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PreprintCC BY 4.0arXiv

The F-signature Function on the Big Cone

  • Seungsu Lee
  • Suchitra Pande
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Abstract

We define and study the F-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety X, we show that the F-signature function defines a continuous function on the big cone of X, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the F-signature of any big ℝ-divisor on X is equivalent to the global F-regularity of X. As applications, we prove a birational transformation rule for the F-signature in terms of the F-signature of certain graded ideal sequences. We also prove that in a reduction modulo p situation, a characteristic independent lower bound for the F-signature of X with respect to any ample divisor L spreads out to a similar bound for arbitrary big divisors on X.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026fsignature,
  title = {{The $F$-signature Function on the Big Cone}},
  author = {Seungsu Lee and Suchitra Pande},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36321},
  eprint = {2609.36321},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36321},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The F-signature Function on the Big Cone
AU  - Seungsu Lee
AU  - Suchitra Pande
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36321
UR  - https://arxiv.org/abs/2609.36321
AB  - We define and study the F-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety X, we show that the F-signature function defines a continuous function on the big cone of X, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the F-signature of any big ℝ-divisor on X is equivalent to the global F-regularity of X. As applications, we prove a birational transformation rule for the F-signature in terms of the F-signature of certain graded ideal sequences. We also prove that in a reduction modulo p situation, a characteristic independent lower bound for the F-signature of X with respect to any ample divisor L spreads out to a similar bound for arbitrary big divisors on X.
ER  -