Polar degree and vanishing cycles
Dirk Siersma, Mihai Tib\u{a}r
TL;DR
This paper establishes a decomposition of the polar degree of singular projective hypersurfaces into sums of local vanishing cycle invariants, providing new lower bounds for their polar degrees.
Contribution
It introduces a novel decomposition of the polar degree into local invariants related to vanishing cycles, applicable to arbitrarily singular hypersurfaces.
Findings
Polar degree decomposes into sums of local vanishing cycle invariants.
Provides lower bounds for the polar degree of singular hypersurfaces.
Applicable to hypersurfaces with arbitrary singularities.
Abstract
We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications