Stein degree of proper morphisms

TL;DR

This paper introduces the Stein degree for proper morphisms in algebraic geometry, exploring its properties, applications in birational geometry, and a related boundedness problem.

Contribution

It provides an expository overview of Stein degree, its role in birational geometry, and discusses a new boundedness problem related to this invariant.

Findings

Stein degree generalizes the notion of degree for proper morphisms.

It interacts with singularities and log Calabi-Yau fibrations.

A boundedness problem for Stein degree is proposed and discussed.

Abstract

The notion of degree begins in field theory as the dimension of a field extension. In algebraic geometry, this idea reappears as the degree of a finite morphism, defined using the induced extension of function fields. For proper morphisms that are not necessarily finite, Stein factorization isolates the finite part of the map and leads to the notion of Stein degree. This invariant is especially useful in birational geometry, where it interacts naturally with singularities of pairs and the study of log Calabi-Yau fibrations. In this article we give an expository introduction to these ideas, discuss motivating examples, and explain a boundedness problem for Stein degree arising in recent work of the author and collaborators.