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PreprintarXiv licencearXiv

A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture

  • Juliette Bruce
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Abstract

We give a tropical perspective on the recent arguments announced by OpenAI for Nagata's conjecture and the Nagata--Biran--Szemberg conjecture. By degenerating to both the normal cone and a semistable model whose dual graph is a cycle, we replace sections on a surface by Laurent polynomials in two variables. In place of theta functions, tropical methods bound the exponents of these Laurent polynomials, and a lift of the tropical Abel--Jacobi class of the normal bundle controls the shape of the resulting exponent diagram. Combining these support bounds with interpolation on a torus proves the Nagata--Biran--Szemberg conjecture with its predicted threshold whenever the corresponding linear series contains an integral curve whose only singularity is an ordinary node that is not a basepoint.

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Cite

BibTeXRIS
BibTeX
@misc{bruce2026tropical,
  title = {{A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture}},
  author = {Juliette Bruce},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41931},
  eprint = {2610.41931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41931},
}
RIS
TY  - UNPB
TI  - A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture
AU  - Juliette Bruce
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41931
UR  - https://arxiv.org/abs/2610.41931
AB  - We give a tropical perspective on the recent arguments announced by OpenAI for Nagata's conjecture and the Nagata--Biran--Szemberg conjecture. By degenerating to both the normal cone and a semistable model whose dual graph is a cycle, we replace sections on a surface by Laurent polynomials in two variables. In place of theta functions, tropical methods bound the exponents of these Laurent polynomials, and a lift of the tropical Abel--Jacobi class of the normal bundle controls the shape of the resulting exponent diagram. Combining these support bounds with interpolation on a torus proves the Nagata--Biran--Szemberg conjecture with its predicted threshold whenever the corresponding linear series contains an integral curve whose only singularity is an ordinary node that is not a basepoint.
ER  -