Dirk SIERSMA – Polar degree in the presence of singularities

Date(s) : 23/01/2020   iCal
14 h 00 min - 15 h 00 min

Dirk SIERSMA (Universiteit Utrecht)

For any projective hypersurface V:={f=0}, the notion of polar degree is defined as the topological degree of the (projectivized) gradient mapping of the homogeneous polynomial f:
Grad(f) : P^{n}Sing(V) –> P^{n}.
Thus pol(V) := Card(Grad(f)^{-1}(a)) for any generic point a.
We will discuss first the history of polar degree and give several examples, e.g. the determinant hypersurface has polar degree 1, which means the gradient map is bi-rational. The hypersurfaces with pol(V)=1 are called homaloidal and are of extra interest.
The case that pol(V)=0 is related to the question what happens if the Hessian of f is identically zero. This was solved by Gordan and Noether in 1876.
Dolgacev classified in 2000 all the projective homoloidal plane curves: a short list. Huh determined in 2014 all homoloidal hypersurfaces with at most isolated singularities.
In this talk we will reprove Huh’s results with methods of singularity theory. Moreover we will prove the conjecture of Huh that his list of polar degree 2 surfaces with isolated singularities is complete! Finally we say something more about hypersurfaces with the non-isolated singularities.

FRUMAM, St Charles


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