Locally non-trivial fibred surfaces with maximal unitary rank
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908475318075392 |
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| author | Stoppino, Lidia |
| author_facet | Stoppino, Lidia |
| contents | Let $f\colon S\to B$ a locally non-trivial fibred surface with fibres of genus $g$. Let $u_f$ be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when $u_f$ is maximal, i.e. $u_f=g-1$. In this case necessarily $g\leq 6$, but examples in genus $5$ and $6$ are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case $g=6$, $u_f=5$. In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally non-trivial fibred surfaces with maximal unitary rank Stoppino, Lidia Algebraic Geometry 14J10, 14D06, 14D07, 14G35, 14H40 Let $f\colon S\to B$ a locally non-trivial fibred surface with fibres of genus $g$. Let $u_f$ be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when $u_f$ is maximal, i.e. $u_f=g-1$. In this case necessarily $g\leq 6$, but examples in genus $5$ and $6$ are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case $g=6$, $u_f=5$. In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor |
| title | Locally non-trivial fibred surfaces with maximal unitary rank |
| topic | Algebraic Geometry 14J10, 14D06, 14D07, 14G35, 14H40 |
| url | https://arxiv.org/abs/2508.00618 |