Tropical Hurwitz loci

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

Enumerative geometry is a part of algebraic geometry in which one counts geometric objects satisfying certain conditions, e.g. plane curves passing through given points with assigned multiplicities. These questions are typically easy to formulate, but hard to solve with elementary techniques. An often succesful 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 symplectic geometry or string theory.Tropical geometry is a recent and quickly growing field in which algebro-geometric objects are degenerated to certain piecewise linear combinatorial objects called tropical varieties. In spite of the degeneration, 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 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 proposal suggests to study questions at the interplay of enumerative geometry and tropical geometry, aiming at results in both fields as well as at a deeper understanding of their connections. The main object of interest are loci of ramified covers of the projective line called (double) Hurwitz loci. A 0-dimensional Hurwitz locus is classically known as Hurwitz number.Tropical analogues of double Hurwitz numbers have been introduced by Cavalieri, Johnson and myself.Double Hurwitz numbers have an interesting piecewise polynomial structure in the entries of the fixed ramification data. The tropical approach was helpful to discover new features of this piecewise polynomial structure.In this proposal, I suggest to extent the picture to higher-dimensional cycles in the moduli space of curves parametrizing certain ramified covers. I intend to study - classical and tropical - Hurwitz loci, their piecewise polynomial structure and wall-crossing formulas.The double-tracked approach - classical and tropical - serves two purposes: first, we can use the whole machinery of both methods to gain maximal outcome. Second, I hope to learn more about the connection between classical and tropical geometry by thouroughly investigating the two sides of the theory of Hurwitz loci. In addition, the study of Hurwitz loci and their connections to their tropical counterparts will advance the study of tropical moduli spaces and their intersection theory in general and might contribute to a more geometric understanding of the wall-crossing formulas for double Hurwitz loci.

• DFG Programme: Research Grants