Hypersurfaces with vanishing hessian via Dual Cayley Trick

Abstract

We present a general construction of hypersurfaces with vanishing hessian, starting from any irreducible non-degenerate variety whose dual variety is a hypersurface and based on the so called Dual Cayley Trick. The geometrical properties of these hypersurfaces are different from the series known until now. In particular, their dual varieties can have arbitrary codimension in the image of the associated polar map.

Symmetry preserving degenerations of the generic symmetric matrix

“…Besides, it will produce additional examples where the codimension of the dual variety of a determinantal hypersurface f in its polar variety can be arbitrarily large when the Hessian determinant h(f ) vanishes. The advantage of these new examples is that the ambient dimension can be smaller than in others previously known examples (such as in [7] and [14]). where T := k[∆ i,j | 2 ≤ i + j ≤ 2m − r, i ≤ j] is a k-subalgebra generated by cofactors.…”

“…A major question is when the Hessian of the degeneration f := det DS vanishes. The general question of the vanishing of a hyperurface has a venerable history since the days of Hesse ( [16], [17]) and ), subsequently studied by several mathematicians of the Italian school ( [24], [10], [11], [25], [26]) and more recently by C. Ciliberto, R. Gondim, F. Russo, G. Staglianò ( [4], [3], [13], [14]). In this paper the focus is on the class of determinantal hypersufaces arising from degenerations of the generic symmetric matrix.…”

Normalised Tangent Bundle, Varieties With Small Codegree and Pseudoeffective Threshold

“…There are very few known examples of smooth projective varieties whose dual variety is a hypersurface with vanishing hessian. Gondim, Russo and Staglianò proved in [12, Corollary 4.5] that the projection from an internal point of is a smooth variety , such that the dual variety is a degree hypersurface with vanishing hessian. It would be very interesting to find more examples.…”

CW-complex Nagata Idealizations

“…In 1876 Gordan and Noether ([9]) proved that Hesse's claim is true for N ≤ 3, and it is false for N ≥ 4. They and Franchetta classified all the counterexamples to Hesse's claim in P 4 (see [9,4,5,7]). In 1900, Perazzo classified cubic hypersurfaces with vanishing Hessian for N ≤ 6 ( [16]).…”