Gaussian maps, the Zak map and projective extensions of singular varieties (Q1428267)

Let \(C\subset {\mathbb P}^n \) be an integral non-degenerate projective variety of codimension \(\geq 2\), and let \(S \subset {\mathbb P}^{n+1}\) be an integral variety such that \(C\) is scheme-theoretically an hyperplane section of \(S\). The authors show that a sufficient condition for \(S\) to be a cone over \(C\) is the surjectivity of a natural map such as a Gaussian map for curves or the Zak map in all dimensions.