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 As an immediate corollary of the main result, we obtain the first proof of the irreducibility of the moduli spaces M g in arbitrary characteristic that involves no reduction to characteristic zero. Indeed, since there is a dense open subset U of V irr g ,d that parameterizes nodal curves, the universal family of curves over U is equinormalizable, and hence induces a map U → M g . However, since for any d large enough, any smooth genus g curve admits a birational immersion as a planar degree d curve, it follows that the map U → M g is dominant. Thus, the irreducibility of M g follows from the Main Theorem in arbitrary characteristic.
1.2. The techniques and the idea of the proof. Our proof of the Main Theorem also follows the general strategy of Severi's original approach, but the tools we use are completely different and combine classical algebraic geometry and tropical geometry.
As a first step, we prove that the closure V ⊆ |O P 2 (d )| of any irreducible component V ⊆ V g ,d contains V 1−d,d (Theorem 5.1). We proceed by induction, i.e., we prove that V contains an irreducible component of V g −1,d , and therefore a component of V 1−d,d . But the variety V 1−d,d is irreducible since it parameterizes unions of d distinct lines, which allows us to conclude that V contains the whole of V 1−d,d . This is the step where the new tools from tropical geometry are involved. To do tropical geometry, we assume that the ground field is the algebraic closure of a complete discretely valued field, which is harmless because the irreducibility property is stable under field extensions, and any field is a subfield of the algebraic closure of the field of Laurent power series with coefficients in the given field.
To prove that V contains an irreducible component of V g −1,d , we consider the intersection Z of V with the space of curves passing through 3d + g − 2 points in general position, chosen such that their tropicalizations are vertically stretched. By a careful analysis of the tropicalization of the corresponding one-parameter family of curves of genus g , we show that there exists [C ] ∈ Z such that C is a reduced curve of geometric genus g −1.
Since such a C passes through 3d + g −2 points in general position, it follows that the intersection V ∩ V g −1,d has the same dimension as V g −1,d itself, and therefore contains one of its irreducible components.
Our degeneration argument is new. It is based on the study of the tropicalizations of general one-parameter families of algebraic curves, and on the investigation of the properties of the induced map from the tropicalization of the base to the moduli space of parameterized tropical curves α : Λ → M trop g ,n,∇ . We prove that α is a piecewise integral affine map, and in good cases, it satisfies the harmonicity and local combinatorial surjectivity properties, see Definition 3.4 and Theorem 4.6. Roughly speaking, what these properties allow us to do is to describe the intersection of the image of α with two types of strata of M trop g ,n,∇ : the nice strata -parameterizing weightless tropical curves whose underlying graph is 3-valent, and the simple walls -parameterizing such curves but with a unique 4-valent vertex. In particular, we show that for a nice stratum, the image of α is either disjoint from it or intersects it along a straight interval, whose boundary does not belong to the stratum. For a simple wall, we show that the image of α is either disjoint from it or intersects all nice strata in the star of the wall.
Next, we use these general properties to reduce the assertion of Theorem 5.1 to a combinatorial game with floor decomposed tropical curves, the latter being a very convenient tool in tropical geometry introduced by Brugallé and Mikhalkin [BM09]. The goal of the game is to prove that the image of α necessarily intersects a nice stratum parameterizing tropical curves having a contracted edge of varying length. Indeed, if this is the case, then Λ contains a leg parameterizing tropical curves for which the length of the contracted edge is growing to infinity. Since the legs of Λ correspond to marked points of the base, it follows that there is a closed point [C ] ∈ Z , such that the generalized tropicalization of the normalization of C has an edge of infinite length, i.e., C has geometric genus less than g . With a minor extra effort we show that C is necessarily reduced, see Step 3 in the proof of Theorem 5.1.
On the tropical side of the game, we show that for a point q ∈ Λ corresponding to a tropical curve passing through 3d +g −1 vertically stretched points, the stratum M Θ of M trop g ,n,∇ containing α(q) is nice, and the boundary of the interval Im(α)∩M Θ belongs to simple walls adjacent to M Θ . We then use the floor-elevator structure of floor decomposed curves to describe the nice strata adjacent to these simple walls, and 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, 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 all possible 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 a 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 the study of families of curves will be found in the near future. 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 has to 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 a majority of 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 simplest example of this sort.
In [CHT22b], we make another step forward, and use a modification of the methods developed in the current paper to deduce the irreducibility of Hurwitz schemes in small positive 4 characteristic. To the best of our knowledge, the latter has been an open question since Hurwitz schemes were introduced by Fulton in 1969 [Ful69]. In particular, we develop new tropical tools that replace monodromy arguments in the proof of the irreducibility of Severi varieties.
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 expected dimension. In Section 3, we recall the notion of parameterized tropical curves, their moduli spaces, and floor decomposed curves.
Section 4 is devoted to the study of 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 result (Theorem 5.1), 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, which provides an explicit description of 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. In particular, we show that in the case of toric surfaces, Severi varieties are locally closed, and their closures are either empty or equidimensional of 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 |: 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 .
such that B ′ → B is finite and surjective, X ′ is an irreducible component of (X × B B ′ ) red dominating X and X ′ → B ′ is a generically equinormalizable family of reduced projective curves.
Remark 2.4. In characteristic zero, any family X → B like in Lemma 2.3 is generically equinormalizable by [DPT80, Page 80] and [CHL06]. This fails in characteristic p as the classic example below shows. Furthermore, the map B ′ → B of Lemma 2.3 may fail to be generically étale, which is one of the pitfalls in a naïve attempt to generalize the known characteristic-zero approaches to Zariski's theorem and Harris' theorem to the case of positive characteristic.
Example 2.5. In characteristic p > 2, consider the family over A 1 K = Spec(K [t ]), given in an affine chart Spec(K [x, y, t ]) by x p + y 2 + t = 0. It has normal total space isomorphic to A 2 K . However, none of its fibers is smooth. In this case, the necessary base change in Lemma 2.3 is given by adjoining p t . In particular, the base change is nowhere étale.
Lemma 2.6. The Severi varieties V irr g ,L and V g ,L are locally closed subsets of |L |.
Proof. Since V irr g ,L is the intersection of V g ,L with the open locus of irreducible curves in |L |, it is sufficient to prove the assertion for V g ,L . Since the universal curve X |L | → |L | is flat and proper, the set U 1 : Let us first show that V g ,L is constructible. We can inductively define a finite stratification of U 1 by geometric genera as follows: Consider the restriction X U 1 → U 1 of the universal curve to U 1 . Assume first, that the total space 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, and 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 that intersects 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 ] Furthermore, if C intersects the boundary divisor non-transversally, then the inequality is strict. Next, let us show the opposite inequality, which will imply 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, 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, |L |) at [X , f ] is bounded from below by χ(X , N f ); cf. [Mor79, Proposition 3]. Since c 1 (N f ) = −L .K S + 2g − 2, it follows from the Riemann-Roch theorem that Consider the projection from a small neighborhood of [X , f ] to |L | given by sending a point [X ′ , f ′ ] to [ f ′ (X ′ )]. Since the fiber over [C ] is finite, it follows that 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 |, and 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.
3.1. Families and parameter spaces for tropical curves. 7 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 set of vertices and edges of G, respectively. We denote by L(G) the set of legs of G and extend ℓ to legs by setting their length 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 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 -an oriented tree with root v, all of whose edges are leaves.
The genus of Γ is defined to be g (Γ) = g (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.
3.1.2. Parameterized tropical curves. A parameterized tropical curve is a pair (Γ, h), where Γ = (G, ℓ) is a tropical curve, and h : Γ → N R is a map such that: (a) for any e ∈ E (G), the restriction h| e is an integral affine function; and (b) (Balancing condition) for any vertex v ∈ V (G), we have e∈Star(v) ∂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 ′ : Γ ′ → N R is an isomorphism of metric graphs ϕ : 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 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 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 • : A family of parameterized tropical curves over Λ will be denoted by h : 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 certain 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 trop g ,n,∇ is constructed as follows. A parameterized tropical curve of type Θ defines naturally a point in N , and the set of parameterized tropical curves of type Θ gets identified with the interior M Θ of a convex polyhedron 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.
Next, notice that an isomorphism α : Θ 1 → Θ 2 of combinatorial types induces an isomorphism N and also respects the integral affine structures. We thus obtain isomorphisms ι α : M Θ 1 → M Θ 2 . In particular, the group Aut(Θ) acts naturally on M Θ .
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.  (1) Suppose α(Star(w )) ⊂ M [Θ] . We say that α is harmonic at w if α | Star(w) lifts to a harmonic map α w : Star(w ) → M Θ , i.e., α w is piecewise integral affine, and e∈Star(w) (2) Suppose α(Star(w )) M [Θ] . We say that α is locally combinatorially surjective at w if for any combinatorial type Θ ′ with an inclusion ι Θ ′ ,E : 3.1.5. Regularity. A parameterized tropical curve (Γ, h), its type Θ, and the corresponding stratum M [Θ] are called regular if M Θ has the expected dimension, where G, as usual, denotes the underlying graph, χ(G) its Euler characteristic, and ov(G) the overvalency of G, i.e., ov(G) := v∈V (Γ) max{0, val(v) − 3}. By [Mik05, Proposition 2.20], the expected dimension provides a lower bound on the dimension of a stratum M Θ . Note that in loc. cit., the expected dimension is stated to be expdim(M Θ ) − c, where c denotes the number of edges contracted by h. This discrepancy in the formulae for the expected dimension appears because in loc. cit. only deformations of the image h(Γ) are taken into account. Remembering also Γ gives one additional parameter for each contracted edge, and these parameters are independent.
The following is Mikhalkin's [Mik05, Proposition 2.23] stated in terms of the current paper: Proposition 3.5. Assume that rank(N ) = 2 and h is an immersion away from the contracted legs. If G is weightless and 3-valent, then (Γ, h) is regular, and dim( 3.1.6. The evaluation map. Consider the natural evaluation map ev : M trop g ,n,∇ → N n R , defined by ev(Γ, h) := (h(l 1 ), . . . , h(l n )). Let Θ be a combinatorial type, (q 1 , . . . , q n ) ∈ N n R any tuple, and ev Θ the composition is a polyhedron cut out in M Θ by an affine subspace, and hence its boundary is disjoint from M Θ unless ev −1 Θ (q 1 , . . . , q n ) is a point. 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, either the codimension of ev −1 Θ (q 1 , . . . , q n ) in M Θ is n · rank(N ), as expected, or ev −1 Θ (q 1 , . . . , q n ) = . 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 Θ of fixed degree and genus, the set of non-general tuples is a nowhere dense closed subset in N n R .
Remark 3.7. There are other ways to define tropical general position in the literature. In particular, we will use Mikhalkin's definition [Mik05,Definition 4.7] in the proof of the next proposition. For rank(N ) = 2 a tuple of points (q 1 , . . . , q n ) ∈ N n R is in general position in the sense of Mikhalkin if for any parameterized tropical curve (Γ, h) of genus g and with n contracted legs l 1 , . . . , l n , such that h(l i ) = q i and with n ≥ |∇| + g − 1 the following holds: (1) The underlying graph of Γ is weightless and 3-valent, all slopes are non-zero, except for the slopes of the contracted legs, and if h(q) = h(q ′ ) for two points q, q ′ ∈ Γ, then neither q nor q ′ is a vertex and there is no third point of Γ mapped to h(q), (2) the only vertex contained in h −1 (q i ) is the one adjacent to l i , and (3) n = |∇| + g − 1.
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 N   Lemma 3.10. Assume that (q 1 , . . . , q n ) ∈ N n R is general, ev −1 Θ (q 1 , . . . , q n ) = , and rank(N ) = 2. Then, 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: 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, in what follows we will call a collection of points as in Proposition 3.13 just 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 an infinite ray attached to a floor. 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
The goal of this section is to construct the tropicalization of certain 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 the section we assume that K is the algebraic closure of a complete discretely valued field F . 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, is a family of curves with marked points over U , then by a model of (Y , σ • ) over Z we mean a family of curves with marked points over Z , whose restriction to U is (Y , σ • ).

Tropicalization for curves.
We first recall the canonical tropicalization construction for a fixed (parameterized) curve over K . This is well established, and we refer to [Tyo12] and [BPR13] 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 . Plainly, {p j ,r } ⊆ {σ i } by definition. 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. 13 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 and m ∈ M we have ∂h ∂ l (m) = −ord σ i f * (x m ), where σ i is the marked point corresponding to l . 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 as a marked point to obtain the curve (X , σ • , η) with marked points {σ i } i ∪ η. We consider the 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 , then h(l ) = 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 is the leg corresponding to σ i , then trop(η) belongs to the leg l . To specify the distance from trop(η) to the vertex w adjacent to l , 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 is given by ν(ψ * (a)).
Proof. The proof is completely analogous to that of Lemma 4.3.

Tropicalization for 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] and [Ran22] for related constructions.   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. Remark 4.10. Under the assumptions of the corollary, the edge of varying length gets necessarily contracted by the map h.
The rest of the section is devoted to the proof of Theorem 4.6 and Corollary 4.9. Since K is the algebraic closure of a complete discretely valued field F as in Section 4.1, any K -scheme 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 a finite extension of F . In the proofs below, we will work over F ′ in order 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, ·) at single monomials m. Thus, we fix an arbitrary m ∈ M for the rest of the proof. We also fix an edge γ ∈ E (G q ) and a vertex u ∈ V (G q ), and 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.
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.
The left picture shows the components of D X 0 . On the right, we indicate the irreducible components of X . 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, ℓ w (ϕ e (γ)) = k w ν(π) = (k b k s + k π )ν(π) = k b ℓ Λ (e) + k π ν(π), (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 , h w (ϕ e (u)) = ν b r b π r π = ν π k s r b +r π = (k s r b + r π )ν(π) = r b ℓ Λ (e) + r π ν(π), 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 ) 19 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 piecewise integral affine, the map 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 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 particular 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)  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.
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 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 M Θ → M Θ ′ there is e ∈ Star(w ) such that α(e) ∩ M [Θ ′ ] = . 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.

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 from Lemma 2.6 and Proposition 2.7 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. The goal of this section is to prove our first Main 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.  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. 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 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.
In order 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 {q i } n−1 i =1 in R 2 . 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, and hence 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: The construction of 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 are going to 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).
The compactification M g ,3d+n−1 admits a finite surjective morphism from a projective scheme 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 by 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).
Lemma 5.5. Let h : Γ Λ → R 2 be the tropicalization of f : X P 2 with respect to X 0 → B 0 . Then there exists a leg l of Λ such that the lengths of all but one edge of G l are constant in the family Γ Λ , and the map h is constant on all vertices of G l , where G l is the underlying graph of the tropical curves parameterized by l .
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 from 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 , maps its generic points to the dense orbit, and the pullback of the boundary divisor is a reduced divisor of degree 3d . Thus, 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.
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 Θ ′ (q 1 , . . . , q n−1 ), 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 ′ , since so is M Θ ′ ∩ev −1 Θ ′ (q 1 , . . . , q n−1 ) 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 ′ . Furthermore, the elevator E adjacent to the top floor of Γ has multiplicity one, cf. Remark 3.14.
Denote the floors from the bottom to the top by F 1 , . . . , F d , and 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 , 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, for any i = k, d , the x-coordinates of all elevators adjacent to F i are fixed in the deformation, 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 . . , 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 .
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.
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.
. 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.
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 2: r = 1 and k > 1. We already checked the case when the multiplicity of E ′ is one in the extended base of induction. 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. 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
The goal of this section is to prove the following theorem describing the local geometry of (2) The branch Br(µ) belongs to V irr g ,d if and only if C 0 \ µ is connected.
Throughout this section, we fix the degree d ≥ 1, the genus 1 − d ≤ g ≤ d−1 2 , and a general union of lines [C 0 ] ∈ V 1−d,d . Recall that for a projective algebraic curve C and a point p ∈ C , the δ-invariant of C at p is defined to be δ(C ; p) := dim (O C ν /O C ) ⊗ O C ,p , where C ν denotes the normalization of C . Plainly, δ(C ; p) = 0 if and only if p is singular. The total δ-invariant of C is then defined to be δ(C ) := dim (O C ν /O C ) = p∈C δ(C ; p). Recall also, that δ(C ) = p a (C ) − p g (C ) is the difference between the arithmetic and the geometric genera; see, e.g., [Ros52,Theorem 8]. We set δ := d−1 2 − g , which is the δ-invariant of an irreducible curve of degree d and geometric . Going down with an elevator E ′ of multiplicity greater than one, and the corresponding wall-crossings.
genus g in P 2 . In order to analyze the local geometry of V g ,d along V 1−d,d , it is convenient to consider the decorated Severi varieties as introduced in [Tyo07].
Definition 6.2. The decorated Severi variety is the incidence locus U d,δ ⊂ |O P 2 (d )| × (P 2 ) δ consisting of the tuples [C ; p 1 , . . . , p δ ], in which C is a reduced curve of degree d , and p 1 , . . . , p δ are distinct nodes of C . The union of the irreducible components U ⊆ U d,δ , for which C is irreducible, is also called a decorated Severi variety, and is denoted by U irr d,δ . The following is a version of [Tyo07, Proposition 2.11]. Proposition 6.3. Let C := [C ; p 1 , . . . , p δ ] ∈ U d,δ be a point, and φ : |O P 2 (d )| × (P 2 ) δ → |O P 2 (d )| the natural projection. Then, (1) The restriction of d φ to the tangent space T C (U d,δ ) is injective; (2) The variety U d,δ is smooth of pure dimension dim(U d,δ ) = d+2 2 − 1 − δ = 3d + g − 1; Proof. The proof is a rather straight-forward computation. After removing a general line from P 2 , we get an open affine plane A 2 = SpecK [x, y] ⊂ P 2 containing all the p l 's. We denote the coordinates x, y on the l -th copy of P 2 in (P 2 ) δ by x l , y l , and identify H 0 (P 2 , O P 2 (d )) with the space of polynomials of degree at most d in the variables x and y. Then the decorated Severi