Date:
Mon, 21/11/202217:00-18:00
Title: Low degree points and subspace configurations
Abstract: The hyperelliptic curve y^2=f(x) has an unusual property: it admits infinitely many quadratic points (a, \sqrt{a}). Conversely, Harris and Silverman show that curves with infinitely many quadratic points are (geometrically) hyperelliptic or bielliptic. Similarly one may wonder which curves over number fields have an infinite collection of points of some fixed degree d. Abramovich and Harris conjectured that, analogously to the quadratic case, such curves admit degree d maps to \mathbb{P}^1 or an elliptic curve, but this simple description turned out to be false for d\geq4.
I will describe recent joint work with Isabel Vogt (arXiv:2208.01067) in which we show how to reduce the general classification problem to a study of curves of low genus. As an application, we obtain a classification for d\leq5. These results are obtained by analyzing discrete geometry of certain configurations of linear subspaces.
Abstract: The hyperelliptic curve y^2=f(x) has an unusual property: it admits infinitely many quadratic points (a, \sqrt{a}). Conversely, Harris and Silverman show that curves with infinitely many quadratic points are (geometrically) hyperelliptic or bielliptic. Similarly one may wonder which curves over number fields have an infinite collection of points of some fixed degree d. Abramovich and Harris conjectured that, analogously to the quadratic case, such curves admit degree d maps to \mathbb{P}^1 or an elliptic curve, but this simple description turned out to be false for d\geq4.
I will describe recent joint work with Isabel Vogt (arXiv:2208.01067) in which we show how to reduce the general classification problem to a study of curves of low genus. As an application, we obtain a classification for d\leq5. These results are obtained by analyzing discrete geometry of certain configurations of linear subspaces.