arXiv subjects

Candace Bethea

Publications and source records attributed to Candace Bethea.

7 recordsLinked to original sources

arXiv2026

Local multiplicities for an equivariantly enriched non-transverse B\'ezout's theorem

We introduce the degree and local degree in equivariant motivic homotopy theory for the purpose of studying equivariant enumerative problems over general fields. Given a finite, tame group scheme $G$ over a field $k$ and an equivariant motivic ring spectrum $E_G$, we define the equivariant motivic degree and a corresponding local degree of a relatively $E_G$-oriented, proper, quasi-smooth morphism of $G$-schemes. We prove a local to global formula expressing the global degree as a sum of local contributions over $G$-orbits. Using these constructions, we define the Euler number of an oriented vector bundle on a quasi-smooth, proper derived stack and show that the Euler number is independent of the choice of section under appropriate hypotheses. In the presence of a finite group action, the equivariant Euler number can be computed as a sum of local equivariant degrees. As an application, we obtain an equivariantly enriched local multiplicity formula for an equivariant non-transverse B\'ezout theorem, expressing an equivariant intersection number as a sum of local equivariant degrees.

math.AG↗

arXiv2025

The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants

Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper offers a historical survey of enumerative geometry, starting with its classical origins and real counterparts, to new advances in quadratic enrichment. We include a brief survey of the paradigm shift initiated by Gromov-Witten theory, whose impact can be seen in recent results in quadratically enriched enumerative geometry. Finally, we conclude with a brief overview of emerging directions including random and equivariant enumerative geometry.

math.AG↗

arXiv2025

The equivariant degree and an enriched count of rational cubics

We define the equivariant degree and local degree of a proper $G$-equivariant map between smooth $G$-manifolds when $G$ is a compact Lie group and prove a local to global result. We show the local degree can be used to compute the equivariant Euler characteristic of a smooth, compact $G$-manifold and the Euler number of a relatively oriented $G$-equivariant vector bundle when $G$ is finite. As an application, we give an equivariantly enriched count of rational plane cubics through a $G$-invariant set of 8 general points in $\mathbb{C}\mathbb{P}^2$, valued in the representation ring and Burnside ring of a finite group. When $\mathbb{Z}/2$ acts by pointwise complex conjugation this recovers a signed count of real rational cubics.

math.AT↗

arXiv2024

Bitangents to symmetric quartics

Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group $G=\mathrm{Aut}(C)$, we prove the $G$-orbits of the bitangents are independent of the choice of $C$, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.

math.AG↗

arXiv2023

An enriched count of nodal orbits in an invariant pencil of conics

This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on $\mathbb{CP}^2$. This is both inspired by and a departure from $R(G)$-valued enrichments such as Roberts's equivariant Milnor number and Damon's equivariant signature formula. Given a $G$-invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in $A(G)$ in terms of the base locus considered as a $G$-set. We show this is true for all finite groups except $\mathbb{Z}/2\times \mathbb{Z}/2$, $A_4$, and $D_8$ and give counterexamples for the exceptional groups.

math.AG↗

arXiv2022

Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex

We show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $χ^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.

math.AG↗

arXiv2020

An Example of Wild Ramification in an Enriched Riemann-Hurwitz Formula

M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.

math.AG↗