# Seminario

#### Seminario Álgebra Conmutativa-Geometría Algebraica y Aritmética UAM-ICMAT

On arc fibers of morphisms of schemes

Ponente:  Christopher Heng Chiu (Technische Universiteit Eindhoven)
Fecha:  martes 14 de junio de 2022 - 12:30
Lugar:  Aula 320, Módulo 17, Departamento de Matemáticas, UAM
Online:  Microsoft Teams: please, contact ana.bravo@uam.es

Resumen:

The arc space of an algebraic variety X is the scheme parametrizing formal arcs on X, first introduced by John Nash and since then studied extensively in the context of birational geometry and singularity theory. Any morphism between varieties induces a map of the corresponding arc spaces. The main goal of this talk is to describe the fibers of this map, which we call arc fibers. We prove that for quasi-finite morphisms the arc fibers are topologically finite and that they are scheme-theoretically finite if and only if the morphism is unramified. If time permits, various applications to invariants such as the embedding (co)dimension and Mather log discrepancies are discussed. The material presented here is joint work with Tommaso de Fernex and Roi Docampo.