Arakelov inequalities for fibered surfaces in positive characteristic

Abstract

Let f:S→ C be a relatively minimal semistable fibration of genus g≥2 over an algebraically closed field of characteristic p>0 with smooth and geometrically connected generic fiber. Let Σ⊂ C denote the set of points over which the fibers are singular, with s=|Σ|. If the total surface S is Hodge Witt and s≥2, we prove the classical Arakelov inequality deg f_*ω_S/C≤ g/2 deg Ω_C¹(logΣ), where ω_S/C denotes the relative dualizing sheaf. For C=ℙ¹, we obtain the stronger estimate deg f_*ω_S/ℙ¹≤ g/2(s−2)−1/2 b₁(S), where b₁(S) is the first Betti number of S. We also construct a genus-$2$ semistable fibration over ℙ¹ with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and deg f_*ω_S/ℙ¹=2=g/2(s−2). Thus Nguyen's lower bound s≥4 and the Hodge--Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in g.

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BibTeX RIS

BibTeX

@misc{sun2026arakelov,
  title = {{Arakelov inequalities for fibered surfaces in positive characteristic}},
  author = {Hao Max Sun and Wan-Yuan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02760},
  eprint = {2610.02760},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02760},
}

RIS

TY  - UNPB
TI  - Arakelov inequalities for fibered surfaces in positive characteristic
AU  - Hao Max Sun
AU  - Wan-Yuan Xu
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02760
UR  - https://arxiv.org/abs/2610.02760
AB  - Let f:S→ C be a relatively minimal semistable fibration of genus g≥2 over an algebraically closed field of characteristic p>0 with smooth and geometrically connected generic fiber. Let Σ⊂ C denote the set of points over which the fibers are singular, with s=|Σ|. If the total surface S is Hodge Witt and s≥2, we prove the classical Arakelov inequality deg f_*ω_S/C≤ g/2 deg Ω_C^1(logΣ), where ω_S/C denotes the relative dualizing sheaf. For C=ℙ^1, we obtain the stronger estimate deg f_*ω_S/ℙ^1≤ g/2(s−2)−1/2 b_1(S), where b_1(S) is the first Betti number of S. We also construct a genus-$2$ semistable fibration over ℙ^1 with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and deg f_*ω_S/ℙ^1=2=g/2(s−2). Thus Nguyen's lower bound s≥4 and the Hodge--Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in g.
ER  -

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