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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 CP2. This can be thought of as spearheading equivariant enumerative enrichments valued in the Burnside Ring, both inspired by and a departure from R(G)-valued enrichments such as Roberts’ 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 terms of the base locus considered as a G-set. We show this is true for all finite groups except Z/2 × Z/2 and D8 and give counterexamples for the two exceptional groups.
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This content will become publicly available on February 16, 2026
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 ℂℙ2, valued in the representation ring and Burnside ring of a finite group. When ℤ/2 acts by pointwise complex conjugation this recovers a signed count of real rational cubics.
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- Award ID(s):
- 2402099
- PAR ID:
- 10583725
- Editor(s):
- NA
- Publisher / Repository:
- arXiv pre-print repository
- Date Published:
- Edition / Version:
- 1
- Volume:
- 1
- Issue:
- 1
- Page Range / eLocation ID:
- 1-42
- Subject(s) / Keyword(s):
- Equivariant homotopy, degree theory, euler characteristic, euler number, rational curves, K-theory, equivariant cohomology
- Format(s):
- Medium: X Size: 416KB Other: 1
- Size(s):
- 416KB
- Sponsoring Org:
- National Science Foundation
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