#
Degree d points on algebraic surfaces

March 18 to March 22, 2024
at the

American Institute of Mathematics,
Pasadena, California

organized by

Nathan Chen and Bianca Viray

## Original Announcement

This workshop will be devoted to the study of degree *d* points on algebraic surfaces over a number field.
The study of degree *d* points on algebraic curves over 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 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 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 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.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Workshop videos