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A higher-order tangent map and a conjecture on the higher Nash blowup of curves

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Abstract

We introduce a higher-order version of the tangent map of a morphism and find a matrix representation. We then apply this matrix to solve a conjecture by Yasuda regarding the semigroup of the higher Nash blowup of formal curves. We first show that the conjecture is true for toric curves. We conclude by exhibiting a family of non-monomial curves where the conjecture fails.

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Acknowledgements

We would like to thank Pedro González Pérez for explaining us some of his results (joint with B. Teissier) from [11]. We also thank Mark Spivakovsky for stimulating discussions. We are very grateful to the referee for his/her careful reading of the paper, for several valuable comments that greatly improved the presentation of some parts of the paper, and for suggesting the study of the independence from the generators of a semigroup of the logarithmic Jacobian ideal of higher order.

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Authors and Affiliations

  1. Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos, Mexico

    Enrique Chávez-Martínez

  2. CONACyT-Universidad Autónoma de Zacatecas, Jardín Juárez 147, Centro Histórico, C.P. 98000, Zacatecas, Zacatecas, Mexico

    Daniel Duarte

  3. Universidad Autónoma de Aguascalientes, Ciudad Universitaria, Av. Universidad 940, C. P. 20131, Aguascalientes, AGS, Mexico

    Arturo Giles Flores

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  1. Enrique Chávez-Martínez
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  2. Daniel Duarte
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  3. Arturo Giles Flores
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Correspondence to Daniel Duarte.

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D. Duarte: Research supported by CONACyT grant 287622.

A. Giles Flores: Research supported by CONACyT grant 221635.

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Chávez-Martínez, E., Duarte, D. & Giles Flores, A. A higher-order tangent map and a conjecture on the higher Nash blowup of curves. Math. Z. 297, 1767–1791 (2021). https://doi.org/10.1007/s00209-020-02579-5

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