The separability of the Gauss map and the reflexivity for a projective surface
Let \(X\subset{\mathbb P}^N\) be a projective variety of dimension \(n\) over an algebraically closed field. The Gauss map of \(X\) is the map associating to every smooth point of \(X\) the embedded tangent space at that point, viewed as an element of the Grassmannian of \(n\)-dimensional linear subspaces of \({\mathbb P}^N\). The variety \(X\) is called reflexive if its conormal variety \(CX\) is isomorphic to the conormal variety of its dual variety \(X^*\). An equivalent condition is that the projection from \(CX\) to \(X^*\) is separable. It is well known that if a variety \(X\) is reflexive, then its Gauss map is separable. The main result of this note is that also the converse holds if \(X\) is a projective surface. This property was already known in the case of curves, but is false if the dimension of \(X\) is bigger than three [see \textit{H. Kaji}, Geom. Dedicata 99, 221--229 (2003; Zbl 1027.14025) and \textit{S. Fukasawa}, Compos. Math. 142, No. 5, 1305--1307 (2006; Zbl 1107.14044)]. In the case of curves, the property is related to the fact that the Gauss map of \(X\) and the natural map \(CX\rightarrow X^*\) always have the same inseparable degree [\textit{S. Kleiman} and \textit{R. Piene}, Contemp. Math. 123, 107--129 (1991; Zbl 0758.14032)]. This does not hold for projective surfaces with \(2\)-dimensional Gauss image, as the authors show by giving a counterexample.
scientific article; zbMATH DE number 5162364
scientific article