Duality and quadratic normality (Q2809902)

We recall that a \textit{congruence of lines} \(B\) of the projective space \(\mathbb P^n\) is an irreducible \((n-1)\)-dimensional subvariety of the Grassmannian \(\mathbb G(1,n)\) of the lines of \(\mathbb P^n\), and its \textit{order} is the number of lines of \(B\) passing through the general point of \(\mathbb P^n\). The most important examples of first order congruences of lines are obtained as families of \(\frac{n-1}{c-1}\)-secant lines to irreducible \(c\)-codimensional subvarieties of \(\mathbb P^n\), thanks to their connections with Hartshorne conjecture for small codimension varieties, Peskine conjecture on \(k\)-normality and its generalizations by Zak, etc.NEWLINENEWLINEThe first important result of this paper (Proposition 2.1) gives the degree of the (projective) dual hypersurface of a smooth subvariety \(X\) of \(\mathbb P^n\) such that its \(k\)-secant lines form a congruence, under some hypothesis on the ideal sheaf of \(X\). This result is then applied to two important classes of first order congruences, namely the linear congruences, which are given by \((n-1)\)-secant lines of varieties of codimension \(2\) and those obtained as the lines \(L\in \mathbb G(1,n)\) such that \(\omega(L)=0\) for a general \(3\)-form \(\omega \in \bigwedge H^0(\mathcal O_{\mathbb P^n}(1)\) (see for example [\textit{P. De Poi} et al., ``Fano congruences of index \(3\) and alternating \(3\)-forms'', Preprint, \url{arXiv:1606.04715}]): in this situation, the congruence is given by the \(\frac{n-1}{2}\)-secant lines of a codimension \(3\) variety \(X\). Finally, he focuses on the geometry of the case \(n=9\) of this last series of congruences: in particular he constructs a component of the double locus of the dual of \(X\) from the Hyper-Kähler \(4\)-fold of \textit{O. Debarre} and \textit{C. Voisin} [J. Reine Angew. Math. 649, 63--87 (2010; Zbl 1217.14028)].