From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912983339237376 |
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| author | Huang, Yonghong |
| author_facet | Huang, Yonghong |
| contents | Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. A particularly interesting point is that we found they can all be obtained by performing a finite number of blow-ups on the second Hirzebruch surface. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes Huang, Yonghong Algebraic Geometry Symplectic Geometry Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. A particularly interesting point is that we found they can all be obtained by performing a finite number of blow-ups on the second Hirzebruch surface. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces. |
| title | From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes |
| topic | Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2509.14812 |