Arithmetic Geometry is ONline in Zoom, Everyone (Agonize)
Welcome to the webpage for the first ever AGONIZE mini-conference. The conference took place on March 25, 2020, from 12pm EST - 5pm EST. Videos can be found below.
Schedule
12pm EST - Wanlin Li (MIT)
The Ceresa class: tropical, topological, and local
Abstract. The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve, which is trivial in the Chow ring when the curve is hyperelliptic. Its image under a certain cycle class map provides a class in étale cohomology called the Ceresa class. There are few examples where the Ceresa class is known for non-hyperelliptic curves. We explain how to define a Ceresa class for a tropical algebraic curve, and also for a Riemann surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). Finally, we explain how these are related to the Ceresa class of a smooth algebraic curve over \(\mathbb{C}((t))\), and show that the Ceresa class for tropical curves is torsion.
115pm EST - Padma Srinivasan (UGA)
Conductors and minimal discriminants of hyperelliptic curves
Abstract. Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In genus one, the Ogg–Saito formula shows that these two invariants are equal, and in genus two, Qing Liu showed that they are related by an inequality. In this talk, we will show that Liu’s inequality extends to hyperelliptic curves of arbitrary genus in odd residue characteristic. This is joint work with Andrew Obus.
230pm EST - Hannah Larson (Stanford)
A refined Brill-Noether theory over Hurwitz spaces
Abstract. The celebrated Brill-Noether theorem says that the space of degree \(d\) maps of a general genus \(g\) curve to \(\mathbb{P}^r\) is irreducible. However, for special curves, this need not be the case. Indeed, for general \(k\)-gonal curves (degree \(k\) covers of \(\mathbb{P}^1\)), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map \(C \rightarrow \mathbb{P}^1\), using the splitting type of push forwards of line bundles to \(\mathbb{P}^1\). In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general \(k\)-gonal curves.
345pm EST - Isabel Vogt (Stanford, UW)
An enriched count of the bitangents to a smooth plane quartic curve
Abstract. Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields, generalizing Segre’s signed count count of elliptic and hyperbolic lines. Their approach using \(\mathbb{A}^1\)-enumerative geometry suggests that other classical enumerative problems should have similar enrichments when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. In this talk, we consider the closely related problem of the 28 bitangents to a smooth plane quartic curve. Subtleties arise because the relevant vector bundle is not relatively orientable.
