An Introduction to $\mathbf{A}^1$-Enumerative Geometry

TL;DR

This paper introduces $ ext{A}^1$-enumerative geometry, a modern approach that extends classical enumerative geometry using $ ext{A}^1$-homotopy theory, providing new tools and examples over various fields.

Contribution

It offers an accessible overview of $ ext{A}^1$-enumerative geometry, including enriched local degrees, $ ext{A}^1$-Milnor numbers, and recent computational techniques.

Findings

Discussion of enriched local degrees of morphisms

Introduction of $ ext{A}^1$-Milnor numbers

Presentation of computational tools and examples

Abstract

We provide an expository introduction to $\mathbb{A}^1$-enumerative geometry, which uses the machinery of $\mathbb{A}^1$-homotopy theory to enrich classical enumerative geometry questions over a broader range of fields. Included is a discussion of enriched local degrees of morphisms of smooth schemes, following Morel, $\mathbb{A}^1$-Milnor numbers, as well as various computational tools and recent examples. Based off lectures delivered by Kirsten Wickelgren at the 2018 LMS-CMI summer school ''Homotopy Theory and Arithmetic Geometry, Motivic and Diophantine Aspects.''