Tropical Hurwitz loci

• Applicant: Professorin Dr. Hannah Markwig
• Subject Area: Mathematics
• Term: from 2009 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 144856147

• Final Report Year: 2016

Final Report Abstract

Enumerative geometry is a part of algebraic geometry in which one counts geometric objects satisfying certain conditions. These questions are typically easy to formulate, but hard to solve. An interesting approach consists in translating an enumerative geometric question into an intersection problem on some appropriate moduli space. This is a reason why enumerative geometry has been very fruitful in pushing new developments in algebraic geometry and in making connections to other fields such as string theory. Tropical geometry is a recent and quickly growing field in which algebro-geometric objects are degenerated to piecewise linear combinatorial objects called tropical varieties. In spite of the degeneration, many properties are preserved and many algebro-geometric invariants carry over to the tropical world. In these cases tropical geometry is an effective computational tool to study problems in algebraic geometry. It also has connections to other fields such as optimization or biomathematics. Tropical geometry has had particular success in the study of enumerative questions. The field of tropical enumerative geometry was pioneered by Mikhalkin with his celebrated Correspondence Theorem relating classical numbers of plane curves to their tropical counterparts. This project dealt with questions at the interplay of enumerative geometry and tropical geometry. The main object of interest were Hurwitz numbers, and their higherdimensional generalizations: loci of ramified covers of P1 called Hurwitz loci. Hurwitz numbers count covers of curves with prescribed ramification data. At the first stage of the project, Hurwitz numbers were the main center of attention. Their tropical counterparts were introduced by myself with coauthors Cavalieri and Johnson. The study of their structural behaviour was initiated by Goulden-Jackson-Vakil and continued, in genus zero, by Shadrin-Shapiro-Vainshtein, leading to interesting wall-crossing formulas. With the tropical approach, we proved conjectures on the structure made by Goulden-Jackson-Vakil, and proved wall-crossing formulas for higher genus. With A. Buchholz, I furthermore studied generalizations of tropical Hurwitz numbers in terms of moduli spaces of tropical covers and a branch map. At the second stage of the project, the attention was turned to higher-dimensional Hurwitz loci. With Bertram and Cavalieri, I investigated their piecewise polynomial structure and their relation to tropical counterparts. My Postdoc Hampe studied the combinatorics of tropical Hurwitz cycles, leading to interesting conjectures about deeper connections to algebraic counterparts. Having completed these tasks, two related problems appeared which were studied in the context of this project. First, with Rau, I investigated real versions of tropical Hurwitz numbers and proved a first correspondence theorem, which leads to questions about the invariance of such real counts. Second, my Postdoc Len focused on tropical covers of degree two and their relation to linear systems on curves.

Publications

• Hyperelliptic graphs and metrized complexes
  Yoav Len
  
• Chamber structure for double Hurwitz numbers. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 227–238, Discrete Math. Theor. Comput. Sci. Proc., AN, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010
  Renzo Cavalieri, Paul Johnson, Hannah Markwig
  
• (1) Renzo Cavalieri, Paul Johnson, Hannah Markwig. Wall crossings for double Hurwitz numbers. Adv. Math. 228 (2011), no. 4, 1894–1937
  Renzo Cavalieri, Paul Johnson, Hannah Markwig
  
• Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles. JCTA 120,7 (2013), 1604–1631
  Aaron Bertram, Renzo Cavalieri, Hannah Markwig
  
• Tropical covers of curves and their moduli spaces. Commun. Contemp. Math. (2013)
  Arne Buchholz, Hannah Markwig
  
• Tropical covers, moduli spaces and mirror symmetry. Dissertation, Unicersität des Saarlandes 2014
  Arne Buchholz
  
• Combinatorics of tropical Hurwitz cycles. J. Algebraic Combin. 42, 4 (2015), 1027–1058
  Simon Hampe
  
• Tropical real Hurwitz numbers. Math. Z. 281, 1 (2015), 501–522
  Hannah Markwig, Johannes Rau