Locally non-trivial fibred surfaces with maximal unitary rank

Lidia Stoppino

doi:10.1007/s40574-025-00493-4 10.1007/s40574-025-00493-4

paper

Locally non-trivial fibred surfaces with maximal unitary rank

arXiv:2508.00618 · doi:10.1007/s40574-025-00493-4 10.1007/s40574-025-00493-4

Abstract

Let $f : S \to B$ 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

15 pages. To appear in Bollettino dell'Unione Matematica Italiana. arXiv admin note: substantial text overlap with arXiv:2406.15909

References in corpus (2)

Cited by in corpus (1)

Fibred surfaces and their unitary rank