Curve singularities with one Puiseux pair and value sets of modules over their local rings

Abstract

In this paper we characterize the value set of the R-modules of the form R+zRfor the local ring R associated to a germ ¿ of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of Kähler differentials attached to ¿,werecoversomeresultsofDelorme.Fromourcharacterizationof weintroduce a proper subset of semimodules over the value semigroup of the ring R. Moreover, we provide a combinatorial algorithm to construct all possible semimodules in this subset for a given value semigroup.

The first named author is partially supported by PID2019-103849GB-I00 and PID2023-146936NB-I00 financed by the Spanish State Agency MCIN/AEI/10.13039/501100011033/FEDER/UE and 2021SGR-00603 financed by the Generalitat de Catalunya. The second named author is supported by Grant RYC2021-034300-I funded by MICIU/AEI/10.13039/501100011033 and by European Union NextGenerationEU/PRTR and during the elaboration of this work was also supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M / AEI /10.13039/501100011033 and by Spanish Ministerio de Ciencia, Innovación y Universidades PID2020-114750GB-C32/AEI/10.13039/501100011033. The third named author is partially funded by MCIN/AEI/10.13039/501100011033, by "ERDF A way of making Europe" (grant PID2022-138906NB-C22) and by Universitat Jaume I, grants UJI-B2021-02 and GACUJIMA/2024/03.

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Postprint (published version)