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arXiv · 2405.16338

Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

Alex Fink·Jeffrey Giansiracusa·Noah Giansiracusa·Joshua Mundinger
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Abstract

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rinc\'on produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

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Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger. 2024-05-25. Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal. https://arxiv.org/abs/2405.16338

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