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A linear bound for Fujita's freeness conjecture

  • Jingjun Han
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Abstract

Let X be a smooth complex projective variety of dimension n, and let L be an ample Cartier divisor. We prove that KX+mL is globally generated for every integer m≥⌈ C0n⌉, where C0=1.77629… is an explicit constant. In particular, KX+2nL is globally generated. We also prove that if Z is a normal projective variety of dimension n with KZ Cartier, nef and big, then |mKZ| defines a birational map for every integer m≥1+⌈ C0n+3√(2en)⌉. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If (X,Δ) is log canonical near a closed point x but is not klt at x, and W is the positive-dimensional minimal log canonical center through x, then 2e̅1(mW,x)≤(dim W−lctx((X,Δ);mx))multxW, where e̅1(mW,x) is the first normal Hilbert coefficient of the maximal ideal of 𝒪W,x. This implies multxW≤ ((a+c)a+c)/aacc, where a:=(dim W−lctx((X,Δ);mx))/2 and c:=edim𝒪W,x−dim W.

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Cite

BibTeXRIS
BibTeX
@misc{han2026linear,
  title = {{A linear bound for Fujita's freeness conjecture}},
  author = {Jingjun Han},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01574},
  eprint = {2609.01574},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01574},
}
RIS
TY  - UNPB
TI  - A linear bound for Fujita's freeness conjecture
AU  - Jingjun Han
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.01574
UR  - https://arxiv.org/abs/2609.01574
AB  - Let X be a smooth complex projective variety of dimension n, and let L be an ample Cartier divisor. We prove that K_X+mL is globally generated for every integer m≥⌈ C_0n⌉, where C_0=1.77629… is an explicit constant. In particular, K_X+2nL is globally generated. We also prove that if Z is a normal projective variety of dimension n with K_Z Cartier, nef and big, then |mK_Z| defines a birational map for every integer m≥1+⌈ C_0n+3√(2en)⌉. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If (X,Δ) is log canonical near a closed point x but is not klt at x, and W is the positive-dimensional minimal log canonical center through x, then 2e̅_1(m_W,x)≤(dim W−lct_x((X,Δ);m_x))mult_xW, where e̅_1(m_W,x) is the first normal Hilbert coefficient of the maximal ideal of 𝒪_W,x. This implies mult_xW≤ ((a+c)^a+c)/a^ac^c, where a:=(dim W−lct_x((X,Δ);m_x))/2 and c:=edim𝒪_W,x−dim W.
ER  -