Title: Low degree points on curves

Abstract: In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

Title: The Brauer-Manin obstruction and the order of Brauer classes that capture it.

Abstracts:The Brauer-Manin obstruction is a powerful and mysterious compatibility that local points on a variety must satisfy in order to approximate a global solution. In the first lecture, we will explain how quadratic reciprocity and higher reciprocity laws encode this compatibility via the Brauer group. We will provide a flavor of the different types of Brauer classes and the methods we have for understanding them, particularly in the case of surfaces. In the second lecture, we focus on a finer understanding of the Brauer-Manin obstruction, namely on the order of elements that capture the obstruction. We present results, joint with Brendan Creutz, characterising these orders in the case of torsors under abelian varieties, Kummer surfaces, and (conditional on the finiteness of Tate-Shafarevich groups) bielliptic surfaces.

Title: Derived Torelli theorems in positive characteristic

Abstract: Broadly speaking, Torelli theorems are concerned with the question of when a variety can be recovered from an invariant associated to it. The global Torelli theorem, proved in the 1970’s, shows that K3 and Enriques surfaces over $\mathbb{C}$ are determined by cohomological data. There is no known equivalent statement that holds generally over algebraically closed fields of positive characteristic. Progress in this direction has been made recently: statements giving Torelli-type theorems in terms of derived categories have been proven. In the first of these talks, I will introduce the notion of lifting from positive characteristic to characteristic 0, which plays a central role in the proofs of these theorems.

Title: Intimations of descent on K3 surfaces

Abstract: The method of descent gives a way to determine, under favorable circumstances, the group of rational points on an elliptic curve. The starting point for the method is an isogeny of elliptic curves. For K3 surfaces, a suitable replacement for an isogeny is a rational isomorphism of Hodge structures. These "K3 isogenies" can be reinterpreted in terms of the Brauer group of a K3 surface. In the first lecture, I will review the classification of algebraic surfaces, and go over material on del Pezzo surfaces and K3 surfaces (and their Brauer groups), in preparation for the second lecture. In the second lecture, I will report on recent work with Jen Berg, which shows how in some cases one can perform a "3-descent" on a K3 surface to prove that it has no rational points.

Title: Counting curves over an arithmetically interesting field

Abstract: Given two general degree \(d\) complex polynomials \(f(x,y)\) and \(g(x,y)\), the equation \(f(x,y) + t g(x,y)\) defines a singular curve for exactly \(3(d-1)^2\) complex values of \(t\). This is a classical result that has been generalized in many ways to results counting complex curves, but the problem of generalizing it by replacing the complex numbers with a more arithmetically interesting field, such as the rational numbers or a finite field or..., has only been taken up recently. I will describe this recent work, especially work of the speaker, Marc Levine, and Kirsten Wickelgren, in my talk.