Cubic hypersurfaces admitting an embedding with Gauss map of rank 0
Let \(X\) be an \(n\)-dimensional projective variety defined over an algebraically closed field \(K\). Each embedding of \(i:\;X\hookrightarrow{\mathbb P}^M\) gives rise to a Gauss map, the rational map that associates to each non-singular point \(x\) of \(X\) the element of the Grassmannian corresponding to the embedded tangent space to \(i(X)\) at \(i(x)\). A projective variety \(X\) is said to satisfy property (GMRZ) if it admits an embedding into projective space such that the rank of the differential of the Gauss map at a general point is \(0\). In the paper [\textit{S. Fukasawa, K. Furukawa} and \textit{H. Kaji}, Adv. Math. 224, No. 6, 2645--2661 (2010; Zbl 1202.14049)] it was shown that a smooth cubic hypersurface of dimension \(n\geq 4\) satisfies (GMRZ) if and only if it is isomorphic to the Fermat cubic hypersurface and the characteristic of the base field \(K\) is \(2\). In the paper under review, this result is extended to cubic threefolds. Moreover, it is shown that every smooth cubic surface over a field of characteristic \(2\) satisfies (GMRZ). To show this, the author proves that, in characteristic \(2\), property (GMRZ) is preserved under blowing up at points. This property is also used to construct further examples of varieties satisfying (GMRZ) in characteristic \(2\).
scientific article; zbMATH DE number 6059024
scientific article