Semistable Reduction of Plane Quartics
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914164899840000 |
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| author | Schwegele, Max |
| author_facet | Schwegele, Max |
| contents | The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite field extension. Computing this model is a central problem in arithmetic geometry. For non-hyperelliptic genus $3$ curves, which are canonically embedded as plane quartics, methods like admissible reduction become challenging in small residue characteristics. This thesis establishes a precise connection between the abstractly defined stable model and computationally accessible GIT-stable plane models. We prove that a GIT-stable plane model of a smooth plane quartic exists if and only if its stable reduction is non-hyperelliptic. When this condition holds, we show that the stable model is the unique minimal semistable model that dominates the GIT-stable model. The corresponding domination morphism is geometrically explicit: it contracts the $1$-tails of the stable reduction to cusps on the special fiber of the GIT-stable model and is an immersion elsewhere. This result provides a geometric framework for computing the stable model by first finding a GIT-stable model and then resolving its cuspidal singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semistable Reduction of Plane Quartics Schwegele, Max Algebraic Geometry Number Theory 14H10, 11G20, 14L24, 14H50, 14H25 The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite field extension. Computing this model is a central problem in arithmetic geometry. For non-hyperelliptic genus $3$ curves, which are canonically embedded as plane quartics, methods like admissible reduction become challenging in small residue characteristics. This thesis establishes a precise connection between the abstractly defined stable model and computationally accessible GIT-stable plane models. We prove that a GIT-stable plane model of a smooth plane quartic exists if and only if its stable reduction is non-hyperelliptic. When this condition holds, we show that the stable model is the unique minimal semistable model that dominates the GIT-stable model. The corresponding domination morphism is geometrically explicit: it contracts the $1$-tails of the stable reduction to cusps on the special fiber of the GIT-stable model and is an immersion elsewhere. This result provides a geometric framework for computing the stable model by first finding a GIT-stable model and then resolving its cuspidal singularities. |
| title | Semistable Reduction of Plane Quartics |
| topic | Algebraic Geometry Number Theory 14H10, 11G20, 14L24, 14H50, 14H25 |
| url | https://arxiv.org/abs/2511.15858 |