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Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal
Alex Fink1, Jeffrey Giansiracusa2, Noah Giansiracusa3
1Queen Mary University of London, London, England, UK.
Research in the Mathematical Sciences
|April 28, 2025
Summary
Researchers introduce the Macaulay tropical ideal, a canonical extension of principal ideals to tropical ideals using transversal matroids. This construction yields non-realizable hypersurface schemes, agreeing with prior work for specific cases.
Area of Science:
- Algebraic Geometry
- Tropical Geometry
- Commutative Algebra
Background:
- Tropical ideals are defined within the idempotent semiring of tropical polynomials.
- These ideals must also satisfy the property of being a tropical linear space, degree by degree.
Purpose of the Study:
- To introduce a canonical construction for extending principal ideals to tropical ideals.
- To establish a universal property for this new construction, termed the Macaulay tropical ideal.
- To investigate the properties of the resulting tropical ideals, particularly in relation to non-realizable schemes.
Main Methods:
- A novel construction based on transversal matroids is employed to extend principal ideals.
- The constructed Macaulay tropical ideal is shown to possess a universal property concerning other extensions.
- The construction is applied to generate specific examples of non-realizable degree d hypersurface schemes.
Main Results:
- The Macaulay tropical ideal is introduced as a canonical extension of any principal ideal to a tropical ideal.
- This construction possesses a universal property: any other valid extension is a weak image of the Macaulay tropical ideal.
- For specific parameters, the construction yields non-realizable degree d hypersurface schemes, aligning with existing results for lines.
Conclusions:
- The Macaulay tropical ideal provides a systematic method for extending principal ideals in tropical geometry.
- The construction offers a new perspective on generating non-realizable algebraic objects within tropical algebraic geometry.
- The comparison with other extension methods, as detailed in the appendix, further clarifies the significance of this construction.
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