Geometry and project plan

First stage: Reading, presenting and discussing

The first main reading material is P. Brändén and J. Huh, Lorentzian polynomials, Annals of Mathematics 192 (2020), 821–891. Botong Wang introduces the geometric background behind our combinatorial object. The discussion starts from a smooth projective (or Kähler) complex surface and its Néron–Severi space.

A key role was played by ample divisors. Using an ample divisor, we consider the intersection pairing on the Néron–Severi space and apply the Hodge index theorem. The theorem says that the intersection form has signature \((1, \rho-1)\), where \(\rho\) is the Picard number of the surface. In particular, there is a unique positive eigenvalue.

The main point is that this “one positive eigenvalue” phenomenon is the geometric origin of the Lorentzian condition. In this sense, the combinatorial structure we study can be viewed as an algebraic abstraction of intersection theory with respect to an ample divisor.

Second stage: Project questions

The second stage is to work on the project questions. Additional reading materials will be provided as the project progresses. The project questions are designed to help participants understand the main paper and explore further directions. Participants are encouraged to work on the questions and discuss their ideas with each other.