Arithmetic counts of bitangents to plane quartics by means of tropical geometry

• Applicant: Professorin Dr. Hannah Markwig
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 504195479

Enumerative geometry is the research area of geometric counting problems. Many enumerative questions can be formulated easily, but remain notoriously hard to answer, in particular when working over a field different from the complex numbers. This intrinsic complexity has contributed to the flourishing of enumerative geometry over the last decades, providing fruitful connections among various areas of mathematics such as algebraic geometry, arithmetic geometry, representation theory, mathematical physics, the theory of random matrices and even to theoretical physics and string theory. Recently, so-called refined enumerative invariants which can be viewed as a counting method in arithmetic geometry offer a universal theory of counting geometric objects. With this proposal, I suggest to enrich the theory of arithmetic counting by introducing a new tool to this theory: the tool of degenerations and tropicalizations.Tropical geometry is a modern area which allows an exchange of methods between algebraic geometry and combinatorics. Through a degeneration process called tropicalization, an algebraic variety is turned into a tropical variety. The latter is a polyhedral complex satisfying certain conditions. Tropical geometry provides connections to many other areas within mathematics, such as symplectic geometry, arithmetic geometry, mathematical physics and optimization, but also to areas in fields of application of mathematics such as economy, machine learning and computational biology.In 2002, following suggestions of Kontsevich, Mikhalkin initiated the use of tropical methods in enumerative geometry by proving the celebrated Mikhalkin correspondence theorem. The project will focus on arithmetic counts of bitangents to smooth plane quartic curves. This is a natural starting point, as the theory of tropical bitangents to a tropical quartic is well understood, not only if we tropicalize over the complex numbers but also if we tropicalize over the real numbers.Already Plücker knew that a plane quartic curve has precisely 28 bitangent lines over the complex numbers. Over the reals however, a quartic can have 4, 8, 16 or 28 bitangents, depending on the topological type of the real curve. Quite recently, Larson and Vogt initiated an arithmetic count of bitangents, which, when specialized to the real numbers, yields a signed count which is invariant under some circumstances, similar to the famous Welschinger signed count of rational plane curves of degree d satisfying point conditions.In this project, we plan to provide tools to compute arithmetic multiplicities of bitangents by means of tropical geometry. We also relate this tropical count to the tropical lines in a cubic surface. As a long-term perspective, we investigate arithmetic counts of plane curves satisfying point conditions.

• DFG Programme: Research Grants