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PreprintCC BY 4.0arXiv

Graded Betti numbers of general curves of large degree

  • JeongDon Lee
  • Li Li
  • Jinhyung Park
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Abstract

Let C be a smooth projective complex curve of genus g and gonality k, and L be a very ample line bundle on C. When L has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups Kp,q(C,L) have been determined previously, but the exact values of the graded Betti numbers κp,q(C, L) remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers κp,q(C, L) when the Brill--Noether locus Wk1(C) has the expected dimension and H1(C, L ⊗ ωC−1)=0. Consequently, we determine the complete Betti table for a general curve when °L ≥ 4g−3 or when °L ≥ 3g−3 and L is general. We also explicitly compute the Boij--Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026graded,
  title = {{Graded Betti numbers of general curves of large degree}},
  author = {JeongDon Lee and Li Li and Jinhyung Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11161},
  eprint = {2609.11161},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11161},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Graded Betti numbers of general curves of large degree
AU  - JeongDon Lee
AU  - Li Li
AU  - Jinhyung Park
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11161
UR  - https://arxiv.org/abs/2609.11161
AB  - Let C be a smooth projective complex curve of genus g and gonality k, and L be a very ample line bundle on C. When L has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups K_p,q(C,L) have been determined previously, but the exact values of the graded Betti numbers κ_p,q(C, L) remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers κ_p,q(C, L) when the Brill--Noether locus W_k^1(C) has the expected dimension and H^1(C, L ⊗ ω_C^−1)=0. Consequently, we determine the complete Betti table for a general curve when °L ≥ 4g−3 or when °L ≥ 3g−3 and L is general. We also explicitly compute the Boij--Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.
ER  -