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Nash multiplicities and resolution invariants

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Abstract

The Nash multiplicity sequence was defined by Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities. Moreover, this indicates that these invariants from constructive resolution are intrinsic to the variety since they can be read in terms of its space of arcs. This result is a first step connecting explicitly arc spaces and algorithmic resolution of singularities.

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Notes

  1. W is just a variable in charge of the degree of the \(I_i\).

  2. Here \(\nu _{\xi }(I)\) denotes the order of the ideal I in the regular local ring \(R_{{\mathcal {M}}_{\xi }}\), where \({\mathcal {M}}_{\xi }\) is the ideal defining the point \(\xi \).

  3. Actually, the Hilbert–Samuel function has to be modified in order to be semicontinuous (see [2]). Here we refer to this modification of the Hilbert–Samuel function.

  4. For simplicity, we will sometimes write \(\xi \) when we refer to the image of \(\xi ^{(n)}\) by most of the maps we use in this article. In particular, we will often write \({\mathrm {ord}}_{\xi }\) meaning \({\mathrm {ord}}_{\xi ^{(n)}}\) or \({\mathrm {ord}}_{\xi ^{(d)}}\).

  5. Note that it is already differentially closed with respect to x.

  6. No larger than the invariant \(\tau \) at \(\xi \), see [1] for more details.

  7. This condition just means that the intersection of \(\mathrm {Ker}(d\beta )\) and the tangent space of \({\mathcal {G}}^{(n)}\) at \(\xi \) is 0. This guarantees that \(\beta \) induces a homeomorphism between \({\mathrm {Sing}}({\mathcal {G}}^{(n)})\) and \(\beta ({\mathrm {Sing}}({\mathcal {G}}^{(n)}))\).

  8. For this, it suffices to have \({\mathcal {G}}^{(n)}\) differentially closed with respect to \(\beta \), that is, closed under the action of the sheaf of relative differential opperators \(\mathrm {Diff}_{V^{(n)}/V^{(n-e)}}\).

  9. Note that we can always assume this situation for any \(\xi \in X\), since one can always consider a neighbourhood of \(\xi \) where this is true.

  10. Therefore, for our purpose, we need to choose arcs in a way such that they are not contained in the set of points of highest multiplicity of X (that is, \(\varphi ^*(<0>)\nsubseteq {\mathrm {\underline{Max}\; mult}}(X)\)).

  11. For simplicity of the notation, we will often identify the points \(\xi _i\) in \(X_i^{(d+1)}\) with their images in \(V_i^{(n+1)}\).

  12. We use the same notation for the image of \(\xi \) by \(p*\) and by \(\bar{\delta }\).

  13. \(C_0\) is a smooth curve in a local ring, and hence a complete intersection.

  14. Note that \({\mathcal {G}}_{\varphi }\) and \({\mathcal {G}}_{\tilde{X}_0}^{(n+1)}\) are differentially closed by definition.

  15. Note that \({\mathcal {G}}_{\varphi }^{(n+1)}\) has order one (see (4.7)).

  16. As we have done already, we will write \(\xi \) for the image of \(\xi \) under most of the morphisms we use, as long as the identification between both points is clear.

  17. Note that \(\mathrm {Ker}(\tilde{\Gamma }_0)=\mathrm {Ker}(\Gamma _0)({\mathcal {O}}_{V^{(n)},\xi }\otimes _{k} K[[t]])_{\xi _0}\).

  18. If \(q\in R\) for a regular local ring R with maximal ideal \({\mathcal {M}}\), then we denote by \(\mathrm {in}_{\xi }(q)\) the initial part of q at the closed point \(\xi \), meaning the equivalence class of q in the quotient \({\mathcal {M}}^n/{\mathcal {M}}^{n+1}\), where n is such that \(q\in {\mathcal {M}}^n\) but \(q\notin {\mathcal {M}}^{n+1}\). Therefore \(\mathrm {in}_{\xi }(q)\in \mathrm {Gr}_{R_{{\mathcal {M}}}}\cong k'[z_1,\ldots ,z_d]\) is a homogeneous polynomial of degree n.

  19. We know that such an arc exists by Remark 5.9.

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Acknowledgements

The authors profited from conversations with F. Aroca, R. Docampo, S. Ishii and H. Kawanoue, as well as from the support of their research team. We also thank C. Abad, J. Conde-Alonso and I. Efraimidis for their help with the presentation, and the referee for many useful suggestions.

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Authors and Affiliations

  1. Depto. Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid and ICMAT (Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM), Cantoblano, 28049, Madrid, Spain

    A. Bravo & B. Pascual-Escudero

  2. Depto. Matemática Aplicada, and IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Paseo de Belén 7, 47011, Valladolid, Spain

    S. Encinas

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  1. A. Bravo
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Correspondence to S. Encinas.

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A. Bravo, S. Encinas and B. Pascual-Escudero were partially supported by MTM2012-35849. B. Pascual-Escudero was supported by BES-2013-062656.

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Bravo, A., Encinas, S. & Pascual-Escudero, B. Nash multiplicities and resolution invariants. Collect. Math. 68, 175–217 (2017). https://doi.org/10.1007/s13348-016-0188-9

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