Projective varieties admitting an embedding with Gauss map of rank zero
Let \(X\) be a projective variety of dimension \(n\) in \({\mathbb P}^N\) defined over an algebraically closed field \(k\). Then the Gauss map associated with this embedding of \(X\) is the rational map that associates with every non-singular point \(x\in X\) the embedded tangent space to \(X\) at \(x\), viewed as a point of the Grassmannian of \(n\)-dimensional subspaces of \({\mathbb P}^N\). When the field \(k\) has characteristic \(0\), a variety whose Gauss map has rank \(0\) is automatically a linear space. The paper under review deals with the case where \(\text{char} k=p>0\). Let us call a variety \(X\) a GMRZ variety if it admits an embedding in projective space such that the rank of the differential of the Gauss map is \(0\) at a general point of \(X\). Then the paper under review shows that this condition imposes strong restrictions on the rational curves on \(X\). In particular, one of the main results is a precise characterization of minimal free rational curves on GMRZ varieties, i.e. rational curves \({\mathbb P}^1\rightarrow X\) such that the pull-back of the tangent bundle \(T_X\) is of the form \({\mathcal O}_{{\mathbb P}^1}(2)\oplus{\mathcal O}_{{\mathbb P}^1}(1)^{d-2}\oplus{\mathcal O}_{{\mathbb P}^1}^{n-d+1}\), in terms of the degree of the pull-back to \({\mathbb P}^1\) of \(K_X\) and of the line bundle defining the GMRZ embedding. Furthermore, a characterization of GMRZ varieties is given inside the classes of Segre and Grassmann varieties, for cubic hypersurfaces and for general hypersurfaces of low degree. As an application, the authors prove that Fermat hypersurfaces of degree \(mp+1\) and dimension \(\geq 2mp\) contain free rational curves but no minimal free rational curves. This is in contradiction with the situation in characteristic zero, where the existence of free rational curves on a variety \(X\), i.e. rational curves such that the pull-back of \(T_X\) is generated by global sections, implies the existence of minimal rational curves; see Theorem IV.2.10 in [\textit{J. Kollár}, Rational curves on projective varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag. (1995; Zbl 0877.14012)].
scientific article; zbMATH DE number 5758038
scientific article