Remarks on GW/PT under del Pezzo transitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913984242778112 |
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| author | Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng |
| author_facet | Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng |
| contents | A projective threefold transition $Y \xrightarrowϕ \bar{Y} \rightsquigarrow X$ is del Pezzo if $ϕ$ contracts a smooth del Pezzo surface to a point. We show that the GW/PT correspondence holds on $Y$ implies that it holds on $X$. In particular, a hypersurface of degree $6$ in $\mathbb{P} (3, 2, 1, 1, 1)$ gives a new example to the correspondence. The main tools are (i) the double point degeneration constructed in arXiv:2508.01374 and (ii) deformations of del Pezzo surfaces into toric surfaces (Proposition 3.12). Applications of the degeneration formulas in GW and PT then reduce the problem to known cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on GW/PT under del Pezzo transitions Lee, Shuang-Yen Wang, Chin-Lung Wang, Sz-Sheng Algebraic Geometry Primary 14N35, Secondary 57R77 A projective threefold transition $Y \xrightarrowϕ \bar{Y} \rightsquigarrow X$ is del Pezzo if $ϕ$ contracts a smooth del Pezzo surface to a point. We show that the GW/PT correspondence holds on $Y$ implies that it holds on $X$. In particular, a hypersurface of degree $6$ in $\mathbb{P} (3, 2, 1, 1, 1)$ gives a new example to the correspondence. The main tools are (i) the double point degeneration constructed in arXiv:2508.01374 and (ii) deformations of del Pezzo surfaces into toric surfaces (Proposition 3.12). Applications of the degeneration formulas in GW and PT then reduce the problem to known cases. |
| title | Remarks on GW/PT under del Pezzo transitions |
| topic | Algebraic Geometry Primary 14N35, Secondary 57R77 |
| url | https://arxiv.org/abs/2508.07715 |