On the infinitesimal Terracini lemma (Q2039139)

Let \(X\subset \mathbb P^r\) be an irreducible non-degenerate projective variety of dimension \(n\) and denote by \(\mathrm{Sec}_k(X)\) the \(k\)-secant variety of \(X\), in other words, the variety obtained by the union of all the \(k\)-planes generated by points \(x_0,\dots,x_{k+1}\in X\). The classical Terracini lemma states that the tangent plane of \(\mathrm{Sec}_k(X)\) at \(x\in \langle x_0,\dots,x_k\rangle\) is given by the sum of the tangent planes of \(X\) at each \(x_i\), more precise \(T_{\mathrm{Sec}_k(X),x}=\langle T_{X,x_0},\dots,T_{X,x_k}\rangle\). The expected dimension of the variety \(\mathrm{Sec}_k(X)\) is \(\min\{r,kn+n+k\}\), the cases where this is not achieved are called defective and \(X\) is said to be \(k\)-defective, the number \(\delta_k(X)=\min\{r,kn+n+k\}-\dim \mathrm{Sec}_k(X)\) is the \(k\)-defect of \(X\). Let \(\mathrm{Osc}_m(X)\) be the variety of \(m\)-osculating spaces to \(X\), \(X\) is said to be \(m\)-osculating regular if \(\dim(\mathrm{Osc}_m(X))=\min\{(m+1)n,r\}\). The author tackles the conjecture of the infinitesimal Terracini Lemma: if \(X\) is also \(m\)-osculating regular and \(r\geq nm+n+m\), then \(X\) is \(m\)-defective if and only if, given the general \(0\)-dimensional curvilinear scheme \(\gamma\) of length \(m+1\) contained in \(X\), the dimension of the linear system of hyperplane sections of \(X\) singular along \(\gamma\) has dimension larger than \(r-(m+1)(n+1)\). The infinitesimal version of Terracini lemma is known to be true for secant lines (\(m=1)\), however the general result is still not proven. The author shows in the Theorem 5.1 that the result holds true also for \(m=2\). Furthermore, the author remarks that the proof indicates that the conjecture should hold in general since no theoretical problems should appear, however extending the proof to its full generality is computationally difficult.