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An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes

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A combinatorial polytope P is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. In this paper, we show that McMullen’s operations preserve not only projective uniqueness but also graphicality. As an application, we show that large families of order polytopes are graphic and thus projectively unique.

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Correspondence to João Gouveia.

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The first and third authors were supported by internal research Grants (INV-2017-51-1453 and INV-2018-48-1373, respectively) from the Faculty of Sciences of the Universidad de los Andes. These Grants allowed them to visit the second author and complete key steps of this project. The third author is also being supported in his doctoral studies, of which this project forms a part, by the Colombian science agency Colciencias. The second author was supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

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Bogart, T., Gouveia, J. & Torres, J.C. An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes. Discrete Comput Geom 67, 462–491 (2022).

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