Equivariant enumerative geometry
| Autor: | Brazelton, Thomas |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Rewrites to the discussions of Pontryagin-Thom transfers, general additions to improve readability. 34 pages, comments welcome! |
| Databáze: | arXiv |
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