Poincaré polynomials of moduli spaces of one-dimensional sheaves on the projective plane
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| Format: | Preprint |
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2025
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| _version_ | 1866915189342863360 |
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| author | Guo, Shuai Wu, Longting Moreira, with an appendix by Miguel |
| author_facet | Guo, Shuai Wu, Longting Moreira, with an appendix by Miguel |
| contents | Let $M_β$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $β$ with Euler characteristic one. We show that the divisibility property of the Poincaré polynomial of $M_β$, proposed by Choi-van Garrel-Katz-Takahashi follows from Bousseau's conjectural refined sheaves/Gromov-Witten correspondence. Since this correspondence is known for $S=\mathbb{P}^2$, our result proves Choi-van Garrel-Katz-Takahashi's conjecture in this case.
For $S=\mathbb{P}^2$, our proof also introduces a novel approach to computing the Poincaré polynomials using Gromov-Witten invariants of local $\mathbb{P}^2$ and a local elliptic curve. Specifically, we compute the Poincaré polynomials of $M_{d}$ with degrees $d\leq 16$ and derive a closed formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 6$ and $i\leq 4d-22$. We also propose a conjectural formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 4$ and $i\leq 6d-20$. In the Appendix (by M. Moreira), a more general conjecture concerning the higher range Betti numbers of $M_{d}$ is presented, along with another conjecture that involves refinements from the perverse/Chern filtration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_05622 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincaré polynomials of moduli spaces of one-dimensional sheaves on the projective plane Guo, Shuai Wu, Longting Moreira, with an appendix by Miguel Algebraic Geometry Let $M_β$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $β$ with Euler characteristic one. We show that the divisibility property of the Poincaré polynomial of $M_β$, proposed by Choi-van Garrel-Katz-Takahashi follows from Bousseau's conjectural refined sheaves/Gromov-Witten correspondence. Since this correspondence is known for $S=\mathbb{P}^2$, our result proves Choi-van Garrel-Katz-Takahashi's conjecture in this case. For $S=\mathbb{P}^2$, our proof also introduces a novel approach to computing the Poincaré polynomials using Gromov-Witten invariants of local $\mathbb{P}^2$ and a local elliptic curve. Specifically, we compute the Poincaré polynomials of $M_{d}$ with degrees $d\leq 16$ and derive a closed formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 6$ and $i\leq 4d-22$. We also propose a conjectural formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 4$ and $i\leq 6d-20$. In the Appendix (by M. Moreira), a more general conjecture concerning the higher range Betti numbers of $M_{d}$ is presented, along with another conjecture that involves refinements from the perverse/Chern filtration. |
| title | Poincaré polynomials of moduli spaces of one-dimensional sheaves on the projective plane |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2501.05622 |