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Tropical and Logarithmic Methods in Enumerative Geometry

  • Textbook
  • © 2023

Overview

Authors:
  1. Renzo Cavalieri

    1. Department of Mathematics, Colorado State University, Fort Collins, USA

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  2. Hannah Markwig

    1. Department of Mathematics, University of Tübingen, Tübingen, Germany

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  3. Dhruv Ranganathan

    1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK

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  • Provides a review and introduction of each of these 3 fields with a particular eye to the interactions
  • Provides hands-on exercises, plenty of examples and pictures
  • The interplay of tropical, logarithmic and enumerative geometry presented is original

Part of the book series: Oberwolfach Seminars (OWS, volume 52)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Toric Geometry and Logarithmic Curve Counting

    1. Front Matter

      Pages 1-7
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    2. Geometry of Toric Varieties

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 9-16

    3. Compactifying Subvarieties of Tori

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 17-24

    4. Points on the Riemann Sphere

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 25-30

    5. Stable Maps and Logarithmic Stable Maps

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 31-38

    6. Cheat Codes for Logarithmic GW Theory

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 39-48

  3. Hurwitz Theory

    1. Front Matter

      Pages 49-52
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    2. Classical Hurwitz Theory and Moduli Spaces

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 53-71

    3. Tropical Hurwitz Theory

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 73-83

    4. Hurwitz Numbers from Piecewise Polynomials

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 85-99

  4. Tropical Plane Curve Counting

    1. Front Matter

      Pages 101-105
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    2. Introduction to Plane Tropical Curve Counts

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 107-123

    3. Lattice Paths and the Caporaso-Harris Formula

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 125-144

    4. The Caporaso-Harris Formula for Tropical Plane Curves and Floor Diagrams

      • Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
      Pages 145-154

  5. Back Matter

    Pages 155-159
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Keywords

About this book

This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.

Authors and Affiliations

  • Department of Mathematics, Colorado State University, Fort Collins, USA

    Renzo Cavalieri

  • Department of Mathematics, University of Tübingen, Tübingen, Germany

    Hannah Markwig

  • Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK

    Dhruv Ranganathan

About the authors

​Renzo Cavalieri completed his PhD at University of Utah in 2005 under the direction of Aaron Bertram. He was a postdoc at University of Michigan under the mentorship of Bill Fulton for the following three years. In 2008, he became faculty at Colorado State University where he is currently a professor in the department of mathematics.

Hannah Markwig completed her PhD in 2006 at the University of Kaiserslautern in Germany, advised by Andreas Gathmann. She was a Postdoc at the Institute of Mathematics and its Applications in Minneapolis and at the University of Michigan in Ann Arbor, before she started a Juniorprofessorship at the University of Göttingen in 2008. In 2011, she moved to the University of the Saarland as a Professor, and in 2016 to the University of Tübingen.


Dhruv Ranganathan completed his PhD at Yale University in 2016 under the direction of Sam Payne. He was a CLE Moore Instructor at MIT and a memberat the Institute for Advanced Study in 2017. Since 2019, he has been at the University of Cambridge, where he is currently a professor of mathematics.


The authors have worked together since 2013, on several projects related to the themes discussed in this book. They have taught several courses, including at MSRI, Stockholm, and of course in Oberwolfach. In addition to their shared love of mathematics, the authors enjoy hiking, cooking, music, and the life-altering card game known as “tichu”.

Bibliographic Information

  • Book Title: Tropical and Logarithmic Methods in Enumerative Geometry

  • Authors: Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan

  • Series Title: Oberwolfach Seminars

  • DOI: https://doi.org/10.1007/978-3-031-39401-0

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Softcover ISBN: 978-3-031-39400-3Published: 02 October 2023

  • eBook ISBN: 978-3-031-39401-0Published: 30 September 2023

  • Series ISSN: 1661-237X

  • Series E-ISSN: 2296-5041

  • Edition Number: 1

  • Number of Pages: XIV, 159

  • Number of Illustrations: 34 b/w illustrations, 11 illustrations in colour

  • Topics: Algebraic Geometry

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