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Creators/Authors contains: "Bethea, Candace"

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  1. ; (, arXiv pre-print repository)
    NA (Ed.)
    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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    Free, publicly-accessible full text available February 16, 2026
  2. ; (, arXiv pre-print repository)
    NA (Ed.)
    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=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. 
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    Accepted Manuscript Full Text Available
  3. (, arXiv pre-print repository)
    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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    Accepted Manuscript Full Text Available
  4. ; ; ; ; (, Topology and its Applications)
    Full Text Available