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Tropicalization and Semistable Reduction of Curves TROPICALIZATION AND STABLE REDUCTION OF CURVES CHRISTOPHER EUR Abstract. The stable reduction theorem implies that Mg, moduli space of curves of genus g, has a compactification to Mg, the Deligne-Mumford stable compactification. The space Mg has a stratification where each stratum corresponds to a combinatorial type of the stable reduction. The combinatorial data of this stratification can be encoded as the moduli of tropical curves. Here we survey a relationship between these tropical curves associated to stable reductions and tropicalizations of curves. Contents 1. The moduli space Mg 1 1.1. What is a moduli space?1 1.2. The spaces Mg; Mg and (semi)stable curves2 1.3. Stratification of Mg and dual graphs3 trop 2. Mg and the moduli space Mg 5 2.1. Moduli of tropical curves5 2.2. Berkovich analyfication and tropicalization6 trop 2.3. Connecting Mg to Mg 7 2.4. Computing the (semi)stable model8 References 10 Notation. Throughout this paper, let k denote an algebraically closed field. For a scheme X, we denote by X(A) := MorSch(Spec A; X) the A-valued points of X.A curve over k is a pure 1-dimensional reduced, separated, and finite type k-scheme. 1. The moduli space Mg 1.1. What is a moduli space? Informally, a moduli space is a geometric space whose points correspond to algebro- geometric objects of specified kind. For example, the space n is a \moduli space of lines PC through the origin in (n + 1)-space over ," since the classical points of n corresponds to C PC lines in Cn+1 through the origin. To consider the notion of moduli space on a more scheme-theoretic and functorial frame- work, we consider the n as an example. The space n is a \moduli space of lines through PZ PZ the origin in (n + 1)-space" in the sense that for any field k, the k-valued points n(k) are PZ n the classical points of Pk. Indeed, this functor of points approach allows one to replace the base scheme Spec k by any scheme B. For example, if B is a k-scheme, then a point of n(B) PZ 1 2 CHRISTOPHER EUR is a map B ! n so that the closed points B(k) parameterize a family of lines through the PZ origin of kn+1: Spec k / B % # n / n Pk PZ Id $ Spec k We thus arrive at the following definition: Definition 1.1.1. A moduli problem consists of two data: (1) What it means to have a family of desired objects over a base scheme B, giving rise to a functor F : Sch ! Sets where F(B) is the set of all families of the desired kind of objects over a base scheme B, and (2) a notion ∼ of when two elements of F(B) are to be considered equivalent from which we obtain a functor M : Sch ! Sets by M(·) := F(·)= ∼. Definition 1.1.2. A scheme M is a fine moduli space for a moduli problem if it represents the functor M(·) by M(·) := MorSch(·; M). As a weaker notion, M is a coarse moduli space if M(·) admits a natural transformation M(·) ! M(·) for which M(Spec k) ! M(k) is a bijection, and is the universal object with this property. Example 1.1.3. The space n is the coarse moduli space of lines through the origin in PZ (n + 1)-space. To be a fine moduli space, one need generalize appropriate the notion of \lines through the origin." Remark 1.1.4. In what follows, we define the moduli problem and the moduli space M over C to make our discussions easier (and due to the author's limited knowledge). However, just as n could be replaced with n to provide full generality, the reader may regard M as a PC PZ coarse moduli space as defined above to a suitable generality. (The author welcomes any comments and/or criticisms regarding this informality). 1.2. The spaces Mg; Mg and (semi)stable curves. Definition/Theorem 1.2.1. The set of smooth, complete, and connected curve of genus g ≥ 2 over C, denoted Mg, is a quasi-projective C-variety of dimension 3g − 3. Notation. We will assume that the genus g is always ≥ 2. That Mg is only quasi-projective calls for a compactification. However, naively taking its closure in the ambient projective space would not be correct, since we would like the compactification Mg itself to be a moduli space. One desire is to have the boundary points Mg n Mg correspond to complete and connected (but necessarily singular) curves of genus g. Deligne and Mumford showed that the suitable singular curves at the boundary are stable curves, defined as follows: Definition 1.2.2. A stable curve C is a complete and connected curve that has only nodes as singularities and has a finite automorphism group. TROPICALIZATION AND STABLE REDUCTION OF CURVES 3 A smooth curve of genus ≥ 2 is indeed stable. A singular stable curve C can be thought of as finitely many curves (irreducible components) with only nodes as singularities con- nected together through normal crossings. With this view, since dim Aut(P1) = 3 whereas dim Aut(C) ≤ 1 for g(C) ≥ 1 (in fact 0 for g(C) ≥ 2), we get the following equivalent definition of stable curves: Proposition 1.2.3. A genus g ≥ 2 curve C with only nodal singularities is stable iff each of its rational components intersects other components at least 3 times. We now come to the main theorems of Deligne and Mumford on the stable compactification of mooduli space of curves: Theorem 1.2.4 (Deligne-Mumford compactification). The space of stable curves over C of genus g ≥ 2, denoted Mg, is a projective variety, containing Mg as an open subset. Theorem 1.2.5 (Stable reduction). Let B be a smooth curve, and 0 a closed point of B, and B∗ := B n f0g. Let X ! B∗ be a flat family of stable curves of genus g ≥ 2. Then there exists a branched cover B0 ! B totally ramified over 0 and a family X0 ! B0 of stable 0 curves extending the fiber product X ×B∗ B . In diagram form: 0 0 X / X ×B∗ B / X 9 % 9 B0 / B Morally, the stable reduction theorem means that given a 1-dimensional parameter of stable curves that limits to a non-stable singular curve, after a small base change the limit curve can be made to a stable curve. The process of finding X0 ! B0 from X ! B is called stable reduction. Remark 1.2.6. In the Theorem 1.2.5, the space X0 may not be smooth. One way to remedy this is to weaken the condition on the allowable curves: a complete and connected curve C of genus ≥ 2 is semistable if each of its rational components meets other components at least 2 times (instead of 3 times). The statement of Theorem 1.2.5 holds verbatim when the word \stable" is replaced with \semistable." The resulting X0 (as a semistable reduction) is now guaranteed smooth. 1.3. Stratification of Mg and dual graphs. S Definition 1.3.1. A space X has a stratification fUigi2I if X = Ui, the subsets Ui's S i are locally closed and disjoint, and I is a poset such that Ui = j≤i Uj. n Example 1.3.2. The projective space Pk, Grassmannians Grk(m; n), and more generally ni flag varieties have an affine stratification (stratification whose Ui's are ' A ). We now describe the stratification of Mg for the case g = 2 in an informal manner by appealing to Riemann surface pictures. A smooth, complete, connected curve over C is a (compact) Riemann surface. Thus, an element of M2 is a Riemann surface of genus 2, whose real picture is: 4 CHRISTOPHER EUR We then consider a 1-dimensional family of smooth genus 2 curves limiting to a singular curve as follows: take a loop around an handle on the Riemann surface, so that the 1- dimensional parameter is the size of the loop, and as the loop shrinks down to size zero, the family limits to a singular Riemann surface. The figure below shows two such families, limiting to two different singular Riemann surfaces when the loops are cinched down to a point: We can continue this process and obtain the following diagram of Riemann surfaces, with a corresponding diagram of complex curves (the first two of the figure below) Figure 1. Illustrations of the stratification of M2 It is a fact that M2 (and Mg in general) has a stratification as drawn in the diagrams above: First, M2 has a (dense) open cell M2 whose points correspond to smooth curves of genus 2 (the top figure in the diagram), and its complement M2 n M2 consists of points corresponding to the singular curves (union of all the figures below). The boundary M2 nM2 itself has two open cells, corresponding to the stable curves that falls into one of two types of singular curves in the second row of the diagram. And the pattern continues. The combinatorial types of stable curves can be encoded as follows: Definition 1.3.3. Let C be a stable curve over k. The dual graph of C is a finite graph with vertex weights ΓC whose vertices are irreducible components of C with weights the TROPICALIZATION AND STABLE REDUCTION OF CURVES 5 genus of the component, and an edge between two vertices for each intersection of the two corresponding components.
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