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Seshadri functions on surfaces

  • Cyril J. Jacob
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Abstract

We study the Seshadri function on the ample cone of a smooth complex projective surface at a fixed point. At a submaximal class, we show that the function is differentiable if and only if there is a unique Seshadri curve. At a maximal class L, differentiability is equivalent to every curve C computing the Seshadri constant satisfying [C]/(multx C)=[L]/√(L2). We also prove that the non-differentiability locus is nowhere dense. For the blow-up of ℙ2 at r≥ 9 very general points, assuming the (−1)-curves conjecture for the blow-up at r+1 very general points, we construct a non-empty open subset of the ample cone on which the Seshadri function at a very general point equals √(L2). Consequently, this function is not piecewise linear. Under the assumption of the Strong SHGH conjecture, we obtain a common non-empty open subset on which maximality holds at every point when r≥36.

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Cite

BibTeXRIS
BibTeX
@misc{jacob2026seshadri,
  title = {{Seshadri functions on surfaces}},
  author = {Cyril J. Jacob},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01138},
  eprint = {2611.01138},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01138},
}
RIS
TY  - UNPB
TI  - Seshadri functions on surfaces
AU  - Cyril J. Jacob
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01138
UR  - https://arxiv.org/abs/2611.01138
AB  - We study the Seshadri function on the ample cone of a smooth complex projective surface at a fixed point. At a submaximal class, we show that the function is differentiable if and only if there is a unique Seshadri curve. At a maximal class L, differentiability is equivalent to every curve C computing the Seshadri constant satisfying [C]/(mult_x C)=[L]/√(L^2). We also prove that the non-differentiability locus is nowhere dense. For the blow-up of ℙ^2 at r≥ 9 very general points, assuming the (−1)-curves conjecture for the blow-up at r+1 very general points, we construct a non-empty open subset of the ample cone on which the Seshadri function at a very general point equals √(L^2). Consequently, this function is not piecewise linear. Under the assumption of the Strong SHGH conjecture, we obtain a common non-empty open subset on which maximality holds at every point when r≥36.
ER  -