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arXiv · 2610.41931

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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Juliette Bruce. 2026-10-30. A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture. https://arxiv.org/abs/2610.41931

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