Abstract
We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus \({\Sigma }\), and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of \({\Sigma }\) and on the effect of the monodromy of the local system over its complement.
Notes
See [2] for details.
The polar locus of a map (h, f) is defined as \(\overline{\mathrm{Sing}\,(h,f) {\setminus }(\mathrm{Sing}\,h \cup \mathrm{Sing}\,f)}\).
We recall from Sect. 5.2 that the notation \(S_q\) depends on whether the point q is considered in Q or in \(Q_i\), namely it takes either the local branches of \({\Sigma }\) at q, or the local branches of at q, accordingly.
Note that no multiplicities but only transversal types are involved in the rank formula.
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The authors express their gratitude to CIRM in Luminy for supporting this research project through the “Recherche en Binôme” program.
Mihai Tibǎr received the support from Labex CEMPI (ANR-11-LABX-0007-01).
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Siersma, D., Tibăr, M. Vanishing homology of projective hypersurfaces with 1-dimensional singularities. European Journal of Mathematics 3, 565–586 (2017). https://doi.org/10.1007/s40879-017-0151-7
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DOI: https://doi.org/10.1007/s40879-017-0151-7