The Gauss map of a \(k\)-dimensional projective variety \(V\subset \mathbb P^N\) is the rational map \(\gamma: V\dashrightarrow \mathbb G(k,N)\) sending a smooth point \(P\in V\) to \(\widetilde T_P(V)\in G(k,N)\), where \(\widetilde T_P(V)\) is the projective tangent space to \(V\) at \(P\). The higher Gauss maps \(\gamma^i: V\dashrightarrow \mathbb G(d_i,N)\) are similarly defined, by replacing the tangent space with the \(i\)th osculating space \(\widetilde T_P(V)^{(i)}\), and where \(d_i\) denotes the dimension of \(\widetilde T_P(V)^{(i)}\) at a general point of \(V\). As the authors remark in their introduction to this interesting paper, ordinary Gauss maps have received much attention, whereas higher-order Gauss maps hardly any.
A Gauss map \(\gamma^i\) is said to be degenerate if the image of \(\gamma^i\) has dimension less than \(k\), i.e., if the fibres are positive-dimensional. Generalizing a result of P. Griffiths and J. Harris [Ann. Sci. Éc. Norm. Supér. (4) 12, 355–452 (1979; Zbl 0426.14019)] for ordinary Gauss maps, the authors describe the higher fundamental forms of varieties with degenerate higher Gauss maps. In particular they show that the fibres of \(\gamma^i\) are \(m\)-dimensional if and only if, at a general point of \(V\), the \(i\)th fundamental form at \(P\), considered as a linear system on \(\mathbb P^{k-1}\), consists of cones over a fixed \(\mathbb P^{m-1}\). They give various applications of this result.
The methods are mainly those of Griffiths and Harris [loc. cit.], consisting in studying Darboux frames adapted to the osculating flags of the variety.