Polar degree of hypersurfaces with 1-dimensional singularities
Dirk Siersma, Mihai Tib\u{a}r
TL;DR
This paper derives a formula for the polar degree of hypersurfaces with 1-dimensional singularities using Milnor data, extending previous isolated singularity results, and explores related deformation properties and classifications.
Contribution
It extends the polar degree formula to hypersurfaces with 1-dimensional singularities and analyzes its semi-continuity and classification of special cases.
Findings
Derived a new formula linking polar degree and Milnor data for 1-dimensional singularities.
Established semi-continuity properties of the polar degree under deformations.
Classified homaloidal cubic surfaces with 1-dimensional singular locus.
Abstract
We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the semi-continuity of the polar degree in deformations, and we classify the homaloidal cubic surfaces with 1-dimensional singular locus. Some open questions are pointed out along the way.
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