Equivariant enumerative geometry
Fuente:
arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913419602427904 |
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| author | Brazelton, Thomas |
| author_facet | Brazelton, Thomas |
| contents | We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin--Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the $S_4$-action on $\mathbb{C}P^3$ are given by $[S_4/C_2]+[S_4/C_2'] + [S_4/D_8]$, where $C_2$ and $C_2'$ denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_08622 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equivariant enumerative geometry Brazelton, Thomas Algebraic Topology Algebraic Geometry We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin--Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the $S_4$-action on $\mathbb{C}P^3$ are given by $[S_4/C_2]+[S_4/C_2'] + [S_4/D_8]$, where $C_2$ and $C_2'$ denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines. |
| title | Equivariant enumerative geometry |
| topic | Algebraic Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2210.08622 |