A note on higher-order Gauss maps (Q2396613)

Let \(X \subset \mathbb{P}^N\) be an irreducible, nondegenerate projective variety (over an algebraically closed field of characteristic zero). The \textit{Gauss map} is a rational map from \(X\) to the corresponding Grassmannian \(\mathbb{G}(\dim(X),N)\) sending each smooth point of \(X\) to its projective tangent space. It naturally generalizes to \textit{the Gauss map of order \(k\)} which is the rational map which sends a general point of \(X\) to its osculating space of order \(k\), which is a \((d_k-1)\)-dimensional linear space. The embedding \(X \subset \mathbb{P}^N\) is said \textit{\(k\)-jet spanned} if the Gauss map of order \(k\) is defined at every point of \(X\) and \(d_k\) is maximal, that is, \(d_k={\dim(X)+k \choose k}\). The first result of the paper under review (see Thm.3.6) states that for a \(k\)-jet spanned embedding of \(X\) smooth complex variety (\(k\geq 1\)), the Gauss maps of order \(k\) is finite unless \(X=\mathbb{P}^n\) embedded by the Veronese embedding of order \(k\). Example 3.7 shows that one cannot expect finiteness without the smoothness hypothesis. Example 3.8 shows that, even in the smooth case, general \(k\)-jet spannedness (\(d_k\) maximal in the general point) is not enough to get a general finite map. Section 4 is devoted to describe combinatorically the Gauss maps of toric varieties, generalizing what occurs for tangential varieties. In fact, see Thm. 1.2 (or Thm. 4.3 for more details) for \(X_A\) toric variety defined by a set of lattice points \(A\) and under the hypothesis of generically \(k\)-jet spannedness, there exists a finite set of laticce points \(B_k\) and a lattice projection \(\pi\) such that the (closure of the) image of the Gauss map of order \(k\) is projectively equivalent to \(X_{B_k}\) and the closure of the irreducible components of the fibers of the Gauss map of order \(k\) is projectively equivalent ot \(X_{\pi(A)}\).