Schedule for: 21w5051 - Geometry via Arithmetic (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will survey several geometric theorems that were first proved using arithmetic and detail progress to give geometric proofs (ongoing joint work with Zhiyu Tian): (1) special cases, first proved by Fried-Jarden, of Ax's conjecture that perfect pseudo-algebraically closed fields are also quasi-algebraically closed, (2) irreducibility of parameter spaces of genus-0 quasi-maps to smooth Fano hypersurfaces, first proved by Browning-Vishe, and Fano complete intersections (thesis of Prithviraj Chowdhury), (3) special cases of the Cohen-Jones-Segal conjecture on low degree homotopy types of spaces of rational curves on Fano hypersurfaces, first proved by Browning-Sawin, and (4) Skinner's proof of weak approximation for smooth low degree complete intersections over global function fields.

Let X be a ramified cover of an abelian variety A over a number field k with A(k) dense. According to Lang's conjecture, the k-rational points of X should not be dense. In joint work with Corvaja, Demeio, Lombardo, and Zannier, we prove a slightly weaker statement. Namely, we show that the complement of the image of X(k) in A(k) is dense, i.e., there are less points on X than there are on A (or: there are more points on A than on X). In this talk I will explain that our proof relies on interpreting this as a special case of a version of Hilbert's irreducibility theorem for abelian varieties.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

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One of the appeals of algebraic geometry is the abundance of miraculous constructions it contains. Examples include ``resolving the quartic''; the existence of 9 flex points on a smooth plane cubic; the Jacobian of a genus g curve; and the 27 lines on a smooth cubic surface. In this talk I will explain some ways to systematize and formalize the idea that such constructions are special: conjecturally, they should be the only ones of their kind. I will state a few of these many (mostly open) conjectures. They can be viewed as forms of rigidity (a la Mostow and Margulis) for various moduli spaces and maps between them.

This talk will survey some results regarding hyperbolicity of varieties of log general type. There are several classical results showing that positivity of the cotangent bundle implies various notions of hyperbolicity for projective varieties coming from algebraic, arithmetic, and differential geometry. The goal of this talk is to review these results and discuss generalizations of these results for log pairs / quasi-projective varieties. This is joint work with K. DeVleming and A. Turchet.

I will report on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia. We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral effective cone, both in characteristic 0 and in characteristic p, for an infinite set of primes p of positive density. As a consequence, we prove that the effective cone of the Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, is not polyhedral if n>=10 in characteristic 0 and in positive characteristic.

Many questions in number theory have a natural analogue, of more geometric nature, formulated in the Grothendieck ring of varieties. For example, Poonen's finite field Bertini theorem has a motivic counterpart due to Vakil and Wood; however, despite the clear similarities between these two results, none of the two can be deduced from the other. The aim of this talk is to describe and motivate a conjectural way of comparing such statements in arithmetic and motivic statistics, by reformulating them in terms of the convergence of zeta functions in different topologies. We will finish by mentioning some concrete settings where our conjectures are satisfied. This is joint work with Ronno Das and Sean Howe.

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Given an abstract curve C, the explicit realizations of C in projective spaces are parameterized by the Brill--Noether loci of C. In this talk, we will explore some natural questions about the geometry and arithmetic of Brill--Noether loci. This will include joint work with Geoff Smith, and with Borys Kadets, and with Eric Larson and Hannah Larson.

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The translation of Manin's program to the context of moduli spaces of morphisms from a curve provides a conjectural framework in which to interpret the asymptotic behaviour of these spaces. More precisely, in many examples, the classes of the spaces of morphisms in a motivic Grothendieck ring converges after renormalisation to a class which may be described as a motivic Tamagawa volume using the work of M. Bilu. The aim of this talk is to give a survey on this geometric setting for Manin's program.

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In this talk I will discuss counting points over function fields in two different contexts: smooth cubic hypersurfaces and the Hilbert scheme of 2 points on P^2. I will provide a geometric interpretation of the results: first by relating the number of F_q(t)-points of bounded height to the number of F_q-rational curves of fixed degree, and in the second part, by connecting the result to a refined version of Manin's conjecture.

I will explain how to formulate a version of Mordell’s conjecture for 1-dimensional 'orbifold pairs' à la Campana. Over number fields, proving such a statement seems out of reach: all we know is that it would follow from the abc conjecture. Over function fields however, such a result can actually be proven, both in characteristic zero and in positive characteristic; I will give an overview of the proof and the techniques involved (joint work with Stefan Kebekus and Jorge Vitoria Pereira), and I will comment on possible generalisations.

A divisibility sequence is a sequence of integers d_n such that, if m divides n, then d_m divides d_n. Bugeaud, Corvaja, Zannier showed that pairs of divisibility sequences of the form a^n-1 have only limited common factors. From a geometric point of view, this divisibility sequence corresponds to a subgroup of the multiplicative group, and Silverman conjectured that a similar behaviour should appear in (a large class of) other algebraic groups. Extending previous works of Silverman and of Ghioca-Hsia-Tucker on elliptic curves over function fields, we will show how to prove the analogue of Silverman’s conjecture over function fields in the case of split semiabelian varieties and some generalizations. The proof relies on some results of unlikely intersections. This is a joint work with F. Barroero and A. Turchet.

Manin's conjecture on rational points suggests a number of different conjectures on moduli spaces of curves on algebraic varieties, of varying levels of generality and strength, that can all be considered its geometric analogue. I will survey a few of these conjectures and then discuss my work with Tim Browning making progress towards one of them.

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