued field K, one can naturally construct its tropicalization, that is, a parametrized tropical curve h:ΓN. Parametrized tropical curves arising this way are called realizable. The tropicalization construction is well known and can be found, e.g., in [7, Section 4.2]. For the convenience of the reader, we recall it in the special case of a non-constant rational function f on a smooth proper marked curve X. Notice that to turn (X,f) into a parametrized curve f:X1, one must first add the zeroes and poles of f to the collection of marked points on X. We always do so when talking about tropicalizations of rational functions, without necessarily mentioning it.

Let f:X1 be a parametrized curve, and X0 the stable model of X over Spec(K0). The underlying graph of Γ:=trop(X) is the dual graph of the central fiber X~, i.e., the vertices correspond to irreducible components of X~, the edges – to nodes, the legs – to marked points, and the natural incidence relation holds. The weight of a vertex is given by the geometric genus of the corresponding irreducible component. Finally, the length of an edge corresponding to a node z is given by the valuation ν(λ), where λK00 is such that étale locally at z, the total space of X0 is given by xy=λ. It remains to define the map h:=trop(f):Γ. Since h is affine along edges and legs, it is sufficient to specify its values at vertices together with its slopes along legs, which are given as follows. For the irreducible component X~v of X~ corresponding to a vertex vV(Γ), let λvK× by such that λvf is an invertible function at the generic point of X~v. Then h(v):=ν(λv). The slope of h along a leg lL(Γ) is defined to be the order of pole of f at the corresponding marked point.

Let λv be as above. The scaled reduction of f at X~v with respect to λv is defined to be the non-zero rational function (λvf)|X~v on X~v. Notice that although scaled reductions of f at X~v depend on λv, their divisors do not. In fact, it is straightforward to check that the divisor of a scaled reduction is given by -eStar(v)𝔴(h,e)pe, where peX~v is either the node of X~ or the reduction of the marked point corresponding to e. In particular, h has integral slopes, satisfies the balancing condition, and therefore (Γ,h) is a parametrized tropical curve, cf. [25, Lemma 2.23].

We say that a parametrized tropical curve h:ΓN is weakly faithful at a vertex vV(Γ) if h does not contract the star of v to a point. If (Γ,h) is the tropicalization of a parametrized curve, then being not weakly faithful means that the scaled reductions of all monomial functions at X~v are constant.

Lemma 3.6.

Let f be a non-constant rational function on a smooth proper marked curve X, and v a vertex of trop(X). Then there is μK such that trop(f-μ) is weakly faithful at v.

Proof.

Let η be a K-point of X at which f is defined and such that the reduction of η lies on the component X~v of the reduction of X corresponding to the vertex v. Set μf(η)K. Since f-μ has a zero at η, every scaled reduction of f-μ at X~v needs to vanish at the reduction of η. However, since f is non-constant, every scaled reduction of f-μ is non-zero. Therefore, the scaled reductions of f-μ at X~v are non-constant, and trop(f-μ) is weakly faithful at v. ∎

A cycle of a graph 𝔾 is a connected subgraph of 𝔾 in which each vertex has valence 2. A cycle O of a parametrized tropical curve h:ΓN is a subgraph of Γ induced by a cycle of the underlying graph 𝔾, possibly containing additional legs that get contracted by h.

Figure 2

From left to right: a flattened cycle, the two versions of an elliptic component, and the two versions of a contracted elliptic tail.

Definition 3.7.

Let h:ΓN be a parametrized tropical curve. We give the following names to certain subgraphs O of Γ (see Figure 2):

  1. O is called a flattened cycle if it is a cycle, whose image in N is a bounded interval, and such that for each endpoint of h(O) there is exactly one vertex of O mapped to it,

  2. O is called an elliptic component if it is either a single vertex of weight 1 or a single vertex of weight 0 with an adjacent loop (which necessarily gets contracted by h),

  3. O is called a contracted elliptic tail if h(O) is a point, and O contains two vertices v1 and v2 connected by a single edge e such that v1 has weight zero, and all edges adjacent to v2 are contained in O. Furthermore, v2 has either weight 1 and is adjacent only to e, or it has weight 0 and is adjacent to an additional loop.

Proposition 3.8.

Let h:ΓNR be a realizable tropical curve. Assume that EΓ is a subgraph that for some choice of coordinates satisfies the three axioms of Definition 3.5 for a basic floor-to-floor elevator. Suppose furthermore, that OE is a subgraph which is either a flattened cycle, an elliptic component, or a contracted elliptic tail. Finally, suppose that EO¯ consists of at most two connected components, each of which is a weightless subgraph of Γ that maps injectively to h(E) away from contracted legs. Then either h(O)=h(E) or the complement of h(O) in h(E) consists of two intervals of the same length.

Proof.

Since h:ΓN is realizable, there is a parametrized curve f:XSΔ, whose tropicalization is h:ΓN. Let X~ be the central fiber of the stable model of X and X~E the subcurve of X~ that corresponds to vertices in E; similarly, let X~u1 and X~u2 be the components corresponding to the (unique) vertices u1,u2E that get mapped to the endpoints of h(E).

Without loss of generality we may assume that E satisfies the three axioms of Definition 3.5 for the standard coordinates in N=2. In particular, h(E) is vertical. Set

m:=(1,0)M=2.

Since h(E) is vertical and the only adjacent edges/legs to E whose slope has non-zero x-coordinate are by assumption adjacent to u1 or u2, we have that any scaled reduction of f*(xm) at components of X~E different from the X~ui has no zeroes or poles – thus, it needs to be constant. In particular, no poles of f*(xm) on X specialize to components of X~E different from the X~ui. On the other hand, and again by assumption, for each ui there are exactly two adjacent edges/legs whose slope has non-zero x-coordinate, and that slope is ±1. Accordingly, any scaled reduction of f*(xm) at X~ui has a simple pole and a simple zero on X~ui – thus, it necessarily has degree 1.

Case 1: O is an elliptic component or a contracted elliptic tail. Let u be the (unique) vertex in O that either has weight 1, or is adjacent to a loop. Let X~u be the component of X~ corresponding to u. Note first that we must have uui since there can be no rational function of degree 1 on a curve of arithmetic genus 1. By Lemma 3.6, we may choose μK such that any scaled reduction of Gf*(xm)-μ at X~u is non-constant. The rational functions f*(xm) and G may have different zeroes on X, and to consider the tropicalization of G, we add the new zeroes as marked points to X. Note, however, that the poles of f*(xm) on X remain unaffected when subtracting μ.

Let LE be the subgraph of E spanned by the edges of E that do not get contracted by h. By the assumptions on E, we can naturally identify L with a bounded line segment. Let γi denote the edge in L adjacent to ui and let piX~ui denote the point corresponding to γi. Let λm,EK be such that λm,Ef*(xm) is regular and invertible at generic points of X~E; such a λm,E exists since h(E) has constant x-coordinate and m=(1,0). The corresponding scaled reduction (λm,Ef*(xm))|X~E has, as explained above, constant value c~ along components of X~E different from the X~ui, and has degree 1 on the X~ui. Furthermore, the restriction of λm,EG to X~E is (λm,Ef*(xm))|X~E-c~. In particular, λm,EG is still a non-zero rational function on the X~ui and we get trop(G)(u1)=trop(G)(u2)=ν(λm,E). Furthermore, the scaled reduction (λm,EG)|X~ui has degree 1 with a simple zero at pi; thus trop(G) has outgoing slope -1 at ui along γi.

We can now deduce the claim from the properties of trop(G), see Figure 3 for an illustration. Let uL be the unique vertex with h(u)=h(u); that is, if O is an elliptic component we set u:=u, and if it is a contracted elliptic tail, then u and u are the two vertices of O. Recall first that trop(G) is harmonic, no poles of G on X specialize to components of X~E different from the X~ui, and the outgoing slope of trop(G) at ui along γi is -1. We conclude that the sum

sv:=γStar(v)Ltrop(G)γ

of outgoing slopes of trop(G) at any vL{u1,u2} along edges of L is non-negative. It follows that trop(G) is convex along L, considered as a line segment. In particular, the outgoing slope of trop(G) along the two edges of L at u are both at most 1. On the other hand, since X~u has arithmetic genus 1 and any scaled reduction G~u of G at X~u is, by the construction of G, not constant, the negative part of div(G~u) needs to have multiplicity at least 2; this means that trop(G) needs to have positive outgoing slopes at u that sum to at least 2. Since no poles of G specialize to X~u or X~u, it follows that trop(G) needs to have outgoing slopes at u along edges contained in L that sum to at least 2. Combining these observations, we get that trop(G) has outgoing slope 1 at u along each of the 2 adjacent edges in L. Therefore, by convexity of trop(G), it is affine on L except at u, and its slope on both connected components of L{u} is 1. Since trop(G)(u1)=trop(G)(u2), this implies that u is the midpoint of L; balancing of h then gives that also h(u)=h(u) is the midpoint of h(E)=h(L), as claimed.

Figure 3

In red, the graph of trop(G), where the numbers indicate outgoing slopes. On the left, Case 1, where O contains a single vertex of weight 1, u=u, and L contains only the two edges γ1 and γ2; if there are contracted legs contained in E, L may contain more edges. On the right, Case 2, with trop(G) constant along both Oi; in general, it is non-constant along at most one of the Oi.

Case 2: O is a flattened cycle. As in Case 1, let u1,u2 be the vertices that get mapped to endpoints of h(E), and let u1 and u2 be the vertices of O that get mapped to the endpoints of h(O). If u1=u1 and u2=u2, there is nothing to show. So suppose u1u1 and we first claim that then also u2u2. Assume to the contrary that u2=u2. Since O is a cycle and u1u1, the closure of X~E(X~u1X~u2) intersects X~u2 in two points p1,p2. Since a scaled reduction (λm,Ef*(xm))|X~E as in Case 1 has constant value c~ along components of X~E different from the X~ui, the scaled reduction (λm,Ef*(xm))|X~u2 at X~u2 needs to achieve the same value c~ at both intersection points p1 and p2; since the scaled reduction has degree 1 on X~u2, this is not possible, and hence u2u2.

The remaining argument is similar to Case 1 and we omit some of the details. Denote by L the subgraph of E spanned by edges that do not get contracted by h; this time, we may identify L with two line segments Li between ui and ui and two line segments O1 and O2 between u1 and u2. By Lemma 3.6, we may choose μ such that the scaled reduction of Gf*(xm)-μ at X~u1 is not constant. Arguing as in Case 1, trop(G)(u1)=trop(G)(u2), trop(G) has outgoing slope -1 at ui along Li, and trop(G) is convex on both line segments L1OiL2. It follows that trop(G) needs to be constant along one of the Oi; thus trop(G)(u1)=trop(G)(u2). Since trop(G) is not constant at u1, it furthermore follows that trop(G) is linear along L1. Combining these observations, we get (L1)(L2) for the lengths of the Li. Repeating the argument for G with scaled reduction at X~u2 non-constant gives (L2)(L1), and therefore (L1)=(L2) as claimed. ∎

Remark 3.9.

In the proof of Proposition 3.8 we considered the tropicalization of the function G=f*(xm)-μ, and one may view the obtained function as an additional tropical coordinate (cf. Figure 3). This is a version of a tropical modification; we refer the interested reader to [14] for a survey and further references.

4 Local liftability results for families of tropical curves

In [7] we introduced the tropicalization of one-parameter families of parametrized curves and studied the properties of the induced moduli maps α:ΛMg,n,trop to the tropical moduli space over certain strata, called nice strata and simple walls. This was then used to locally lift the family of tropical curves, an essential ingredient in the degeneration argument for 2. For the convenience of the reader, we recall this construction in Section 4.1 and the properties of α in Section 4.2.

For the remainder of this section, we fix a family of parametrized curves f:𝒳SΔ such that 𝒳B admits a split stable model 𝒳0B0 over Spec(K0), where B0 is a prestable model of the projective curve with marked points (B,τ), cf. Section 2. We assume, in addition, that the irreducible components of the reduction B~ of B0 are smooth.

4.1 Tropicalization for one-parameter families

The tropicalization of the family of parametrized curvesf:𝒳SΔ with respect to a fixed model 𝒳0B0 is a family of parametrized tropical curves h:ΓΛN as in Section 2.4. We now sketch its construction, and refer to [7, Section 4] for details.

The base Λ of the tropical family is the tropicalization of the base curve (B,τ) with respect to the model B0. For a vertex vV(Λ) denote by B~v the irreducible component of the reduction B~ corresponding to v. Let ηB(K) be a K-point, whose reduction sη~ is a non-special point of B~v, i.e., s is neither a node of B~, nor a marked point. Then the fiber hv:ΓvN of the family h:ΓΛN over v is defined to be the tropicalization of f|𝒳η:𝒳ηSΔ. As long as sB~v is non-special, one checks that the parametrized tropical curves obtained this way are isomorphic for different choices of η. Furthermore, the splitness of the model 𝒳0B0 ensures that they can be identified in a canonical way. More generally, if eE¯(Λ) and qe is a ν(K×)-rational point, then we can refine the model B0 of B, such that q becomes a vertex in the tropicalization of the base with respect to the refined model. The fiber hq:ΓqN over q now is defined as before. One can check that the fibers over the ν(K×)-rational points of Λ satisfy the five axioms of Section 2.4, and therefore can be extended uniquely by continuity to a family over the whole of Λ.

To verify the first axiom, recall that the slopes of hq along the legs of Γq are given by the order of pole along the marked points of the pullbacks of monomials (f|𝒳η)*(xm). Since marked points of 𝒳η are restrictions of marked points σ of 𝒳, the slopes of hq along the legs of Γq are globally determined by the order of pole of f*(xm) along the σ, and hence independent of q.

If sB~v is a special point corresponding to an edge/leg e of Λ, the underlying graph 𝔾e of fibers over e is by construction the dual graph of the fiber 𝒳~s over s. In particular, it is constant along e, which verifies the second axiom. Furthermore, the family 𝒳0B0 prescribes a one-parameter degeneration of stable curves in a neighborhood of s in B~v. Such a degeneration naturally induces an edge contraction φe:𝔾e𝔾v between the dual graphs, as required by the third axiom. It remains to check that the edge lengths (γ) and the images of vertices h(u) in the fibers are given by integral affine functions along the edges/legs e of Λ.

Let us show how to verify this in the case of (γ) over an edge e, as the remaining cases can be verified in a similar way. Consider the toroidal embedding B(iτi)B0, whose boundary B~(iτi) contains all points over which fibers of 𝒳0B0 fail to be smooth. All models are defined over some discretely valued subfield FK with a uniformizer πF00. Let sB~vB~w be the node corresponding to e. Then in an étale neighborhood U of s we have ab=πks, where a=0 and b=0 define the components B~v and B~w, respectively. Thus, the length of e in Λ is given by Λ(e)=ksν(π).

Let z be the node of 𝒳~s corresponding to γ. Then, possibly after shrinking U, an étale neighborhood of z in the total space 𝒳0 is given by xy=gz for some regular function gz𝒪U(U). Let ψ:Spec(F)B be an F-point with image η and reduction s. Then the length of γ in trop(𝒳η) satisfies (γ)=ν(ψ*(gz)) by the tropicalization construction, cf. Section 3.3. Since gz vanishes only along B~vB~w in U, there are ka,kb,kπ such that a-kab-kbπ-kπgz is regular and invertible on U. Thus,

ν(ψ*(gz))=kaν(ψ*(a))+kbν(ψ*(b))+kπν(π).

Since ab=πks, it follows that ν(ψ*(a))+ν(ψ*(b))=ksν(π), and therefore

ν(ψ*(gz))=(ka-kb)ν(ψ*(a))+kbΛ(e)+kπν(π).

By the construction, ν(ψ*a) is nothing but the distance from trop(η)e from the vertex w, and kaΛ(e)+kπν(π) and kbΛ(e)+kπν(π) are the lengths of γ in 𝔾v and 𝔾w, respectively, identified via the contraction maps described above. Thus, the length of γ along e is given by an integral affine function of slope ka-kb, as claimed.

Remark 4.1.

Note that ka-kb is the vanishing order at s of the scaled reduction of gz at B~w since (ab)-kbπ-kπgz is regular and invertible at the generic point of B~w. Thus, the slope (γ)/e is given by the vanishing order at s of a scaled reduction of gz at B~w. Similarly, for a leg l of Λ adjacent to w and corresponding to a marked point τB, the slope (γ)/l is the vanishing order of gz at τ, which coincides with the vanishing order at τ~ of a scaled reduction of gz at B~w.

The slopes of h(u)/e can be described similarly. Let s be a special point of B~w corresponding to an edge/leg e. Then a vertex u of 𝔾e corresponds to an irreducible component 𝒳~s,u of the fiber 𝒳~s. There is a unique irreducible component 𝒳~u of the surface 𝒳~|B~w that contains 𝒳~s,u. Then for mM, the slope h(u)(m)/e is the order of pole along 𝒳~s,u of a scaled reduction of f*(xm) at 𝒳~u, which in the case of a leg e is nothing but the order of pole of f*(xm) along the irreducible component of the fiber 𝒳τ that contains 𝒳~s,u.

4.2 Simple walls and nice strata

Let α:ΛMg,n,trop be the map induced by tropicalizing a one-parameter family, let v be a vertex of Λ, and let Θ be the combinatorial type such that α(v)M[Θ]. The map α is harmonic at v if α lifts to a map to MΘ locally at v and the slopes α/e along outgoing edges/legs e in Star(v) sum to zero. The map α is locally combinatorially surjective at v, if the image α(Star(v)) intersects all strata M[Θ] adjacent to M[Θ]. The following is Case 1 of Step 3 in the proof of [7, Theorem 4.6]:

Lemma 4.2.

Let h:ΓΛNR be the tropicalization of a family of parametrized curves XSΔ with respect to X0B0. Let vΛ be a vertex corresponding to an irreducible component B~v of the reduction B~. Suppose that the moduli map χ:B0M¯g,n+|| induced by the family X0B0 contracts B~v. Then the induced map α:ΛMg,n,trop is harmonic at v.

Sketch of the proof.

If χ(B~v)=pM¯g,n+||, then every fiber of 𝒳0B0 over B~v has the same dual graph 𝔾. This implies that the combinatorial type Θ of h:ΓΛΛ is locally constant around v, and thus α lifts to a map Star(v)MΘ. As explained in Section 2.4, we may view MΘ naturally as a subset of |E(𝔾)|+2|V(𝔾)|, and harmonicity can be checked coordinate-wise.

We first consider the γ-coordinate for an edge γE(𝔾). The locus in M¯g,n+|| in which the node z corresponding to γ persists, is cut out locally at p by a single function. Since we can pull this function back from M¯g,n+|| to B0, the node z is given by xy=gz in the total space of 𝒳0, where gz is a regular function in a neighborhood of B~v in B0 (and not just in a neighborhood of a closed point of B~v as it is in general). By Remark 4.1, the slopes of α at v in the γ-coordinate are the orders of vanishing of a scaled reduction of gz at B~v, and hence their sum vanishes.

Let uV(𝔾) be a vertex, and consider the slopes of α at v in the h(u)-coordinates. Let 𝒳~uB~v be the surface corresponding to u. Since the model is split and fibers of 𝒳~uB~v are isomorphic, we have (after a finite base change in B~v) 𝒳~u𝒞u×B~v for some curve 𝒞u. Fix a general section {c}×B~v. By Remark 4.1, the slopes of h(u)(m) for mM are given by the orders of pole along the 𝒳~s,u{s}×B~v of a scaled reduction of f*(xm) at 𝒳~u𝒞u×B~v for the special points sB~v. The latter coincide with the orders of pole of the restriction of the scaled reduction to {c}×B~v at the points {c}×{s}, and hence again have to sum up to zero for any mM. ∎

Remark 4.3.

Suppose in the setting of Lemma 4.2 that vΛ is a vertex, such that the fiber Γv has weightless and 3-valent underlying graph 𝔾. Since the stratum M𝔾 of curves with dual graph 𝔾 in M¯g,n+|| is a single point, the lemma in this case implies that α is harmonic at v.

Recall that a stratum MΘ is called nice if it is regular (that is, of the expected dimension), and the underlying graph of Θ is weightless and 3-valent; MΘ is a simple wall if it is regular, and the underlying graph is weightless and 3-valent except for a unique 4-valent vertex. Note that each simple wall MΘ is contained in the closure of exactly three strata, all of them nice. They correspond to the three ways of splitting the 4-valent vertex into two 3-valent vertices. The following lemma is a part of [7, Theorem 4.6] and applies in particular to simple walls and nice strata.

Lemma 4.4.

Let h:ΓΛNR be the tropicalization of a family of parametrized curves XSΔ with respect to X0B0. Let vΛ be a vertex with α(v)M[Θ], where the underlying graph of Θ is weightless and 3-valent except for at most one 4-valent vertex. Then the induced map α:ΛMg,n,trop is either harmonic or locally combinatorially surjective at v.

Sketch of the proof.

Consider the moduli map χ:B0M¯g,n+|| induced by the family 𝒳0B0. If χ contracts B~v, then α is harmonic at v by Lemma 4.2. Otherwise, the underlying graph 𝔾 of Θ must contain a unique 4-valent vertex, since strata M𝔾 in M¯g,n+|| corresponding to weightless, 3-valent dual graphs 𝔾 have dimension 0. Thus, M𝔾 is one-dimensional, and χ maps B~v surjectively onto the closure M¯𝔾M¯g,n+||. Now one notes that M[Θ] is contained in the closure of three strata in Mg,n,trop, which correspond to the three splittings of the 4-valent vertex, and M¯𝔾 contains as its boundary the three strata parametrizing curves with the corresponding dual graphs; hence local combinatorial surjectivity follows in this case. ∎

4.3 Flattened cycles

The first case we need, that was not covered in [7], is a vertex v of Λ, such that the fiber over v contains two 4-valent vertices that form the endpoints of a flattened cycle. In this case, we do not get local combinatorial surjectivity of α at v, as will become clear in the proof of the next lemma; instead, we get that α is either harmonic at v or α(Star(v)) intersects precisely the maximal adjacent strata in Mg,n,trop.

Lemma 4.5.

Let h:ΓΛNR be the tropicalization of a family of parametrized curves XSΔ with respect to X0B0. Let vΛ be a vertex with α(v)M[Θ], where the underlying graph of Θ is weightless and 3-valent, except for the two endpoints of a flattened cycle, which are 4-valent. Assume that for some choice of coordinates, the flattened cycle satisfies the three axioms of Definition 3.5 for a basic floor-to-floor elevator. Then either α is harmonic at v; or for every M[Θ] whose closure contains M[Θ] and where the underlying graph of Θ is 3-valent, there is an edge/leg e in Star(v) such that α(e)M[Θ].

Proof.

Let χ:B0M¯g,n+|| be the moduli map induced by the family 𝒳0B0. If χ contracts B~v, then α is harmonic at v by Lemma 4.2. Thus, we assume χ does not contract B~v. Let u,u be the endpoints of the flattened cycle E in Θ; that is, h(E) is the line segment between h(u) and h(u). Any combinatorial type Θ as in the claim of the lemma is obtained by splitting u and u each into two 3-valent vertices.

Since 𝒳0B0 is split, there are irreducible components 𝒳~uB~v and 𝒳~uB~v of 𝒳~ that restrict over non-special points of B~v to the irreducible component of the fiber corresponding to u and u, respectively. Furthermore, both 𝒳~uB~v and 𝒳~uB~v have four marked points, corresponding to the four adjacent edges/legs to u and u, respectively. Thus, we get maps χu:B~vM¯0,4 and χu:B~vM¯0,4 induced by the two families. Since χ does not contract B~v and a non-special fiber of 𝒳0B0 over B~v has moduli only at the components corresponding to u and u, at least one of the maps χu and χu is non-constant, and hence surjective. We assume this to be the case for χu.

Surjectivity of the map χu implies that there is an edge or leg eStar(v) such that α(e)M[Θ′′] for any Θ′′ obtained by splitting u: each splitting of u corresponds to a boundary point of M¯0,4. Let γ1,γ2 be the two edges adjacent to u that are contained in E and γ1,γ2 the two edges adjacent to u contained in E (possibly identical with the γi); see Figure 4. If γ1 and γ2 are not adjacent to the same vertex of the underlying graph of Θ′′, also u has to split into two 3-valent vertices in Θ′′ by balancing, and the claim of the lemma follows for such Θ′′.

Figure 4

On the left, the flattened cycle in a curve of combinatorial type Θ. In the middle, a splitting of u in which γ1 and γ2 are not adjacent to the same vertex. On the right, the splitting in which they remain adjacent.

Thus, it remains to show the following. If γ1,γ2 are adjacent to the same vertex in the underlying graph of Θ′′, then the vertex u necessarily splits in Θ′′. If not, then the curves of type Θ′′ are not realizable by Proposition 3.8, and therefore cannot appear in the tropicalization of a family of parametrized curves. ∎

4.4 Elliptic components and contracted elliptic tails

We will next describe, how to develop a loop or a contracted edge from a 2-valent, weight 1 vertex u in the tropicalization ΓΛN of a family of parametrized curves. If Θ is the combinatorial type containing such a vertex u, we will consider the following combinatorial types obtained from Θ (see also Definition 3.7 and Figure 2): let Θ be obtained from Θ by developing a loop based at u, replacing u with a weightless vertex u; let Θ′′ be obtained from Θ by developing a contracted edge adjacent to a 1-valent vertex u′′ of weight 1; and let Θ′′′ be obtained from Θ′′ by developing a loop at u′′, replacing u′′ with a weightless vertex.

Lemma 4.6.

Let ΓΛNR be the tropicalization of a family of parametrized curves XSΔ with respect to X0B0. Let vΛ be a vertex with α(v)M[Θ], where Θ is 3-valent and weightless, except for one 2-valent vertex u of weight 1. Then, with Θ,Θ′′ and Θ′′′ as above, the following holds:

  1. Suppose char(K~) does not divide the multiplicity of h along the two edges adjacent to u. Then α is harmonic at v, or there is an edge/leg e in Star(v) such that α(e)M[Θ].

  2. Suppose the multiplicity of h along the two edges adjacent to u is a power of char(K~). Then α is harmonic at v, or there is an edge/leg e in Star(v) such that α(e)M[Θ′′].

  3. Let wΛ be a vertex. If α(w)M[Θ] or α(w)M[Θ′′′], then α is harmonic at w. If α(w)M[Θ′′], then α is harmonic at w, or there is an edge/leg e in Star(w) such that α(e)M[Θ′′′].

Proof.

As in the proof of Lemma 4.5, let 𝒳~u be the irreducible component of 𝒳~ corresponding to u. Let σ1 and σ2 be the two sections of 𝒳~uB~v corresponding to the two edges γ1 and γ2 adjacent to u (as before, they are sections since 𝒳0B0 is split). Let γi be the orientation of γi away from u; then by balancing we have h/γ1=-h/γ2. Let nN be a primitive vector for the line spanned by h/γ1; that is, h/γi is ±n times the multiplicity κ𝔴(h,γi) of h along the γi. Let mM be a primitive lattice element such that (m,n)=0, where (,) is the pairing M×N defining M and N as dual lattices.

We consider the rational function

G(λm,uf*(xm))|𝒳~u

on 𝒳~u, where λm,uK is chosen such that G is regular and invertible at the generic point of 𝒳~u. Over non-special points sB~v, the function G restricts to a rational function G|𝒳~s,u on the fiber 𝒳~s,u. We have that G|𝒳~s,u is regular and invertible away from the image of s under the two sections σ1 and σ2. More precisely, we get

div(G|𝒳~s,u)=κσ1(s)-κσ2(s).

Viewing 𝒳~s,u as an elliptic curve with identity σ2(s), we thus get that σ1(s) is a point of order dividing κ. Possibly replacing κ by a number that divides it, we may assume that σ1(s) has order κ. Note that this replacement preserves the divisibility condition – char(K~)κ or κ=char(K~)l – in the assumptions of the lemma. Since σ2(s)σ1(s) and u is of weight 1, we must have κ>1. Furthermore, this order is constant along smooth, non-supersingular fibers of 𝒳~uB~v. Let χ:B0M¯g,n+|| as before be the moduli map induced by the family 𝒳0B0. If χ contracts B~v, then α is harmonic at v by Lemma 4.2. Thus we assume χ does not contract B~v. Since the underlying graph of Θ is 3-valent and weightless except for u, this implies that the family 𝒳~uB~v contains non-isomorphic fibers.

For the first claim assume char(K~)κ. We will show that in this case 𝒳~uB~v necessarily contains a singular, irreducible fiber, which immediately implies that there is an element eStar(v) satisfying the assertion of the first claim. Let X1(κ)Spec(K~) be the compactification of the modular curve Y1(κ) as constructed by Deligne and Rapoport [11], where Y1(κ) parametrizes elliptic curves together with a point of exact order κ. Let 𝒞1(κ)X1(κ) be the universal curve. The fibers of 𝒞1(κ) over boundary points of X1(κ) are Néron polygons; they are semistable curves with dual graph a weightless cycle with an additional group structure. In the case of X1(κ), the Néron polygons that appear as fibers of 𝒞1(κ) are precisely the ones for which the length of the dual graph divides κ (this is an immediate consequence of [11, Construction IV.4.14, p. 224]). In particular, there are fibers of 𝒞1(κ) that are Néron polygons of length 1, or in other words, 𝒞1(κ) contains a singular irreducible fiber.

By what we showed above, 𝒳~uB~v induces a rational moduli map χu,κ:B~vX1(κ) defined at points where the fibers of 𝒳~uB~v are smooth. Furthermore, over this locus 𝒳~uB~v is the pullback of 𝒞1(κ). Recall next that because char(K~)κ, X1(κ) is irreducible by [11, Corollaire IV.5.6, p. 227]. Since 𝒳~uB~v contains non-isomorphic fibers and X1(κ) has dimension 1, this implies that the moduli map χu,κ is dominant. The curve B~v is smooth and X1(κ) is proper. Thus, after replacing B~v with a finite covering, the map χu,κ extends to the whole curve. To ease the notation, we will assume that χu,κ is defined on all of B~v.

Let bB~v be any point such that the fiber Cb of 𝒞1(κ)X1(κ) over χu,κ(b) is singular and irreducible. Since 𝒞1(κ)X1(κ) is a family of stable curves over a neighborhood of χu,κ(b), so is its pullback to B~v. But the family 𝒳~uB~v is also stable, and coincides with the pullback over a dense open subset of B~v. Therefore, its fiber over b is Cb by the uniqueness of the stable model. Hence, the family 𝒳~uB~v contains irreducible singular fibers as claimed.

For the second claim, assume now that κ=char(K~)l. As before, we obtain a rational map χu,κ:B~vY1(κ), defined at points where the fibers of 𝒳~uB~v are smooth; it dominates an irreducible component of Y1(κ). Since for any irreducible component of Y1(κ) the forgetful morphism to M1,1 is surjective, there is in particular sB~v such that χu,κ(s) parametrizes a supersingular elliptic curve. Since κ=char(K~)l, the only point of order κ on such a supersingular curve is the origin. Thus the images of the two points σ1(s) and σ2(s) in the universal family over Y1(κ) need to coincide. Since 𝒳~uB~v is a family of stable curves and the σi sections, we thus get that the fiber of 𝒳~uB~v over s has two irreducible components: a rational curve containing σ1(s) and σ2(s), which is attached to an elliptic curve along a single node. This is clearly equivalent to the second claim.

For the third claim, note first that the underlying graph of Θ′′′ is weightless and 3-valent; thus if α(w)M[Θ′′′], then α is harmonic at w by Remark 4.3. We next consider Θ. Recall that u denotes the 4-valent vertex of the underlying graph of Θ that is adjacent to the loop, and let wΛ be a vertex such that α(w)M[Θ]. As before, let 𝒳~uB~w be the irreducible component of 𝒳~B~ corresponding to u. Up to an isomorphism, we may assume that the general fiber of 𝒳~uB~w is 1 with 0 and identified, and one of its two marked points is 1. Arguing as for the first claim, we get that the second marked point needs to be a κ-th root of unity in 1. In particular, all fibers of 𝒳~uB~w are isomorphic. Since general fibers of (𝒳~)|B~w have by assumption moduli only at 𝒳~u, this implies that the global moduli map χ:B0M¯g,n+|| contracts B~w. Thus α is harmonic at w by Lemma 4.2. Finally, we consider Θ′′. Recall that u′′ denotes the vertex of weight and valence 1. As before, u′′ corresponds to a family 𝒳~u′′B~w of elliptic curves and induces a moduli map χu′′:B~wM¯1,1. If this map has constant image, the global moduli map χ:B0M¯g,n+|| contracts B~w, since general fibers of (𝒳~)|B~w have by assumption moduli only at 𝒳~u′′. In this case α is harmonic at w by Lemma 4.2. Otherwise, χu′′ is dominant, and the closure of its image contains the boundary point of M¯1,1. Again by uniqueness of the stable model, this implies that the family 𝒳~u′′B~w contains a singular, irreducible fiber, and the claim follows. ∎

5 Degeneration of curves on polarized toric surfaces

In this section, we prove the main result, Theorem 5.6. We work over an arbitrary algebraically closed field K and consider h-transverse polygons Δ. For a line bundle on SΔ, recall that the Severi varietiesVg, and Vg,irr are the following loci of the linear system ||:

Vg,={[C]||:C is reduced, torically transverse, and pg(C)=g},

where pg(C) denotes the geometric genus of C, and torically transverse means that C contains no zero-dimensional orbits of the torus action. Similarly,

Vg,irr={[C]Vg,:C is irreducible}.

When =Δ, we will write Vg,Δ:=Vg,Δ and Vg,Δirr:=Vg,Δirr.

Remark 5.1.

The definition of Severi varieties given here is slightly more restrictive than the one in [7, Definition 2.1] since we require the curves to avoid all zero-dimensional orbits in SΔ. This restriction is necessary to obtain well-behaved tangency profiles, which will be introduced below. One can check that the closures of the loci defined in the two different ways coincide, thus the modification is harmless for our purposes.

5.1 Tangency conditions

In Theorem 5.6 we will allow for some prescribed tangencies with the toric boundary of SΔ, and we establish the setup necessary for this statement next. Denote by D1,,DsSΔ for i=1,2,,s the closures of the codimension-one orbits in SΔ corresponding to the sides Δi of Δ. A tangency profile of(SΔ,) is a collection of multisets

d¯:=({di,j}1jai)1is

with ai0 and di,j>0 such that Di=1jaidi,j. We say that a parametrized curve f:XSΔ with f(X) torically transverse has tangency profile d¯, if for all i we have f*(Di)=Σ1jaidi,jpi,j as an effective divisor on X, where pi,j are distinct points.

Definition 5.2.

For a tangency profile d¯ of (SΔ,), we define Vg,d¯, (respectively, Vg,d¯,irr) to be the locus of Vg, (respectively, Vg,irr) consisting of curves C such that the normalization CνC induces a parametrized curve f:CνSΔ with tangency profile d¯.

As before, we set Vg,d¯,Δ:=Vg,d¯,Δ and Vg,d¯,Δirr:=Vg,d¯,Δirr for the line bundle Δ associated to Δ. Notice that in this case

Di=1jaidi,j

is the integral length of Δi. We say that a tangency profile d¯ is trivial on Di if di,j=1 for all 1jai. If a tangency profile is trivial on all orbits Di we simply call it trivial, and denote it by d¯0; that is, d¯0 is the tangency profile for (SΔ,) with ai=Di and di,j=1 for all i,j.

Remark 5.3.

If SΔ=2 and d¯ is trivial on two of the codimension-one orbits, then both Vg,d¯,Δ and Vg,d¯,Δirr are generalized Severi varieties in the sense of [4]. On the other hand, in [4] the authors also discuss the loci obtained by fixing the point of given tangency with the toric boundary, which we do not consider here.

Remark 5.4.

Clearly we have Vg,d¯0,irrVg,irr. On the other hand, Vg,irrV¯g,d¯0,irr by [7, Proposition 2.7]. Therefore

V¯g,irr=V¯g,d¯0,irr

and we are free to switch between Vg,d¯0,irr and Vg,irr in the study of degenerations of irreducible curves in SΔ.

We write |d¯|:=iai; that is, |d¯| denotes the number of distinct points in the preimage under f of the boundary divisor of SΔ for any parametrized curve f:XSΔ with tangency profile d¯.

Proposition 5.5.

Let d¯ be a tangency profile of (SΔ,L). Then Vg,d¯,L and Vg,d¯,Lirr are either empty or constructible subsets of |L| of pure dimension |d¯|+g-1.

Proof.

By [7, Lemma 2.6], the Severi variety Vg, is a locally closed subset in the linear system ||, and by [7, Proposition 2.7], it has pure dimension -KSΔ+g-1. To estimate the codimension of Vg,d¯, in Vg, we follow the approach of [5].

Let VVg, be an irreducible component, ηV its generic point, η¯ the corresponding geometric point, and Cη¯ the corresponding curve. Since the geometric genus of Cη¯ is g, it follows that there exists an irreducible finite covering FV:UV such that the geometric genus of the fiber over F-1(η) is g. Therefore, the pullback CUU of the universal curve to U is equinormalizable by [6, Theorem 4.2], since normalization commutes with localization. Set W to be the disjoint union of varietes U for different irreducible components. Then F:WVg, is a finite surjective map, and the pullback CWW of the universal curve to W is equinormalizable.

We consider the relative Hilbert scheme of points HW on the normalized family CWνW of degree KSΔ, and the natural section ψ:WH that associates to wW the pullback of the boundary divisor Di under the map CwνCwSΔ. Since the (relative) Hilbert scheme of points on a smooth (relative) curve is smooth, the standard dimension-theoretic arguments provide a lower bound on the dimensions of the irreducible components of Vg,d¯, as follows. The locus ZH of subschemes of the form i,jdi,jpi,j is locally closed of pure codimension i,j(di,j-1). Therefore, ψ-1(Z) is locally closed and has codimension at most

i,j(di,j-1)=-KSΔ-|d¯|

at any point. Finally, since Vg,d¯, is a union of components of Fψ-1(Z) and F is finite, it follows that Vg,d¯, is constructible and the dimension of any of its components is at least

(-KSΔ+g-1)-(-KSΔ-|d¯|)=|d¯|+g-1.

On the other hand, let V be an irreducible component of Vg,d¯,, and [C]V a general point. Suppose C has m irreducible components C1,,Cm. Set

gk:=pg(Ck),k:=𝒪SΔ(Ck),

and let d¯k be the tangency profile of the parametrized curve induced by normalizing Ck. Consider the map k=1mVgk,d¯k,kVg,d¯,. Its fiber over [C] is finite, and the map is locally surjective. Thus,

dim[C](V)=k=1mdim[Ck](Vgk,d¯k,k).

However, dim[Ck](Vgk,d¯k,k)|d¯k|+gk-1 by [26, Theorem 1.2 (1)], and hence

dim[C](V)k=1m(|d¯k|+gk-1)=|d¯|+g-1.

Thus, V has dimension |d¯|+g-1, which completes the proof for Vg,d¯,. The assertion for Vg,d¯,irr now follows too, since Vg,d¯,irr is a union of components of Vg,d¯,. ∎

5.2 The degeneration result

We are now ready to state our main result. Denote by Δ the interior of the polygon Δ and recall that we denote by 𝚠(Δ) the width of Δ. Recall furthermore, that a smooth curve in the linear system |Δ| has genus #|ΔM|.

Theorem 5.6.

Let Δ be an h-transverse polygon and suppose that char(K)=0 or char(K)>12w(Δ). Let 0g#|ΔM| be an integer and d¯ a tangency profile of (SΔ,LΔ), which is trivial on toric divisors corresponding to non-horizontal sides of Δ. Then V0,d¯,ΔirrV for every irreducible component V of V¯g,d¯,Δirr. In particular, each irreducible component of V¯g,Δirr contains V0,Δirr.

The proof of Theorem 5.6 occupies much of the remainder of this section. It proceeds by induction on g. In Lemma 5.12, we prove that V0,d¯,Δirr is non-empty and irreducible. To prove the induction step, we show that V must contain an irreducible component of the Severi variety Vg-1,Δirr. To do so, we impose dim(V)-1 general point constraints on curves in V and reduce its dimension to 1. We consider a one-parameter family of parametrized curves 𝒳B associated to an irreducible component of the obtained locus and analyze its tropicalization ΓΛN by investigating the induced map α from Λ to the moduli space of parametrized tropical curves.

Lemma 5.10 provides us with good control over the image of α. In particular, it is used in Lemma 5.11 to show that V must contain a codimension-one locus of curves [C], whose tropicalization contains a cycle O satisfying combinatorial properties, that are tailored to be able to develop a contracted edge or loop from it. We complete the proof in Section 5.2.3, by showing that for an appropriate choice of point constraints, Λ necessarily contains a leg parametrizing tropical curves with a fixed image in N and having a contracted edge or loop, whose length grows to infinity. Furthermore, one of the components of the complement of this edge/loop is a parametrized tropical curve of genus g-1 satisfying dim(V)-1 general point constraints. This allows us to prove that V must contain a codimension-one locus of curves of genus g-1.

To use tropicalizations, we have to work over a valued field. To this end, note that the statement of Theorem 5.6 is compatible with base field extension. Hence, by passing to the algebraic closure of K((t)), we work in the proof over the algebraic closure of a complete DVR, whose residue field is algebraically closed and has characteristic either 0 or greater than 12𝚠(Δ).

5.2.1 Point constraints

Let be a tropical degree dual to Δ; see Section 2.3. Then induces a tangency profile d¯ of (SΔ,Δ) as follows. For each 1is, let ai be the number of slopes in corresponding to the side iΔΔ, and {di,1,,di,ai} the multiset of multiplicities of these slopes. Vice versa, d¯ determines a tropical degree dual to Δ uniquely up-to ordering of the slopes. Notice also that if f:XSΔ is a parametrized curve of tangency profile d¯, then the degree of its tropicalization induces the tangency profile d¯ by the very construction of the tropicalization. We set once and for all

n:=|d¯|+g-1=||+g-1.

Denote by ev:Mg,k,tropNk the evaluation map

ev:(Γ,h)(h(l1),,h(lk)),

where h:ΓN is a parametrized tropical curve with contracted legs l1,,lk. If Θ is a combinatorial type with k contracted legs, we define evΘ to be the composition

M¯ΘMg,k,tropNk.

Then for k points q1,,qk in N, the preimage evΘ-1(q1,,qk) is a polyhedron cut out in M¯Θ by an affine subspace.

Definition 5.7.

For k>0 fix points p1,,pkSΔ and q1,,qkN. Denote by Hi|Δ| the hyperplane parametrizing curves that pass through the point pi.

  1. Let [C]Vg,d¯,Δirr pass through all of the points pi. We say that the points pi are in general position with respect to[C] if the intersection Vg,d¯,Δirr(i=1kHi) has dimension n-k at [C].

  2. Let (Γ,h)Mg,k,trop. We say that the points qi are in tropical general position with respect to(Γ,h) if the preimage ev-1(q1,,qk) has dimension n-k at (Γ,h).

Remark 5.8.

Suppose f:XSΔ is a parametrized curve with C:=f(X) passing through points p1,,pk and [C]Vg,d¯,Δirr. Assume that its tropicalization (Γ,h) is weightless, 3-valent, and immersed except at contracted legs. Set qi:=trop(pi). If q1,,qk are in general position with respect to (Γ,h), then so are p1,,pk with respect to [C]. Indeed, let MΘ be the stratum of Mg,0,trop that contains the curve obtained from (Γ,h) by forgetting the contracted legs and stabilizing. Then MΘ is maximal and has dimension n by [21, Proposition 2.23]. Thus we may choose n-k points qk+1,,qn contained in h(Γ), such that (Γ,h) is an isolated point in ev-1(q1,,qn)Mg,n,trop. Let pk+1,,pnC be n-k points such that trop(pi)=qi. We claim that then also [C] is an isolated point in Vg,d¯,Δirr(i=1nHi), and therefore Vg,d¯,Δirr(i=1kHi) has dimension n-k at [C], since Vg,d¯,Δirr is equidimensional of dimension n. Indeed, any irreducible curve

BVg,d¯,Δirr(i=1nHi)

containing [C] gives rise to a one-dimensional family of parametrized tropical curves ΓΛN (cf. Construction 5.9 below). The family ΓΛN then induces a map α:ΛMg,n,trop satisfying (Γ,h)α(Λ)ev-1(q1,,qn). Since the curves over B dominate SΔ, their tropicalizations cover a dense subset of N and α is not constant. Since Λ and hence also α(Λ) is connected, this contradicts the fact that (Γ,h) is an isolated point in ev-1(q1,,qn).

Let VV¯g,d¯,Δirr be an irreducible component, [C]V a curve, and p1,,pn-1SΔ points in general position with respect to [C]. Pick an irreducible component

ZV(i=1n-1Hi)

containing [C], and let us briefly explain how to associate a family of parametrized curves to Z, which we then can study by investigating its tropicalization. We refer to Step 2 of the proof of [7, Theorem 5.1] for technical details.

Construction 5.9.

First, we pull back the tautological family of curves from Z to the normalization Zν. After replacing Zν with a finite covering, we may assume that the resulting family is generically equinormalizable by [10, Section 5.10]. Normalizing its total space, and restricting the obtained family to an open dense subset of the base, we get a family f:𝒳SΔ of genus g parametrized curves over the complement of finitely many points τ in a smooth projective base curve B. Next, after replacing B with a finite covering, we may assume that the marked points σ of 𝒳 are the points that are mapped to {pi}i=1n-1 and to the boundary divisors of SΔ. Furthermore, we may assume that the family 𝒳Bτ extends to a split stable model 𝒳0B0, where B0 is a prestable model of (B,τ) over K0. In particular, for bB that gets mapped to a general [C]Z under the natural projection BZ, the fiber 𝒳b of 𝒳B is the normalization of C, equipped with the natural map to SΔ.

Recall from Section 4.2 that we consider two types of regular strata MΘ in Mg,n-1,trop: nice strata and simple walls. The former have the expected dimension 2n-1, and the latter 2n-2. Since both are automorphism free, we have MΘ=M[Θ] and may consider MΘ as a subset of Mg,n-1,trop.

Lemma 5.10.

Let α:ΛMg,n-1,trop be the moduli map induced by the tropicalization of the family f:XSΔ given by imposing n-1 general point constraints pi on V as in Construction 5.9. Set qi=trop(pi). Then α is not constant and the following hold:

  1. Suppose α maps a vertex vV(Λ) to a simple wall MΘ, and

    MΘevΘ-1(q1,,qn-1)=α(v)

    is a single point. Then there exists a vertex wV(Λ) such that α(w)=α(v) and the map α is locally combinatorially surjective at w.

  2. Suppose MΘ is a nice stratum, α(Λ)MΘ and

    dim(MΘevΘ-1(q1,,qn-1))=1.

    Then MΘevΘ-1(q1,,qn-1)=α(Λ)MΘ is an interval, whose boundary in M¯Θ is disjoint from MΘ.

Proof.

By construction, α is obtained by tropicalizing a family of parametrized curves f:𝒳SΔ over a curve B. Since f is dominant onto SΔ, the image of ΓΛN covers a dense subset of N. Hence the image of Λ in Mg,n,trop is not constant.

(1) Since α is not constant and Λ connected, we can find a vertex wV(Λ) such that α(w)=α(v) and α is non-constant at w. Since α(Λ)ev-1(q1,,qn-1), we have α(Λ)MΘ=α(v). It follows that α does not induce a map Star(w)MΘ. By definition, α is thus not harmonic at w. Hence by Lemma 4.4, α is locally combinatorially surjective at w.

(2) Since MΘ is a nice stratum of Mg,n-1,trop, the underlying graph 𝔾 of Θ in particular is weightless and 3-valent. Thus by Remark 4.3, α is harmonic at vertices of Λ that map to MΘ. On the other hand, since MΘevΘ-1(q1,,qn-1) is the intersection of finitely many (open) half-spaces and hyperplanes, it is an interval. Since α(Λ)MΘ is contained in MΘevΘ-1(q1,,qn-1), the harmonicity of α shows that

MΘevΘ-1(q1,,qn-1)=α(Λ)MΘ,

and its boundary in M¯Θ is disjoint from MΘ. Compare to [7, Lemma 3.9 (3)]. ∎

5.2.2 Cycles in floor decomposed curves

We specify some notation that is convenient for the later proofs. Given a stable, 3-valent, and floor decomposed curve (Γ,h), and a cycle O in Γ, we denote by n(O) the number of basic floor-to-floor elevators of Γ that are contained in O, and call it the vertical complexity of O. Let n(Γ) be the minimal vertical complexity among all cycles of Γ, which we call the vertical complexity of Γ. A point in h(Γ) is called special if it is the image of a vertex or a contracted leg on a floor.

Recall that V denotes an irreducible component of V¯g,d¯,Δirr. Since curves [C] in Vg,d¯,Δirr are torically transverse, we can view the map CνSΔ from the normalization of C as a parametrized curve by marking the preimage of the toric boundary. We denote by

trop(C)=(ΓC,hC)

the tropicalization of this parametrized curve. Let

VFDVVg,d¯,Δirr

be the locus of curves [C] such that the tropicalization trop(C) is floor decomposed, weightless, 3-valent, and immersed except at contracted legs (in particular, contained in a nice stratum in the moduli of parametrized tropical curves by [21, Proposition 2.23]). Given [C]VFD, we define the vertical complexity of C to be the vertical complexity of trop(C). Note that by the construction of VFD, all basic floor-to-floor elevators of trop(C) have genus 0, and hC is injective on any basic floor-to-floor elevator of ΓC away from contracted legs. The following lemma provides the first part of the degeneration we construct in order to prove Theorem 5.6.

Lemma 5.11.

Suppose V is a component of V¯g,d¯,Δirr with g1 and d¯ is a tangency profile trivial on all toric divisors of SΔ corresponding to non-horizontal sides of Δ as in Theorem 5.6. Then there exists [C]VFD whose tropicalization trop(C)=(ΓC,hC) contains two elevators E and E adjacent to the same floor F and contained in the same cycle O of ΓC such that either

  1. hC/E=-hC/E, where E and E are oriented outwards from F, or

  2. O has vertical complexity 2.

Proof.

We first show that VFD is not empty. Since Δ is h-transverse, it follows that for any n vertically stretched points q1,,qnN that are in general position, all curves in ev-1(q1,,qn)Mg,n,trop are weightless, 3-valent, floor decomposed and immersed by [3, Section 5.1]. Note that while in [3, Section 5.1] only tropical degrees associated to trivial d¯ are considered, the argument there works also in our case. For the coordinate-wise tropicalization map trop:TN on the dense torus TSΔ, the preimage trop-1(qi) is Zariski dense (though not algebraic) in T. Thus, we can choose pi in SΔ, such that trop(pi)=qi and there are curves in VVg,d¯,Δirr that pass through all of the pi. Since any such curve is contained in VFD, it follows in particular that VFD is not empty.

Pick [C]VFD with minimal vertical complexity; we will show that C satisfies the claim of the lemma. Let 𝚑 be the height of Δ, namely 𝚑=maxqΔyq-minqΔyq. Denote the floors of the tropicalization trop(C) from the bottom to the top by F1,,F𝚑. Pick points q1,,qnN such that (1) each floor and elevator of trop(C) contains exactly one of the qi in its image and (2) the preimage xi=hC-1(qi)ΓC is a single point contained in the interior of an edge or leg.

For each 1in, let Θi be the combinatorial type in Mg,n-1,trop obtained from trop(C) by adding one contracted leg at xj for each ji. Note that each contracted leg is either on a floor or adjacent to two elevators, and MΘi is a nice stratum. In this way, we can view [trop(C)] as a point in

Mi:=MΘievΘi-1(q1,,q^i,,qn).

As ΓC{xi}1in is a disjoint union of trees, each containing exactly one non-contracted leg, the points (q1,,q^i,,qn) are in general position with respect to trop(C) and dimMi=1 (compare to [21, Lemma 4.20]).

Step 1: The reduction to the case of vertically stretched points. For each 1in, pick piC such that trop(pi)=qi. It follows from Remark 5.8 that, for each i, the points {pj}ji are in general position with respect to [C]V. Let αi:ΛiMg,n-1,trop be induced by the family of parametrized tropical curves obtained from Construction 5.9 with respect to the n-1 points (p1,,p^i,,pn). We have Miαi(Λi) by Lemma 5.10 (2), and the curves in Mi are obtained from trop(C) by either moving the image of a floor vertically or moving the image of an elevator horizontally within a suitable distance (until some two adjacent vertices collapse); see Figure 5. Since trop(C) is floor decomposed, the curves in Mi have the same vertical complexity as trop(C).

Figure 5

The curves in Mi. The left is when qi is contained in the image of a floor of trop(C), while on the right qi is contained in the image of an elevator.

Assume now that qi is contained in the image of Fi for 1i𝚑. We deform all the points {qi}1in so that they are vertically stretched and in general position with respect to a curve that has the same vertical complexity as trop(C), as follows. We start with the top floor F𝚑. On any upward elevator adjacent to F𝚑, we may move the point qj it contains up the elevator for any distance; that is, any point qj=qj+t(0,1) with t0 is still contained in hC(ΓC). By first moving the qj up an appropriate distance, we then can move up F𝚑 any distance, without changing the combinatorial type of trop(C); that is, we consider the family of curves, starting with trop(C), that is cut out by replacing q𝚑 with q𝚑+t(0,1) for varying t0. By definition, the curves obtained in this way are contained in Mj or M𝚑. Repeating this construction for each floor Fi, with i going down from 𝚑 to 1, we see that we can increase the vertical distance between any two points in {qi}1in arbitrarily while preserving their horizontal distance. In particular, we can move the {qi}1in until they are vertically stretched.

As the curves obtained by moving the points qi as above remain in Mi and Miαi(Λi), the parametrized tropical curve we obtain after stretching the qi is still the tropicalization of a curve C in VFD. Furthermore, C has the same vertical complexity as C, and the image hC(ΓC) contains n vertically stretched points qi that are in general position with respect to trop(C). To ease notation, we assume that C already satisfies these properties.

Step 2: The curve C satisfies the assertion of the lemma. Recall that qi is contained in the image of Fi for 1i𝚑. Let O be a cycle in trop(C) with minimal vertical complexity. Let Fk be the highest floor containing an edge in O and pick an elevator EO that is adjacent to Fk. Let Fs be the other floor adjacent to E. Suppose O is oriented from E towards an edge in Fs, and let EO be the first elevator adjacent to Fs that appears after E; see Figure 6. Reordering {qi}i=1n if necessary, we may assume that E contains qn. As before, pick p1,,pn-1C such that trop(pi)=qi, and consider trop(C) as a curve in the family of parametrized tropical curves h:ΓΛN obtained from Construction 5.9 for the points p1,,pn-1. Let Θ be the combinatorial type of trop(C). Then MΘ is a nice stratum of Mg,n-1,trop by assumption.

Figure 6

Moving E towards E. The left column is trop(C). In the top row passing through qs, in the bottom row through a vertex in Fs. The case of passing through qk or a vertex in Fk is analogous. The edges in O are colored green.

We claim that, by moving E towards E and passing through the special points on the way, the problem reduces to the case where there is no special point in FkFs with x-coordinate between the x-coordinates of E and E. To prove this, assume without loss of generality that the x-coordinate of E is less than that of E, and there are exactly r special points in FkFs with x-coordinates between those of E and E. We will proceed by induction on r. The base case is trivial, so we assume r>0. We will describe the inductive step only for the case that, among the r special points, the one with x-coordinate closest to that of E is the marked point qsFs, the image of a contracted leg ls. The argument for the other cases is similar, see Figure 6.

Recall that α:ΛMg,n-1,trop is the map induced by the family of parametrized tropical curves ΓΛN. Moving E horizontally towards E, until its lower vertex hits qs, we get a tropical curve represented by a boundary point of MΘevΘ-1(q1,,qn-1), which is contained in a simple wall MΘs such that E is adjacent to the leg ls contracted to qs. By Lemma 5.10 (2), α(Λ) contains MΘevΘ-1(q1,,qn-1), hence also contains this boundary point. Let Θs′′′Θ be the other nice stratum in the star of MΘs such that E and the leg ls contracted to qs are not adjacent. Hence in Θs′′′ the x-coordinate of E is greater than that of qs (whereas in Θ it is the other way around). By Lemma 5.10 (1), there is a vertex w of Λ such that α(w)MΘsevΘs-1(q1,,qn-1) and α is locally combinatorially surjective at w. Thus, there is an edge e of Λ such that α(e)MΘS′′′evΘs′′′-1(q1,,qn-1), where e denotes the interior of e. Now we may replace trop(C) with any curve represented by a point in α(e), which contains r-1 special points on FkFs with x-coordinates between those of E and E, and has the same vertical complexity as trop(C) by construction. This completes the inductive step.

Now assume that there is no special point of trop(C) on FkFs with x-coordinate between those of E and E. Let qj be the marked point contained in hC(E), and E′′ the second elevator for which qjhC(E′′). Let Fs be the floor adjacent to E′′. Since there is no special point in FkFs with x-coordinate between those of E and E, there is a boundary point of MΘevΘ-1(q1,,qn-1) that is contained in the simple wall MΘ in which E and E get adjacent to the same 4-valent vertex u. There are two cases to consider:

  1. s<s, i.e., E lies below Fs, and

  2. s>s, i.e., E lies above Fs.

See Figure 7 and Figure 8, respectively, for an illustration.

Figure 7

Moving E down the elevators E and E′′ with 𝔴(E)>𝔴(E) (Case 1 in the proof of Lemma 5.11). The edges in the cycle O are colored green.

Figure 8

Moving E up through the elevators E and E′′ (Case 2 in the proof of Lemma 5.11). The edges in the cycle O are colored green.

In both cases, let MΘ′′′ denote the nice stratum in the star of MΘ in which the elevators E and E get attached to the same 3-valent vertex u obtained from splitting u into two 3-valent vertices. By Lemma 5.10 (1), there is a vertex wV(Λ) and an edge eE(Λ) adjacent to w, such that α(w)MΘ and α(e)MΘ′′′.

Case 1: s<s. If 𝔴(E)=𝔴(E), then assertion (1) is satisfied and we are done. Otherwise, we may assume, without loss of generality, that 𝔴(E)<𝔴(E).

It follows from the balancing condition that u lies below Fs, see Figure 7. We keep using Lemma 5.10, to deduce that all curve types illustrated in Figure 7 belong to the image of α. Starting with the nice cone MΘ′′′ and applying Lemma 5.10 (2), we conclude that there is a vertex of Λ corresponding to the simple wall illustrated on the top right of Figure 7 and parametrizing curves with a 4-valent vertex u attached to E,E,E′′, and to the contracted leg corresponding to qj. As before, by Lemma 5.10 (1), there exists an edge eE(Λ) parametrizing curves illustrated on the bottom left of Figure 7, and therefore, there also exists a vertex of Λ corresponding to the simple wall illustrated on the bottom middle of Figure 7 and parametrizing curves with a 4-valent vertex u attached to E,E, and the two edges on the floor Fs. Finally, applying once again Lemma 5.10 (1), we conclude that the image of α contains curves (Γ,h) of the desired type illustrated on the bottom right of Figure 7.

The cycle in (Γ,h) has vertical complexity n(O)-1 and is obtained from O by replacing the elevators {E,E,E′′} in trop(C) with the elevator E in Γ (note that E and E′′ together form a basic floor-to-floor elevator in trop(C), and thus contribute 1 to the vertical complexity). This gives a contradiction, since we chose the curve C with minimal vertical complexity in VFD.

Case 2: s>s. If s=k, then {E,E,E′′} induces a cycle of vertical complexity 2, which is assertion (2) and we are done. Otherwise, by a similar argument as in Case 1, we get a tropical curve in α(Λ) with vertical complexity less than trop(C). The only difference is that we are moving up E along E and E′′ instead of moving down. See Figure 8 for an illustration. This provides a contradiction and we are done. ∎

5.2.3 The proof of Theorem 5.6

We now complete the proof of Theorem 5.6 by induction on g. The base of induction follows from the first assertion of the following lemma.

Lemma 5.12.

Let Δ and d¯ be as in Theorem 5.6, and [C]V0,d¯,Δirr. Pick a parametrization of the normalization P1Cν, and let f:P1SΔ be the composition

1CνCSΔ.

Then:

  1. The Severi variety V0,d¯,Δirr is non-empty and irreducible, regardless of the characteristic. Furthermore, if [C] is general, then f is injective on the preimage of the boundary of SΔ.

  2. If [C] is general and either char(K)=0 or char(K)>𝚠(Δ), then f is locally an immersion. In particular, the singular locus of C is contained in the maximal torus.

Proof.

(1) Let DiSΔ be the irreducible toric divisors for 1is, and niN the primitive inner normal vectors to the corresponding sides Δi of Δ. Let pi,j1 be the points such that f*(Di)=1jaidi,jpi,j. Without loss of generality, we may assume that pi,j for all i and j. Let t be the coordinate on 1{}, and set ci,j:=t(pi,j). Since C is torically transverse of tangency profile d¯, it follows that the pullback under f of the monomial functions is given by

f*(xm)=χ(m)i,j(t-ci,j)di,j(ni,m),

where (,) is the pairing M×N on the dual lattices and χ:MK× is a character.

Vice versa, for any choice of a character χ and distinct points pi,j in 1{}, the formula above defines a morphism f:1SΔ, whose image is torically transverse. It remains to show that for a general choice of the points pi,j, the map f:1SΔ is injective on the preimage of the boundary of SΔ. Indeed, by the assumption, the tangency profile d¯ is trivial on the non-horizontal sides of Δ. Therefore, the injectivity of f over the boundary of SΔ implies that f is birational onto its image, and hence the image of f belongs to V0,d¯,Δirr. Thus, the variety V0,d¯,Δirr is non-empty and is dominated by an open subset of Hom(M,K×)×(1)|d¯|. The irreducibility now follows.

Let us show the asserted injectivity. Since f*(Dk)=1jakdk,jpk,j, it is sufficient to show that for each k and each j1j2, there exists mM such that

f*(xm)(ck,j1)f*(xm)(ck,j2).

Let mkM be a primitive integral vector along the side Δk. Then (nk,mk)=0, and hence the function f*(xmk) is a non-trivial rational function in t that does not depend on the parameters ck,j1 and ck,j2. Therefore, for a general choice of parameters ck,j, we have

f*(xm)(ck,j1)f*(xm)(ck,j2),

as claimed.

(2) To prove that f is locally an immersion, we shall analyze the differential of f. For any mM, set

(5.1)mlog(f):=1f*(xm)d(f*(xm))dt=1is1jaidi,j(ni,m)t-ci,j

to be the log-derivative of f*(xm).

Let us first show that for each k, the log-derivative mklog(f) is a non-constant rational function in t that does not depend on the parameters ck,j. Indeed, since (nk,mk)=0, the function mklog(f) contains no term(s) with {ck,j}1jak. On the other hand, we claim that we can pick a side Δl such that 0<|(nl,mk)|w(Δ). If this is true, then by the assumptions on the characteristic of K, we get in particular, that (nl,mk) is not divisible by char(K), and neither is dl,j. It follows that mklog(f) contains a non-trivial summand dl,j(nl,mk)t-cl,j in expansion (5.1) for any 1jal. Thus, mklog(f) depends non-trivially on the parameters cl,j, and hence, for a general choice of the parameters ci,j, it is a non-constant rational function in t.

Let us now explain how to choose the side Δl as above. If Δk is horizontal, then any non-horizontal side works. Otherwise, let y=a be a horizontal line in M with non-integral y-coordinate that intersects Δk. Let Δk be the other side of Δ that intersects this line. If Δk is parallel to Δk, let Δl be a side of Δ that is adjacent to Δk; otherwise let Δl:=Δk. Then |(nl,mk)| is twice the area of the lattice triangle with vertices (0,0),mk and ml, which is non-zero in each case. Furthermore, since Δ is h-transverse, this lattice triangle has a horizontal edge, it is of height 1, and can be embedded into Δ via a translation; hence its area does not exceed 12𝚠(Δ). Therefore, |(nl,mk)|𝚠(Δ). This justifies our choice of Δl.

Finally, we are ready to prove that f is locally an immersion. Let t01 be any point. We shall show that there exists an element mM such that mlog(f)(t0)0. Assume first that t0=ck,j. Since the parameters ci,j are general and the function mklog(f) is non-constant and does not depend on ck,j, it follows that t=ck,j is not a zero of mklog(f) as needed. Assume next that t0ci,j for all i and j. In particular, f*(xm) is defined at t0, and we have f*(xm)(t0)0 for all mM. Pick 1ks, and let mkM be such that (nk,mk)=1. We will show that either mklog(f)(t0)0 or mklog(f)(t0)0. We have seen above that mklog(f) is not identically zero, and does not depend on the parameters ck,j. Thus, the zero set Z1 of mklog(f) is finite and is independent of the choice of the parameters ck,j. On the other hand, the log derivative mklog(f) contains a non-trivial summand 1t-ck,jdk,j(nk,mk) for all 1jak. Thus, no point of Z is a zero of mklog(f) since the parameters ck,j are general. Hence either mklog(f)(t0)0 or mklog(f)(t0)0, and we are done. ∎

Proof of Theorem 5.6.

Recall that we assume that K is the algebraic closure of a complete DVR whose residue field K~ is algebraically closed and has characteristic either 0 or greater than 12𝚠(Δ). As mentioned, we prove Theorem 5.6 by induction on g. The base case g=0 follows from Lemma 5.12 (1), and we can assume g>0 in the sequel. By the inductive hypothesis, it is enough to show that each irreducible component V of V¯g,d¯,Δirr contains an irreducible component of V¯g-1,d¯,Δirr, which further reduces to showing that the locus of curves of geometric genus g-1 in V has dimension n-1=|d¯|+g-2, the dimension of V¯g-1,d¯,Δirr by Proposition 5.5.

Let [C]VFD be a curve as in Lemma 5.11. As in Step 1 of the proof of Lemma 5.11, after replacing [C] with another curve in VFD, we can find some vertically stretched points q1,,qnhC(ΓC) that are in general position with respect to trop(C); here we view trop(C) as an element of Mg,n,trop by adding contracted legs over the qi, as before. Pick p1,,pnC such that trop(pi)=qi. Then p1,,pn are in general position with respect to the curve C by Remark 5.8. Let E and E be the elevators of trop(C) as in Lemma 5.11, and assume without loss of generality that qnhC(E).

Let h:ΓΛN be the family of parametrized tropical curves over Λ obtained from Construction 5.9 for the points p1,,pn-1 and the curve C. Let α:ΛMg,n-1,trop be the induced map. Recall that this family is obtained from tropicalizing a family of parametrized curves f:𝒳SΔ over a base (B,τ), where B admits a finite cover to a component Z[C] of the locus in V of curves passing through p1,,pn-1. To prove the theorem, it suffices to show that there is a τB(K) such that [f(𝒳τ)]V is an integral curve of geometric genus g-1, since varying the p1,,pn-1 then produces an (n-1)-dimensional locus of such curves in V.

By the definition of a family of parametrized tropical curves (cf. Section 2.4), for any edge e of Λ the underlying graph of the fiber of ΓΛ over any point in the interior e of e is a fixed graph 𝔾e.

Claim 5.13.

Λ contains a leg l such that the map h is constant on all vertices of Gl along l and one of the following holds:

  1. The lengths of all but one edge γ of 𝔾l are constant in the family ΓΛ, and the graph obtained from 𝔾l by removing γ is connected.

  2. The length of all edges of 𝔾l is constant in the family ΓΛ, except for at least one edge of a contracted elliptic tail O.

Let l be as in Claim 5.13, and τB(K) the marked point corresponding to l. We will show that τ is the desired point, namely that [f(𝒳τ)]V is an integral curve of geometric genus g-1. Set D𝒳0:=𝒳~(i𝒳τi)(jσj), where τi are the marked points of B and σj are the marked points of 𝒳B. First, let us show that f is defined on 𝒳τ.

To see that the rational map f is defined at the generic points of 𝒳τ, consider the pullbacks f*(xm) for mM. Let u be a vertex of 𝔾l such that the corresponding component of 𝒳τ~ belongs to the closure of a given generic point η𝒳τ. By Remark 4.1, the order of pole of f*(xm) at η is equal to the slope h(u)(m)/l, which has to vanish since h(u) is constant along l. Therefore, f is defined at the generic points of 𝒳τ and maps them to the dense orbit of SΔ. Next, let us show that f is defined on all of 𝒳τ. Notice that f*(xm) is regular and invertible away from D𝒳0, and it is regular and invertible in codimension one on 𝒳τ. Since 𝒳0 is normal, it follows that f is defined on 𝒳τ(jσj). Pick any σj, and let us show that f is defined at σj(τ)𝒳τ, too. Let SjSΔ be the affine toric variety consisting of the dense torus orbit and the orbit of codimension at most one containing the image of σj(b) for a general bB. Then the pullback of any regular monomial function xm𝒪Sj(Sj) is regular in codimension one in a neighborhood of σj(τ), and hence regular at σj(τ) by normality of 𝒳. Thus, f is defined at σj(τ), and maps it to Sj.

To complete the proof, assume that assertion (2) of Claim 5.13 is satisfied. The case of assertion (1) can be treated in a similar way. Let γE(𝔾l) be an edge that corresponds to a node z𝒳τ~. Then 𝒳0 is given étale locally near z by xy=gz for some regular function gz defined on a neighborhood of τ~B0. If the length of γ varies over l, then gz vanishes along τ by Remark 4.1, and thus z does not get smoothed in 𝒳τ.

If the only edge of varying length in O is a loop, then 𝒳τ is irreducible, since 𝔾l is still connected even if we remove this loop. By the argument above, f is defined on 𝒳τ and maps its generic point to the dense orbit. Moreover, the image of 𝒳τ intersects the boundary divisor at |d¯| distinct points, and some of them are points of simple intersection by the assumption on d¯. Thus, f|𝒳τ is birational onto its image, and [f(𝒳τ)]V is an integral curve of geometric genus g-1. Assume next that O contains a non-loop edge of varying length. Then 𝒳τ is a union of a smooth irreducible component of genus g-1 and an irreducible component of arithmetic genus 1, and without marked points. Denote the components by 𝒳τ and E, respectively. Then f(E) is disjoint from the toric boundary, and therefore f contracts E to a point in the dense orbit of SΔ. In particular, f(𝒳τ)=f(𝒳τ) is irreducible. On the other hand, we get arguing as above, that f|𝒳τ is birational onto its image. Thus, the normalization of f(𝒳τ) is equal to 𝒳τ, and [f(𝒳τ)]V is an integral curve of geometric genus g-1. ∎

Proof of Claim 5.13.

Recall that C satisfies the assertion of Lemma 5.11. The points q1,,qn are in general position with respect to trop(C)=(ΓC,hC) and vertically stretched. The points p1,,pnC are points such that trop(pi)=qi. The edges E and E are the elevators of trop(C) contained in a cycle O and adjacent to a floor F as in Lemma 5.11, and qnhC(E). Moreover, h:ΓΛN is the family of parametrized tropical curves obtained from Construction 5.9 for p1,,pn-1, and α:ΛMg,n-1,trop the induced map.

As before, we may consider trop(C) as a curve in the family h:ΓΛN by marking the preimages of q1,,qn-1 in ΓC. Then E contains no marked point and E contains one marked point which we denote by qj. As in the proof of Lemma 5.11, we may assume that on the floors adjacent to E there is no special point that has x-coordinate between those of E and E. Let MΘMg,n-1,trop be the nice stratum that contains trop(C). According to the two conditions of Lemma 5.11, there are two cases to consider.

Case 1: hC/E=-hC/E, where E and E are oriented away from F. As in the proof of Lemma 5.11, we move E towards E till we hit a simple wall MΘ that parametrizes curves (Γ,h) containing a 4-valent vertex u adjacent to both E and E. See Figure 9 for an illustration. As before, all parametrized tropical curves along this deformation belong to α(Λ), and hence [(Γ,h)]=α(w) for a vertex w of Λ. Furthermore, by Lemma 5.10 (1), we may assume that α is locally combinatorially surjective at w. Let MΘ′′′ be the nice stratum in the star of MΘ in which E and E are adjacent to the same vertex u of the splitting of u into two 3-valent vertices. Let eStar(w) be an edge such that α(e)MΘ′′′. Since 𝔴(E)=𝔴(E), it follows from the balancing condition that the third edge adjacent to u gets contracted by the parametrization map. Thus, the locus MΘ′′′evΘ′′′-1(q1,,qn-1) is an unbounded interval. Furthermore, the images of all curves in this locus agree with h(Γ).

Figure 9

Developing a contracted edge in Case 1 of the proof of Claim 5.13.

By Lemma 5.10 (2), MΘ′′′evΘ′′′-1(q1,,qn-1)=α(Λ)MΘ′′′, and thus there exists a leg l of Λ such that α(l)MΘ′′′evΘ′′′-1(q1,,qn-1) is not bounded. The leg l satisfies the assertion of the claim: the graph 𝔾l is the underlying graph of Θ′′′, and the edge γ in the claim is just the contracted edge adjacent to u as above. The connectivity of 𝔾lγ follows from the fact that E and E are contained in the same cycle, and hence assertion (i) of Claim 5.13 is satisfied.

Case 2: O has vertical complexity 2. See left picture in Figure 10. We may assume that E lies above qj. Let E′′ be the other elevator that contains qj. By moving E towards E as before, we find a boundary point (Γ,h) of MΘevΘ-1(q1,,qn-1) that is contained in a stratum MΘ. In this case, MΘ is not a simple wall. Instead, (Γ,h) is weightless and 3-valent except for two 4-valent vertices u0 and u0′′, with u0 adjacent to E and E, and u0′′ adjacent to E and E′′; in other words, E,E,E′′ form a flattened cycle between u0 and u0′′. By Lemma 5.10 (2), α(Λ) contains MΘevΘ-1(q1,,qn-1), hence also contains (Γ,h).

Figure 10

Developing a contracted loop in Case 2 of the proof of Claim 5.13.

We keep deforming the obtained tropical curve by shrinking the flattened cycle, cf. Figure 10: let MΘ′′′ be the stratum in the star of MΘ in which both u0 and u0′′ split into two 3-valent vertices such that each pair of elevators adjacent to u0 (respectively, u0′′) remains adjacent to vertices u1 (respectively, u1′′); that is, the flattened cycle is preserved. Denote the new edges adjacent to u1 and u1′′ by E1 and E1′′, respectively. Then α(Λ)MΘ′′′ is non-empty by Lemma 4.5. Note that, unlike before, MΘ′′′evΘ′′′-1(q1,,qn-1) has dimension 2: one can perturb the images of u1 and u1′′ independently. Nevertheless, the locus of realizable curves in MΘ′′′evΘ′′′-1(q1,,qn-1) is one-dimensional by Proposition 3.8; more precisely, we get that α(Λ)MΘ′′′ is contained in the sublocus LΘ′′′ of MΘ′′′evΘ′′′-1(q1,,qn-1) consisting of curves such that (E1)=(E1′′), where () denotes the edge lengths. Plainly, LΘ′′′ is a line segment in MΘ′′′. Since the underlying graph of Θ′′′ is weightless and 3-valent, α is harmonic at vertices of Λ that are mapped to MΘ′′′ by Remark 4.3. It follows that α(Λ) contains LΘ′′′.

By the balancing condition, we have 𝔴(E1)=𝔴(E1′′). This means that, along LΘ′′′, the images of u1 and u1′′ are moving (one downwards and the other upwards) with the same speed. We may assume that qj lies strictly below the center of the flattened cycle. When u1′′ reaches qj, we obtain a boundary point of LΘ′′′ that is not contained in MΘ′′′; namely, a curve which contains a 4-valent vertex u2 with the following adjacent edges/legs: E1′′, the leg lj contracted to qj, and the two elevators that lie above it. Let Ξ be the combinatorial type of this curve. Note that this parametrized tropical curve has a non-trivial automorphism induced by switching the two elevators above u2 when these two elevators have the same multiplicity.

Let Ξ be the combinatorial type obtained from Ξ by splitting u2 into two 3-valent vertices, such that lj and E1′′ remain incident. This shrinks the flattened cycle further, and we denote by u2 its lower endpoint and by E2 the downward elevator adjacent to u2. By Lemma 4.4, α(Λ)M[Ξ], and, arguing as above, α(Λ) contains (the image in M[Ξ] of) the line segment LΞMΞevΞ-1(q1,,qn-1) consisting of curves with

(E1)=(E2)+(E1′′).

The second boundary point of LΞ, obtained by shrinking the flattened cycle to a point, contains a 2-valent vertex u3 of weight 1 adjacent to E1 and E2. Let Υ be its combinatorial type. Recall that char(K~) is either 0 or greater than 12𝚠(Δ).[1] By Lemma 3.4, the multiplicity of the edge E1 (and, by balancing, of E2) is at most 𝚠(Δ). Thus either the multiplicity is not divisible by char(K~) or equal to char(K~). In the first case we develop a contracted loop, and in the second case a contracted elliptic tail as follows.

First, assume that 𝔴(E1) is not divisible by char(K~). Let M[Υ] be a stratum in the star of M[Υ] such that Υ is obtained from Υ by adding a contracted loop at u3 and setting the weight of u3 to 0. By Lemma 4.6 (1) we have that α(Λ)M[Υ] is non-empty. By Proposition 3.8, we have that α(Λ)M[Υ] is contained in (the image of) the ray LΥMΥevΥ-1(q1,,qn-1) consisting of curves such that (E1)=(E2)+(E1′′). By Lemma 4.6 (3), α is harmonic at vertices of Λ that are mapped to M[Υ], and hence, similar as above, LΥ is contained in α(Λ). Thus, there is a leg l of Λ such that α(l)LΥ is not bounded. Then l satisfies the assertion of the claim: the graph 𝔾l of the claim is the underlying graph of Υ, and the edge γ of the claim is the contracted loop at u3 (in particular, 𝔾lγ is connected). Thus assertion (i) of Claim 5.13 is satisfied.

Finally, suppose that 𝔴(E1)=char(K~). In this case, let Υ′′ be the combinatorial type obtained from Υ by adding a contracted edge adjacent to u3, now 3-valent and of weight 0, and u3, a new vertex of weight 1 and valence 1; let Υ′′′ be the combinatorial type obtained from Υ′′ by developing a loop at u3 (that is, u3, u3, the edge between them and possibly the loop form a contracted elliptic tail; cf. Figure 2). By Lemma 4.6 (2) we have that α(Λ)M[Υ′′] is non-empty. By Lemma 4.6 (3), for a vertex wΛ with α(w)M[Υ′′], α is harmonic at w, or there is an edge/leg e adjacent to w such that α(e)M[Υ′′′]. If α is harmonic at all vertices mapped to M[Υ′′], we argue as above to show that assertion (ii) of Claim 5.13 is satisfied; that is, we deduce from Proposition 3.8 that α(Λ) contains the ray LΥ′′, defined analogous to LΥ. Otherwise, we get that α(Λ)M[Υ′′′] is not empty. Since α is harmonic at vertices mapped to M[Υ′′′] by Lemma 4.6 (3), we can argue again as above to obtain that assertion (ii) of Claim 5.13 is satisfied; that is, we deduce from Proposition 3.8 that α(Λ) contains a ray in the intersection of M[Υ′′′] with the plane given by the condition (E1)=(E2)+(E1′′). ∎

5.3 A counterexample in low characteristic

Figure 11

The case k=k=3.

We conclude our discussion of degenerations on SΔ with an example that shows that Theorem 5.6 may fail without the assumption on the characteristic of K. Recall that we defined kites in Example 3.2 following [20] as polygons with vertices (0,0),(k,±1) and (k+k,0) for non-negative integers k,k with kk and k>0. As we noted above, kites are h-transverse, and by definition we have 𝚠(Δ)=k+k.

Proposition 5.14.

Suppose char(K)=p>0 and let Δ be the kite with

k=k={3if p=2,p𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Consider V1,Δirr parametrizing irreducible curves of geometric genus 1 on SΔ with trivial tangency profile. Then there is an irreducible component V of V¯1,Δirr such that VV0,Δirr=.

Proof.

We assume k=k=p>2, the argument for p=2 and k=k=3 is completely analogous. We first follow the description in [20, Section 3] to obtain some properties of the curves parametrized by V1,Δirr. For a general point [C]V1,Δirr, let f:ESΔ be the parametrized curve given by normalizing C. Let O,P,Q,R be the preimages of the toric divisors Di corresponding to the sides of Δ with outer normals (-1,-p),(-1,p),(1,-p),(1,p), respectively. Then the divisors on E of the pullbacks of the monomials x and y are given by div(f*(y))=-pO+pP-pQ+pR and div(f*(x))=-O-P+Q+R. Using these equations, we get in the group law of the elliptic curve (E,O):

(5.2)P=R+Qand2pR=0.

Conversely, any smooth genus 1 curve E and choice of distinct points O,P,Q,R satisfying the two conditions above gives an element in V1,Δirr.

For l2 let X1(l) denote the modular curve parametrizing generalized elliptic curves together with an ample Drinfeld /l-structure as in [9, Definitions 2.4.1 and 2.4.6]; see also [11] and [16]. Denote by Y1(l) the open subset of X1(l) over which the elliptic curves are smooth. We will consider the forgetful map from V1,Δirr to Y1(2p) induced by (E,O,R). More precisely, by what we discussed above, we may choose a point [C]V1,Δirr such that for the triple (E,O,R), R is a point of exact order 2p – that is, the subgroup scheme generated by multiples of R is isomorphic to /2p. Denote by V the irreducible component of V1,Δirr containing [C]. Possibly after pulling back along a finite surjective map, we may assume that the universal family over V is equinormalizable by [10, Section 5.10]. Thus the normalization of the total space of the universal family induces a map VY1(2p), whose image is contained in an irreducible component Y of Y1(2p). Denote by YX1(2p) its closure.

Now let [C0] be a point in the boundary of V=V¯ and assume contrary to the claim that [C0]V0,Δirr. Then choose a discretely valued field F with valuation ring F0 and a map Spec(F0)V such that the generic point of Spec(F0) maps to a point in V, and the special point to [C0]. Pulling back the normalization of the total space of the universal family over V, we get a smooth genus 1 curve Spec(F) with the four marked points O,P,Q,R. Up to a finite extension of F, we may assume that this family extends to a stable model 0Spec(F0). Then f extends on the central fiber ~ of 0 to a map f~:~SΔ with C0=f~(~). Since we assume that C0 is torically transverse, C0 intersects the toric boundary in 4 distinct points. Since ~ is stable of arithmetic genus 1, and C0 is irreducible, this implies the following:

  1. at least three marked points lie on an irreducible component E0 of ~ of geometric genus 0.

On the other hand, consider the map φ:Spec(F)Y induced by (,O,R) as above. As X1(2p) is proper, the map extends uniquely to a map φ:Spec(F0)Y. Denote by [X0]Y the image of the special point under this extension, corresponding to a generalized elliptic curve X0. Let O0,P0,Q0 and R0 denote the points on X0 that extend, respectively, φ(O), φ(P), φ(Q), and φ(R) to X0. Recall that we have two natural maps φ2:X1(2p)X1(p) and φp:X1(2p)X1(2), both over M¯1,1. They map a level 2p structure to a level p structure (respectively, a level 2 structure) via multiplication by 2 (respectively, p); on a singular curve, they contract irreducible components, that no longer intersect the level structure, see [9, Lemma 4.2.3 (ii)].

Suppose first X0 is smooth. Since p2, the torsion points parametrized by Y1(2) all are distinct from the origin on the elliptic curves. Since R has exact order 2p, this implies that the map on universal families induced by φp maps R0 and O0 to distinct points. In particular, R0 and O0 are distinct points on X0. On the other hand, by (5.2), we have Q0=O0+Q0 and P0=R0+Q0. Thus also Q0 and P0 are distinct points on X0. This implies that the stable model of the curve with marked points (φ(),φ(O),φ(P),φ(Q),φ(R)) cannot contain a rational component with 3 marked points; by uniqueness of the stable model, this contradicts (). Now suppose X0 is singular. By [11, Proposition 2.3, p. 246], X1(p) has 2 irreducible components and one of them, Y, parametrizes, away from supersingular curves, generalized elliptic curves together with a fixed inclusion of /p as a subgroup scheme. Since the multiplicative torus 𝔾m does not contain a subgroup scheme isomorphic to /p in characteristic p, the universal family over Y does not contain singular, irreducible fibers. Since we chose R such that it generates a subgroup scheme isomorphic to /2p, we have that φ2(Y)Y. Thus X0 is not irreducible. By the definition of an ample Drinfeld /2p structure, it follows that O0 and R0 lie on different irreducible components of X0. Arguing similar as before, we obtain that also Q0 and P0 lie on different irreducible components of X0 by (5.2). This contradicts () by the uniqueness of the stable model. ∎

6 Zariski’s Theorem

As a consequence of Theorem 5.6, we prove in this section Zariski’s Theorem for SΔ. Recall that we denote the sides of Δ by Δi and the corresponding toric divisors of SΔ by Di, where 1is.

Theorem 6.1 (Zariski’s Theorem).

Let Δ be an h-transverse polygon and suppose either char(K)=0 or char(K)>w(Δ). Let 0g#|ΔM| be an integer and d¯ a tangency profile of (SΔ,LΔ), which is trivial on toric divisors corresponding to non-horizontal sides of Δ. Let VVg,d¯,Δ be a subvariety. Then:

  1. dimV|d¯|+g-1, and

  2. if dimV=|d¯|+g-1, then for a general [C]V, the curve C is nodal, and its singular locus is contained in the maximal torus.

Lemma 6.2.

Under the assumptions of Theorem 6.1, suppose that Δ has height 2 and no horizontal sides. Then for a general point [C]V0,Δirr, the curve C is nodal.

Proof.

After a translation and a change of coordinates, we may assume that Δ has vertices (0,0), (a,1),(a+b,0),(0,-1), where a,b0 and a+b>0. We may further assume that 𝚠(Δ)2 as otherwise C is smooth. Therefore, char(K)>2. Let us label the sides of Δ as illustrated in Figure 12. Note that the case where Δ is a triangle is included by setting a=0.

Figure 12

As in the proof of Lemma 5.12, a general curve [C]V0,Δirr is represented as the image of a map f:1SΔ. We may assume that

f-1(D1)=c,f-1(D2)=1,f-1(D3)=0,f-1(D4)=.

Then

f*(x)=αt(t-c)(t-1)andf*(y)=βtb(t-1)a,

where α,βK×.

By Lemma 5.12, the curve C has no unibranch singularities and is smooth on the boundary divisor of SΔ. On the other hand, since f*(x) is the quotient of two polynomials of degree at most 2, each singularity of C has multiplicity at most 2. Thus it suffices to show that a general curve [C]V0,Δirr does not contain a (generalized) tacnode that lies in the maximal torus.

Suppose to the contrary that p(K×)2 is a tacnode of C, and f-1(p)={t1,t2}. Then

f*(x)(t1)=f*(x)(t2),

from which we get t1t2=c. On the other hand, since (df*(x)dt|ti,df*(y)dt|ti) gives the same tangent direction for i=1,2, we have

(6.1)df*(x)dt|t1df*(y)dt|t2=df*(x)dt|t2df*(y)dt|t1.

Let us compute the derivatives

df*(x)dt=f*(x)(1t-1t-c-1t-1)anddf*(y)dt=f*(y)(at-1+bt).

Substituting these in equation (6.1), we get

(a+b)(t1-c)(t2-c)+a(c-1)t1t2+bc(t1-1)(t2-1)=0.

Since t1t2=c0, we can rearrange terms to obtain

(a+2b)(t1+t2)=(2a+2b)c+2b.

As c is general and char(K)>max{2,a+b}, this equation cannot hold if a+2b=0. Hence we may assume a+2b0, which gives

t1+t2=(2a+2b)c+2ba+2b.

It follows that t1 and t2 are the two roots of the quadratic polynomial

F(t):=t2-(2a+2b)c+2ba+2bt+c.

On the other hand, since f*(y)(t1)=f*(y)(t2), t1 and t2 are also two roots of

Gμ(t):=tb(t-1)a-μ

for some μK×. Therefore, in order to get a contradiction, it suffices to show that for a general c1, there is no μK× such that F(t) divides Gμ(t).

Let R(t) be the residue of tb(t-1)a divided by F(t). We can write

R(t)=A(c)t+B(c),

where A and B are polynomials in c. Now F(t) not dividing Gμ(t) for any μK (including μ=0) is equivalent to A(c)0. It is therefore sufficient to show that A is a non-zero polynomial, which further reduces to showing that F(t) does not divide Gμ(t) for any μK when c=0. In this case the two roots of F(t) are 0 and 2b/(a+2b). If a=0, then t=0 and t=2b/(a+2b)=1 cannot be the roots of Gμ=tb-μ at the same time. If a0 and b=0 then F=t2 and there is a term ±at in Gμ, hence F does not divide Gμ. If a,b0 and μ=0 then t=2b/(a+2b)0,1 is not a root of Gμ=tb(t-1)a; if a,b,μ0 then t=0 is not a root of Gμ. In each case Gμ is not divisible by F, hence we are done. ∎

Proof of Theorem 6.1.

The first part of the claim follows immediately from Proposition 5.5. We need to show that if V is an irreducible component of Vg,d¯,Δ, then for a general point [C]V, the curve C is nodal, and all its singular points are contained in the maximal torus. As in the proof of Proposition 5.5, suppose C has m irreducible components C1,,Cm. Set gk:=pg(Ck), k:=𝒪SΔ(Ck), and let d¯k be the tangency profile of the parametrized curve induced by the normalization of Ck. Then V is dominated by an irreducible component k=1mVk of k=1mV¯gk,d¯k,kirr, where Vk is an irreducible component of V¯gk,d¯k,kirr.

Let ΔkM be the polygon that defines the line bundle k. If Δk is degenerate, then gk=0, d¯k is trivial, |d¯k|=1, and Vk=V¯0,d¯k,kirr=|k|. Otherwise it is easy to check that Δk is still h-transverse, and we have a natural map πk:SΔSΔk obtained from a sequence of toric blow-ups. Note that πk induces a natural map

πk:VkV¯gk,d¯k,Δkirr.

By Proposition 5.5, we have

dimVk=dimV¯gk,d¯k,Δkirr.

Since a general point of Vk represents a curve in SΔ that does not intersect the exceptional divisors of πk, the map πk maps Vk birationally to an irreducible component Vk of V¯gk,d¯k,Δkirr. Now by Theorem 5.6, Vk contains V0,d¯k,Δkirr, hence Vk contains V0,d¯k,kirr, and k=1mVk contains k=1mV0,d¯k,kirr. Since the locus of nodal curves is open in k=1mVk by [29, Tag 0DSC], and similarly the locus of curves with singular locus contained in the maximal torus of SΔ is open, it remains to prove the following claim. ∎

Claim 6.3.

Let C=1kmCk be a curve such that [Ck]V0,d¯k,Lkirr is general. Then C is nodal, and its singular locus is contained in the maximal torus.

Proof.

According to Lemma 5.12, the singular locus of each Ck is contained in the maximal torus of SΔ. On the other hand, as each V0,d¯k,kirr admits a torus action, we can move Ck so that it misses the points of intersection of the other components of C with the boundary of SΔ. As a result, the intersections of components of C are also contained in the maximal torus, hence the singular locus of C is contained in the maximal torus. It remains to show that C is nodal.

Let d¯=({di,j}1jai)1is and CDi={pi,1,,pi,ai} for all 1is. Assume to the contrary that C has a singular point p that is not a node. By Lemma 5.12, each Ci contains no unibranch singularities, and thus there are the following two cases to consider.

Case 1: There are at least three branches of C that pass through p. Again, since the variety V0,d¯k,kirr admits a torus action, and Ck is general, these branches must come from the same component of C, say C1. Then Δ1 is non-degenerate and has height at least 2. In particular, we have |d¯1|4 as d¯1 is trivial on toric divisors corresponding to non-horizontal sides of Δ. Let U|Δ| be the locus of curves C=1kmCk such that