Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices
Abstract
Let ℱ be a canonical foliation of general type on a smooth projective surface X, and write vol(ℱ):=vol(Kℱ). Let G⊆Bir(X,ℱ) be a finite subgroup, and let 𝒢:=ℱ/G be the quotient foliation in the birational sense. For a canonical foliation ℋ, define its r-th pluricanonical index by δr(ℋ) := min { m∈ℤ>0 | h0(mKℋ)≥ r }, where min∅:=∞. For an arbitrary foliation, these indices are computed on any canonical birational model. If κ(𝒢)≥0, we prove |G| ≤ 4δ1(𝒢) vol(ℱ), &κ(𝒢)=0, [1mm] 4/3δ2(𝒢) vol(ℱ), &κ(𝒢)=1, [3mm] δ2(𝒢)2 (1+δ2(𝒢)) vol(ℱ), &κ(𝒢)=2. Since Bir(X,ℱ) is finite, one may in particular take G=Bir(X,ℱ). When κ(𝒢)=0 or $1$, the effective bounds δ1(𝒢)≤12 and δ2(𝒢)≤42 give |G|≤48 vol(ℱ) and |G|≤56 vol(ℱ), respectively. The main new ingredient is a cluster formula for adjoint volumes, which yields index-dependent lower bounds for tangency-free foliated surface pairs whose underlying foliation has Kodaira dimension zero or one.
Related
- Bounds on automorphism groups of surfaces of general type with negative c_2
- Algebraic entropy of birational automorphism groups in families of hyper-Kähler manifolds
- Freeness of Pluricanonical Linear System on Smooth Minimal n-folds of General Type
- Foliations on abelian surfaces in positive characteristic
- Class VII surfaces with b2=3 and two foliations are Kato
BibTeX
@misc{xu2026birational,
title = {{Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices}},
author = {Shi Xu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29900},
eprint = {2608.29900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29900},
}
RIS
TY - UNPB
TI - Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices
AU - Shi Xu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29900
UR - https://arxiv.org/abs/2608.29900
AB - Let ℱ be a canonical foliation of general type on a smooth projective surface X, and write vol(ℱ):=vol(K_ℱ). Let G⊆Bir(X,ℱ) be a finite subgroup, and let 𝒢:=ℱ/G be the quotient foliation in the birational sense. For a canonical foliation ℋ, define its r-th pluricanonical index by δ_r(ℋ) := min { m∈ℤ_>0 | h^0(mK_ℋ)≥ r }, where min∅:=∞. For an arbitrary foliation, these indices are computed on any canonical birational model. If κ(𝒢)≥0, we prove |G| ≤ 4δ_1(𝒢) vol(ℱ), &κ(𝒢)=0, [1mm] 4/3δ_2(𝒢) vol(ℱ), &κ(𝒢)=1, [3mm] δ_2(𝒢)^2 (1+δ_2(𝒢)) vol(ℱ), &κ(𝒢)=2. Since Bir(X,ℱ) is finite, one may in particular take G=Bir(X,ℱ). When κ(𝒢)=0 or $1$, the effective bounds δ_1(𝒢)≤12 and δ_2(𝒢)≤42 give |G|≤48 vol(ℱ) and |G|≤56 vol(ℱ), respectively. The main new ingredient is a cluster formula for adjoint volumes, which yields index-dependent lower bounds for tangency-free foliated surface pairs whose underlying foliation has Kodaira dimension zero or one.
ER -