Introduction to Tropical Geometry Diane Maclagan Bernd Sturmfels Graduate Studies in Mathematics Volume 161 American Mathematical Society Introduction to Tropical Geometry https://doi.org/10.1090//gsm/161 Introduction to Tropical Geometry Diane Maclagan Bernd Sturmfels Graduate Studies in Mathematics Volume 161 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 14T05; Secondary 05B35, 12J25, 13P10, 14M25, 15A80, 52B20. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-161 Library of Congress Cataloging-in-Publication Data Maclagan, Diane, 1974– Introduction to tropical geometry / Diane Maclagan, Bernd Sturmfels. pages cm. — (Graduate studies in mathematics ; volume 161) Includes bibliographical references and index ISBN 978-0-8218-5198-2 (alk. paper) 1. Tropical geometry—Study and teaching (Graduate) 2. Geometry, Algebraic—Study and teaching (Graduate) I. Sturmfels, Bernd, 1962– II. Title. III. Tropical geometry. QA582.M33 2015 516.35—dc23 2014036141 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. 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Curve Counting 31 §1.8. Compactifications 34 §1.9. Exercises 39 Chapter 2. Building Blocks 43 §2.1. Fields 43 §2.2. Algebraic Varieties 52 §2.3. Polyhedral Geometry 58 §2.4. Gr¨obner Bases 65 §2.5. Gr¨obner Complexes 74 §2.6. Tropical Bases 81 §2.7. Exercises 89 Chapter 3. Tropical Varieties 93 §3.1. Hypersurfaces 93 vii viii Contents §3.2. The Fundamental Theorem 102 §3.3. The Structure Theorem 110 §3.4. Multiplicities and Balancing 118 §3.5. Connectivity and Fans 128 §3.6. Stable Intersection 133 §3.7. Exercises 149 Chapter 4. Tropical Rain Forest 153 §4.1. Hyperplane Arrangements 153 §4.2. Matroids 161 §4.3. Grassmannians 170 §4.4. Linear Spaces 182 §4.5. Surfaces 192 §4.6. Complete Intersections 201 §4.7. Exercises 214 Chapter 5. Tropical Garden 221 §5.1. Eigenvalues and Eigenvectors 222 §5.2. Tropical Convexity 228 §5.3. The Rank of a Matrix 243 §5.4. Arrangements of Trees 255 §5.5. Monomials in Linear Forms 268 §5.6. Exercises 273 Chapter 6. Toric Connections 277 §6.1. Toric Background 278 §6.2. Tropicalizing Toric Varieties 281 §6.3. Orbits 291 §6.4. Tropical Compactifications 297 §6.5. Geometric Tropicalization 309 §6.6. Degenerations 322 §6.7. Intersection Theory 334 §6.8. Exercises 346 Bibliography 351 Index 361 Preface Tropical geometry is an exciting new field at the interface between algebraic geometry and combinatorics with connections to many other areas. At its heart it is geometry over the tropical semiring, which is R ∪{∞}with the usual operations of addition and multiplication replaced by minimum and addition, respectively. This turns polynomials into piecewise-linear func- tions and replaces an algebraic variety by an object from polyhedral geom- etry, which can be regarded as a “combinatorial shadow” of the original variety. In this book we introduce this subject at a level that is accessible to beginners. Tropical geometry has become a large field, and only a small selection of topics can be covered in a first course. We focus on the study of tropical varieties that arise from classical algebraic varieties. Methods from commutative algebra and polyhedral geometry are central to our approach. This necessarily means that many important topics are left out. These in- clude the systematic development of tropical geometry as an intrinsic geome- try in its own right, connections to enumerative and real algebraic geometry, connections to mirror symmetry, connections to Berkovich spaces and ab- stract curves, and the more applied aspects of max-plus algebra. Luckily most of these topics are covered in other recent or forthcoming books, such as [BCOQ92], [But10], [Gro11], [IMS07], [Jos], [MR], and [PS05]. Prerequisites. This text is intended to be suitable for a class on trop- ical geometry for first-year graduate students in mathematics. We have attempted to make the material accessible to readers with a minimal back- ground in algebraic geometry, at the level of the undergraduate text book Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea [CLO07]. ix x Preface Essential prerequisites for this book are mastery of linear algebra and the material on rings and fields in a first course in abstract algebra. Since tropical geometry draws on many fields of mathematics, some additional background in geometry, topology, or number theory will be beneficial. Polyhedra and polytopes play a fundamental role in tropical geometry, and some prior exposure to convexity and polyhedral combinatorics may help. For that we recommend Ziegler’s book Lectures on Polytopes [Zie95]. Chapter 1 offers a friendly welcome to our readers. It has no specific prerequisites and is meant to be enjoyable for all. The first three sections in Chapter 2 cover background material in abstract algebra, algebraic geom- etry, and polyhedral geometry. Enough definitions and examples are given that an enthusiastic reader can fill in any gaps. All students (and their teachers) are strongly urged to explore the exercises for Chapters 1 and 2. Some of the results and their proofs will demand more mathematical ma- turity and expertise. Chapters 2 and 3 require some commutative algebra. Combinatorics and multilinear algebra will be useful for studying Chapters 4 and 5. Chapter 6 assumes familiarity with modern algebraic geometry. Overview. We begin by relearning the arithmetic operations of addition and multiplication. The rest of Chapter 1 offers tapas that can be enjoyed in any order. They show a glimpse of the past, present, and future of tropical geometry and serve as an introduction to the more formal contents of this book. In Chapter 2, the first half covers background material, while the second half develops a version of Gr¨obner basis theory suitable for algebraic varieties over a field with valuation. The highlights are the construction of the Gr¨obner complex and the resulting finiteness of tropical bases. Chapter 3 is the heart of the book. The two main results are the Fun- damental Theorem 3.2.3, which characterizes tropical varieties in seemingly different ways, and the Structure Theorem 3.3.5, which says that they are connected balanced polyhedral complexes of the correct dimension. Stable intersections of tropical varieties reveal a hint of intersection theory. Tropical linear spaces and their parameter spaces, the Grassmannian and the Dressian, appear in Chapter 4. Matroid theory plays a foundational role. Our discussion of complete intersections includes mixed volumes of Newton polytopes and a tropical proof of Bernstein’s Theorem for n equations in n variables. We also study the combinatorics of surfaces in 3-space. Chapter 5 covers spectral theory for tropical matrices, tropical convexity, and determinantal varieties. It also showcases computations with Bergman fans of matroids and other linear spaces. Chapter 6 concerns the connec- tion between tropical varieties and toric varieties. It introduces the tropical approach to degenerations, compactifications, and enumerative geometry. Preface xi Teaching possibilities. A one-semester graduate course could be based on Chapters 2 and 3, plus selected topics from the other chapters. One possibility is to start with two or three weeks of motivating examples selected from Chapter 1 before moving on to Chapters 2 and 3. A course for more advanced graduate students could start with Gr¨obner bases as presented in the second half of Chapter 2, cover Chapter 3 with proofs, and end with a sampling of topics from the later chapters. Students with an interest in combinatorics and computation might gravitate toward Chapters 4 and 5. An advanced course for students specializing in algebraic geometry would focus on Chapters 3 and 6. Covering the entire book would require a full academic year or an exceptionally well-prepared group of participants. We have attempted to keep the prerequisites low enough to make parts of the book appropriate for self-study by a final-year undergraduate. The sec- tions in Chapter 2 could serve as first introductions to their subject areas. A simple route through Chapter 3 is to focus in detail on the hypersurface case, and to discuss the Fundamental Theorem and Structure Theorem without proofs.