221129 Dirk Siersma

November 29, 2022

Title: Polar degree of projective hypersurfaces in the presence of singularities (MI-talk)

Speaker: Dirk Siersma

For any hypersurface V in projective n-space P, given by f=0, the notion of polar degree is defined as the topological degree of the (projectivized) gradient mapping of the homogeneous polynomial f. This is a map from P-V to P.

We will discuss first the history of polar degree and give several examples, e.g. the determinant hypersurface has polar degree 1. The hypersurfaces with polar degree are called homaloidal and are of extra interest because the gradient map is bi-rational.

Polar degree zero is related to the question of what happens if the Hessian of f is identically zero. This was solved by Gordan and Noether in 1876.

After a long period of algebraic studies, recently topological methods gave some interesting results. Dolgacev classified in 2000 all the projective homoloidal plane curves: a short list. Huh determined in 2014 all homoloidal hypersurfaces in P with at most isolated singularities.

In this talk we will reprove Huh's results with methods of singularity theory. Moreover we will prove the Huh's conjecture 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.

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