Corpus ID: 208006103

# $\mathbb{A}^{1}$-Local Degree via Stacks

@article{Kobin2019mathbbA1LocalDV,
title={\$\mathbb\{A\}^\{1\}\$-Local Degree via Stacks},
author={Andrew Kobin and Libby Taylor},
journal={arXiv: Algebraic Geometry},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Algebraic Geometry
We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of… Expand
2 Citations
The trace of the local $\mathbf{A}^1$-degree
• Mathematics
• 2019
We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over theExpand
An arithmetic enrichment of B\'ezout's Theorem
The classical version of Bezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass andExpand

#### References

SHOWING 1-10 OF 32 REFERENCES
An enriched count of the bitangents to a smooth plane quartic curve
• Mathematics
• 2019
Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that otherExpand
A "bottom up" characterization of smooth Deligne-Mumford stacks
• Mathematics
• 2015
In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquelyExpand
An Arithmetic Count of the Lines on a Smooth Cubic Surface
• Mathematics
• 2017
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the numberExpand
Intrinsic signs and lower bounds in real algebraic geometry
• Mathematics
• 2011
A classical result due to Segre states that on a real cubic surface in ${\mathbb P}^3_\R$ there exists two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines areExpand
An arithmetic enrichment of Bézout’s Theorem
The classical version of Bézout’s Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass andExpand
3264 and All That: A Second Course in Algebraic Geometry
• Mathematics
• 2016
Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. SingularExpand
Artin-Schreier Root Stacks
We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve,Expand
The canonical ring of a stacky curve
• Mathematics
• 2015
Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Grobner basis. We work in a generalExpand
A1-algebraic topology
We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationshipExpand
THE CLASS OF EISENBUD–KHIMSHIASHVILI–LEVINE IS THE LOCAL A-BROUWER DEGREE
Given a polynomial function with an isolated zero at the origin, we prove that the local A-Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by DavidExpand