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Torical modification of Newton non-degenerate ideals

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

We give a definition of Newton non-degeneracy independent of the system of generators defining the variety. This definition extends the notion of Newton non-degeneracy to varieties that are not necessarily complete intersection. As in the previous definition of non-degeneracy for complete intersection varieties, it is shown that the varieties satisfying our definition can be resolved with a toric modification. Using tools of both toric and tropical geometry we describe the toric modification in terms of the Gröbner fan of the ideal defining the variety. The first part of the paper is devoted to introducing the classical concepts and the proof for the hypersurface case.

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Notes

  1. Since \(G_\sigma \) is a Gröbner basis, it is in particular a normalised standard basis and the equality holds always (see for example [15, 16] or, for the analytic case [2]).

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Acknowledgments

M. Gómez-Morales would like to thank Carles Bibia-Ausina and Dmitry Kerner, for clarifying discussions on the subject and, to Meral Tosun, for the example in Sect. 11. K. Shabbir would like to thank A. Jensen for answering several questions by e-mail. In particular for the proof of Proposition 10.6.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Mexico

    Fuensanta Aroca

  2. Mathematics Institute, University of Warwick, Coventry, UK

    Mirna Gómez-Morales

  3. Department of Mathematics, GC University Lahore, Lahore, Pakistan

    Khurram Shabbir

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  1. Fuensanta Aroca
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  2. Mirna Gómez-Morales
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  3. Khurram Shabbir
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Correspondence to Fuensanta Aroca.

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To Professor Heisuke Hironaka on the occasion of his 80th birthday

Research partially supported by PAPIIT-UNAM, CONACYT (Mexico) Grants 55084, 162340 and 117110. K. Shabbir was partially supported by CONACYT (Mexico) and TWAS (Italy) Grant FR 3240223595. He wants to thank these institutions for their support.

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Aroca, F., Gómez-Morales, M. & Shabbir, K. Torical modification of Newton non-degenerate ideals. RACSAM 107, 221–239 (2013). https://doi.org/10.1007/s13398-012-0100-8

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