Inflectional loci of scrolls (Q2471026)

If \(X\subset \mathbb P^n\) is a non-degenerate smooth projective variety of dimension \(n\), with hyperplane bundle \(\mathcal L=\mathcal O_{\mathbb P^N}(1)\mid X\), for every \(x\in X\) the \(k\)th osculating space to \(X\) at \(x\), \(\text{Osc}_x^k(X)\), is defined as \(\mathbb P(\text{Im}(j_{k,x}))\), where \(j_k\) is the natural sheaf homomorphism sending a section \(s\) in \(H^0(X, \mathcal L)\) to its \(k\)th jet. For general \(X\), the following relation holds: \(\dim \text{Osc}_x^k(X)\leq {{k+n}\choose{n}}-1\). In the paper under review, the authors investigate the special case of scrolls over a curve \(C\) of genus \(g\), where always \(\dim \text{Osc}_x^k(X)\leq kn\). Assuming that the generic rank of \(j_k\) is the expected one, i.e. \(kn+1\), the \(k\)th inflectional locus \(\Phi_k\) of \(X\) is the set of points of \(X\) where \(\text{Im}(j_k)\), which is a subsheaf of the \(k\)th principal parts bundle \(\mathcal P_X^k(\mathcal L)\), is not a vector bundle. The main results of the article are that, if \(k\) is the largest integer such that \(kn\leq N\) and if \(\Phi_k\) has the expected dimension, then it is naturally a Cohen-Macaulay scheme and its cohomology can be explicitly computed: \[ [\Phi_k]=L{N+1-kn }+k(d+(n(k-1)+2(N+1-kn))(g-1)L^{N-kn}F, \] where \(L=c_1(\mathcal L)\), \(F\) is the class of a fiber of the scroll and \(d\) is the degree of \(X\). In particular, the only uninflected scroll in the projective space of dimension \(nk+l-1\), with \(1\leq l\leq n\), is the balanced rational normal scroll of degree \(gn\) in \(\mathbb P^{(k+1)n-1}\).