Examples of Enumerative Problems for Arbitrary Fields

Abstract: Over the complex numbers the solutions to enumerative problems are invariant: the number of solutions of a polynomial equation or polynomial system, the number of lines or curves in a surface, etc. Over the real numbers such invariance fails. However, the signed count of solutions may lead to numerical invariants: Descartes' rule of signs, Poincaré-Hopf theorem, real curve-counting invariants.

Since many of these problems have a geometric nature, one may ask the same problems for arbitrary fields. Motivic homotopy theory allows to do enumerative geometry over an arbitrary base, leading to additional arithmetic and geometric information.

The goal of this talk is to illustrate a generalized notion of sign that allows us to state a motivic version of classical problems: the number of lines on cubic surfaces, the Bézout theorem, and the curve-counting invariants.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html