Gromov-Witten/Hurwitz wall-crossing
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866908399898198016 |
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| author | Nesterov, Denis |
| author_facet | Nesterov, Denis |
| contents | For a target variety $X$ and a nodal curve $C$, we introduce a one-parameter stability condition, termed $ε$-admissibility, for maps from nodal curves to $X\times C$. If $X$ is a point, $ε$-admissibility interpolates between moduli spaces of stable maps to $C$ relative to some fixed points and moduli spaces of admissible covers with arbitrary ramifications over the same fixed points and simple ramifications elsewhere on $C$. Using Zhou's entangled tails, we prove wall-crossing formulas relating invariants for different values of $ε$. If $X$ is a surface, we use this wall-crossing in conjunction with author's quasimap wall-crossing to show that the relative Pandharipande-Thomas/Gromov-Witten correspondence of $X\times C$ and Ruan's extended crepant resolution conjecture of the pair $X^{[n]}$ and $[X^{(n)}]$ are equivalent up to explicit wall-crossings. We thereby prove the crepant resolution conjecture for 3-point genus-0 invariants in all classes, if $X$ is a toric del Pezzo surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00889 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gromov-Witten/Hurwitz wall-crossing Nesterov, Denis Algebraic Geometry For a target variety $X$ and a nodal curve $C$, we introduce a one-parameter stability condition, termed $ε$-admissibility, for maps from nodal curves to $X\times C$. If $X$ is a point, $ε$-admissibility interpolates between moduli spaces of stable maps to $C$ relative to some fixed points and moduli spaces of admissible covers with arbitrary ramifications over the same fixed points and simple ramifications elsewhere on $C$. Using Zhou's entangled tails, we prove wall-crossing formulas relating invariants for different values of $ε$. If $X$ is a surface, we use this wall-crossing in conjunction with author's quasimap wall-crossing to show that the relative Pandharipande-Thomas/Gromov-Witten correspondence of $X\times C$ and Ruan's extended crepant resolution conjecture of the pair $X^{[n]}$ and $[X^{(n)}]$ are equivalent up to explicit wall-crossings. We thereby prove the crepant resolution conjecture for 3-point genus-0 invariants in all classes, if $X$ is a toric del Pezzo surface. |
| title | Gromov-Witten/Hurwitz wall-crossing |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2208.00889 |