An enriched count of nodal orbits in an invariant pencil of conics
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910925116669952 |
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| author | Bethea, Candace |
| author_facet | Bethea, Candace |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An enriched count of nodal orbits in an invariant pencil of conics Bethea, Candace Algebraic Geometry 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. |
| title | An enriched count of nodal orbits in an invariant pencil of conics |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2310.08980 |