Normal bundle of rational curves and Waring decomposition (Q405825)

The present paper is devoted to a classical subject in the theory of algebraic curves -- a classification of rational curves \(C \subset \mathbb{P}^{m}\) of degree \(n \geq m\) by the splitting type of their normal bundles \(N_{C;\mathbb{P}^{m}}\) and restricted tangent bundles \(T\mathbb{P}^{m}|_{C}\).NEWLINENEWLINEIn the note the author consider the projective situation (in order to see these degree \(n\) curves as projections of rational normal curves \(C_{n} \subset \mathbb{P}^{n}\) from \(L = \mathbb{P}^{k-1}\)) under the additional assumption that \(C\) has at worst ordinary singularities, which brings about that by \(N_{C;\mathbb{P}^{m}}\) and \(T\mathbb{P}^{m}|_{C}\) one means \(p^{*}(N_{C;\mathbb{P}^{m}})\) and \(p^{*}(T\mathbb{P}^{m}|_{C})\) respectively, where \(p: \mathbb{P}^{1} \rightarrow C\) is the composition \(p = p_{L} \circ v_{n}\) of the \(n\)-th Veronese and the projection from \(L\).NEWLINENEWLINEFor every \(n \in \mathbb{N}\) one defines the number \(\rho_{r}^{n,k}\) which is either equal to \(n-3k+r-1\) when \(3k < n-1, r \leq 2k-1\) or \(r\) when \(3k \geq n-1, 1 \leq r \leq n-k.\)NEWLINENEWLINENow we can recall some main results of the paper.NEWLINENEWLINETheorem.NEWLINENEWLINEIf \(\rho_{r}^{n,k} \leq (n-k+1)/3\), then the following conditions are equivalent: {\parindent=6mm \begin{itemize} \item[i)] the curve \(C = \pi_{L}(C_{n}) \subset \mathbb{P}^{n-k}\) projected from \(L\) has the splitting NEWLINE\[NEWLINEN_{\pi_{L}(C_{n});\mathbb{P}^{n-k}} \cong \mathcal{O}_{\mathbb{P}^{1}}(n+2)^{\rho_{r}^{n,k}} \oplus \mathcal{F},NEWLINE\]NEWLINE where \(\text{deg}(\mathcal{F}^{\vee}(n+2)) = -2k\) and \(\mathcal{F} \cong \bigoplus_{i}^{n-1-\rho_{r}^{n,k}-k}\mathcal{O}_{\mathbb{P}^{1}}(l_{i})\) with \(l_{i} \geq n+3\), \item [ii)] the center of projection \(L\) is contained in the base locus of a linear series \(\Phi\) generated by \(\rho_{r}^{n,k}\) linearly independent \(\mathbb{P}^{n-3}\)'s which intersect \(C_{n}\) in degree \(n-2\). NEWLINENEWLINE\end{itemize}} Another result characterizes the parameterizing variety for a certain class of curves.NEWLINENEWLINETheorem.NEWLINENEWLINEFor any \(0 \leq \alpha \leq n-k-2\) the variety parameterizing rational curves \(C\) of degree \(n\) in \(\mathbb{P}^{n-k}\) whose normal bundles \(N_{C, \mathbb{P}^{n-k}}\) has the summand \(\mathcal{O}_{\mathbb{P}^{1}}(n+2)^{\alpha}\) is irreducible.NEWLINENEWLINEIt is worth to point out that the summand \(\mathcal{O}_{\mathbb{P}^{1}}(n+2)\) appearing in the splitting type of normal bundles has the minimal possible degree for curves \(C\) having ordinary singularities.NEWLINENEWLINEThe author uses Waring decomposition tools, for instance Apolarity Lemma and decompositions of binary forms, which is a new and interesting approach.