Summer Learning/Research Seminar

Abstract: It is by now widespread knowledge that the McKay correspondence predicts that all crepant resolutions of a given singular variety have equivalent derived categories of coherent sheaves. Interestingly, this viewpoint doesn't appear in the original paper by McKay at all. This talk will be an introduction to the actual McKay correspondence, which doesn't feature derived categories, but rather finite subgroups of SO(3) or SU(2) and extended Dynkin diagrams of type ADE.

Given a del Pezzo surface (dPS), such as a cubic surface, what is the smallest degree among its closed points? For which degrees d can we say that all degree-d zero-cycles are effective? With these questions in mind, we will first review both classical and recent methods for constructing effective 0-cycles on surfaces. We will see where I am currently stuck using these methods via explicit computations on a cubic surface.

This motivates the second part of the talk, where I will present (and ask questions about) a recent paper by Claire Voisin, in which she uses rank-2 vector bundles to construct "new" effective 0-cycles on dPS of degree 3, 2, and 1. In this way, she obtains new effectivity results for dPS in characteristic zero, and improves Corray's result on the Cassels--Swinnerton-Dyer conjecture. If time permits, I will also mention the other topic of the paper: the unirationality and stable rationality of the third symmetric powers of cubic hypersurfaces. 

A parallelogram height inequality relates the "arithmetic complexities" (heights) of four arithmetic objects linked by morphisms arranged in a parallelogram-shaped diagram. For instance, given an elliptic curve E over a number field, and two finite subgroups G, H of E, Rémond's inequality relates the Faltings heights of the quotients of E by G, H, G \cap H, and G + H. This "parallelogram inequality" is but a special case of a theorem of Rémond, valid for abelian varieties of arbitrary dimension over any number field. This inequality has interesting Diophantine consequences. More broadly, it complements a classical theorem of Faltings which relates the heights of E and E/G, and contributes to our understanding of how heights vary under isogenies.

In recent work with Le Fourn and Pazuki, we prove a parallelogram inequality for elliptic curves over function fields, where the role of the Faltings height is played by the differential height (our proof is actually written in the context of higher-dimensional abelian varieties). In another project with Baker and Pazuki, we prove a parallelogram inequality in the analogous setting of Drinfeld modules over a function field of positive characteristic.

In this talk, I will introduce the relevant notions, explain the general questions around understanding how maps interact with heights. I will also sketch the key parts of the proof of the parallelogram inequality. Time permitting, I will mention applications of the inequality.

This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, highlighting the analogy between number fields and function fields. I will discuss two finiteness theorems for étale divisors on smooth proper models of algebraic curves. The number field case is joint work with F. Janbazi.

We show that, for a fixed degree, there are only finitely many relative étale divisors over the base, up to the symmetries of the ambient space. These results recover several classical finiteness theorems, including those of Faltings, Siegel, and Birch–Merriman, and provide a common perspective from which they may be understood.

Together, these theorems illustrate both the similarities and the differences between the arithmetic of number fields and function fields, contributing to a broader understanding of finiteness phenomena on curves.

I will conclude by discussing a natural extension of these results and some of the challenges that arise in pursuing it.

Abstract: Let Gr(2,4) be the Grassmannian of 2-planes in the 4-dimensional vector space of binary quartics over the complex numbers. Under this identification, Gr(2,4) is the space of pencils of binary cubics, and it is endowed with a natural PGL2 action. Thus, we may study intersections of PGL2-orbit closures in Gr(2,4), which are governed by the PGL2-equivariant Chow ring of Gr(2,4). In this talk, I will give a definition of equivariant Chow rings and explain their importance for equivariant intersection theory. Then, I will give an overview of computations of the PGL2-equivariant Chow ring of Gr(2,4)^s, the open subset of Gr(2,4) consisting of all PGL2-stable points in Gr(2,4). Finally, I will discuss some apparent difficulties with computing the full PGL2-equivariant Chow ring of Gr(2,4).

The localization property allows us to associate certain semi-orthogonal decompositions to open-closed decompositions of geometric objects such as schemes. This property was first established for the motivic homotopy of schemes by Morel and Voevodsky in 1999 after which it has been reproven in various settings and used to prove important results such as cdh descent and establish 6-functor formalisms.

After introducing this property and going over its history, we will study some general strategies for proving it, and explore some of its recent extensions outside of the algebro-geometric setting. If time permits, we will also discuss some unfinished ideas for proving localization in new settings.

I want to explain Alex's counterexample to the period-index conjecture of Colliot-Thélène.

Abstract: A central problem in Diophantine geometry is to determine which algebraic varieties satisfy potential density of integral points, that is, which varieties admit a Zariski dense set of integral points over some ring of S-integers of a number field. A broad conjectural picture, developed by several authors such as Bombieri, Campana, Lang and Vojta, predicts that this arithmetic property should depend only on the geometry of the variety. The problem is solved for curves, due to Siegel and Faltings' finiteness theorems, and substantial progress has been made for special classes of higher-dimensional varieties, notably subvarieties of semiabelian varieties. Nevertheless, already in dimension two the general picture remains largely open.

In this talk I will give a brief overview of the main conjectures, results, and techniques surrounding potential density of integral points. I will then exhume some old work of mine concerning open subvarieties of elliptic ruled surfaces, explain some partial results on potential density in this setting, and discuss the many questions that remain open.