Abstract
In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall.
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Notes
By an abuse of notation, we will also denote \(\mathcal {J}\left( \pmb {\mathfrak {a}}^{{\pmb {c}}}\right) \) its stalk at O so we will omit the word “sheaf” if no confusion arises.
The order on the set of points \(\{\mathbf{c}_1, \ldots , \mathbf{c}_k\}\) is given by the values of the parameter \(\mu \). We order the hyperplanes \(V_1,\ldots , V_k\) accordingly.
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All four authors are partially supported by Spanish Ministerio de Economía y Competitividad MTM2015-69135-P. VGA is partially supported by ERC StG 279723 Arithmetic of algebraic surfaces (SURFARI). MAC and JAM are also supported by Generalitat de Catalunya SGR2017-932 Project and they are with the Barcelona Graduate School of Mathematics (BGSMath, MDM-2014-0445).
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Alberich-Carramiñana, M., Àlvarez Montaner, J., Dachs-Cadefau, F. et al. Multiplicities of jumping points for mixed multiplier ideals. Rev Mat Complut 33, 325–348 (2020). https://doi.org/10.1007/s13163-019-00309-y
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DOI: https://doi.org/10.1007/s13163-019-00309-y