at the
American Institute of Mathematics, Pasadena, California
organized by
Nathan Chen and Bianca Viray
The study of degree $d$ points on algebraic curves over $\Bbb Q$ is a rich and mature area of research, with the Abel-Jacobi map and the Mordell-Lang conjecture providing powerful tools for exploration. However, for higher dimensional varieties there is no such approach that works in general. Because of this, we lack even a conjectural framework for understanding which higher dimensional varieties over $\Bbb Q$ should have "many" degree $d$ points.
The workshop will focus on questions aimed at addressing this dearth, concentrating on the case of algebraic surfaces. For instance, what does it mean for a surface over $\Bbb Q$ to have "many" degree $d$ points? What are some geometric constructions that give rise to abundant degree $d$ points? Are these related to geometric measures of irrationality? If $Hilb^d_X$ has a Zariski dense set of $\Bbb Q$-points for some small $d$, does that yield any arithmetic or geometric consequences for $X$? If $X$ embeds into its Albanese, can we obtain results analogous to that of curves?
Participants will be researchers from a broad array of backgrounds (e.g., arithmetic of surfaces, geometry of Hilbert schemes of surfaces, geometric measures of irrationality, arithmetic of 0-cycles, to name a few), ideally with a curiosity and interest in arithmetic questions.
The workshop schedule.
A report on the workshop activities.
A list of open problems.