{'fraction_non_alphanumeric': 0.06367514834836505, 'fraction_numerical': 0.012090218040828805, 'mean_word_length': 3.3289418275882814, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 54, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi's original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.


INTRODUCTION
In the current paper, we study the geometry of Severi varieties over an algebraically closed field K of arbitrary characteristic. Recall that the Severi variety V g ,d is defined to be the locus of degree d reduced plane curves of geometric genus g in the complete linear system |O P 2 (d )|. We denote by V irr g ,d the union of the irreducible components parameterizing irreducible curves. The main goal of this paper is to prove that the Severi varieties V irr g ,d are either empty or irreducible. These varieties were introduced by Severi in 1921 in order to prove the irreducibility of the moduli space M g of genus g compact Riemann surfaces [Sev21]. He noticed that for d large enough, there exists a natural surjective map from V irr g ,d to M g since any Riemann surface of genus g admits an immersion as a plane curve of degree d . Therefore, the irreducibility of M g follows once one proves that V irr g ,d itself is irreducible. The irreducibility (or rather connectedness) of M g was first asserted by Klein in 1882, who deduced it from the connectedness of certain spaces parameterizing branched coverings of the Riemann sphere, nowadays called Hurwitz schemes, see [Kle82,Hur91,EC85,Ful69]. Severi's We then use the floor-elevator structure of floor decomposed curves to describe the nice strata adjacent to these simple walls. We show that by traveling along the nice strata and by crossing the corresponding simple walls, we can find a leg of Λ such that α maps it to a nice stratum parameterizing tropical curves having a contracted edge of varying length, see Lemma 5.5, and the figures within its proof.
To complete the proof of the Main Theorem we follow the ideas of [Tyo07], and consider the so-called decorated Severi varieties U d,δ that parameterize reduced curves with δ := d−1 2 − g marked nodes. We prove that these varieties are smooth and equidimensional and that the natural map U d,δ → |O P 2 (d )| is generically finite and has V g ,d as its image. Finally, we show that the variety U irr d,δ parameterizing irreducible reduced curves of degree d with δ marked nodes is irreducible (Theorem 7.1), which implies the Main Theorem.
The analysis of the decorated Severi varieties allows us to generalize the well-known description of the local geometry of the closure of the Severi variety V g ,d along V 1−d,d to the case of arbitrary characteristic (Theorem 6.1). We show that as in the case of characteristic zero, at a general [C 0 ] ∈ V 1−d,d , the variety V g ,d consists of smooth branches parameterized by the subsets µ of δ nodes of C 0 . Furthermore, the intersections of branches are reduced, have expected dimension, and the branch corresponding to µ belongs to V irr g ,d if and only if C 0 \ µ is connected.
1.3. Generalizations and applications. The ideas and the techniques introduced in this paper admit a variety of generalizations and applications, some of which we investigate in our followup papers. In particular, our Key Lemma (Lemma 5.5) is a new tool providing good control over the skeletons of curves and the way they vary as curves move in tropically general one-parameter families. Therefore, we expect that further applications of our tropical machinery to studying families of curves will be found soon. Let us describe two generalizations we already explored.
A natural generalization considered in [CHT22a] is the case of polarized toric surfaces corresponding to h-transverse polygons, e.g., Hirzebruch surfaces. These are the types of polygons for which the technique of floor-decomposed curves applies. It turns out that if a polygon ∆ is htransverse, then any component of the Severi variety V irr g ,∆ contains V irr 0,∆ in its closure. However, unlike in the planar case, to prove this in full generality, one must deal with walls corresponding to elliptic components. This brings the geometry of the moduli spaces of elliptic curves with a level structure into play and restricts the validity of the results for the general h-transverse polygons to characteristic zero (or large enough characteristic). In some cases, such as Hirzebruch surfaces, one can avoid elliptic walls and prove the degeneration result in arbitrary characteristic. The degeneration result combined with the recent monodromy result of Lang [Lan20] gives rise to the irreducibility of the Severi varieties on most such surfaces. Notice, however, that there are examples of reducible Severi varieties even on polarized toric surfaces corresponding to certain h-transverse polygons; see [Tyo14] for a simple example.
In [CHT22b], we develop further the tropical tools introduced in the current paper, and deduce the irreducibility of Hurwitz schemes in small positive characteristic. Remarkably, our tools allow to replace the monodromy argument with a purely tropical one. To the best of our knowledge, this problem has been open since Hurwitz schemes were introduced by Fulton in 1969 [Ful69].
1.4. The structure of the paper. The first two sections are preliminaries from algebraic and tropical geometry. In Section 2, we define the Severi varieties and prove that they are equidimensional of the expected dimension. In Section 3, we recall the notion of parameterized tropical curves, their moduli spaces, and floor decomposed curves.
In Section 4, we study the tropicalization of a general one-parameter family of smooth curves with a map to the projective plane. The results of this section are the main technical tool in the proof of the Degeneration Theorem, which is proved in Section 5 together with the generalization of Zariski's theorem to the case of positive characteristic (Corollary 5.3).
The main result of Section 6 is Theorem 6.1 describing the local geometry of V g ,d along V 1−d,d . We also define and study the decorated Severi varieties in this section. Finally, in Section 7, we prove the irreducibility of decorated Severi varieties and deduce the Main Theorem.
Acknowledgements. We are grateful to Michael Temkin for helpful discussions and to an anonymous referee for many insightful comments that helped us to improve the presentation.

SEVERI VARIETIES
In this section, we recall the notion of Severi varieties on algebraic surfaces and prove their basic properties. We show that in the case of toric surfaces, Severi varieties are locally closed, and their closures are either empty or equidimensional of the expected dimension. Let S be a projective surface over an algebraically closed ground field K , and L a line bundle on S. We write [C ] ∈ |L | for the K -point in the complete linear system |L | parameterizing a curve C .
Definition 2.1. The Severi varieties V g ,L and V irr g ,L are defined to be the following loci in |L |: V g ,L := [C ] ∈ |L | |C is reduced, C ∩ S sing = , p g (C ) = g , where p g (C ) denotes the geometric genus of C , and V irr g ,L := [C ] ∈ V g ,L |C is irreducible .
We denote by V g ,L and V irr g ,L the closures in |L | of V g ,L and V irr g ,L , respectively. Notice that since the locus of integral curves is open in the linear system, the variety V irr g ,L is a union of irreducible components of V g ,L . If (S, L ) = P 2 , O P 2 (d ) , we will often use the classical notation V g ,d := V g ,L and V irr g ,d := V irr g ,L .
Definition 2.2. A family of curves X → B is called generically equinormalizable if the normalization of the total space, X ν → B , is a family of curves with smooth generic fiber. has constant geometric genus. Since the map U ′ 1 → U 1 is finite, the image Z of the complement of V ′ 1 is closed and nowhere dense in U 1 . Set V 1 := U 1 \ Z . Then V 1 is open and dense in U 1 . Furthermore, X U 1 → U 1 has constant geometric genus over V 1 since so does X ′ U 1 → U ′ 1 over V ′ 1 , and the pullback of V 1 is a subset of V ′ 1 . If the total space X U 1 is reducible, we apply the above reasoning to each irreducible component of X U 1 separately and define V 1 to be the intersection of the dense open sets obtained in this way.
For an irreducible component U 2 of the complement U 1 \V 1 , the same procedure gives rise to a dense open subset V 2 ⊂ U 2 such that the restriction of X U 2 → U 2 to V 2 has fibers of constant geometric genus. Repeating this process for all irreducible components of the U i \ V j , gives a finite stratification of U 1 by the locally closed V j 's. By construction, each V g ,L is a union of V j 's, hence a constructible subset of |L | as asserted.
It remains to prove that the constructible subsets g ′ ≤g V g ′ ,L and g ′ <g V g ′ ,L are closed in U 1 , since then V g ,L is locally closed. Let R be a discrete valuation ring, and φ : Spec(R) → U 1 a map taking the generic point η ∈ Spec(R) to [C η ] ∈ V g ′ ,L for some g ′ ≤ g . After a quasifinite base change, we may assume that the normalization of C η → η is smooth and admits a stable model over Spec(R). Let [C ] ∈ U 1 be the image of the closed point of Spec(R). Then C is a reduced curve dominated by a semi-stable reduction of the normalization C ν η . Thus, p g (C ) ≤ p g (C η ) = g ′ . Therefore g ′ ≤g V g ′ ,L and g ′ <g V g ′ ,L are closed in U 1 by the valuative criterion [Sta20, Tag 0ARK], and we are done.
In general, Severi varieties may have components of different dimensions; see [CC99]. However, in the case of toric surfaces, they are equidimensional, as the following proposition shows: Proposition 2.7. If S is a toric surface, then the Severi varieties V irr g ,L and V g ,L are either empty or equidimensional of dimension −L .K S + g − 1. Furthermore, a general [C ] ∈ V g ,L corresponds to a curve C intersecting the boundary divisor transversally.
Proof. Since V irr g ,L is a union of irreducible components of V g ,L , it is sufficient to prove the assertion for V g ,L . Let V be an irreducible component of V g ,L , and [C ] ∈ V general. Suppose C has m irreducible components C 1 , . . . ,C m . Set g i := p g (C i ) and L i := O S (C i ), and consider the map m i =1 V g i ,L i → V g ,L . Its fiber over [C ] is finite, and the map is locally surjective. Thus, dim [C ] Theorem 1.2], and hence Furthermore, if C intersects the boundary divisor non-transversally, then the inequality is strict. Next, let us show the opposite inequality, implying both assertions of the proposition.
Let X be the normalization of C , and f : X → S the corresponding map. It defines a point [X , f ] in the Kontsevich moduli space M g ,n (S, |L |) of stable maps to S with image in |L |; see [Kon95] and [dJHS11,Theorem 5.7]. Since f is birational onto its image, it is automorphism-free, and hence M g ,n (S, |L |) is a quasi-projective scheme in a neighborhood of [X , f ]. Let (R, m) be the formal completion of the algebra of functions on M g ,n (S, |L |) at [X , f ]. By [Ill71, Théorème III.2.1.7], the tangent space to M g ,n (S, |L |) at [X , f ] is given by m/m 2 * = H 0 (X , N f ), where is the normal sheaf to f , and the obstruction space is a subspace in H 1 (X , N f ).
Let us express the algebra (R, m) as a quotient of a ring of formal power series (A, n) by an ideal I ⊂ n such that n/n 2 ≃ m/m 2 . Then, by a standard argument in deformation theory, the ideal I is generated by at most h 1 (X , N f ) elements, and hence the dimension of any local germ of M g ,n (S, Consider the projection from a small neighborhood of [X , f ] to |L | given by sending a point and we are done. Remark 2.8. If M is the lattice of monomials of S, and (S, L ) is the polarized toric surface corresponding to a convex lattice polygon ∆ ⊂ M R , then −L .K S = |∂∆ ∩ M |. Hence the dimension of the Severi varieties can be expressed combinatorially in terms of ∆ and g . In particular, dim(V g ,d ) = 3d + g − 1.

TROPICAL CURVES
In this section, we review the theory of (parameterized) tropical curves and recall the notions of floor decomposed curves and moduli spaces of parameterized tropical curves. We mainly follow [Mik05,GM07,BM09,Tyo12,ACP15], to which we refer for further details. We also discuss families of parameterized tropical curves, cf. [CCUW20,Ran22], and introduce the notions of harmonicity and local combinatorial surjectivity of the induced map to the moduli space. Throughout the section, M and N denote a pair of dual lattices.

Families and parameter spaces for tropical curves.
3.1.1. Abstract tropical curves. The graphs we consider have half-edges, called legs. A tropical curve is a weighted metric graph Γ = (G, ℓ) with ordered legs, i.e., G is a (connected) graph with ordered legs equipped with a weight (or genus) function g : V (G) → Z ≥0 , and a length function ℓ : E (G) → R >0 . Here V (G) and E (G) denote the sets of vertices and edges of G, respectively. We denote by L(G) the set of legs of G and extend ℓ to it by declaring the lengths of the legs to be infinite. Set E (G) := E (G) ∪ L(G). We will view tropical curves as polyhedral complexes by identifying the edges of G with bounded closed intervals of corresponding lengths in R and identifying the legs of G with semi-bounded closed intervals in R.
For e ∈ E(G) we denote by e • the interior of e, and use e to indicate a choice of orientation on e. If e ∈ L(G) is a leg, then it will always be oriented away from the vertex. Bounded edges will be considered with both possible orientations. For v ∈ V (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 valency of v and is denoted by val(v). By abuse of notation, we will often identify Star(v) with its geometric realization -the vertex v with germs of the oriented edges e ∈ Star(v) attached to it.
The genus of Γ is defined to be g (Γ) = g (G) := 1−χ(G)+ v∈V (G) g (v), where χ(G) := b 0 (G)−b 1 (G) denotes the Euler characteristic of G. Finally, a tropical curve Γ = (G, ℓ) is called stable if so is G, that is, if every vertex of weight zero has valency at least three, and every vertex of weight one has valency at least one. ∂h ∂ e = 0. Note that the slope ∂h ∂ e ∈ N 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. In particular, ∂h ∂ e = 0 if and only if h contracts e. A parameterized 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 e ∈ E(G). We define the extended degree ∇ to be the sequence of slopes ∂h ∂ l l ∈L(G) , and the degree ∇ to be the subsequence ∂h ∂ l i , where l i 's are the legs not contracted by h. The (extended) degree clearly depends only on the combinatorial type Θ. An isomorphism of parameterized tropical curves h : Γ → N R and h ′ : Similarly, an isomorphism of combinatorial types Θ and Θ ′ is an isomorphism of the underlying graphs respecting the order of the legs, the weight function, and the slopes of the edges and legs. We denote the group of automorphisms of a combinatorial type Θ by Aut(Θ), and the isomorphism class of Θ by [Θ].
3.1.3. Families of tropical curves. Next we define families of (parameterized) tropical curves; cf.
[CCUW20, §3.1] and [Ran22, §2.2]. We will adapt these more general definitions to the special case we need -families of (parameterized) tropical curves over a tropical curve Λ without stacky structure. Since tropical curves are nothing but parameterized tropical curves with N = 0, we discuss only families of parameterized tropical curves. Let Λ = (G Λ , ℓ Λ ) be a tropical curve without loops. Loosely speaking, a family of parameterized tropical curves over Λ is a continuous family such that for any e ∈ E (G Λ ), the combinatorial type of the fiber is constant along e • and the lengths of the edges of the fibers, as well as the images of the vertices of the fibers in N R , form integral affine functions on e. To define this notion formally, consider a datum ( †) consisting of the following: • an extended degree ∇; • a combinatorial type Θ e = G e , ( ∂h ∂ γ ) of extended degree ∇ for each e ∈ E(G Λ ); • an integral affine function ℓ(γ, ·) : e → R ≥0 for each e ∈ E (G Λ ) and γ ∈ E (G e ); • an integral affine function h(u, ·) : e → N R for each e ∈ E (G Λ ) and u ∈ V (G e ); • a parameterized tropical curve h w : Γ w → N R of extended degree ∇ for each w ∈ V (G Λ ); • a weighted contraction ϕ e : G e → G w preserving the order of the legs for each w ∈ V (G Λ ) and e ∈ Star(w ).
For any e ∈ E(G Λ ) and q ∈ e • , set Γ q := (G e , ℓ q ), where ℓ q : E (G e ) → R ≥0 is the function defined by ℓ q (γ) := ℓ(γ, q). Denote by h q : Γ q → N R the unique piecewise affine map for which h q (u) = h(u, q) for all u ∈ V (G e ) and ∂h q ∂ l = ∇. 8 Definition 3.1. Let Λ be a tropical curve without loops. We say that a datum ( †) is a family of parameterized tropical curves over Λ if the following compatibilities hold for all w ∈ V (G Λ ), e ∈ Star(w ), and q ∈ e • : (1) (Γ q , h q ) is a parameterized tropical curve of combinatorial type Θ e ; (2) the length of ϕ e (γ) in Γ w is ℓ(γ, w ) for all γ ∈ E (G e ); (3) h w (ϕ e (u)) = h(u, w ) for all u ∈ V (G e ).
A family of parameterized tropical curves over Λ will be denoted by h : Γ Λ → N R .
3.1.4. The parameter space. We denote by M trop g ,n,∇ the parameter space of isomorphism classes of genus g stable parameterized tropical curves of degree ∇ with exactly n contracted legs. We assume that the contracted legs are l 1 , . . . , l n . Similarly to the case of the parameter space of abstract tropical curves studied in [ACP15, Section 2], we consider M trop g ,n,∇ as a generalized polyhedral complex, i.e. M trop g ,n,∇ is glued from "orbifold" quotients of polyhedra M Θ factored by finite groups of automorphisms Aut(Θ). In particular, the integral affine structure on the quotient is determined by that of M Θ , and the "new" faces of the geometric quotient M Θ /Aut(Θ) are not considered to be faces of the orbifold quotient; see loc. cit. for more details. More explicitly, M is either empty or can be identified naturally with M Θ E , where Θ E is the type of degree ∇ and genus g , obtained from Θ by the weighted edge contraction of E ⊂ E (G). Plainly, the corresponding inclusion ι Θ,E : M Θ E → M Θ respects the integral affine structures.
The space M trop g ,n,∇ is defined to be the colimit of the diagram, whose entries are M Θ 's for all combinatorial types Θ of genus g , degree ∇ curves having exactly n contracted legs l 1 , . . . , l n ; and arrows are the inclusions ι Θ,E 's and the isomorphisms ι α 's described above. By the construction, for each type Θ of degree ∇ and genus g with exactly n contracted legs l 1 , . . . , l n , we have a finite map Definition 3.2. Let Λ be a tropical curve, and α : Λ → M trop g ,n,∇ a continuous map. We say that α is piecewise integral affine if for any e ∈ E (Λ) the restriction α | e lifts to an integral affine map e → M Θ for some combinatorial type Θ.
Definition 3.6. We say that a tuple (q 1 , . . . , q n ) ∈ N n R is general with respect to ∇ and g if for any combinatorial type Θ of degree ∇ and genus g with exactly n contracted legs, the intersection ev −1 Θ (q 1 , . . . , q n ) ∩ M Θ is either empty, or has codimension n · rank(N ) in M Θ .
In what follows, tuples of points will be called general, without mentioning ∇ and g , which we tacitly assume to be fixed. Since there are only finitely many isomorphism classes of types of fixed degree and genus, the set of non-general tuples is a nowhere dense closed subset in N n R . Remark 3.7. Other notions of general position can be found in the literature. For instance, Mikhalkin uses a different definition in case rank(N ) = 2, [Mik05, Definitions 4.2 and 4.7]. Let n = |∇|+g −1 and suppose that (q 1 , . . . , q n ) ∈ N n R are in general position in the sense of Mikhalkin. Then, by Mikhalkin's definition, any curve (Γ, h) of genus g and degree ∇ passing through the q i 's is necessarily 3-valent, weightless, and all slopes of h are non-zero (except for the slopes of the contracted legs). In our setting, similar properties hold true as the following proposition shows.
Proposition 3.8. Assume that rank(N ) = 2, n = |∇|+ g −1, and (q 1 , . . . , q n ) ∈ N n R is a general tuple. Let Θ be a combinatorial type of degree ∇ and genus g with n contracted legs and let G be its underlying weighted graph. If ev −1 Θ (q 1 , . . . , q n ) = , then G is 3-valent and weightless, and all slopes are non-zero, except for the slopes of the contracted legs.
Proof. Let ξ ∈ ev −1 Θ (q 1 , . . . , q n ) be any point. Since weightless 3-valent graphs are not contractions of other stable weighted graphs, we may assume that ξ ∈ M Θ . The map ev Θ is the restriction of a linear projection By definition of points in general position, ev −1 Θ (q 1 , . . . , q n ) has codimension 2n in M Θ . Since N n R has dimension 2n, it follows that ev Θ is a submersion at any point of ev −1 Θ (q 1 , . . . , q n )∩M Θ , hence also in a small open neighborhood of ξ. By [Mik05,Corollary 4.12], the tuples of points in tropically general position in the sense of Mikhalkin are dense, and hence there exists a small perturbation is a tuple in tropically general position in the sense of Mikhalkin. Thus, the assertion holds by definition (see Remark 3.7).
is regular, and the underlying graph G is weightless and 3-valent. A stratum M [Θ] is called a simple wall if it is regular, and G is weightless and 3-valent except for a unique 4-valent vertex.
Proof. Assertion (1) follows from Proposition 3.8. Thus, we assume that n = |∇|+g −2. If M [Θ] is a simple wall, then it is regular, and the graph G is weightless and has overvalency one. Therefore, is nice, then the dimension count shows that ev −1 Θ (q 1 , . . . , q n ) is an interval, which implies assertion (3). 3.2. Floor decomposed curves. Next, we recall the notion of a floor decomposed curve as introduced by Brugallé and Mikhalkin [BM09]. We will follow the presentation in [BIMS15,§4.4], see also [Rau17,§2.5]. For the rest of this section, we assume M = N = Z 2 and ∆ ⊂ M R = R 2 is an h-transverse polygon (cf. [BM09,§2]), e.g., the triangle ∆ d with vertices (0, 0), (0, d ) and (d , 0). We associate to ∆ the reduced degree ∇ of tropical curves dual to ∆, i.e., ∇ consists of primitive outer normals to the sides of ∆, and the number of slopes outer normal to a given side is equal to its integral length. FIGURE 1. A collection of vertically stretched points, and a floor decomposed cubic tropical curve passing through them and having the reduced tropical degree associated to ∆ 3 . Definition 3.11. Let λ ∈ R be a positive real number. A point configuration q 1 , . . . , q n in R 2 is called vertically λ-stretched if for any pair of distinct points q i = (x, y) and q j = (x ′ , y ′ ) in the configuration the following holds: |y − y ′ | > λ · |x − x ′ |.
Definition 3.12. A parameterized tropical curve (Γ, h) is called floor decomposed if for any edge e of Γ, the slope ∂h ∂ e is either (±1, * ) or (0, * ). Let (Γ, h) be a floor decomposed parameterized tropical curve. If the image of a non-contracted edge (resp. leg) is vertical, then the edge (resp. leg) is called an elevator. After removing the interiors of all elevators in Γ, we are left with a disconnected graph. The non-contracted connected components of this graph are called the floors of Γ. In Figure 1, the elevators are the red edges, and the blue components are the floors. By abuse of language, we call a collection of points as in Proposition 3.13 vertically stretched.
Remark 3.14. If ∆ = ∆ d , then there is a unique elevator adjacent to the top floor, and, by the balancing condition, for every floor, there is a downward elevator adjacent to it. The elevator adjacent to the top floor and the downward elevators adjacent to the bottom floor -all have multiplicity one. If, in addition, the points q 1 , . . . , q n belong to the line given by y = −µx for µ ≫ 1, then the x-coordinate of the marked point belonging to the image of the top floor is smaller than the x-coordinate of any other q i .
Remark 3.15. In the original definition of [BM09], an elevator either connects two floors or is a leg. In our convention, an elevator of [BM09] may split into a couple of elevators separated by a vertex mapped to one of the q i 's.

TROPICALIZATION
In this section, we construct the tropicalization of one-parameter families of algebraic curves with a map to a toric variety (Theorem 4.6). As a result, we obtain a tropical tool (Corollary 4.9) for studying algebraic degenerations that plays a central role in the proof of the Main Theorem. 4.1. Notation and terminology. Throughout this section, we assume that K is the algebraic closure of a complete discretely valued field. We denote the valuation by ν : K → R ∪ {∞}, the ring of integers by K 0 , and its maximal ideal by K 00 . We fix a pair of dual lattices M and N and a toric variety S with lattice of monomials M . In this paper, we will mostly be interested in the case S = P 2 , but the tropicalization we are going to describe works in general.
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; cf. [Sta20, Tag 0E6T]. It is called (semi-)stable if so are its geometric fibers. A prestable curve with marked points 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; cf. [dJ96, § 2.22]. If U ⊂ Z is open, and (Y , σ • ) is a family of curves with marked points over U , then a model of (Y , σ • ) is a family of curves with marked points over Z , whose restriction to U is (Y , σ • ).

Tropicalization of curves.
We first recall the canonical tropicalization construction for a fixed (parameterized) curve over K . This is well established, and we refer to [BPR13,Tyo12] for details. The normalization of signs is chosen such that the algebraic definition is compatible with the standard tropical pictures. A parameterized curve in S is a smooth projective curve with marked points (X , σ • ) and a map f : X → S such that f (X ) does not intersect orbits of codimension greater than one, and the image of X \( i σ i ) under f is contained in the dense torus T ⊂ S.
Let f : X → S be a parameterized curve, and X 0 → Spec(K 0 ) a prestable model. Denote by X the fiber of X 0 over the closed point of Spec(K 0 ). As usual, a point s ∈ X is called special if it is either a node or a marked point of X . Let D = Σ j D j be the boundary divisor of S. Set f * (D j ) := Σ r d j ,r p j ,r with d j ,r ∈ N and p j ,r ∈ X . By definition, {p j ,r } ⊆ {σ i }. The collection of multisets {d j ,r } r for each j is called the tangency profile of f : X → S. We say that the tangency profile is trivial if d j ,r = 1 for all j and r .
The tropicalization trop(X ) of X with respect to the model X 0 is the tropical curve Γ = (G, ℓ) defined as follows: The underlying graph G is the dual graph of the central fiber X , i.e., the vertices of G correspond to irreducible components of X , the edges -to nodes, the legs -to marked points, and the natural incidence relation holds. For a vertex v of G, its weight is defined to be the geometric genus of the corresponding component of the reduction X v . As for the length function, if e ∈ E (G) is the edge corresponding to a node z ∈ X , then ℓ(e) is defined to be the valuation of λ, where λ ∈ K 00 is such that étale locally at z, the total space of X 0 is given by x y = λ. Although λ depends on the étale neighborhood, its valuation does not, and hence the length function is well-defined. Finally, notice that the order on the set of marked points induces an order on the set of legs of Γ. By abuse of notation, we will not distinguish between the tropical curve trop(X ) and its geometric realization.
Next, we explain how to construct the parameterization h : trop(X ) → N R . Let X v be an irreducible component of X . Then, for any m ∈ M , the pullback f * (x m ) of the monomial x m is a non-zero rational function on X 0 , since the preimage of the big orbit is dense in X . Thus, there is λ m ∈ K × , unique up to an element invertible in K 0 , such that λ m f * (x m ) is an invertible function at the generic point of X v . The function h(v), associating to m ∈ M the valuation ν(λ m ), is clearly linear, and hence h(v) ∈ N R . The parameterization h : trop(X ) → N R is defined to be the unique piecewise integral affine function with values h(v) at the vertices of trop(X ), whose slopes along the legs satisfy the following: for any leg l i and m ∈ M we have ∂h σ i is the marked point corresponding to l i . Then h : trop(X ) → N R is a parameterized tropical curve, by [Tyo12, Lemma 2.23]. The curve trop(X ) (resp. h : trop(X ) → N R ) is called the tropicalization of X (resp. f : X → S) with respect to the model X 0 . Plainly, the tropical curve trop(X ) is independent of the parameterization and depends only on X 0 . If the family X → S is stable and X 0 is the stable model, then the corresponding tropicalization is called simply the tropicalization of X (resp. f : X → S). Plainly, trop(X ) (resp. h : trop(X ) → N R ) is stable in this case. Next, let us recall the tropicalization map trop : X (K )\ i σ i → trop(X ) from K -points of the curve to the tropicalization of the curve. The image trop(η) of a K -point η ∈ X (K )\ i σ i is defined as follows. We temporarily add η to the collection of marked points of X and consider the 13 minimal modification (X 0 ) ′ → X 0 such that (X 0 ) ′ is a prestable model of (X , σ • , η). Then trop(X ) ′ is obtained from trop(X ) by attaching a leg either to an existing vertex or to a new two-valent vertex splitting an edge or a leg of trop(X ). Either way, the geometric realization of trop(X ) ′ is obtained from trop(X ) by attaching a leg to some point q ∈ trop(X ). The tropicalization map trop : X (K )\ i σ i → trop(X ) sends η to the point q.
Remark 4.2. The tropicalization defined above is compatible with the natural tropicalization of the torus in the sense that the following diagram is commutative: (p)) . In particular, in the notation of Remark 4.1, if h contracts l i , then h(l i ) = trop( f (σ i )).
Below we will need a more explicit description of trop(η) ∈ trop(X ). Let ψ : Spec(K ) → X be the immersion of the point η. Since X 0 → Spec(K 0 ) is proper, ψ admits a unique extension ψ 0 : Spec(K 0 ) → X 0 with image η. Let s ∈ Spec(K 0 ) be the closed point. Its image in X under the map ψ 0 is called the reduction of η, and is denoted by red(η). If the reduction of η is a nonspecial point of a component X v , then (X 0 ) ′ = X 0 and trop(η) = v. If the reduction of η is a node z ∈ X , then trop(η) belongs to the edge e corresponding to z, and we shall specify the distance from trop(η) to the two vertices v and w adjacent to e. Let λ ∈ K 00 be such that X 0 is given by ab = λ étale locally at z, and let X v , X w be the branches of X associated to the vertices v and w , respectively. Without loss of generality, X v is given locally by a = 0 and X w by b = 0. Proof. The model (X 0 ) ′ is a blow up of X 0 at the point z, and its local charts are given by at = µ and bt −1 = λµ −1 . In the first chart, the exceptional divisor is given by a = 0 and X w by t = 0. Furthermore, ψ * (t ) ∈ K 0 is invertible since η specializes to a non-special point of the exceptional divisor. Thus, ν(ψ * (a)) = ν(µ) is the distance from trop(η) to w in e.
Finally, if the reduction of η is the specialization of a marked point σ i , and l i is the leg corresponding to σ i , then trop(η) belongs to the leg l i . To specify the distance from trop(η) to the vertex w adjacent to l i , let a = 0 be an étale local equation of σ 0 i in X 0 around ψ 0 (s). Then, Lemma 4.4. The distance from trop(η) to w in l i is given by ν(ψ * (a)).
Proof. The proof is completely analogous to that of Lemma 4.3.

Tropicalization of one-parameter families of curves.
In this section, we describe the tropicalization procedure for one-dimensional families of stable curves with a map to a toric variety. We expect that the main existence statement -the first part of Theorem 4.6 -is known to experts; see, in particular, [ACGS20, CCUW20, Ran22] for related constructions. We use the local (toroidal) structure of the moduli space of pointed curves and the universal curve over it; cf. Remark 4.8. Without the splitness assumption, the dual graphs G w for w ∈ V (Λ) can be defined only up to an automorphism. In general, one needs to consider families Γ Λ → Λ with a stacky structure, following [CCUW20]. (1) X τ has exactly one node. In particular, the geometric genus of X τ is one less than the geometric genus of a general fiber of X → B ; (2) The rational map f : X S is defined on the fiber X τ . Furthermore, f maps the generic point(s) of X τ to the dense orbit T ⊂ S, and f : X τ → S has the same tangency profile as the general fiber f : X b → S. The rest of the section is devoted to the proof of Theorem 4.6 and Corollary 4.9. Recall that K is the algebraic closure of a complete discretely valued field, which we denote by F . Any Kscheme of finite type is defined over a finite extension F ′ of F , which is also a complete discretely valued field since so is F . Thus, we may view a K -scheme of finite type as the base change of a scheme over F ′ . In the proofs below, we will work over F ′ to have a well-behaved total space of the families we consider. To ease the notation, we will assume that all models and points we are interested in are defined already over F 0 but will not assume that the valuation of the uniformizer π is one.
In particular, we may assume that B 0 , X 0 , X 0 → B 0 , and η are all defined over F 0 . In the proof, we will use the following notation: ψ : Spec(F ) → B will denote the immersion of the point η, s := red(η) ∈ B its reduction, and X s the corresponding fiber. We set D B 0 := B ∪ ( i τ i ) and D X 0 := X ∪ i X τ i ∪ j σ j , where X τ i are the fibers over the marked points τ i ∈ B (F ). Then the pullback of any monomial function f * (x m ) is regular and invertible on X 0 \ D X 0 by the assumptions of the theorem.

Proof of Theorem 4.6.
Step 1: The tropicalization h η : trop(X η ) → N R depends only on trop(η) ∈ Λ. Set q := trop(η). After modifying B 0 , we may assume that s ∈ B is non-special and, by definition, this modification depends only on trop(η), cf. Section 4.2. Let B w be the component containing s, and w = q the corresponding vertex of Λ. By definition, the underlying graph of trop(X η ) is the dual graph G s of X s . Let z be a node of X s . Then there exist an étale neighborhood U of s in B 0 and a function g z ∈ O U (U ) vanishing at s such that the family X 0 × B 0 U is given by x y = g z étale locally near z. After shrinking U , we may assume that the latter is true in a neighborhood of any node of X s over U . Furthermore, since s ∈ B is non-special, we may choose U such that the pullback of D B 0 to U is B w .
By assumption, X is smooth over B ′ , and hence each g z is invertible on the complement of D B 0 , i.e., away from B w . However, U is normal, and g z (s) = 0 for all nodes z. Thus, all the g z 's vanish identically along B w , which implies that the dual graph G s of X s is étale locally constant over B w . We claim that the lengths of the edges of G s are also étale locally constant. Indeed, pick a node z ∈ X s . Since g z is invertible away from B w , and π vanishes to order one along B w , there exists k w ∈ N such that π −k w g z is regular and invertible in codimension one, and hence, by normality, it is regular and invertible on U . Thus, the length of the edge of G s corresponding to z is given by ν(ψ * g z ) = k w ν(π), cf. Section 4.2, which is étale locally constant around s. Finally, since X | B w → B w is split, the identifications of graphs G s with the dual graph G w of the generic fiber is canonical. Hence the tropicalization trop(X η ) depends only on q, and we set Γ q := trop(X η ).
It remains to check that the parameterization h η also depends only on q. Since the tangency profile of X η → S is independent of η, so are the slopes of the legs of trop(X η ), cf. Remark 4.1. Let u be a vertex of G w , and X u ⊂ X the component corresponding to u that dominates B w . Pick any m ∈ M . Since f * (x m ) is regular and invertible away from D X 0 , X 0 is normal, and π vanishes to order one along X u , it follows that there is an integer k u ∈ Z such that π k u f * (x m ) is regular and invertible at the generic point of X u . Thus, h η (u)(m) = k u ν(π), and hence h η depends only on q. We set h q := h η , and obtain a parameterized tropical curve h q : Γ q → N R that is canonically isomorphic to the tropicalization of f : X η → S for any K -point η ∈ B ′ satisfying trop(η) = q.
Step 2: h : Γ Λ → N R is a family of parameterized tropical curves. We need to show that the fiberwise tropicalizations constructed in Step 1 form a family of parameterized tropical curves; that is, we need to specify a datum ( †) as in § 3.1.3 that satisfies the conditions of Definition 3.1. Since the tropicalizations of K -points give rise only to rational points in the tropical curve, we will work with rational points Λ Q ⊂ Λ, and in the very end, extend the family by linearity to the non-rational points of Λ. By Step 1, the extended degree of h q : Γ q → N R is independent of q ∈ Λ, and will be denoted by ∇. Furthermore, if q is an inner point of some e ∈ E (G Λ ), and trop(η) = q, then s is the special point of B corresponding to e, and the underlying graph of Γ q is the dual graph of the reduction X s . Therefore, the underlying graph of Γ q depends only on e and will be denoted by G e . We will see below that in this case, the slopes of the bounded edges of Γ q , also depend only on e. Hence so does the combinatorial type of (Γ q , h q ).
Next, we specify the contraction maps. Let s ′ ∈ B w be a node of B corresponding to an edge e of Λ, and G e := G s ′ the corresponding weighted graph. Any degeneration of stable curves corresponds to a weighted edge contraction on the level of dual graphs. Thus, for the étale local branch of B w at s ′ corresponding to e ∈ Star(w ), we get the contraction ϕ e : G e → G w between the associated dual graphs. Analogously, for the reduction s ′ ∈ B w of a marked point with corresponding leg l ∈ L(Λ), we obtain the desired contraction ϕ l : G l → G w .
To finish the proof of Step 2, it remains to show that the functions ℓ(γ, ·) : e ∩ Q → R ≥0 and h(u, ·) : e ∩Q → N R are restrictions of integral affine functions. Indeed, if this is the case, then the family extends to the irrational points of e. Furthermore, it follows that the slope of h q along any γ ∈ E (G e ) is continuous on e • , and obtains values in N at the rational points q ∈ e • ∩Q. Therefore, it is necessarily constant on e • , and hence so is the combinatorial type.
Notice that for the function h, the assertion can be verified separately for the evaluations h(u, ·)(m) of h(u, ·) for each m ∈ M . For the rest of the proof, we fix an m ∈ M , an edge γ ∈ E (G q ), and a vertex u ∈ V (G q ). Let z ∈ X s denote the node corresponding to γ and X s,u the irreducible component of X s corresponding to u; see Figure 2. There are two cases to consider.
The left picture shows the components of D X 0 . On the right, we indicate the irreducible components of X .
Case 1: s ∈ B is a node. Let e be the edge of Λ corresponding to s. Then q ∈ e • and G q = G e . After shrinking U , we may assume that it is given by ab = π k s for some positive integer k s , and B has two components B v and B w in U given by a = 0 and b = 0, respectively. Thus, the length ℓ Λ (e) is given by k s ν(π).
We start with the function ℓ. Since g z is regular and vanishes only along B v ∪ B w in U , there exist k a , k b , k π ∈ N such that a −k a b −k b π −k π g z is regular and invertible on U . Therefore, By Lemma 4.3, ν(ψ * (a)) is the distance of q from w in e. Thus, (4.1) defines an integral affine function of q on e with slope k a − k b ; see Figure 3 for an illustration.
It remains to show that the value of (4.1) for ν(ψ * (a)) = 0 is ℓ w (ϕ e (γ)), where the orientation on e is such that w is its tail. By Step 1, ℓ w (ϕ e (γ)) = k w ν(π). Since a is invertible at the generic point of B w , the order of vanishing k w of g z at the generic point of B w is equal to the order of vanishing of b k b π k π , which in turn is the order of vanishing of π k b k s +k π . Thus, (4.2) as needed.
Next we consider the function h. For the two orientations e and e on e, let ϕ e : G e → G w and ϕ e : G e → G v be the weighted edge contractions defined above. As in Step 1, we have irreducible components X ϕ e (u) and X ϕ e (u) of X ; they are the components of X containing X s,u and supported over B v and B w , respectively; see Figure 2. Since f * (x m ) is regular and invertible outside D X 0 , there exist r a , r b , r π ∈ Z such that a r a b r b π r π f * (x m ) is regular and invertible at the generic points of X ϕ e (u) and X ϕ e (u) , and hence, by normality of X , it is regular and invertible in a neighborhood of the generic point of X s,u . Therefore, h(u, q)(m) = r a ν(ψ * (a)) + r b ν(ψ * (b)) + r π ν(π) = (r a − r b )ν(ψ * (a)) + r b ℓ Λ (e) + r π ν(π). (4.3) By Lemma 4.3, ν(ψ * (a)) is the distance of q from w in e. Thus, (4.3) defines an integral affine function on e with slope r a − r b . Furthermore, the value of this function for ν(ψ * (a)) = 0 is h w (ϕ e (u)). Indeed, since a is regular and invertible at the generic point of B w , it is regular and invertible at the generic point of X ϕ e (u) . Thus, by the definition of h w , as needed. Similarly, the value of (4.3) for ν(ψ * (a)) = ℓ Λ (e) is h v (ϕ e (u)).
Case 2: s ∈ B is the reduction of a marked point. Let τ ∈ B be the marked point with reduction s, l the associated leg of Λ, and B w the component of B containing s. Then q ∈ l • and G q = G l . After shrinking U , we may assume that B ∪ τ in U is given by πa = 0.
Again, we start with the function ℓ. Since g z is regular in U and vanishes only along B ∪τ, there exist k w , k τ ∈ N such that π −k w a −k τ g z is regular and invertible on U . The length ℓ(γ, q) = ν(ψ * g z ) is thus given by ℓ(γ, q) = k w ν(π) + k τ ν(ψ * (a)).
(4.4) By Lemma 4.4, ν(ψ * (a)) is the distance of q from w in l . Thus, (4.4) defines an integral affine function on l with slope k τ and, by Step 1, its value at ν(ψ * (a)) = 0 is k w ν(π) = ℓ w (ϕ l (γ)), as required; see Figure 4 for an illustration. We proceed with the function h. Let ϕ l : G l → G w be the edge contraction defined above. As before, we have a unique component X ϕ l (u) of X containing X s,u and supported over B w .
Similarly, there is a unique component X τ,u of X τ containing X s,u in its closure. Since f * (x m ) is regular and invertible away from D X 0 , there exist r w , r τ ∈ Z such that π r w a r τ f * (x m ) is regular and invertible on U . Therefore, h(u, q)(m) = r τ ν(ψ * (a)) + r w ν(π). (4.5) By Lemma 4.4, ν(ψ * (a)) is the distance of q from w in l . Thus, (4.5) defines an integral affine function on l with slope r τ , and the value h w (ϕ l (u))(m) at w , since w is given by ν(ψ * (a)) = 0, cf.
Step 3: The harmonicity and the local combinatorial surjectivity of the map α. Let w ∈ V (Λ) be a vertex, Θ the combinatorial type of α(w ), and C → M g ,n+|∇| the universal curve. Consider the natural map to the coarse moduli space χ : B 0 → M g ,n+|∇| . There are two cases to consider: Case 1: χ contracts B w . Set p := χ( B w ). Since X → B is split, it follows that the restriction of X to B w is the product B w × C p . Furthermore, G w is the dual graph of C p , and α maps Star(w ) to M [Θ] . Since α is a piecewise integral affine map, it lifts to a map to M Θ , which we denote by α w . Let us show that α is harmonic at w , i.e., The latter equality can be verified coordinatewise. Recall that the integral affine structure on M Θ is induced from N |V (G w )| R ×R |E(G w )| . Let γ ∈ E (G w ) be an edge corresponding to a node z ∈ C p , and assume for simplicity that γ is not a loop. The case of a loop can be treated similarly, and we leave it to the reader. Let u, u ′ ∈ V (G w ) be the vertices adjacent to γ, C u and C u ′ the corresponding components of C p , and X u , X u ′ , Z the pullbacks of C u , C u ′ , z to X , respectively. The universal curve C is given étale locally at z by x y = m z , where m z is defined on an étale neighborhood of p and vanishes at p. Thus, X 0 is given by x y = g Z := χ * m z in an étale neighborhood of Z . Notice that we constructed a function defined in a neighborhood of the whole family of nodes Z , which globalizes the local construction of Step 1 in the case we consider here.
To prove harmonicity with respect to the coordinate x γ corresponding to γ, we now consider (π −k w g Z )| B w , where, as in Step 1, k w denotes the order of vanishing of g Z at the generic point of B w . Thus, (π −k w g Z )| B w is a non-zero rational function on B w , and hence the sum of orders of zeroes and poles of this function is zero. We claim that this is precisely the harmonicity condition we are looking for. Indeed, since X 0 is smooth over B ′ , the function g Z is regular and invertible away from D B 0 , and hence, as usual, (π −k w g Z )| B w has zeroes and poles only at the special points of B w . For e ∈ Star(w ), let s be the corresponding special point. If s is a node of B, and B v is the second irreducible component containing s, then pick k s ∈ N as in Case 1 of Step 2, i.e., such that B 0 is given étale locally at s by ab = π k s . Then, using Case 1 of Step 2, which is the order of vanishing of (π −k w g Z )| B w at s. Similarly, if s is the specialization of a marked point, ∂x γ ∂ e is again the order of vanishing of (π −k w g Z )| B w at s. Next, let us show harmonicity with respect to a coordinate n u for u ∈ V (G w ), that is, we need to show that for any m ∈ M . Let k u ∈ Z be such that π k u f * (x m ) is regular and invertible at the generic point of X u . Then the divisor D u,m of (π k u f * (x m ))| X u has horizontal components supported on the special points of the fibers of X → B , and vertical components supported on the preimages of the special points of B w . Pick a general point c ∈ C u , and consider the horizontal curve B w,c := B w × {c} ⊂ X u . It intersects no horizontal components of D u,m , and intersects its vertical components transversally. Let s ∈ B w be a special point, and X s,u the corresponding vertical component of D u,m . Its multiplicity in D u,m is equal to the multiplicity of the point (s, c) in the divisor of (π k u f * (x m ))| B w,q , and hence the sum over all special points of B w of these multiplicities vanishes. On the other hand, we claim that the multiplicity of X s,u in D u,m is nothing but ∂h(u,·)(m) ∂ e , where e ∈ Star(w ) corresponds to the special point s. Indeed, if s is the specialization of a marked point τ, then ∂h(u,·)(m) ∂ e = r τ , where r τ is the order of pole of f * (x m ) at τ as in Case 2 of Step 2. Since π does not vanish at τ, we conclude that ∂h(u,·)(m) ∂ e is the multiplicity of the reduction X s,u of X τ in D u,m , as asserted. A similar computation shows that if s is a node of B, then ∂h(u,·)(m) ∂ e is again the multiplicity of X s,u in D u,m . We leave the details to the reader.
Case 2: χ does not contract B w . By the assumptions of the theorem, the graph G w is weightless and 3-valent except for at most one 4-valent vertex. Since rational curves with three special points have no moduli, it follows that G w has a 4-valent vertex, which we denote by u ∈ G w . In this case we will show that the map α is locally combinatorially surjective at w . Consider X u as above. By construction, X u → B w is a family of rational curves and since X → B is split, the four marked points in each fiber define sections of the family. Let ξ : B w → M 0,4 ≃ P 1 be the induced map. Since χ does not contract B w , it follows that ξ is not constant, and hence surjective. We . An illustration for Case 2 of Step 3 in the proof of Theorem 4.6. Since Γ Λ is not embeddable in R 3 , the picture is a "cartoon". The valency of w may be greater than three, but for each resolution of the 4-valent vertex, there is at least one germ in Star(w ) as in the picture.
conclude that B w contains points, the fibers over which have dual graphs corresponding to the three possible splittings of the 4-valent vertex u into a pair of 3-valent vertices joined by an edge. In particular, α does not map Star(w ) into M [Θ] , and we need to show that for each Θ ′ with an inclusion To see this, notice that any such polyhedron M Θ ′ corresponds to one of the three possible splittings of the 4-valent vertex u since, by balancing, the slope of the new edge is uniquely determined. 4.5. Proof of Corollary 4.9. Set s ′ := red(τ) ∈ B , and let z be the node of the fiber X s ′ corresponding to the edge of varying length γ ∈ E (G l ). Let B w be the component of B containing s ′ . Recall that we set D X 0 := X ∪ i X τ i ∪ j σ j , where σ j are the marked points of X → B .
(1) As in Case 2 of Step 2 in the proof of Theorem 4.6, in an étale neighborhood U of s ′ , B ∪τ is given by πa = 0, and the family X 0 → B 0 over U is given étale locally near z by x y = g z for some g z ∈ O U (U ). The length of γ is an integral affine function on l , and by (4.4), its slope is given by the order of vanishing k τ of g z at τ. Since the slope of ℓ(γ, ·) along l is not constant, k τ > 0, i.e., g z vanishes at τ. Thus, X τ has a node as asserted. Vice versa, any node of X τ specializes to some node z ′ of X s ′ corresponding to an edge γ ′ ∈ E (G l ). Then étale locally at z ′ , the family X 0 → B 0 is given by x y = g z ′ , and g z ′ vanishes at τ. By (4.4) we conclude that the slope of ℓ(γ ′ , ·) is not constant along l . Thus, γ = γ ′ by the assumption of the corollary, and hence z ′ = z.
(2) Let X ′ τ ⊆ X τ be an irreducible component. First, we show that for any m ∈ M , the pullback f * (x m ) is regular and invertible at the generic point of X ′ τ . Hence the rational map f is defined at the generic point of X ′ τ and maps it to the dense orbit T . Pick a vertex u of G l such that X s,u belongs to the closure of X ′ τ . By (4.5), the slope of h(u, ·)(m) along l is given by the order of pole of f * (x m ) along X ′ τ . However, h(u, ·)(m) is constant along l by the assumptions of the corollary, and hence f * (x m ) has neither zero nor pole at the generic point of X ′ τ . Second, notice that since X 0 is normal, f * (x m ) is regular and invertible away from D X 0 , and it is regular and invertible in codimension one on X τ , it follows that f is defined on X τ \ j σ j . Pick any σ j , and let us show that f is defined at σ j (τ) ∈ X τ , too. Let S j ⊆ S be the affine toric variety consisting of the dense torus orbit and the orbit of codimension at most one containing the image of σ j (b) for a general b ∈ B . Then the pullback of any regular monomial function x m ∈ O S j (S j ) is regular in codimension one in a neighborhood of σ j (τ), and hence regular at σ j (τ) by normality of X 0 . Thus, f is defined at σ j (τ), and maps it to S j . The last assertion is clear. 21

DEGENERATION VIA POINT CONSTRAINTS
In this section, (S, L ) = P 2 , O P 2 (d ) , M = N = Z 2 , and ∇ = ∇ d is the reduced degree of tropical curves associated to the triangle ∆ d , i.e., ∇ consists of 3d vectors: (1, 1), (−1, 0), (0, −1), each appearing d times. Recall that for an integer 1 − d ≤ g ≤ d−1 2 , the Severi variety parameterizing curves of degree d and geometric genus g is a locally closed subset V g ,d ⊆ |O P 2 (d )| of pure dimension 3d + g − 1 by Lemma 2.6 and Proposition 2.7. In this section, we prove the Degeneration Theorem and its corollary, which generalizes Zariski's Theorem to arbitrary characteristic. Remark 5.2. If the geometric genus of a degree d curve C is 1 − d , then C is necessarily a union of d lines. Therefore, V 1−d,d is dominated by (P 2 ) * d . In particular, V 1−d,d is irreducible, and its general element corresponds to a nodal curve. Remark 5.4. Assertion (1) in arbitrary characteristic and for any toric surface was already proved in [Tyo13, Theorem 1.2] by the third author. Examples of toric surfaces for which assertion (2) fails can also be found loc. cit.
The rest of the section is devoted to the proof of Theorem 5.1, which proceeds by induction on (d , g ) with the lexicographic order. The base of induction, (d , g ) = (1, 0), is clear. To prove the induction step, let (d , g ) > (1, 0) be a pair of integers such that 1 − d ≤ g ≤ d−1 2 , and assume that for all (d ′ , g ′ ) < (d , g ) the assertion is true. Let us prove that it holds true also for the pair (d , g ). If g = 1 − d , then there is nothing to prove since the variety V 1−d,d is irreducible by Remark 5.2. Thus, we may assume that g > 1 − d .
Step 1: The reduction to the case of irreducible curves. We claim that it is enough to prove the assertion for V ⊆ V irr g ,d . Indeed, let [C ] ∈ V be a general point and assume that C is reducible. Denote the degrees of the components by d i , and their geometric genera by g i . Then d i < d for Our goal is to prove that the locus of curves of geometric genus g − 1 in V has dimension 3d +g −2, since then V necessarily contains a component of V g −1,d by Proposition 2.7, and hence 22 also V 1−d,d by the induction assumption. We proceed as follows: set n := 3d +g −1, and pick n −1 points {p i } n−1 i =1 ⊂ P 2 in general position. For each i , let H i ⊂ |O P 2 (d )| be the hyperplane parameterizing curves passing through the point p i . Since dim(V ) = n and the points {p i } n−1 i =1 ⊂ P 2 are in general position, the intersection Z := V ∩ n−1 i =1 H i has pure dimension one and a general [C ] ∈ Z corresponds to an integral curve C of geometric genus g that intersects the boundary divisor transversally by Proposition 2.7. It is sufficient to show that there exists [C ′ ] ∈ Z such that C ′ is reduced and has geometric genus g − 1. Indeed, such a C ′ passes through a general collection of n − 1 points {p i } n−1 i =1 ⊂ P 2 , and therefore the locus of curves of geometric genus g − 1 in V has dimension n − 1 = 3d + g − 2.
To apply the results of the previous section, we assume that the field K is the algebraic closure of a complete discretely valued field, which we may do by the Lefschetz principle. Furthermore, we may assume that the points {p i } n−1 i =1 ⊂ P 2 tropicalize to distinct vertically stretched points This assumption allows us to work on the tropical side with floor decomposed curves, which are very convenient for controlling the degenerations of curves parameterized by Z . The rest of the proof proceeds as follows. In Step 2, we modify the base curve Z and the family of curves over it so that both Z and the general fiber of the family become smooth. Hence, the tropicalization results of Section 4.3 apply. And in Step 3, we investigate the tropicalization of the modified family and prove that it necessarily contains a tropical curve that corresponds to an algebraic fiber of genus g − 1.
Step 2: Constructing a family of parameterized curves. This step follows the ideas and the techniques of de Jong [dJ96] based on the results of Deligne [Del85]. The goal is to construct a family f : X P 2 of parameterized curves over a smooth base curve (B, τ • ), a prestable model B 0 of (B, τ • ) whose reduction B has smooth irreducible components, and a finite morphism B → Z that satisfy the following properties: (i) (X → B, σ • ) extends to a split family of stable marked curves over B 0 , and (ii) for a general [C ] ∈ Z , the fibers of X → B over the preimages of [C ] are the normalization X of C equipped with the natural map to P 2 and with 3d +n −1 marked points such that the first n−1 of them are mapped to p 1 , . . . , p n−1 , and the rest -to the boundary divisor; cf. Definition 4.5.
Let us start with the normalization B → Z and with the pullback X → B of the tautological family to B equipped with the natural map f : X → P 2 . We will replace B with finite coverings, dense open subsets, and compactifications several times, but to simplify the presentation, we will use the same notation B, X , and f . First, we apply Lemma 2.3. After replacing B with B ′ and X with X ′ as in the lemma, we may assume that the family X → B is a generically equinormalizable family of projective curves. After shrinking B , we may further assume that X → B is equinormalizable, the pullback of the boundary divisor of P 2 on each fiber is reduced, and the preimages of p i 's are smooth points of the fibers.
Second, we label the points of X that are mapped to {p i } n−1 i =1 and to the boundary divisor, which results in a finite covering of the base curve B . After replacing B with this covering, and X with the normalization of the pullback, we equip the family X → B with marked points σ • such that the first n − 1 of them are mapped to p 1 , . . . , p n−1 , and the rest -to the boundary divisor. In particular, we obtain a 1-morphism B → M g ,3d+n−1 that induces the family (X → B, σ • ). Notice that the natural morphism f : X → P 2 satisfies Property (ii).
By [dJ96, §2.24], the compactification M g ,3d+n−1 admits a finite surjective morphism from a projective scheme M. Thus, after replacing B with an irreducible component of B × M g ,3d+n−1 M, we may extend the family (X → B, σ • ) to a family of stable curves with marked points over a smooth projective base, i.e., we may assume that B is projective. Furthermore, the family over 23 B is the pullback of the universal family over M along B → M. Plainly, the induced rational map f : X P 2 still satisfies Property (ii). Let B st be the stable model of B , and B 0 the closure of the graph B → M in B st × M. It is a projective integral model of B over which the family (X → B, σ • ) naturally extends. Furthermore, the extension is obtained by pulling back the universal family from M. After replacing B 0 with a prestable model dominating B 0 , whose reduction has smooth irreducible components, we obtain a model of the base over which the family extends to a family of stable marked curves satisfying Property (ii). It remains to achieve splitness. To do so, we proceed as in [dJ96, § 5.17].
We begin by adding sections such that the nodes of the geometric fibers are contained in the sections. Such sections exist by [dJ96,Lemma 5.3]. We temporarily add these sections as marked points, replace B with an appropriate finite covering, and construct a prestable model of the base over which the family of curves with the extended collection of marked points admits a stable model. By construction, the irreducible components of the geometric fibers of the new family are smooth. By applying [dJ96, Lemmata 5.2 and 5.3] once again, we may assume that the new family admits a tuple of sections such that for any geometric fiber, any component contains a section, and any node is contained in a section. This implies that the new family is split; cf. [dJ96, § 5.17]. Finally, we remove the temporarily marked points σ l 's and stabilize. The obtained family is still split.
To summarize, we constructed a projective curve with marked points (B, τ • ), its integral model B 0 , a family of marked curves (X → B, σ • ), and a rational map f : X P 2 that satisfy Properties (i) and (ii).
Step 3: The conclusion. We postpone the proof of the lemma and first deduce the theorem. Let τ ∈ B (K ) be the marked point corresponding to the leg l as in Lemma 5.5. By Corollary 4.9, the geometric genus of X τ is g − 1. Furthermore, the map f is defined on X τ and maps its generic points to the dense orbit.
We claim that f | X τ is birational onto its image, and hence [ f (X τ )] ∈ Z is a reduced curve of genus g − 1, which completes the proof of Theorem 5.1.
To show the birationality of f | X τ , first note that f does not contract irreducible components of X τ . Indeed, the tropicalization of a component must contain a non-contracted leg by Lemma 5.5 and therefore, its image intersects the boundary divisor. However, its generic point is mapped to the dense orbit, which implies that the component is not contracted by f . If f | X τ is not birational onto its image, then f (X τ ) (with the reduced structure) is a curve of geometric genus at most g − 1 and degree at most d − 1. However, f (X τ ) passes through the n − 1 = 3d + g − 2 points in general position p 1 , . . . , p n−1 , which is a contradiction by Propostion 2.7.
Proof of Lemma 5.5. To prove the lemma, we shall first analyze the image of the induced map α : Λ → M trop g ,n−1,∇ . The strata of the moduli space we will deal with admit no automorphisms. Thus, throughout the proof, one can think about M trop g ,n−1,∇ as a usual polyhedral complex rather than a generalized one. In particular, we will use the standard notion of star of a given stratum in a polyhedral complex.
Notice that α is not constant, since by the construction of B , for any p ∈ P 2 , there exists b ∈ B such that the curve f (X b ) passes through p, and hence h(Γ trop(b) ) contains trop(p), which can be any point in Q 2 ⊂ R 2 , cf. Remark 4.2. We will also need the following key properties of α, which we prove next: (a) if α maps a vertex v ∈ V (Λ) to a simple wall M Θ , see Definition 3.9, then there exists a vertex w ∈ V (Λ) such that α(w ) = α(v) and the map α is locally combinatorially surjective at w ; and (b) if M Θ ′ is nice and α(Λ) ∩ M Θ ′ = , then where ev Θ ′ denotes the evaluation map defined in Section 3.1.6. In particular, α(Λ) ∩ M Θ ′ is an interval, whose boundary is disjoint from M Θ ′ by Lemma 3.10 (3). We start with property (a). Notice that M Θ ∩ ev −1 Θ (q 1 , . . . , q n−1 ) is a point by Lemma 3.10 (2). However, α is not constant, and hence there exists a vertex w such that α(Star(w )) M Θ and α(w ) = α(v). Thus, α is not harmonic at w , and hence it is locally combinatorially surjective at w by Theorem 4.6.
Pick a point p n ∈ P 2 with tropicalization q n ∈ R 2 such that the collection {p i } n i =1 is in general position, and the configuration {q i } n i =1 is vertically stretched, see Section 3.2. Assume further that the points {q i } n i =1 belong to a line defined by y = −µx for some µ ≫ 1. Let b ∈ B (K ) be a point such that f (X b ) passes through p n . Then the tropicaliztion (Γ, h) of (X b , f ) is a floor decomposed curve by Proposition 3.13. By Proposition 3.8, if we marked the point p n on X b , the tropicalization trop(X b ; p 1 , . . . , p n ) would be weightless and 3-valent, and the map to R 2 would be an immersion away from the contracted legs. The tropicalization trop(X b ; p 1 , . . . , p n−1 ) is obtained from trop(X b ; p 1 , . . . , p n ) by removing the leg contracted to q n and stabilizing. Thus, α(trop(b)) belongs to a nice stratum M Θ ′ .
Denote the floors from the bottom to the top by F 1 , . . . , F d and the elevator adjacent to the top floor by E . Recall that E has multiplicity one, cf. Remark 3.14. Let F k , k < d , be the non-top floor adjacent to E . We may assume that q n belongs to the image of the elevator E , and q i to the image of the floor F i for all 1 ≤ i ≤ d . Indeed, pick a permutation σ : {1, . . . , n} → {1, . . . , n} such that q σ(n) ∈ h(E ) and q σ(i ) ∈ h(F i ) for all 1 ≤ i ≤ d . Set p ′ i := p σ(i ) , and consider the curve B ′ associated to {p ′ i } n−1 i =1 and the corresponding family of parameterized curves f ′ : X ′ P 2 . By . It remains to replace B with B ′ , (X , f ) with (X ′ , f ′ ), and p i 's with p ′ i 's. Denote the x-coordinate of E ∩ F k by x and let E ′ be the downward elevator adjacent to F k whose x-coordinate x ′ is the closest to x. Notice that by Remark 3.14, the x-coordinate of the marked point q d does not belong to the interval joining x and x ′ . Without loss of generality, we may assume that x ′ > x. We will call a point of F k special if it is either a marked point or a vertex. Denote the x-coordinates of the special points of F k that belong to the interval [x, x ′ ] by x = x 0 < x 1 < · · · < x r = x ′ . If x i belongs to an elevator, then the elevator will be denoted by E i . Plainly, E = E 0 and E r = E ′ .
Set q n (t ) := q n + t (x 1 − x 0 , 0), 0 ≤ t ≤ 1, and consider the continuous family of tropical curves (Γ t , h t ) ∈ M Θ ′ ∩ ev −1 Θ ′ (q 1 , . . . , q n−1 ), where (Γ 0 , h 0 ) = (Γ, h), and such that q n (t ) ∈ h t (E ) for all t . Since q 1 , . . . , q n−1 , q n (t ) are vertically λ-stretched for a very large value of λ, the floors of the curve remain disjoint in the deformation (Γ t , h t ). Notice also that since the points q 1 , . . . , q n−1 are fixed, the x-coordinate of each elevator except E remains the same in the deformation. Furthermore, the x-coordinates of all elevators adjacent to F i are fixed in the deformation for any i = k, d , as well as the position of the marked point q i on F i . Thus, the restriction of the parameterization h to F i is also fixed. Finally, since the x-coordinate of q d does not belong to [x, x ′ ], it follows that (Γ t , h t ) ∈ M Θ ′ ∩ ev −1 Θ ′ (q 1 , . . . , q n−1 ) for all 0 ≤ t ≤ 1. Furthermore, the curve (Γ 1 , h 1 ) belongs to a simple wall M Θ , in which the elevator E = E 0 gets adjacent to a 4-valent vertex on the floor F k together with either the elevator E 1 or the leg contracted to the marked point q k . Let us describe Star(M Θ ) explicitly. It consists of three nice strata M Θ ′ , M Θ ′′ , M Θ ′′′ , cf. Case 2 of Step 3 in the proof of Theorem 4.6. The stratum M Θ ′′ parameterizes curves in which the elevator E = E 0 has x-coordinate larger than that of E 1 (resp. q k ), and M Θ ′′′ parameterizes curves in which E 0 and E 1 (resp. q k ) get adjacent to a common vertex u in the perturbation of the 4-valent vertex of (Γ 1 , h 1 ); see Figure 6. By Property (b), there exists a vertex v of Λ, such that α(v) ∈ M trop g ,n−1,∇ is the isomorphism class of (Γ 1 , h 1 ), and hence by Property (a), there exists a vertex w such that α(w ) = α(v) and α is locally combinatorially surjective at w .
. On the left, the elevator E reaches a special point. On the rightthe local pictures of typical elements in the three nice cones in Star(M Θ ). The top row illustrates the case when the special point is a vertex, and the bottomwhen it is a marked point.
We proceed by induction on (k, r ) with the lexicographic order and start with the extended base of induction: r = 1 and the multiplicity of E ′ is one. Since ∇ is reduced, this is the case in the actual base of induction (k, r ) = (1, 1). Let e ∈ Star(w ) be an edge such that α(e) ⊂ M Θ ′′′ . Since the elevators E and E ′ both have multiplicity one, it follows from the balancing condition that the third edge adjacent to u gets contracted by the parameterization map h; see Figure 7.
Thus, the locus M Θ ′′′ ∩ ev −1 Θ ′′′ (q 1 , . . . , q n−1 ) is an unbounded interval. Furthermore, all curves in this locus factor through (Γ 1 , h 1 ). By Property (b), M Θ ′′′ ∩ ev −1 Θ ′′′ (q 1 , . . . , q n−1 ) = α(Λ) ∩ M Θ ′′′ . Hence there exists a leg l ∈ L(Λ) such that α(l ) ⊂ M Θ ′′′ ∩ ev −1 Θ ′′′ (q 1 , . . . , q n−1 ) is not bounded. The leg l satisfies the first assertion of the lemma. Note that here the edge γ l is just the third edge adjacent to u (considered as a vertex of G l ) other than E and E ′ , mentioned above. As γ l connects F k and u, and u is connected to the top floor F d by E , each component of G l \{γ l } contains either the non-contracted legs in F k or the non-contracted legs in F d . This fulfils the second assertion. Next, let us prove the induction step. We distinguish between two cases.
The case in which the 4-valent vertex is adjacent to two elevators of multiplicity one. The right picture illustrates the fact that in this case, the curves parameterized by M Θ ′′′ ∩ ev −1 Θ ′′′ (q 1 , . . . , q n−1 ) have constant image in R 2 and contain an edge of varying length contracted to u.
Case 1: r > 1. Let e ∈ Star(w ) be an edge such that α(e) ⊂ M Θ ′′ , and (Γ 1+ǫ , h 1+ǫ ) the tropical curve parameterized by any point in e • . Then the corresponding invariant is (k, r −1), and hence, by the induction assumption, there exists a leg l ∈ L(Λ) satisfying the assertion of the lemma. See Figure 8 for an illustration. Case 2: r = 1 and k > 1. We already checked the case when the multiplicity of E ′ is one in the extended induction base. Thus, assume that the multiplicity w(E ′ ) of E ′ is greater than one. Let q j be the marked point contained in h 1 (E ′ ), and E ′′ the second elevator for which q j ∈ h 1 (E ′′ ). Let F k ′ be the floor adjacent to E ′′ . Then k ′ < k. Let e ∈ Star(w ) be an edge such that α(e) ⊂ M Θ ′′′ , where M Θ ′′′ denotes the nice cone in the star of M Θ in which the elevators E and E ′ get attached to a 3-valent vertex u obtained from the perturbation of the 4-valent vertex of (Γ 1 , h 1 ). We denote the two elevators corresponding to E ′ in this cone by E ′ 1 and E ′ 2 , see Figure 9 for an illustration.
. Going down with an elevator E ′ of multiplicity greater than one, and the corresponding wall-crossings.
It follows from the balancing condition that the slope of E ′ 1 in Star(u) is (0, w(E ′ ) − 1). Thus, the locus M Θ ′′′ ∩ ev −1 Θ ′′′ (q 1 , . . . , q n−1 ) is a bounded interval, whose second boundary point belongs to another simple wall M Ξ , which parameterizes curves with a 4-valent vertex u ′ adjacent to the elevators E , E ′ , E ′′ and to the leg l j contracted to q j .
As before, there exists a vertex w ′ of Λ such that α(w ′ ) ∈ M Ξ and α is locally combinatorially surjective at w ′ by Property (b). Let M Ξ ′ ∈ Star(M Ξ ) be the nice stratum in which the 4-valent vertex u ′ is perturbed to a 3-valent vertex adjacent to E and the downward elevator E ′′ 2 , and a 3-valent vertex adjacent to E ′ and l j . By Property (a), the image of α intersects M Ξ ′ non-trivially. Furthermore, M Ξ ′ ∩ ev −1 Ξ ′ (q 1 , . . . , q n−1 ) is a bounded interval, whose second boundary point belongs to the simple wall M Υ parameterizing curves with a 4-valent vertex u ′′ adjacent to the floor F k ′ and to the elevators E and E ′′ . As usual, M Ξ ′ ∩ ev −1 Ξ ′ (q 1 , . . . , q n−1 ) belongs to the image of α, and hence all three nice strata of Star(M Υ ) intersect the image of α non-trivially. In particular, there exists an edge e ′ of Λ parameterizing curves with invariant (k ′ , * ) < (k, 1), and hence, by the induction assumption, there exists a leg l ∈ L(Λ) satisfying the assertion of the lemma, which completes the proof.

THE LOCAL GEOMETRY OF SEVERI VARIETIES
In this section, we describe the local geometry of V g ,d and V irr g ,d along V 1−d,d .