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Division formulas on projective varieties

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Abstract

We introduce a division formula on a possibly singular projective subvariety X of complex projective space \({\mathbb {P}}^N\), which, e.g., provides explicit representations of solutions to various polynomial division problems on the affine part of X. Especially we consider a global effective version of the Briançon–Skoda–Huneke theorem.

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Notes

  1. As does the ingenious formula in [10] in case with Nullstellensatz data and \(X={\mathbb {P}}^n\).

  2. The optimal result for case \(V={\mathbb {C}}^n\) was proved by Kollár, [20], for \(d\ge 3\); see [21] and [19] for \(d=2\).

  3. Usually ”z in \({\mathbb {P}}^N\)” means that \(z\in {\mathbb {C}}^{N+1}{\setminus }\{0\}\) and the point in question is \(\pi (z)\) in \({\mathbb {P}}^N\)”.

  4. The initial minus sign here is because we have \(\delta _{w-z}\) rather than \(\delta _{z-w}\) as in [3].

  5. These mappings \(\delta _f\) are thus instances of the mappings \(\varphi _j\) in Sects. 2.1 and 2.2.

  6. Since \(R^E\) has even degree this is consistent with (3.2), cf. Remark 6.1.

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Acknowledgments

We are grateful for the referee’s careful reading and suggestions to improve the presentation.

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Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, 412 96, Göteborg, Sweden

    Mats Andersson & Lisa Nilsson

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  1. Mats Andersson
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  2. Lisa Nilsson
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Correspondence to Mats Andersson.

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The first author was partially supported by the Swedish Research Council. The second author was supported by a postdoc grant from the Swedish Research Council.

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Andersson, M., Nilsson, L. Division formulas on projective varieties. Math. Z. 284, 575–593 (2016). https://doi.org/10.1007/s00209-016-1667-0

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