The Hessian map of hypersurfaces

Let F be a degree d form in n+1 variables. The Gauss map sends each smooth point of the hypersurface defined by F to the evaluation of the Jacobian of F at the point. The image of the Gauss map is the dual variety of the hypersurface. After applying the Gauss map again, the original hypersurface is recovered. In this talk we study the analogous map for the Hessian matrix. The Hessian map of F is a rational map that sends a point of the hypersurface defined by F to the evaluation of the Hessian matrix of F at the point. We study some properties of this morphism as the birationality and the smoothness. In the case d=3, we introduce the variety of k--planes containing the span of gradients of degree 3 forms. The language of symmetric tensor provides a dictionary between this variety and the Hessian map that allows us to derive an effective method for recovering the initial hypersurface from its image through the Hessian map.