Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915390172430336 |
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| author | McDuff, Dusa Siegel, Kyler |
| author_facet | McDuff, Dusa Siegel, Kyler |
| contents | We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing McDuff, Dusa Siegel, Kyler Algebraic Geometry Symplectic Geometry 53D, 14H, 14T We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces. |
| title | Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing |
| topic | Algebraic Geometry Symplectic Geometry 53D, 14H, 14T |
| url | https://arxiv.org/abs/2412.00561 |