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Groebner fans and embedded resolutions of ideals on toric varieties

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Abstract

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the “Groebner fan” of such an ideal is actually a polyhedral fan and that a subvariety defined by a Newton non-degenerate ideal on a toric variety \(X_\sigma \) admits a toric embedded resolution of singularities \(Z\longrightarrow X_\sigma .\)

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Acknowledgements

Fuensanta Aroca is partially supported by UNAM-DGAPA projects IN108320 and IN113323. Hussein Mourtada would like to thank the UNAM, Mexico City, for its hospitality during the preparation of this article and the UMI LaSol which has funded the visit. The authors would like to thank the referee for his careful reading and comments.

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Authors and Affiliations

  1. Unidad Cuernavaca, Universidad Nacional Autónoma de México, A.P. 273-3, CP 62251, Cuernavaca, MOR, Mexico

    Fuensanta Aroca

  2. ENES, Unidad Morelia, Universidad Nacional Autónoma de México, Antigua Carretera a Pátzcuaro No. 8701 Col. Ex Hacienda de San José de la Huerta, CP 58190, Morelia, Michoacán, Mexico

    Mirna Gómez-Morales

  3. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité, Sorbonne Université, CNRS, 75013, Paris, France

    Hussein Mourtada

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  1. Fuensanta Aroca
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Aroca, F., Gómez-Morales, M. & Mourtada, H. Groebner fans and embedded resolutions of ideals on toric varieties. Beitr Algebra Geom 65, 217–228 (2024). https://doi.org/10.1007/s13366-023-00684-0

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