Tropical Geometry of Curves by Madeline V Brandt A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Bernd Sturmfels, Chair Martin Olsson Alessandro Chiesa Spring 2020 Tropical Geometry of Curves Copyright 2020 by Madeline V Brandt 1 Abstract Tropical Geometry of Curves by Madeline V Brandt Doctor of Philosophy in Mathematics University of California, Berkeley Bernd Sturmfels, Chair Algebraic geometry is a classical subject which studies shapes arising as zero sets of polynomial equations. Such objects, called varieties, may be quite complicated but many aspects of their geometry are governed by discrete data. In turn, combinatorial structures arising from particularly meaningful varieties, such as moduli spaces, are interesting in their own right. In recent years tropical geometry has emerged as a robust tool for studying varieties. As a result, many rich connections between algebraic geometry and combinatorics have developed. Tropical geometry associates polyhedral complexes, like tropical varieties and skeletons, to algebraic varieties. These encode information about the variety or the equations they came from, providing insight in to the underlying combinatorial structure. In this thesis, I develop tropical geometry of curves from the perspectives of divisors, moduli spaces, computation of skeletons, and enumeration. Already, the realm of curves is rich to explore using the tools of tropical geometry. This thesis is divided into five chapters, each focusing on different aspects of the tropical geometry of curves. I begin by introducing algebraic curves, tropical curves, and non-Archimedean curves. The ways in which these objects interact will be a common theme throughout the thesis. The picture on the previous page expresses the idea that tropical curves are projections of non-Archimedean curves. Berkovich analytic spaces are heavenly abstract objects which can be viewed by earthly beings through their tropical shadows. In the second chapter, I develop divisors on tropical curves and tropicalize algebraic divisors. Many constructions for classical curves related to divisors carry over to the tropical world. This will include a tropical Jacobian, a tropical version of the Riemann-Roch theorem, and a tropical Abel-Jacobi map. I first define and compute these objects. Then, I focus on the symmetric power of a curve, because this functions as a moduli space for effective divisors on the curve. I prove that the non-Archimedean skeleton of the symmetric power of a curve is equal to the symmetric power of the non-Archimedean skeleton of the curve. Using this, I prove a realizable version of the tropical Riemann-Roch Theorem. In the third chapter I focus on moduli spaces. A recurring phenomenon in tropical geometry is that the non-Archimedean skeleton of an algebraic moduli space gives a tropical 2 one. I will develop detailed examples of this, following [1, 38, 99, 119]. Then, I define a divisorial motivic zeta function for marked stable curves, and prove that it is rational. In the fourth chapter I compute abstract tropicalizations or non-Archimedean skeletons of a curve. In genus one [46, 77] and two [51, 70, 111] there are known methods for computing these tropicalizations. I develop an algorithm for computing the abstract tropicalizations of hyperelliptic and superelliptic curves. In higher genus, these are the only known results for computing abstract tropicalizations of curves. In the final chapter I study enumerative problems, following [90, 91, 96]. Tropical geom- etry has proven to be a very useful tool in counting curves in the plane. I turn my attention to surfaces in space, and develop tropical counting techniques in this domain. This leads to a preliminary count of binodal cubic surfaces. i Contents List of Figures ii List of Tables v 1 Introduction 1 1.1 Algebraic Curves . 1 1.2 Tropical Curves . 7 1.3 Non-Archimedean Curves . 15 1.4 Contributions in this Dissertation . 27 2 Divisors 30 2.1 Divisors on Tropical Curves . 30 2.2 Tropicalizing Divisors . 42 2.3 Tropicalizing the Symmetric Power . 44 3 Moduli Spaces 70 3.1 Pointed Rational Curves . 72 3.2 Other Tropical Moduli Spaces . 76 3.3 Divisorial Motivic Zeta Functions . 80 4 Skeletons 90 4.1 Genus One . 92 4.2 Genus Two . 93 4.3 Higher Genus . 99 4.4 Hyperelliptic Curves . 102 4.5 Superelliptic Curves . 106 5 Enumeration 127 5.1 Counting Curves in the Plane . 128 5.2 Counting Surfaces in Space . 132 Bibliography 152 ii List of Figures 1.1 Semistable reduction in Example 1.1.13. 5 1.2 Blowing up in Example 1.1.13. 6 1.3 Base changes in Example 1.1.13. 6 1.4 The special fiber of the semistable model obtained in Example 1.1.13 . 7 1.5 Examples and non-examples of polyhedral fans. 8 1.6 An example of a polyhedral complex. 8 1.7 The square lifted by the weight vector w, viewed from below by a tropical geometer. 9 1.8 The square with subdivision induced by w...................... 9 1.9 The tropical line from Example 1.2.9. 10 1.10 The subdivision in Example 1.2.19 . 12 1.11 The tropical curve in Example 1.2.19 . 12 1.12 The tree T with 12 infinite leaves from Example 1.2.21 and the hyperelliptic tropical curve Γ of genus 5 which admissibly covers T by φ. 14 1.13 A special fiber . 15 1.14 A dual graph . 15 1 an 1.15 The space (AK ) when K is algebraically closed and trivially valued. 19 1.16 The Berkovich projective line when K is non-trivially valued. 22 1.17 The Berkovich analytification of an elliptic curve. 25 1.18 The Newton polygon from Example 1.3.30 with its regular subdivision on the left, and the embedded tropicalization on the right. 26 1.19 The strata in iv+(C) in Example 1.4.1. 28 D d 2.1 A graph and the complete linear system of its canonical divisor. 31 2.2 The metric graph and edge orientation used in Example 2.1.10. 34 2.3 The weight function induced by the quadratic form in Example 2.1.12 on the left, and the corresponding Delaunay subdivision on the right. 36 2.4 Delaunay decompositions of R2 (solid lines) and their associated Voronoi decom- positions (dotted lines). 36 2.5 The left figure shows the Delaunay subdivision by tetrahedra and a dual permu- tohedron in grey. The right figure illustrates a tiling of R3 by permutohedra. 37 iii 2.6 Each vertex of the permutohedron corresponds to a divisor supported on the vertices of Γ. The square faces correspond to divisors supported on the interiors of edges of Γ which do not meet in a vertex. Each hexagonal face corresponds to divisors which are supported on edges of Γ which are adjacent to a fixed vertex. Then, the edges correspond to keeping one point of the divisor fixed, and moving the other point along an edge of Γ. The grey curve depicted above represents the embedding of Γ into its Jacobian under the Abel-Jacobi map, which, under the identifications, is again K4. ............................. 38 2.7 The graph Γ and the model G for Example 2.1.18. 39 2.8 The cell decomposition for J(Γ) in Example 2.1.18. 40 2.9 Delaunay subdivisions for g =2............................ 42 2.10 The graph G for Example 2.3.6 . 49 2.11 The space G2 in Example 2.3.6 . 49 2.12 The space G2 in Example 2.3.6 . 49 1 2.13 We see (S )2 is a M¨obiusstrip through cutting and rearranging the pieces. 49 2.14 The poset Ξ(G) for the dumbell graph G and d =2................. 50 2.15 A chain of loops with edge lengths labeled. 65 3.1 The universal family over ............................ 72 M0;3 3.2 The universal family over 0;4............................ 72 3.3 Stable pointed curves. .M . 72 3.4 Points in the boundary of ............................ 73 M0;4 3.5 Combinatorial types of curves in the boundary of 0;5. 73 3.6 A nodal labelled curve and its dual graph. .M . 73 3.7 The tropicalization of 0;5.............................. 76 tropM 3.8 The moduli space M2 of genus 2 tropical curves. 77 3.9 Let (X; p) be a smooth curve with genus g 1 and one marked point p. In this + ≥ case, Div2 (X; p) has four strata, corresponding to the pictured combinatorial types of marked stable curves and divisors. 84 3.10 Let X be a curve with two smooth components each having genus larger than + one meeting in a node. In this case, Div2 (X) has seven strata, corresponding to the pictured combinatorial types of stable curves and divisors. 84 3.11 Stable pair in Proposition 3.3.9 . 86 3.12 Stable pair in Proposition 3.3.10 . 86 4.1 The Newton polygon from Example 4.0.1 together with its unimodular triangu- lation on the upper left, the embedded tropicalization on the upper right, and the metric graph at the bottom. 91 4.2 Tropicalizations of elliptic curves in normal form. 92 4.3 Tropicalization of an elliptic curve in honeycomb form. 93 4.4 The seven types of genus two tropical curves. 94 4.5 The seven types of trees with six leaves . 95 iv 4.6 The tropicalization of the curve in Example 4.2.12 . 99 4.7 The weighted graph in Example 4.3.2. 101 4.8 Arrangement of curves in P2.