Tropical geometry of logarithmic schemes

Transcription

1 Tropical geometry of logarithmic schemes By Martin Ulirsch M.S., Brown University Dipl.-Math., University of Regensburg A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Mathematics Department at Brown University PROVIDENCE, RHODE ISLAND May 2015

2 c Copyright 2015 by Martin Ulirsch

3 This dissertation by Martin Ulirsch is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Dan Abramovich, Advisor Recommended to the Graduate Council Date Sam Payne, Reader Date Joseph H. Silverman, Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii

4 Vita Martin Ulirsch was born on 11. August 1986 in Vilshofen an der Donau as the only child of Katharina and Josef Ulirsch. He received his elementary school education at Grundschule Aicha vorm Wald and his secondary school education at Gymnasium Vilshofen. From 2005 until 2010 he was a student at the Department of Mathematics at the University of Regensburg earning his undergraduate degree in July During this time he spent the academic year as a visiting graduate student at Brandeis University and wrote a thesis under the guidance of Prof. Klaus Künnemann with the title Semistabile O K -Moduln und die (Ko-)Homologie von GL n (O K ). From 2010 until now he was a graduate student at the Mathematics Department at Brown University, where he has been working under the guidance of Prof. Dan Abramovich. In May 2012 he received a Master of Science in Mathematics from Brown University. iv

5 Acknowledgements I would like to thank my advisor Dan Abramovich, whose role in my development as a mathematician cannot be overstated. Not only did he constantly provide support, practical advice, and encouragement, he also steered me towards a very rich area of mathematical investigation, sharpened my mathematical senses with his constructive criticism, and taught me much of what I know about mathematics in, as it seems, uncountably many meetings we had over the last few years. Thanks are also due to Sam Payne; in addition to serving on my thesis committee and providing the main source of inspiration for this thesis through his work, he also always had an open ear for my questions and gave well thought-out advice on issues of mathematical and practical importance. Moreover, I would like to thank Joseph Silverman for serving on my thesis committee and teaching me a lot about mathematics and, in particular, arithmetic geometry during the last few years. Mathematics would only be half as fulfilling, if it was not for the abundance of intellectually stimulating conversations that arise from it. I would therefore like to thank the following people for many mathematical discussions we had and hopefully will continue to have: Kenny Ascher, Matt Baker, Dori Bejleri, Aaron Bertram, Renzo Cavalieri, Melody Chan, Qile Chen, Mandy Cheung, Angelica Cueto, Lorenzo Fantini, Jeffrey Giansiracusa, Noah Giansiracusa, Angela Gibney, Walter Gubler, Simon Hampe, Klaus Künnemann, Diane MacLagan, Steffen Marcus, Yoav Len, Samouil Molcho, Johannes Nicaise, Jennifer Park, Joseph Rabinoff, Dhruv Ranganathan, Farbod Shokrieh, Michael Temkin, Amaury Thuillier, Amos Turchet, Jonathan Wise, and Tony Yue Yu. I also owe particular thanks to Brian Conrad for finding an inaccuracy in an earlier version of Proposition IV.3.2 and for subsequently suggesting the statement of Lemma IV.3.3, to Lorenzo Fantini for catching many typos in an earlier version of Chapter I, to Walter Gubler for suggesting the technique used in the proof of Lemma II.3.10, to Martin Olsson v

6 vi ACKNOWLEDGEMENTS for sharing a draft of his upcoming book [Ols14], to Michael Temkin for his advice concerning the descent theory of non-archimedean analytic spaces, to Jenia Tevelev for sharing his unpublished notes [Tev09], as well as to Amaury Thuillier, who originally suggested the use of Kato fans to define tropicalization maps, and to Jonathan Wise, whose ideas related to Artin fans heavily influenced this work. Parts of the research culminating in this thesis have been carried out while enjoying the hospitality of Hebrew University, Jerusalem, and the University of Regensburg (twice). For the latter I would in particular like to thank my generous hosts Walter Gubler and Klaus Künnemann. Finally, I would to thank my extended family, in particular my parents Josef and Katharina Ulirsch, for their support as well as my friends in Germany and the United States for helping me stay sane throughout the years. This thesis is dedicated to my beloved Rama Srinivasan, who makes me a complete human being.

7 Contents Vita Acknowledgements iv v Introduction 1 1. Non-Archimedean geometry à la Berkovich 1 2. Tropicalization 3 3. Toric varieties 5 4. Tropical compactification 8 5. Moduli spaces 11 Overview 15 Chapter I. Functorial tropicalization of logarithmic schemes: The case of constant coefficients Introduction Monoids, cones, and monoidal spaces Kato fans and extended cone complexes Logarithmic geometry Non-Archimedean analytification Tropicalization Examples 52 Chapter II. Tropical compactification in log-regular varieties Introduction Log-regular varieties Tropicalization Tropical compactification Dual complexes and the cohomology of links 75 vii

8 viii CONTENTS Chapter III. Tropical geometry of moduli spaces of weighted stable curves Introduction Weighted stable tropical curves and their moduli Dual graphs and the boundary of M g,a Deformation retraction onto the non-archimedean skeleton Tropical tautological maps Variations of weight data Losev-Manin spaces 104 Chapter IV. A geometric theory of non-archimedean analytic stacks Introduction Étale analytic spaces and analytic stacks Presentations and Morita equivalence Analytification Topology of analytic stacks Tropicalization as a stack quotient 133 Chapter V. Tropicalization is the analytification of Artin fans Introduction Preliminaries Stacky generic fibers over trivially valued fields Artin fans Analytification of Artin fans 149 Bibliography 154

9 Introduction Tropical geometry is a theoretical framework that aims to unite many of the combinatorial methods of algebraic geometry in one consistent language. Its basic idea is to treat the combinatorial data associated to degenerations or compactifications of a variety X as if it was an algebraic geometric object itself, a so called tropical variety Trop(X). The approach of tropical geometry is to describe the combinatorial geometry of Trop(X) and, subsequently, to extract information about the algebraic geometry of the original algebraic variety X itself. As recently observed in [Gub07], [EKL06], [Gub13], [BPR11], and [GRW14], there is an intricate relationship between non-archimedean analytic geometry in the sense of Berkovich [Ber90] and tropical geometry. In particular, in many interesting situations, the tropicalization of an algebraic variety X can be regarded as a natural deformation retract of the Berkovich analytic space X an associated to X, a so called skeleton of X an. The purpose of this expository chapter is to introduce the reader to some aspects and features of tropical and non-archimedean geometry that motivate the developments in the following chapters. Parts of Sections 1 and 3 are going to be part of the upcoming survey [ACM + 15]. 1. Non-Archimedean geometry à la Berkovich Ever since [Tat71] it has been known that affinoid algebras are the correct coordinate rings for defining non-archimedean analogues of complex analytic spaces. Working with affinoid algebras as coordinate rings alone is enough information to build an intricate theory with many applications. The work of V. Berkovich [Ber90] and [Ber93] beautifully enriches this theory by providing us with an alternative definition of non-archimedean analytic spaces, which naturally come with underlying topological spaces that have many of the favorable properties of complex analytic spaces, such as being locally path-connected and locally compact. 1

10 2 INTRODUCTION Let k be a non-archimedean field, i.e. suppose that k is complete with respect to a non- Archimedean absolute value.. We explicitly allow k to carry the trivial absolute value. If U = Spec A is an affine scheme of finite type over k, as a set the analytic space U an associated to U is equal to the set of multiplicative seminorms on A that restrict to the given absolute value on k. We usually write x for a point in U an and. x, if we want to emphasize that x is thought of as a multiplicative seminorm on A. The topology on U an is the coarsest that makes the maps U an R x f x continuous for all f A. There is a natural continuous structure morphism ρ : U an U given by sending x U an to the kernel of. x : A R. A morphism f : U V between affine schemes U = Spec A and V = Spec B of finite type over k, given by a k-algebra homomorphism f # : B A, induces a natural continuous map f an : U an V an. It is defined by associating to x U an the point f(x) V an given as the multiplicative seminorm. f(x) =. x f # on B. The association f f an is functorial in f. Let X be a scheme that is locally of finite type over k. Choose a covering U i = Spec A i of X by open affines. Then the analytic space X an associated to X is defined by glueing the U an i over the open subsets ρ 1 (U i U j ). It is easy to see that this construction does not depend on the choice of a covering and that it is functorial with respect to morphisms of schemes that are locally of finite type over k. We refrain from describing the structure sheaf on X an, since, for now, we are only going to be interested in the topological properties of X an, and refer the reader to [Ber90], [Ber93], and [Tem10] for further details. Example Suppose that k is algebraically closed and endowed with the trivial absolute value. Let t be a coordinate on the affine line A 1 = Spec k[t]. One can classify the points in (A 1 ) an as follows: For every a k and r [0, 1) we have the seminorm. a,r on k[t] that is uniquely determined by and t a a,r = r,

11 2. TROPICALIZATION 3 for r [1, ) we have the seminorm.,r uniquely determined by t a,r = r Noting that for all a k. lim r 1. a,r =.,1 for all a k we can visualize (A 1 ) an as follows: η (A 1 ) an 0 A 1 (k) In particular, we can embed the closed points A 1 (k) into (A 1 ) an by the association a. a,0 for a k, and by associating to the generic point η of A 1 to the Gauss point.,1. One can give an alternative description of X an as the set of equivalence classes of non- Archimedean points of X. A non-archimedean point of X consists of a pair (K, φ) where K is a non-archimedean extension of k and φ : Spec K X is a k-morphism. non-archimedean points (K, φ) and (L, ψ) of X are equivalent, if there is a common non- Archimedean extension Ω of both K and L that makes the diagram commute. Spec Ω Spec L ψ Spec K φ X Two 2. Tropicalization Let k be a non-archimedean field, let T be a split algebraic torus with character lattice M and cocharacter lattice N, and set N R = N R as well as M R = M R = Hom(N, R).

12 4 INTRODUCTION Denote the duality pairing between both N and M as well as N R and M R by.,.. Following [EKL06], [Gub07], and [Gub13] there is a natural continuous tropicalization map trop : T an N R, whose image is determined by trop(x), m = log χ m x for all m M, where χ m denotes the character of m in k[m]. Definition Let Y be a closed subset of T. The continuous projection Trop(Y ) = trop(y an ) of the analytic space Y an onto N R is said to be the tropical variety associated to Y. The first result describing the structure of Trop(Y ) is due to Bieri and Groves and, somewhat surprisingly, predates the advent of tropical geometry as well as the work of Berkovich (see [BG84, Theorem A] and [EKL06, Theorem 2.2.3]). Theorem ([BG84] Theorem A). For a closed subset Y of T the tropical variety Trop(Y ) has the structure of a rational polyhedral complex in N R. If Y can be chosen to be equidimensional of dimension d, then Trop(Y ) is of dimension d as well. Moreover, if k is endowed with the trivial absolute value, then Trop(Y ) even has the structure of a rational polyhedral fan. The topological properties of the tropicalization map, starting with its continuity, allow us to exhibit close ties between the Euclidean topology of tropical varieties and the topology on Berkovich analytic spaces. A first application of this relationship can be found in the following Theorem. Theorem ([EKL06] Theorem 2.2.7). Suppose that the closed subset Y of T is connected. Then the tropical variety Trop(Y ) is connected as well. Proof. Since Y is connected, the analytic space Y an is connected by [Ber90, Theorem (iii) and Theorem (iii)]. The continuity of the tropicalization map therefore implies that Trop(Y ) = trop(y an ) is connected, too.

13 3. TORIC VARIETIES 5 3. Toric varieties In this section we give a detailed explanation of the relationship between skeletons of non-archimedean analytic spaces and tropicalization in the simplest possible case, that of toric varieties. Again, let k be a non-archimedean field and let T be a split algebraic torus with character lattice M and cocharacter lattice N. Consider a T -toric variety X = X( ) that is defined by a rational polyhedral fan in N R. We refer the reader to [Ful93] and [CLS11] for the standard notation and further details of this beautiful theory. Kajiwara [Kaj08] and, independently, Payne [Pay09] construct a tropicalization map trop : X an N R ( ) associated to X, whose codomain is a partial compactification of N R, uniquely determined by (also see [PPS13, Section 4] and [Rab12, Section 3]). For a cone σ in set N R (σ) = Hom(S σ, R), where S σ denotes the toric monoid σ M and write R for the additive monoid (R { }, +). pointwise convergence. Endow N R (σ) with the topology of Lemma ([Rab12] Proposition 3.4). (i) The space N R (σ) has a stratification by locally closed subsets isomorphic to the vector spaces N R / Span(τ) for all faces τ of σ. (ii) For a face τ of σ the natural map S σ S τ induces the open embedding N R (τ) N R (σ) that identifies N R (τ) with the union of strata in N R (σ) corresponding to faces of τ in N R (σ). So one can think of N R (σ) as a partial compactification of N R given by adding a vector space N R / Span(τ) at infinity for every face τ 0 of σ. The partial compactification N R ( ) of N R is defined to be the colimit of the N R (σ) for all cones σ in. Since the stratifications on the N R (σ) are compatible, the space N R ( ) is a partial compactification of N R that admits a stratification by locally closed subsets isomorphic to N R / Span(σ) for every cone σ in. On the T -invariant open affine subset U σ = Spec k[s σ ] the tropicalization map trop σ : U an σ N R (σ)

14 6 INTRODUCTION is defined by associating to an element x X an the homomorphism s log χ s x in N R (σ) = Hom(S σ, R). Lemma The tropicalization map trop σ is continuous and, for a face τ of σ, the natural diagram commutes. Uτ an U an σ trop τ NR (τ) trop σ NR (σ) Therefore we can glue the trop σ on local T -invariant patches U σ and obtain a global continuous extended tropicalization map trop : X an N R ( ). that restricts to the trop σ on T -invariant open affine subsets U σ. Its restriction to a T -orbits is the usual tropicalization map in the sense of Section 2. The following Proposition is well-known among experts and is a special case of the much more general results in [Ber99]. It can be found in [Thu07, Section 2] in the constant coefficient case, i.e. the case that k is trivially valued. Proposition The tropicalization map trop : X an N R ( ) has a continuous section J and the composition p = J trop : X an X an defines a strong deformation retraction. The deformation retract S(X) = J ( NR ( ) ) = p (X an ) is said to be the non-archimedean skeleton of X an. Proof Sketch of Proposition Consider a T -invariant open affine subset U σ = Spec k[s σ ]. We may construct the section J σ : N R (σ) U an σ by associating to u N R (σ) = Hom(S σ, R) the seminorm J σ (u) defined by J σ (u)(f) = max s S σ a s exp ( u(s) )

15 3. TORIC VARIETIES 7 for f = s a sχ s k[s σ ]. A direct verification shows that J σ is continuous and fulfills trop σ J σ = id NR (σ). The construction of J σ is compatible with restrictions to T -invariant affine open subsets and we obtain a global continuous section J : N R ( ) X an of the tropicalization map trop : X an N R ( ). Since J is a section of trop, the continuous map p = J trop : X an X an is a retraction map. On U σ the image p σ (x) of x U an σ is the seminorm given by p σ (x)(f) = max s S σ a s χ s x for f = s a sχ s k[s σ ]. The arguments in [Thu07, Section 2.2] generalize to this situation and show that p is, in fact, a strong deformation retraction (see in particular [Thu07, Lemme 2.8 (1)]). Example The skeleton of A 1 is given by half open line connecting 0 to, i.e. for trivially valued k we have S(X) = { 0,r r [0, 1) } {,r r [1, ) } in the notation of Example Definition Given a closed subset Y of X = X( ) we define its associated extended tropical variety by Trop(Y, ) = trop (Y an ). Suppose that Y T is non-empty and dense in Y. Since the tropicalization map is proper, the extended tropical variety Trop(Y, ) is nothing but the closure of Trop(Y T ) in N R ( ). For a variety Y as well as a closed toric embedding i : Y X, we write Trop(Y, i) for the associated extended tropical variety. This procedure is functorial with respect to equivariant morphisms: Let f : X X be a torus-equivariant morphism between the toric varieties X = X( ) and X = X ( ) such that i = f i is a closed embedding. Then there is an induced continuous map Trop(f) : N R ( ) N R ( )

16 8 INTRODUCTION that respects the stratifications and in this case we have Trop(Y, i ) = Trop(f) ( Trop(Y, i) ). Using this terminology, we can state the following Theorem that illustrates how the topology of a non-archimedean analytic space is determined by the topology of all its tropicalizations. Theorem ([Pay09] Theorem 4.2). Let X be a quasiprojective variety. extended tropicalization maps induce a homeomorphism Then the Y an lim Trop(Y, i), where the projective limit on the right is taken over all closed toric embeddings i : Y X( ). In Theorem the condition that Y is quasiprojective can be weakened to Y admitting a toric embeddings (see [FGP14]). 4. Tropical compactification Let Y be a closed subset of a split algebraic torus T over k of dimension n. In the above Section 3 we have seen that the tropicalization of Y is nothing but the continuous projection of the analytic space Y an onto the skeleton S(T ) of T. The skeleton S(T ) itself is a hybrid that, in some sense, naturally lives over both the generic point of T and all toric compactifications of T. Therefore one can hope to retrieve information about the (partial) compactifications of Y obtained by taking closures Y in T -toric varieties X = X( ). This approach is known as the study of tropical compactifications, which has originally been introduced by Tevelev in [Tev07] in the case of constant coefficients. Extensions of this theory to tropical compactifications in toric schemes over discrete valuation rings, or more general valuations rings of rank one, can be found in [LQ11], [HK12], and [Gub13]. From now on, however, assume that k is endowed the trivial absolute value. Theorem ([Tev07] ). Let Y T be a closed subset and X = X( ) a T -toric variety defined by a rational polyhedral fan in N R. (i) The closure Y of Y in X is complete if and only if Trop(Y ).

17 4. TROPICAL COMPACTIFICATION 9 (ii) The intersection of Y with every torus orbit O of X is non-empty and has the expected dimension if and only if Trop(Y ). The main ingredient for the proof of Theorem (i) is the following Lemma 0.4.2, which is also known as Tevelev s Lemma. Lemma (Tevelev s Lemma [Tev07]). The closure Y of Y in X intersects a T -orbit O σ corresponding to a cone σ in if and only if Trop(Y ) intersects the relative interior of σ. For a proof of Lemma we refer the reader to Lemma II.4.1 below, which is a generalization of this result. Proof of Theorem Part (i): Let X = X ( ) be a T -equivariant compactification of X, i.e. a compactification that is given by choosing a complete fan in N R containing as a subfan. Such a compactification exists by Sumihiro s equivariant version [Sum74] of Nagata s embedding theorem [Nag62]. Suppose that the closure Y of Y in X is not complete. Then the closure Y of Y in X is complete and will intersect a torus orbit O σ that is not contained in X, or equivalenty with σ /. By Tevelev s Lemma the intersection of Trop(Y ) with the relative interior of σ is non-empty and therefore Trop(Y ). Conversely suppose that Y is already complete. Then Y = Y and therefore Y only intersects torus orbits contained in X. Thus Tevelev s Lemma implies Trop(Y ). Definition A pair (Y, X) consisting of a closed subset Y of a split algebraic torus T and a T -toric variety X is said to be a tropical pair, if (i) the closure Y of Y in X is complete, and (ii) the T -operation µ : T Y X is faithfully flat. Suppose that (Y, X) is a tropical pair. In this case we refer to Y as a tropical compactification of Y. Since µ is faithfully flat, the intersection of Y with a torus orbit O σ is always non-empty and of the expected dimension. Corollary For a tropical pair (Y, X) we have Trop(Y ) =.

18 10 INTRODUCTION Proof. Apply Theorem (i) to find that Trop(Y ). Suppose now that this inclusion is strict. By subdividing we can assume that there is a cone σ in whose relative interior is not contained in Trop(Y ). Note that this makes use of Trop(Y ) being a rational polyhedral fan (see Theorem above). Tevelev s Lemma implies that in this case Y does not intersect the T -orbit O σ. The faithful flatness of m, however, is preserved under toric blow-ups and we have arrived at a contradiction. A crucial ingredient for the proof of Theorem (ii) is the following version of the Raynaud-Gruson flattening theorem [RG71], due to Tevelev [Tev09]. For a T -toric variety X we write µ : T X X for the T -operation on X and π : T X X for the projection to the second coordinate. Theorem ([Tev09] Section 2.4). Let F be a coherent sheaf on a toric variety X. Then there is a projective T -equivariant birational morphism X X such that the strict transform of π F is flat over X via the multiplication map µ : T X X. We refrain from proving Theorem and refer the reader to [Tev09]. Corollary Let Y T be a closed subset and X a T -toric variety. Then there is a projective T -equivariant birational morphism X X such that the T -operation µ : T Y X on the closure Y of Y in X is flat. Proof. Apply Theorem to the ideal sheaf I Y O X defining Y. Corollary Let Y be a closed subset of T. Then there is a T -toric variety X = X( ) such that the closure Y of Y in X is a tropical compactification. Proof. Choose a complete T -toric variety X = X ( ) defined by a complete rational polyhedral fan in N R. By Corollary we can find a projective T -equivariant birational morphism X X such that µ : T Y X is flat. Since X is complete, the closure Y of Y in X is complete as well. Now in order to obtain the T -toric variety X remove those T -orbits O σ in X whose intersection with Y is empty. The closure Y of Y in X is equal to Y. Therefore Y is complete and the

19 5. MODULI SPACES 11 T -operation µ : T Y X on Y into X is faithfully flat. Proof of Theorem Part (ii): Let Y be a closed subset of T and X be a T - toric variety. Suppose that Trop(Y ). Take a T -equivariant compactification X of X and apply the construction from Corollary to find a toric variety X such that that (Y, X ) is a tropical pair. Since the intersection of Y with all T -orbits of X is non-empty, the intersections of Y with all T -orbits of X are non-empty, too. Let O σ be a T -orbit in X of codimension e. Since X is Cohen-Macaulay, every T - orbit O σ of X is set-theoretically cut out by a regular sequence of length e and therefore codim(y O σ ) e. Since the closure of Y O σ in X is complete, the tropical variety Trop(Y O σ ) is contained in the normal fan of in N R (σ) = N R / Span(σ). Therefore Trop(Y O σ ) has dimension n e and, since tropicalization preserves dimension by Theorem 0.2.2, this implies codim(y O σ ) e. The converse implication involves more convex geometry than we dare to include in this introductory chapter. We refer the reader to [Gub13, Proposition 14.8] for the essential part of this proof. M trop g,n 5. Moduli spaces Much like the moduli space M g,n of n-marked stable algebraic curves the moduli space of stable tropical curves is an object of fundamental importance in tropical geometry. It has originally been introduced by Mikhalkin (see [Mik06] and [Mik07]) and further developed in [GKM09], [BMV11], [CMV13], and [Cap13]. In this section we shed light on this construction and explain how the recent results of [ACP12] fit into the picture. Let us first begin by introducing tropical curves and their moduli spaces. weighted graph G is a tuple ( V (G), F (G), r, i, m, h ) that consists of the following data: a finite set of vertices V (G), A finite a finite set of flags F (G) as well as root map r : F (G) V (G) associating to a flag of G the vertex it emanates from,

20 12 INTRODUCTION an involution i : F (G) F (G) that gives rise to a decomposition of F (G) into set of fixed points L(G) under i, sometimes referred to as the legs of G, and a finite union of pairs of points, the edges of G, a marking of L(G), i.e. a bijection {1,..., n} L(G) given by L(G) = {l 1,..., l n }, and a weight function h : V (G) N that associates to every vertex its genus h(v). We hereby let E(G) denote the set of edges of G. The genus g(g) of G is given by g(g) = b 1 (G) + h(v), v V where b 1 (G) = dim Q H 1 (G, Q) = #E(G) #V (G)+1 is the first Betti number of G. A finite weighted graph G of genus g with n legs is said to be stable, if for every vertex v V (G) we have 2h(v) 2 + v > 0, where we write v for the number of flags emanating from v. Definition A tropical curve Γ is a tuple (G, l) consisting of a finite weighted graph G and a length function l : E(G) R 0. If the length function is allowed to take values in R 0 = R 0 { }, we say that Γ = (G, l) is an extended tropical curve. A tropical curve Γ = (G, l) is said to be stable, if the underlying finite weighted graph G is stable. Fix a finite weighted graph G. The rational polyhedral cone σ G = R E(G) 0 is a set-theoretic parameter space for tropical curves, whose underlying graph is G. canonical compactification σ G = R E(G) 0 Its parametrizes extended tropical curves with underlying graph G. There is an operation of the automorphism group Aut(G) of G on σ G and σ G by permuting the entries. A weighted graph contraction π : G G is a composition of edge contractions that preserves the weight function in the sense that h (v ) = g ( π 1 (v ) )

21 5. MODULI SPACES 13 for all vertices v V = V (G ). If G is a stable graph, the contracted graph G is stable as well. A weighted edge contraction G G induces the embedding of a face σ G σ G and this embedding is equivariant with respect to automorphisms of graphs. Definition The moduli space M trop g,n of stable tropical curves of genus g with n legs is given as the colimit M trop g,n = lim σ G taken over all stable finite weighted graphs of type (g, n) with face maps induced by weighted edge contractions and automorphisms of graphs. Similarly, the moduli space M trop g,n of extended stable tropical curves of genus g with n legs is given as the colimit M trop g,n = lim σ G taken over all stable finite weighted graphs of type (g, n) with face maps induced by weighted edge contractions and automorphisms of graphs. Fix an algebraically closed ground field k that is endowed with the trivial absolute value. Following [BPR11], [Viv12, Section 2.2.3], and [ACP12, Section 1.1] there is a naive set-theoretic tropicalization map trop g,n : M an g,n M trop g,n from the non-archimedean analytic stack M an g,n onto the extended tropical moduli space M trop g,n defined as follows. A point x M an g,n can be represented by a morphism x : Spec K M g,n for a non- Archimedean extension K of k. Since M g,n is proper, there is a finite extension K of K such that x extends to a morphism Spec R M g,n for the valuation ring R of K. This datum in turn corresponds to a family C of stable curves over Spec R. The image trop g,n (x) = [Γ x ] M trop g,n is the point parametrizing the tropical curve Γ x, whose underlying finite weighted graph is the weighted dual graph G x of the special fiber C s of C and the edge lengths are given by l(e) = val(t e ), where xy = t e, with t e R, describes the node p e in C corresponding to e in formal coordinates around p e. Let X 0 X be a toroidal embedding, i.e. an open immersion that is étale locally isomorphic to the embedding of a big torus into a toric variety. Suppose that X is proper.

22 14 INTRODUCTION In [Thu07, Section 3] Thuillier lifts the skeleton construction for toric varieties to all proper toroidal embeddings and obtains a deformation retraction p X : X an X an onto a non- Archimedean skeleton S(X) of X an that is canonically associated to the toroidal structure of X 0 X. This construction can be extended to proper toroidal Deligne-Mumford stacks (see [ACP12, Section 6] and Chapter IV Section 1.3). Finally, the moduli stack M g,n of smooth n-marked algebraic curves admits a natural compactification by the moduli stack M g,n of stable n-marked curves. The boundary M g,n M g,n has (stack-theoretically) normal crossing and therefore the open embedding M g,n M g,n is toroidal. So it makes sense to consider the skeleton S(M g,n ) of M an g,n. Theorem ([ACP12] Theorem 1.2.1). There is a natural isomorphism J g,n : M trop g,n S(M g,n ) that makes the diagram p Mg,n M an g,n trop g,n g,n S(M g,n ) M trop J g,n commutative. Theorem implies that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the skeleton of M an g,n. Therefore, in particular, the set-theoretic tropicalization map trop g,n is well-defined and continuous. The main role of Theorem is that it relates the dual graph of a stable degeneration of an algebraic curve to the combinatorial description of a compactification the moduli space of curves. Its proof depends on the modularity of the moduli stack M g,n and the description of the boundary of M g,n in terms of dual graphs of the parametrized curves. Analogous results have been found for the moduli space of admissible covers (see [CMR14]) and Hassett s moduli spaces of weighted stable curves (see [CHMR14] in the genus 0 case and Chapter III below for the general case).

23 Overview Chapter I: Functorial tropicalization of logarithmic scheme: The case of constant coefficients. The purpose of this chapter is to develop foundational techniques from logarithmic geometry needed in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach uses the theory of fans in the sense of K. Kato and generalizes Thuillier s retraction map onto the non-archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as introductory treatments of logarithmic geometry and the theory of Kato fans are included. This chapter has appeared as [Uli13a]. Chapter II: Tropical compactification in log-regular varieties. In this chapter we follow Chapter I and define a natural tropicalization procedure for closed subsets of log-regular varieties in the case of constant coefficients and study its basic properties. This framework allows us to generalize some of Tevelev s results on tropical compactification as well as Hacking s result on the cohomology of the link of a tropical variety to log-regular varieties. This chapter has appeared as [Uli13b]. Chapter III: Tropical geometry of moduli spaces of weighted stable curves. Hassett s moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this chapter we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces. This chapter has appeared as [Uli14b]. 15

24 16 OVERVIEW Chapter IV: A geometric theory of non-archimedean analytic stacks. The purpose of this chapter is to develop a basic theory of geometric stacks over the category of Berkovich analytic spaces. Among the foundational topics discussed here are analytic groupoids and their quotients, Morita equivalence of analytic groupoids, the process of analytification, and the topology of analytic stacks. As an application of this theory we give a reinterpretation of the well-known Kajiwara-Payne tropicalization map of a toric variety as a stack quotient. This chapter has appeared as [Uli14a]. Chapter V: Tropicalization is the analytification of Artin fans. The Artin fan of a logarithmic scheme X is a logarithmically étale algebraic stack A X that encodes the combinatorics of the characteristic monoid of X. In this chapter we show that the analytification of a natural strict morphism X A X is equal to the tropicalization map of X, as constructed in Chapter I.

25 CHAPTER I Functorial tropicalization of logarithmic schemes: The case of constant coefficients 1. Introduction Let k be a field that is endowed with a (possibly trivial) non-archimedean absolute value. k, let N be a finitely generated free abelian group of dimension n, and write M for its dual Hom(N, Z) as well as.,. for the duality pairing between N and M Tropicalization is a process that associates to a closed subset Y of the split algebraic torus T = Spec k[m] a subset Trop(Y ) in N R = N R that can be endowed with the structure of a rational polyhedral complex. It is called the tropical variety associated to Y in N R. Following [EKL06] as well as [Gub07] and [Gub13], one of the many ways of defining Trop(Y ) is by taking it to be the projection of the non-archimedean analytic space 1 Y an associated to Y into N R via the natural continuous tropicalization map trop : T an N R. The image trop(x) of a point x T an, given by a seminorm. x : k[m] R extending. k, is hereby uniquely determined by the condition trop(x), m = log χ m x for all m M, where χ m denotes the basis element in k[m] corresponding to m M Kajiwara [Kaj08, Section 1] and, independently, Payne [Pay09, Section 3] define a natural continuous extension of the above tropicalization map to a T -toric variety X = X( ) defined by a rational polyhedral fan in N R. For standard notation and general background on toric varieties we refer the reader to [Ful93]. text. 1 See Section 4 in this chapter for details on the two analytification functors (.) an and (.) ℶ used in this 17

26 18 I. FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES The codomain of their tropicalization map is a partial compactification N R ( ) of N R determined by, a detailed construction of which can be found in [Rab12, Section 3]. For a torus-invariant open affine subset U σ = Spec k[s σ ] of X, defined by the semigroup S σ = σ M for a cone σ, the compactification N R (σ) is given by Hom(S σ, R), where R = R { } with the natural additive monoid structure, and the tropicalization map trop : U an σ N R (σ) sends x U an σ to the element trop (x) Hom(S σ, R) that is determined by trop (x)(s) = log χ s x for all s S σ. In the literature this tropicalization map has also appeared under the name non-archimedean moment map. See for example [BGPS14, Section 4.1] Suppose from now on that. k is the trivial absolute value, i.e. a = 1 for all a k. In this case Thuillier [Thu07, Section 2] constructs a closely related strong deformation retraction p : X ℶ X ℶ, from X ℶ onto the non-archimedean skeleton S(X) of X using the natural action of the analytic group T ℶ on X ℶ. By [Thu07, Proposition 2.9] there is a natural embedding i : S(X) N R ( ) such that the diagram X ℶ p S(X) i X an trop N R ( ) is commutative and the image of S(X) in N R ( ) is the closure of in N R ( ). Note that on a torus-invariant open affine subset U σ = Spec k[s σ ] of X, for a cone σ, the image of i is given by σ = Hom(S σ, R 0 ) Hom(S σ, R), which is known as the canonical compactifcation of the cone σ = Hom(S, R 0 ).

27 1. INTRODUCTION Suppose now that X 0 X is a toroidal embedding, i.e. an open and dense embedding that is étale locally isomorphic to the open embedding of a big algebraic torus into a toric variety. Using formal torus actions Thuillier [Thu07, Section 3] is able to lift his construction and obtains a strong deformation retraction p : X ℶ X ℶ onto the non-archimedean skeleton S(X) of X. We refer to [KKMSD73, Chapter 2] and the beginning of [Thu07, Section 3] for the basic theory of toroidal embeddings. In [KKMSD73] the authors work with formal instead of étale neighborhoods, but by [Den13, Section 2] both approaches are equivalent. In [ACP12] Abramovich, Caporaso, and Payne explain how S(X) can be endowed with the structure of a generalized extended cone complex Σ X. By [Thu07, Proposition 3.15], if X 0 X has no self-intersection in the terminology of [KKMSD73], then Σ X is the canonical compactification of the rational polyhedral cone complex Σ X associated to the toroidal embedding as constructed in the end of [KKMSD73, Section 2.1]. These rational polyhedral cone complexes are also known as abstract rational fans in the convex geometry literature. Note hereby that other authors also use the adjectives simple or strict to denote toroidal embeddings without self-intersection Let now X be a fine and saturated logarithmic scheme that is of finite type over a trivially valued field k. The goal of this chapter is to construct a continuous tropicalization map trop X : X ℶ Σ X into a generalized extended cone complex Σ X that is naturally associated to the logarithmic structure (M X, ρ X ) on X. For our definition to be reasonable we require it to fulfill the following two properties: (i) The tropicalization map trop X is functorial. (ii) In the logarithmically smooth case trop X recovers Thuillier s retraction map. To be precise, the following two theorems have to hold: Theorem I.1.1. A morphism f : X X of fine and saturated logarithmic schemes of finite type over k induces a morphism Σ(f) : Σ X Σ X that makes the natural diagram X ℶ f ℶ trop X ΣX Σ(f) (X ) ℶ trop X Σ X

28 20 I. FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES commutative. The association f Σ(f) is functorial in f. Suppose now that X is logarithmically smooth over k. Then X has the structure of a toroidal embedding (see Sections 4.2 and 4.4). Theorem I.1.2. In this case the tropicalization map trop X has a section, i.e. there is a homeomorphism J X : Σ X S(X) such that the diagram p S(X) X ℶ J X trop X Σ X is commutative Consider now a closed subset Y of X. It is a natural choice to define the tropical variety Trop X (Y ) associated to Y relative to X as the continuous projection of Y ℶ onto Σ X via trop X, i.e. Trop X (Y ) = trop X (Y ℶ ) Σ X. Corollary I.1.3. If X is logarithmically smooth over k, then the equality Trop X (X) = Σ X holds. We are going to see in Section 7.1 of this chapter that, if X = X( ) is a toric variety, then Trop X (Y ) is nothing but Trop(Y ), where Trop(Y ) denotes the extended tropical variety in the sense of [Kaj08, Section 1] and [Pay09, Section 3]. Note in particular that our definition of a tropical variety Trop X (Y ) depends on the choice of the surrounding logarithmic scheme X. We refer to Corollary II.1.5 for an interesting consequence of this dependence in the case of schön varieties Applications of our tropicalization procedure include the tropical geometry of modular normal crossing compactifications X of moduli spaces. The idea is that the local modular degeneration data of the boundary divisors typically can be arranged in a global set-theoretic tropical moduli space that can be identified with Σ X. The first examples of this approach can be found in [ACP12] treating the moduli spaces of marked stable curves,

29 1. INTRODUCTION 21 in [CMR14] treating the moduli space of admissible covers, as well as in [CHMR14], and Chapter III treating Hasset s moduli spaces of weighted stable curves. A further application of this tropicalization procedure, namely a generalization of Tevelev s [Tev07] results on tropical compactification to logarithmically regular varieties, is given in Chapter II below. As explained in [Thu07, Section 4], if D is a normal crossing divisor on a smooth variety X and the logarithmic structure (M X, ρ X ) on X is the one associated to D, the extended generalized cone complex Σ X contains the dual complex (D) associated to D as its link. These dual complexes have attracted a significant amount of attention as invariants associated to logarithmic resolutions of singularities and logarithmic compactifications and also serve as model spaces for weight zero parts of the cohomology of X D, in case X is defined over the complex numbers. For these recent developments we refer the reader to [ABW13], [dfkx12], [Kol12], [Pay13], and [Ste06] Our approach to the construction of trop X is a global version of the local tropicalization map defined by Popescu-Pampu and Stepanov in [PPS13, Section 6]. Fix a morphism α : P A from a monoid P into the multiplicative monoid of an algebra A of finite type over k and set X = Spec A. Then Hom(P, R 0 ) is the canonical compactification σ P of the rational polyhedral cone σ P = Hom(P, R 0 ) and there is a natural continuous tropicalization map trop α : X ℶ σ P that is defined by sending x X ℶ to the homomorphism trop α (x) σ P = Hom(P, R 0 ) given by p log α(p) x for p P. In the global case we associate to a fine and saturated Zariski logarithmic scheme X without monodromy an essentially unique characteristic morphism φ X : (X, O X ) F X into a fine and saturated Kato fan F X. Kato fans are locally monoidal spaces, which have been introduced by K. Kato in [Kat94, Section 9], that extend the dual of the category of monoids in the same way the category of schemes extends the dual of the category of rings. Note that in Kato s theory there is, in particular, the notion of a spectrum Spec P of a monoid P in analogy with the spectrum of a ring in the category of schemes.

30 22 I. FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES The set F X (R 0 ) of R 0 -valued points of F carries the structure of an extended cone complex Σ X and the natural continuous tropicalization map trop X : X ℶ Σ X is defined by sending a point x X ℶ, represented by a morphism x : Spec R (X, O X ), to the point trop X (x) Σ X = F X (R 0 ) that is given by the composition Spec R 0 val # Spec R x (X, O X ) φ X FX, where val # denotes the morphism induced by the valuation val : R R 0 on R. In the special case that F X = Spec P is affine, the extended cone complex Σ P = Hom(P, R 0 ) is nothing but the canonical compactification of the cone Σ P = Hom(P, R 0 ). The tropicalization map trop X restricts to the local tropicalization map of Popescu-Pampu and Stepanov [PPS13] on affine patches of X and F X. For a general fine and saturated logarithmic scheme X of finite type over k, the tropicalization map trop X : X ℶ Σ X is defined by taking colimits over all tropicalization maps trop X : X ℶ Σ X taken over all simple strict étale coverings X X, where X is a fine and saturated Zariski logarithmic scheme without monodromy. In this case Σ X is, in general, not an extended cone complex, but a generalized extended cone complex, a topological spaces that arises as a colimit of the strict diagram of extended cone complexes Σ X In [AW13, Section 2] and [ACMW14, Section 2], based on ideas already present in [Ols03, Section 5], the authors develop the notion of an Artin fan, a generalization of the notion of a Kato fan in the category of logarithmic algebraic stacks. By [AW13, Proposition 2.1.2] there is an essentially unique morphism ψ X : X A X into an algebraic stack A X that admits a canonical étale and representable morphism into Olsson s [Ols03] stack LOG k of logarithmic structures over k. The stack A X is called the Artin fan associated to X. It is étale locally isomorphic to a toric stack [X/T ], where T is the big torus in a toric variety X. It is now natural to suspect that the induced morphism ψ ℶ : X ℶ A ℶ X is a stacky version of the tropicalization map trop X : X ℶ Σ X. In fact, there is a natural homeomorphism µ X : A ℶ X ΣX from the topological space A ℶ X underlying the non-archimedean analytic stack A ℶ X with Σ X making the diagram

31 2. MONOIDS, CONES, AND MONOIDAL SPACES 23 ψ ℶ X A ℶ X X ℶ µ X trop X Σ X commutative. This procedure also gives every rational polyhedral cone complex Σ as well as its canonical compactification Σ canonically a structure of a non-archimedean analytic stack. Details on this result can be found in Chapter V below Our approach to the tropicalization of logarithmic schemes is also related to the construction in [GS13, Appendix B], where Gross and Siebert consider the space ( x X )/ Hom(M X,x, R 0 ). The equivalence relation is hereby induced by the duals of the generization maps M X,x M X,x, whenever x is a specialization of x in X. This space is an analogue of the tropical part of an exploded manifold in the sense of [Par12] and can be canonically identified with the generalized cone complex Σ X, whose canonical compactification is Σ X An alternative approach to the tropicalization of subvarieties of toric varieties, that is similar in spirit to our construction, can be found in [GG13]. As one of the crucial ingredients the authors use the notion of a toric variety over the field with one element F 1, which is closely related to fine and saturated Kato fans. In fact, let N be a free finitely generated abelian group and consider a rational polyhedral fan in N R. Then defines a toric variety X over F 1. Instead, however, of considering the R 0 -valued points of X, the authors of [GG13] work with the base change X F1 T, where T denotes the semi-ring of tropical numbers. 2. Monoids, cones, and monoidal spaces 2.1. Monoids. A monoid P is a commutative semigroup with an identity element. All monoids will be written additively, unless noted otherwise. The non-negative real numbers together with addition form a monoid that is denoted by R 0. Its monoid structure naturally extends to R 0 = R 0 { } by setting x + = for all x R 0.

32 24 I. FUNCTORIAL TROPICALIZATION OF LOGARITHMIC SCHEMES An ideal I in a monoid P is a subset I P such that p + I I for all p P. Every monoid P contains a unique maximal ideal m P = P P. An ideal p in P is called prime if its complement P p in P is a submonoid, or equivalently, if p 1 + p 2 p already implies p 1 p or p 2 p for all p i P. The complement of a prime ideal in P is referred to as a face of P. The localization of a monoid P with respect to a submonoid S is given by S 1 P = {p s p P, s S}, where p s denotes an equivalence class of pairs (p, s) P S under the equivalence relation (p, s) (p, s ) t S such that p + s + t = p + s + t. If S is the set N f for an element f P we write P f for the localization S 1 P and if S is the complement of a prime ideal p in P we denote S 1 P by P p. A monoid P is called fine, if it is finitely generated and the canonical homomorphism into the group P gp = P 1 P = {p q p, q P } is injective. It is said to be saturated if, whenever p P gp, the property n p P for some n N >0 already implies p P. An element p P is called a torsion element, if n p = 0 for some n N >0 ; it is called a unit, if there is q P such that p + q = 0. Denote the subgroup of torsion elements in P by P tors and the subgroup of units by P. A fine and saturated monoid P is said to be toric, if P tors = 0; an arbitrary monoid P is said to be sharp, if P = 0. We denote the category of fine and saturated by fs Mon and the full subcategory of toric monoids by tor Mon. Lemma I.2.1. Let P be a fine and saturated monoid. (i) There is a toric submonoid P of P such that P = P P tors. (ii) There exists a sharp submonoid P of P such that P = P P. For the convenience of the reader we provide proofs of these two well-known statements. Proof of Lemma I.2.1. The abelian group Q = P gp is finitely generated. So we can find a finitely generated free abelian subgroup Q of Q such that Q = Q Q tors. Note that hereby Q tors = P tors, since n q = 0 P for q Q and some n N >0 already implies q P using that P is saturated. Set P = P Q. Every p P can be uniquely written as p + t with p Q and t P tors and we have p = p t P. Thus P = P P tors. This proves part (i).

33 2. MONOIDS, CONES, AND MONOIDAL SPACES 25 In view of (i), we may assume P = 0 for the proof of part (ii). Given q Q such that n q P for some n N >0, we already have q P, since P is saturated. Therefore P is a saturated abelian subgroup in Q, i.e. Q/P is free, and we can find a subgroup Q of Q such that Q = Q P. So every element p P can be uniquely written as p + u with p Q and u P. Set P = P Q. Since p = p u P, this implies P = P P. In a slight abuse of notation, we write P for the toric monoid P/P tors and P for the sharp monoid P/P Rational polyhedral cones. A strictly convex rational polyhedral cone (or short: a rational polyhedral cone) is a pair (σ, N) consisting of a finitely generated free abelian group N and a strictly convex rational polyhedral cone σ N R = N R, i.e. a finite intersection of half spaces H i = { u N R u, vi 0 }, where v i M such that σ does not contain any non-trivial linear subspaces. We refer to [Ful93, Section 1.2] and [Gub13, Appendix A] for the essential background on these notions. Note hereby that Gubler [Gub13] calls rational polyhedral cones pointed integral polyhedral cones. We denote the relative interior of a rational polyhedral cone σ, i.e. the interior of σ in its span in N R, by σ. A morphism f : (σ, N) (σ, N ) of rational polyhedral cones is given by an element f Hom(N, N ) such that f(σ) σ. We denote the category of rational polyhedral cones by RPC. Consider now the functor Σ : fs Mon op RPC that associates to a fine and saturated monoid P the rational polyhedral cone Σ P = (σ P, N P ) given by N P = Hom(P gp, Z) and σ P = Hom(P, R 0 ) = { u Hom(P gp, R) u(p) 0 for all p P } (N P ) R.