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PreprintCC BY 4.0arXiv

Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions

  • Meng Chen
  • Yong Hu
  • Chen Jiang
  • Yang Liu
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Abstract

For a smooth projective $3$-fold V of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume Ivol(V) and the geometric genus pg(V). Specifically, for a smooth projective $3$-fold V of Kodaira dimension $2$, we show that Ivol(V)≥ 3/4pg(V)−5/4; while if the Kodaira dimension of V is $1$, then Ivol(V)≥ pg(V)−1. Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of V induces the Iitaka fibration and the canonical system of V is composed with a pencil of surfaces whose general member has Iitaka volume 1/2 provided that pg(V)≥ 4.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions}},
  author = {Meng Chen and Yong Hu and Chen Jiang and Yang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02689},
  eprint = {2610.02689},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02689},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions
AU  - Meng Chen
AU  - Yong Hu
AU  - Chen Jiang
AU  - Yang Liu
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02689
UR  - https://arxiv.org/abs/2610.02689
AB  - For a smooth projective $3$-fold V of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume Ivol(V) and the geometric genus p_g(V). Specifically, for a smooth projective $3$-fold V of Kodaira dimension $2$, we show that Ivol(V)≥ 3/4p_g(V)−5/4; while if the Kodaira dimension of V is $1$, then Ivol(V)≥ p_g(V)−1. Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of V induces the Iitaka fibration and the canonical system of V is composed with a pencil of surfaces whose general member has Iitaka volume 1/2 provided that p_g(V)≥ 4.
ER  -